Grothendieck Duality and Q-Gorenstein Morphisms 101

arXiv:1612.01690v1 [math.AG] 6 Dec 2016 government(MSIP)(No.2013006431). edo n hrceitci adt be to said is characteristic any of field a leri aite ncaatrsi eo omlagbacvariet algebraic normal A zero: characteristic in varieties algebraic oinwtotteChnMcua odto per n[4 o exa for [24] in appears condition r Cohen–Macaulay is the condition without Cohen–Macaulay the notion where (0.12.e)], [49, in Gorenstein” oiieinteger positive X odfietecnnclsnuaiyi 4,Df 11] n enmdth named he and (1.1)], Def. [48, in singularity canonical the define to sadtoal eurdt eChnMcua.Ri sdti oines notion this used Reid Cohen–Macaulay. be to required additionally is 2010 e od n phrases. and words Key h rtnmdato sprl upre yteNFo oe f Korea of NRF the by supported partly is author named first The .Rltv aoia sheaves 6. canonical Relative duality5. Grothendieck 4. .Relative 3. 7. Serre’s 2. Introduction 1. h oinof notion The pedxA oebscpoete nshm hoy105 theory scheme in properties basic Some References A. Appendix Q Q ahmtc ujc Classification. Subject Mathematics Abstract. ola odto,oe ed ysuiso relative on studies By field. a over Kollár condition, hnepoete,vlal eut r rvdfor proved are results valuable properties, change n rltv)cnnclsevs hs oe l h previ the dual all cover by of These schemes sheaves. Noetherian canonical (relative) locally and for introduced are phism et n oforth. so and ment, hc nld nntsmlciein autv criterio valuative criterion, infinitesimal include which Grnti schemes -Gorenstein Grnti morphisms -Gorenstein RTEDEKDAIYAND DUALITY GROTHENDIECK Q Grnti leri ait n of and variety algebraic -Gorenstein S k S r Q -condition where , 2 cniinadflatness and -condition Grnti ait sipratfrtemnmlmdlter of theory model minimal the for important is variety -Gorenstein h oin of notions The OGA E N OOUNAKAYAMA NOBORU AND LEE YONGNAM rtedekduality, Grothendieck K X tnsfrtecnncldvsrof divisor canonical the for stands 1. Q MORPHISMS Grnti ceeadof and scheme -Gorenstein Introduction Contents rmr 42;Scnay1J0 40,14A15. 14F05, 14J10, Secondary 14B25; Primary 1 Q Q Q Grnti if -Gorenstein Grnti morphism. -Gorenstein Grnti eomto satisfying deformation -Gorenstein Q Q Grnti morphisms, -Gorenstein n, -GORENSTEIN S 2 Q ul nw notions known ously cniinadbase and -condition rK Q Grnti refine- -Gorenstein Grnti mor- -Gorenstein zn complexes izing X X ne yteKorean the by unded sCrirfrsome for Cartier is nsm papers, some In . y X qie.The equired. pe nthe In mple. oin“ notion e endover defined sentially 108 Q 63 41 24 72 82 9 1 - 2 YONGNAMLEEANDNOBORUNAKAYAMA minimal model theory of algebraic varieties of dimension more than two, we must deal with varieties with mild singularities such as terminal, canonical, log-terminal, and log-canonical (cf. [24, 0-2] for the definition). The notion of Q-Gorenstein is § hence frequently used in studying the higher dimensional birational geometry. The notion of Q-Gorenstein deformation is also popular in the study of degen- erations of normal algebraic varieties in characteristic zero related to the minimal model theory and the moduli theory since the paper [30] by Kollár and Shepherd- Barron. Roughly speaking, a Q-Gorenstein deformation C of a Q-Gorenstein X → normal algebraic variety X is considered as a flat family of algebraic varieties over a smooth curve C with a closed fiber being isomorphic to X such that rKX /C is Cartier and rK rK for some r > 0, where K stands for the relative X /C|X ∼ X X /C canonical divisor. We call such a deformation “naively Q-Gorenstein” (cf. Defini- tion 7.1 below). This is said to be “weakly Q-Gorenstein” in [15, 3], or satisfying § Viehweg’s condition (cf. Property V[N] in [21, 2]). We say that C is a § X → Q-Gorenstein deformation if (mK ) (mK ) OX X /C ⊗OX OX ≃ OX X for any integer m. This additional condition seems to be considered first by Kollár [27, 2.1.2], and it is called the Kollár condition; A similar condition is named as Property K in [21, 2] for example. A typical example of Q-Gorenstein deforma- § tion appears as a deformation of the weighted projective plane P(1, 1, 4): Its versal deformation space has two irreducible components, in which the one-dimensional component corresponds to the Q-Gorenstein deformation and its general fibers are P2 (cf. [45, 8]). The Q-Gorenstein deformation is also used in constructing some § simply connected surfaces of general type over the complex number field C in [33]. The authors have succeeded in generalizing the construction to the positive characteristic case in [32], where a special case of Q-Gorenstein deformation over a mixed characteristic base scheme is considered. During the preparation of the joint paper [32], the authors began generalizing the notion of Q-Gorenstein morphism to the case of morphisms between locally Noetherian schemes. The purpose of this article is to give good definitions of Q-Gorenstein scheme and Q-Gorenstein morphism: We define the notion of “Q-Gorenstein” for locally Noetherian schemes admitting dualizing complexes (cf. Definition 6.1 below) and define the notion of “Q-Gorenstein” for flat morphisms locally of finite type between locally Noetherian schemes (cf. Definition 7.1 below). So, we try to define the notion of “Q-Gorenstein” as general as possible. We do not require the Cohen–Macaulay condition for fibers, which is assumed in most articles on Q-Gorenstein deformations, and we allow all locally Noetherian schemes as the base scheme of a Q-Gorenstein morphism.

Q-Gorenstein schemes and Q-Gorenstein morphisms. The definition of Q- Gorenstein scheme in Definition 6.1 below is interpreted as follows (cf. Lemma 6.4(3)): A locally Noetherian scheme X is said to be Q-Gorenstein if and only if it satisfies Serre’s condition S , • 2 it is Gorenstein in codimension one, • GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 3

there exists a dualizing sheaf locally on X, and the double dual of ⊗r • L L is invertible for some integer r> 0 locally on X. Here, we consider a dualizing sheaf (cf. Definition 4.13) as the zero-th cohomology 0( •) of an ordinary dualizing complex •, which is a dualizing complex of H R R special type and exists for any locally equi-dimensional (cf. Definition 2.2(3)) and locally Noetherian schemes admitting dualizing complexes (cf. Lemma 4.14). The dualizing sheaf of the Gorenstein locus U = Gor(X) is isomorphic to up to L|U tensor product with invertible sheaves, and is a reflexive -module with an L OX isomorphism j ( ) for the open immersion j : U ֒ X. This definition L ≃ ∗ L|U → generalizes the usual definition of Q-Gorenstein normal algebraic varieties over a field (cf. Example 6.6). On the other hand, in order to define Q-Gorenstein morphisms, we need to discuss the relative canonical sheaf of an S2-morphism. An S2-morphism is defined as a flat morphism of locally Noetherian schemes which is locally of finite type and every fiber satisfies Serre’s condition S2 (cf. Definition 2.30). By Conrad [7, Sect. 3.5] and Sastry [50], we have a good notion of the relative canonical sheaf for Cohen–Macaulay morphisms. For an S -morphism f : Y T of locally Noetherian 2 → schemes, the relative Cohen–Macaulay locus Y ♭ = CM(Y/T ) is an open subset (cf.

Definition 2.28 and Fact 2.29), and we define the relative canonical sheaf ωY/T as the ♭ direct image by the open immersion Y Y of the relative canonical sheaf ω ♭ ⊂ Y /T of the Cohen–Macaulay morphism Y ♭ T (cf. Definition 5.3). The sheaf ω is → Y/T coherent (cf. Proposition 5.5), and it is reflexive when every fiber is Gorenstein in [m] ⊗m codimension one (cf. Proposition 5.6). We set ωY/T to be the double dual of ωY/T for m Z, and we define Q-Gorenstein morphisms as follows: A flat morphism ∈ f : Y T locally of finite type between locally Noetherian schemes is called a → Q-Gorenstein morphism (cf. Definition 7.1) if and only if every fiber is a Q-Gorenstein scheme, and • ω[m] satisfies relative S over T for any m Z. • Y/T 2 ∈ Note that f is an S2-morphism and every fiber is Gorenstein in codimension one, by the first condition. The second condition corresponds to the Kollár condition. A weaker notion: naively Q-Gorenstein morphism is defined by replacing the second condition to: ω[m] is invertible for some m locally on Y . • Y/T This condition corresponds to Viehweg’s condition. By our definition above, we can consider Q-Gorenstein deformations f : Y T → of a Q-Gorenstein scheme X defined over a field k. Here, f is a Q-Gorenstein −1 morphism, T contains a point o with residue field k, and the fiber Yo = f (o) is isomorphic to X over k. The scheme X is not necessarily assumed to be reduced nor normal, and f is not necessarily a morphism of k-schemes. The Q-Gorenstein deformations of non-normal schemes have been treated in articles in some special cases: Hacking [15] and Tziolas [54] consider Q-Gorenstein deformations of slc surfaces, which are not normal in general, over C. The work of Abramovich–Hassett [1] covers also non-normal reduced Cohen–Macaulay algebraic schemes over a fixed 4 YONGNAMLEEANDNOBORUNAKAYAMA

field. By further studies of Q-Gorenstein morphisms, we may have a well-defined theory of infinitesimal Q-Gorenstein deformations, which is now in progress in the authors’ joint work.

Notable results on Q-Gorenstein morphisms. Some expected properties on Q-Gorenstein morphisms can be verified by standard methods. For example, we prove that Q-Gorenstein morphisms are stable under base change and composition (cf. Propositions 7.22(5) and 7.23(3)). Besides such elementary properties, we have notable results in the topics below, which show that our definition of Q-Gorenstein morphism is reasonable and widely applicable: (1) A sufficient condition for a virtually Q-Gorenstein morphism to be Q- Gorenstein; (2) Infinitesimal and valuative criteria for a morphism to be Q-Gorenstein; (3) Some conditions on fibers related to Serre’s S3-condition which are sufficient for a morphism to be Q-Gorenstein; (4) The existence of Q-Gorenstein refinement. We shall explain results on these topics briefly. (1): The virtually Q-Gorenstein morphism is introduced in Section 7.2 as a weak form of Q-Gorenstein morphism (cf. Definition 7.13). This is inspired by the definition [15, Def. 3.1] by Hacking on Q-Gorenstein deformation of an slc surface in characteristic zero: His definition is generalized to the notion of Kollár family of Q-line bundles in [1]. Hacking defines the Q-Gorenstein deformation by the property that it locally lifts to an equivariant deformation of an index-one cover. This definition essentially coincides with our definition of virtually Q-Gorenstein morphism (cf. Lemma 7.16 and Remark 7.17). A Q-Gorenstein morphism is always a virtually Q-Gorenstein morphism. The converse holds if every fiber satisfies S3; It is proved as a part of Theorem 7.18. This theorem is derived from Theorems 3.16 and 5.10 on criteria for certain sheaves to be invertible, and from a study of the relative canonical dualizing complex in Section 5.1. By Theorem 7.18, we can study infinitesimal Q-Gorenstein deformations of a Q-Gorenstein algebraic scheme over a field satisfying S3 via the equivariant deformations of the index one cover. (2): The infinitesimal criterion says that, for a given flat morphism f : Y → T locally of finite type between locally Noetherian schemes, it is a Q-Gorenstein morphism if and only if the base change f : Y = Y Spec A Spec A is a Q- A A ×T → Gorenstein morphism for any morphism Spec A T for any Artinian local ring A. → The valuative criterion is similar but T is assumed to be reduced and A is replaced with any discrete valuation ring. These criteria and some variants are proved in Theorems 7.25 and 7.29 and Corollaries 7.26 and 7.27. The proof of these criteria uses Proposition 3.19 on infinitesimal and valuative criteria for a reflexive sheaf on Y to satisfy relative S2 over T . (3): Theorem 7.30 proves that, for a morphism f : Y T in (2) above, it is −1 → Q-Gorenstein along a fiber Yt = f (t) if Yt is Q-Gorenstein and Gorenstein in codimension two and if ω[m] = ω[m] Yt/k(t) Yt/ Spec k(t) GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 5 satisfies S for any m Z. Here, k(t) denotes the residue field of . 3 ∈ OT,t (4): For an S -morphism f : Y T of locally Noetherian schemes such that 2 → every fiber is Q-Gorenstein, the Q-Gorenstein refinement is defined as a monomorphism S T satisfying the following universal property (cf. Definition 7.31): For → a morphism T ′ T from another locally Noetherian scheme T ′, the base change → Y T ′ T ′ is a Q-Gorenstein morphism if and only if T ′ T factors through ×T → → S T . Theorem 7.32 proves the existence of Q-Gorenstein refinement, for exam- → ple, in the case where f is proper and ω[m] is invertible for a constant m> 0 for Yt/k(t) any fiber Y . In this case, S T is shown to be separated and locally of finite type. t → Similar results are given as Theorem 7.34 for a local version and as Theorem 7.35 for naively Q-Gorenstein morphisms. Kollár’s result [29, Cor. 25] is stronger than Theorem 7.32 when f is a projective morphism.

The role of our key proposition. The deep results above on the topics (1)– (4) and some basic properties of S2-morphisms and Q-Gorenstein morphisms are obtained by applying our key proposition (= Proposition 3.7). It proves that, for a flat morphism Y T of locally Noetherian schemes and for an exact sequence → 0 0 1 0 →F→E →E → G → of coherent -modules satisfying suitable conditions, the relative S -condition for OY 2 over T is equivalent to the flatness of over T . For example, if f is an S - F G 2 morphism, then a reflexive sheaf on Y admits such an exact sequence locally F on Y when is locally free in codimension one on each fiber (cf. Lemma 3.14). F Therefore, the relative S -condition for over T can be studied by the flatness of 2 F another sheaf defined locally on Y . This is useful, since the sheaves ω[m] are G Y/T reflexive. For the relative canonical sheaf ω of an S -morphism Y T , we Y/T 2 → can show in the proof of Proposition 5.5 that Y is locally embedded into an affine smooth T -scheme P as a closed subscheme and ωY/T admits such an exact sequence as on P . F Applying the local criterion of flatness (cf. Proposition A.1) and the valuative criterion of flatness (cf. [12, IV, Th. (11.8.1)]) for , we have infinitesimal and G valuative criteria for to satisfy relative S over T in Proposition 3.19. This is F 2 applied to the reflexive sheaf = ω[m] in the topic (2) above. The flattening F Y/T stratification theorem by Mumford in [38, Lect. 8] and the representability theorem of unramified functors by Murre [40] applied to yield Theorem 3.26 on the relative G S refinement for (cf. Definition 3.20). This is defined as a monomorphism S T 2 F → satisfying the following universal property: For a morphism T ′ T from a locally → Noetherian scheme T ′ and for the induced morphisms p: Y ′ Y and Y ′ T ′ → → from the fiber product Y ′ = Y T ′, the double dual of p∗ satisfies relative S ×T F 2 over T ′ if and only if T ′ T factors through S T . When Y T is projective, → → → Theorem 3.26 is similar to Kollár’s result [29, Th. 2] on “hulls and husks.” We also have a “local version” of the relative S refinement for as Theorem 3.28 by 2 F applying to theorems on local universal flattening functor by Raynaud–Gruson G [47, Part 1, Th. (4.1.2)] or Raynaud [46, Ch. 3, Th. 1]. Applying the results on 6 YONGNAMLEEANDNOBORUNAKAYAMA relative S refinement for = ω[m] , we have theorems on Q-Gorenstein refinement 2 F Y/T mentioned in the topic (4) above. The implication (b′) (b) in Proposition 3.7 is important. It is essential in the ⇒ proof of Corollary 3.10, and it is used in the proofs of Theorem 3.16 on a criterion for a certain sheaf to be invertible, mentioned in the explanation of (1) above and of Theorem 7.30 in (3) above. Corollary 3.10 is similar to a special case of [28, Th. 12], but to which we have found a counterexample (cf. Example 3.12).

Various other results and remarks. Most parts of Sections 2 and 4 are surveys: Section 2 discusses basic properties on Sk-conditions and relative Sk-conditions, and Section 4 discusses the dualizing complex and the relative dualizing complex in a little detail. Even in the surveys, we present some results and remarks, which seem to be new or not well known. These are listed as follows: Some general properties on reflexive modules over a locally Noetherian • F scheme X are presented in Lemmas 2.21, 2.33, 2.34, and Corollary 2.22. These are related to the S -conditions and the relative S -conditions for 2 2 F and for X. Similar properties can be found in [21, 3]. Note that reflexive § modules are well understood over an integral domain (cf. [6, VII, 4]). § Every S -morphism (cf. Definition 2.30) of locally Noetherian schemes lo- • 2 cally has pure relative dimension (cf. Lemma 2.38(1)). For a locally Noetherian scheme X admitting a dualizing complex and for • a coherent sheaf on it, the S -locus S ( ), the Cohen–Macaulay locus F k k F CM( ), and the Gorenstein locus Gor(X) are open (cf. Proposition 4.11). F Let X be a locally equi-dimensional and locally Noetherian scheme admit- • ting a dualizing complex. In this case, X has an ordinary dualizing complex • and the dualizing sheaf = 0( •) in the sense of Definition 4.13 (cf. R L H R Lemma 4.14). Then, satisfies S , and om ( , ) is an -algebra L 2 H OX L L OX defining the S2-ification of X (cf. Proposition 4.21 and its Remark). (cf. Corollary 4.38) For a flat separated morphism Y T of finite type of • → Noetherian schemes and for the canonical inclusion morphism ψ : Y Y t t → from the fiber Y = f −1(t) over a point t T , there is a quasi-isomorphism t ∈ Lψ∗(f ! ) ω• t OT ≃qis Yt/k(t) ! of complexes of Yt -modules, where f T is the twisted inverse image of O • O T by f (cf. Example 4.33), and ω is the canonical dualizing complex O Yt/k(t) of the algebraic scheme Y over the residue field k(t) of (cf. Defini- t OT,t tion 4.28). When is regular, the assertion has been proved by [44, OT,t Prop. 3.3(1)]. In the other sections 3, 5, 6, and 7, we can find the following interesting results and remarks which are not listed above as notable ones: A counterexample of Kollár’s theorem [28, Th. 12] is given in Example 3.12. • Using another example produced there, in Remark 3.13, we explain a wrong assertion on the commutativity of projective limit and the direct image by open immersions for quasi-coherent sheaves. GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 7

For an S -morphism f : Y T of locally Noetherian schemes and for • 2 → the open immersion j : Y ♭ ֒ T from the relative Cohen–Macaulay locus ♭ → Y = CM(Y/T ), the pushforward j∗ωY ♭/T is coherent, and it is isomorphic to −d(f ! ) locally on Y for the relative dimension d and the twisted H OT inverse image f ! (cf. Proposition 5.5). Moreover, this sheaf is reflexive OT if every fiber is Gorenstein in codimension one (cf. Proposition 5.6). In Section 6.2, we discuss affine cones X of connected polarized projective • schemes (S, ) over a field k. Here, the projective scheme S is not assumed A to be reduced nor irreducible. In the sequel, we give useful conditions for [r] ωX/k to satisfy Sk and for X to be Q-Gorenstein, in several situations of (S, ) (cf. Proposition 6.13, Corollaries 6.14 and 6.15). A By Lemma 7.8 and Example 7.9, we present a new example of naively Q- • Gorenstein morphisms which is not Q-Gorenstein in the case of morphisms of algebraic varieties over an algebraically closed field of characteristic zero. For known examples, see Fact 7.7. For a naively Q-Gorenstein morphism f : Y T and a point t T , the • → −1 ∈ relative Gorenstein index of f along the fiber Yt = f (t) coincides with the Gorenstein index of Yt under suitable conditions (cf. Proposition 7.11). This covers [30, Lem. 3.16], but whose proof has problems explained in Remark 7.12. In Remark 7.28, applying Corollary 7.27, we verify the unboundedness of • r for Kollár’s example of naively Q-Gorenstein morphisms over the spec- { n} tra of Artinian rings, explained in [16, 14.7] and [31, Exam. 7.6]. There, the proof is left to the reader, but we are afraid that the expected proof might have a problem similar to the second problem in Remark 7.12. For a famous example (Example 7.4) of deformations of the weighted projec- • tive plane P(1, 1, 4) which is not Q-Gorenstein, its Q-Gorenstein refinement is determined in Example 7.33 by using Lemma 7.5.

Organization of this article. In Section 2, we recall some basic notions and properties related to Serre’s Sk-condition. Section 2.1 recalls basic properties on dimension, depth, and the Sk-condition. The relative Sk-condition is explained in Section 2.2. In Section 3, we proceed the study of relative S2-condition and give some criteria for this condition. Section 3.1 is devoted to prove the key proposition (Proposition 3.7) and related properties. Some applications of Proposition 3.7 are given in Section 3.2: Theorem 3.16 on a criterion for a certain sheaf to be invertible, Proposition 3.19 on infinitesimal and valuative criteria, Theorem 3.26 on the relative S2 refinement, and its local version: Theorem 3.28. The theory of Grothendieck duality is surveyed briefly in Section 4 with a few original results. In Sections 4.1 and 4.2, we recall some well-known properties on the dualizing complex based on arguments in [17] and [7]. The twisted inverse image functor is explained in Section 4.3 with the famous Grothendieck duality theorem for proper morphisms (cf. Theorem 4.30). The base change theorem for the relative dualizing sheaf for a Cohen–Macaulay morphism is mentioned in Section 4.4. In 8 YONGNAMLEEANDNOBORUNAKAYAMA

Section 5, we give some technical base change results for the relative canonical sheaf of an S2-morphism. Section 5.2 is devoted to proving Theorem 5.10 on a criterion for a certain sheaf related to the relative canonical sheaf to be invertible. This technical theorem also gives sufficient conditions for the base change homomorphism of the relative canonical sheaf to the fiber to be an isomorphism, and it is applied to the proof of Theorem 7.18 on virtually Q-Gorenstein morphism (cf. the topic (1) above). In Section 6, we study Q-Gorenstein schemes. The definition and its basic properties are given in Section 6.1. As an example of Q-Gorenstein schemes, in Sec- tion 6.2, we consider the case of affine cones over polarized projective schemes over a field. In Section 7, we study Q-Gorenstein morphisms, and two variants: naively Q- Gorenstein morphisms and virtually Q-Gorenstein morphisms. The Q-Gorenstein morphism and the naively Q-Gorenstein morphism are defined in Section 7.1, and their basic properties are discussed. The virtually Q-Gorenstein morphism is defined in Section 7.2, and we prove Theorem 7.18 on a criterion for a virtually Q- Gorenstein morphism to be Q-Gorenstein (cf. the topic (1) above). In Section 7.3, several basic properties including base change of Q-Gorenstein morphisms and of their variants are discussed. Theorems mentioned in the topics (2)–(4) above are proved in Section 7.4. Some elementary facts on local criterion of flatness and base change isomorphisms are explained in Appendix A for the readers’ convenience. In this article, we try to cite references kindly as much as possible for the readers’ convenience and for the authors’ assurance. We also try to refer to the original article if possible.

Acknowledgements. The first named author would like to thank Research In- stitute for Mathematical Sciences (RIMS) in Kyoto University for their support and hospitality. This work was initiated during his stay at RIMS in 2010 and is continued to his next stay in 2016. The second named author expresses his thanks to Professor Yoshio Fujimoto for his advices and continuous encouragement. Both authors are grateful to Professors János Kollár and Sándor Kovács for useful infor- mation.

Notation and conventions. i (1) For a complex K• = [ Ki d Ki+1 ] in an abelian category and ···→ −→ → · · · for an integer q, we denote by τ ≤q(K•) (resp. τ ≥q(K•)) the “truncation” of K•, which is deﬁned as the complex

q−2 [ Kq−2 d Kq−1 Ker(dq) 0 ] ···→ −−−→ → → → · · · q+1 (resp. [ 0 Coker(dq−1) Kq+1 d Kq+2 ]) ···→ → → −−−→ → · · · (cf. [10, D´ef. 1.1.13]). The complex K•[m] shifted by an integer m is deﬁned di as the complex L• = [ Li L Li+1 ] such that Li = Ki+m and ···→ −−→ → · · · di = ( 1)mdi+m for any i Z. It is known that the mapping cone of the L − ∈ natural morphism τ ≤q(K•) K• is quasi-isomorphic to τ ≥q+1(K•) for → any q Z. ∈ GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 9

(2) For a complex K• in an abelian category (resp. for an object K• of the derived category), the i-th cohomology of K• is denoted usually by Hi(K•). For a complex • of sheaves on a scheme, the i-th cohomology is a sheaf K and is denoted by i( •). H K (3) The derived category of an abelian category A is denoted by D(A). More- over, we write D+(A) (resp. D−(A), resp. Db(A)) for the full subcategory consisting of bounded below (resp. bounded above, resp. bounded) complexes. (4) An algebraic scheme over a field k means a k-scheme of finite type. An algebraic variety over k is an integral separated algebraic scheme over k. (5) For a scheme X, a sheaf of -modules is called an -module for simplic- OX OX ity. A coherent (resp. quasi-coherent) sheaf on X means a coherent (resp. quasi-coherent) -module. The (abelian) category of -modules (resp. OX OX quasi-coherent -modules) is denoted by Mod( ) (resp. QCoh( )). OX OX OX (6) For a scheme X and a point x X, the maximal ideal (resp. the residue ∈ field) of the local ring is denoted by m (resp. k(x)). The stalk of OX,x X,x a sheaf on X at x is denoted by . F Fx (7) For a morphism f : Y T of schemes and for a point t T , the fiber → ∈ f −1(t) over t is defined as Y Spec k(t) and is denoted by Y . For an ×T t -module , the restriction to the fiber Y is denoted by OY F F⊗OY OYt t F(t) (cf. Notation 2.24). (8) The derived category of a scheme X is defined as the derived category of Mod( ), and is denoted by D(X). The full subcategory consisting of OX complexes with quasi-coherent (resp. coherent) cohomology is denoted by D (X) (resp. D (X)). For = +, , b and for = qcoh, coh, we set qcoh coh ∗ − † D∗(X)= D∗(Mod( )) and D∗(X)= D∗(X) D (X). OX † ∩ † (9) For a sheaf on a scheme X and for a closed subset Z, the i-th local F cohomology sheaf of with support in Z is denoted by i ( ) (cf. [18]). F HZ F (10) For a morphism X Y of schemes, Ω1 denotes the sheaf of relative → X/Y one-forms. When X Y is smooth, Ωp denotes the p-th exterior power → X/Y pΩ1 for integers p 0. X/Y ≥

V 2. Serre’s Sk-condition We shall recall several fundamental properties on locally Noetherian schemes, which are indispensable for understanding the explanation of dualizing complex and Grothendieck duality in Section 4 as well as the discussion of relative canonical sheaves and Q-Gorenstein morphisms in Sections 5 and 6, respectively. In Section 2.1, we recall basic properties on dimension, depth, Serre’s Sk-condition especially for k = 1 and 2, and reﬂexive sheaves. The relative Sk-condition is discussed in Section 2.2.

2.1. Basics on Serre’s condition. The Sk-condition is deﬁned by “depth” and “dimension.” We begin with recalling some elementary properties on dimension, codimension, and on depth. 10 YONGNAMLEEANDNOBORUNAKAYAMA

Property 2.1 (dimension, codimension). Let X be a scheme and let be a quasi- F coherent -module of finite type (cf. [12, 0 , (5.2.1)]), i.e., is quasi-coherent and OX I F locally finitely generated as an -module. Then, Supp is a closed subset (cf. OX F [12, 0I, (5.2.2)]). (1) If Y is a closed subscheme of X such that Y = Supp as a set, then F dim = dim = codim( y , Y ) Fy OY,y { } for any point y Y , where dim is considered as the dimension of the ∈ Fy closed subset Supp of Spec (cf. [12, IV, (5.1.2), (5.1.12)]). Fy OX,y (2) The dimension of , denoted by dim , is defined as dim Supp (cf. [12, F F F IV, (5.1.12)]). Then, dim = sup dim x X F { Fx | ∈ } (cf. [12, IV, (5.1.12.3)]). If X is locally Noetherian, then dim = sup dim x is a closed point of X F { Fx | } by [12, IV, (5.1.4.2), (5.1.12.1), and Cor. (5.1.11)]. Note that the local dimension of at a point x, denoted by dim , is just the infimum of F x F dim for all the open neighborhoods U of x. F|U (3) For a closed subset Z X, the equality ⊂ codim(Z,X) = inf dim z Z { OX,z | ∈ } holds, and moreover, if X is locally Noetherian, then codim (Z,X) = inf dim z Z, x z x { OX,z | ∈ ∈ { }} for any point x X (cf. [12, IV, Cor. (5.1.3)]). Note that codim( ,X) = ∈ ∅ + and that codim (Z,X)=+ if x Z. Furthermore, if Z is locally ∞ x ∞ 6∈ Noetherian, then the function x codim (Z,X) is lower semi-continuous 7→ x on X (cf. [12, 0IV, Cor. (14.2.6)(ii)]). Definition 2.2 (equi-dimensional). Let X be a scheme and a quasi-coherent F -module of finite type. Let A be a ring and M a finitely generated A-module. OX (1) We call X (resp. ) equi-dimensional if all the irreducible components of F X (resp. Supp ) have the same dimension. F (2) We call A (resp. M) equi-dimensional if all the irreducible components of Spec A (resp. Supp M) have the same dimension, where Supp M is the closed subset of Spec A defined by the annihilator ideal Ann(M). Note that Supp M equals Supp M ∼ for the associated quasi-coherent sheaf M ∼ on Spec A. (3) We call X (resp. ) locally equi-dimensional if the local ring (resp. F OX,x the stalk as an -module) is equi-dimensional for any point x X. Fx OX,x ∈ Property 2.3 (catenary). A scheme X is said to be catenary if codim(Y,Z) + codim(Z,T ) = codim(Y,T ) for any irreducible closed subsets Y Z T of X (cf. [12, 0 , Prop. (14.3.2)]). A ⊂ ⊂ IV ring A is said to be catenary if Spec A is so. Then, for a scheme X, it is catenary GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 11 if and only if every local ring is catenary (cf. [12, IV, Cor. (5.1.5)]). If X is a OX,x locally Noetherian scheme and if is catenary for a point x X, then OX,x ∈ codim (Y,X) = dim dim x OX,x − OY,x for any closed subscheme Y of X containing x (cf. [12, IV, Prop. (5.1.9)]).

Property 2.4 (depth). Let A be a Noetherian ring, I an ideal of A, and let M be a

ﬁnitely generated A-module. The I-depth of M, denoted by depthI M, is deﬁned as the length of any maximal M-regular sequence contained in I when M = IM, 6 and as + when M = IM. Here, an element a I is said to be M-regular if a is ∞ ∈ not a zero divisor of M, i.e., the multiplication map x ax induces an injection 7→ M M, and a sequence a ,a ,...,a of elements of I is said to be M-regular if → 1 2 n ai is Mi-regular for any i, where Mi = M/(a1,...,ai−1)M. The following equality is well known (cf. [18, Prop. 3.3], [14, III, Prop. 2.4], [36, Th. 16.6, 16.7]): depth M = inf i Z Exti (A/I,M) =0 . I { ∈ ≥0 | A 6 } If A is a local ring and if I is the maximal ideal mA, then depthI M is denoted simply by depth M; In this case, we have depth M dim M when M = 0 (cf. [12, ≤ 6 0IV, (16.4.5.1)], [36, Exer. 16.1, Th. 17.2]).

Definition 2.5 (Z-depth). Let X be a locally Noetherian scheme and a coherent F -module. For a closed subset Z of X, the Z-depth of is defined as OX F depth = inf depth z Z Z F { Fz | ∈ } (cf. [18, p. 43, Def.], [12, IV, (5.10.1.1)], [2, III, Def. (3.12)]), where the stalk of Fz at z is regarded as an -module. Note that depth 0=+ . F OX,z Z ∞ Property 2.6 (cf. [18, Th. 3.8]). In the situation above, for a given integer k 1, ≥ one has the equivalence: depth k i ( )=0 forany i < k. Z F ≥ ⇐⇒ HZ F Here, i ( ) stands for the i-th local cohomology sheaf of with support in Z HZ F F (cf. [18], [14]). In particular, the condition: depth 1 (resp. 2) is equiva- Z F ≥ ≥ lent to that the restriction homomorphism j ( ) is an injection (resp. F → ∗ F|X\Z :isomorphism) for the open immersion j : X Z ֒ X. Furthermore, the condition \ → depth 3 is equivalent to: j ( ) and R1j ( ) = 0. Z F ≥ F ≃ ∗ F|X\Z ∗ F|X\Z Remark (cf. [18, Cor. 3.6], [2, III, Cor. 3.14]). Let A be a Noetherian ring with an ideal I and let M be a finitely generated A-module. Then,

∼ depthI M = depthZ M for the closed subscheme Z = Spec A/I of X = Spec A and for the coherent - OX module M ∼ associated with M.

Remark 2.7 (associated prime). Let be a coherent -module on a locally Noe- F OX therian scheme X. A point x X is called an associated point of if the maximal ∈ F ideal m is an associated prime of the stalk (cf. [12, IV, D´ef. (3.1.1)]). This x Fx 12 YONGNAMLEEANDNOBORUNAKAYAMA condition is equivalent to: depth = 0. We denote by Ass( ) the set of associ- Fx F ated points. This is a discrete subset of Supp . If an associated point x of is F F not a generic point of , i.e., depth = 0 and dim > 0, then x is called the F Fx Fx embedded point of . If X = Spec A and = M ∼ for a Noetherian ring A and for F F a ﬁnitely generated A-module M, then Ass( ) is just the set of associated primes F of M, and the embedded points of are the embedded primes of M. F Remark 2.8. Let φ: j ( ) be the homomorphism in Property 2.6 and set F → ∗ F|X\Z U = X Z. Then, φ is an injection (resp. isomorphism) at a point x Z, i.e., the \ ∈ homomorphism φ : (j ( )) x Fx → ∗ F|U x of stalks is an injection (resp. isomorphism), if and only if

depth ′ 1 (resp. 2) Fx ≥ ≥ for any x′ Z such that x x′ . In fact, φ is identical to the inverse image ∈ ∈ { } x p∗(φ) by a canonical morphism p : Spec X, and it is regarded as the x x OX,x → restriction homomorphism of p∗ ( ) to the open subset U = p−1(U) via the base x F x x change isomorphism p∗(j ( )) j ((p∗ ) ) x ∗ F|U ≃ x∗ xF |Ux .cf. Lemma A.9 below), where jx stands for the open immersion Ux ֒ Spec X,x) −1 → O For the complement Zx = px (Z) of Ux in Spec X,x, by Property 2.6, we know ∗ O that px(φ) is an injection (resp. isomorphism) if and only if depth p∗( ) 1 (resp. 2). Zx x F ≥ ≥ This implies the assertion, since Z is identical to the set of points x′ Z such that x ∈ x x′ . ∈ { } We recall Serre’s condition Sk (cf. [12, IV, Déf. (5.7.2)], [2, VII, Def. (2.1)], [36, p. 183]): Definition 2.9. Let X be a locally Noetherian scheme, a coherent -module, F OX and k a positive integer. We say that satisfies S if the inequality F k depth inf k, dim Fx ≥ { Fx} holds for any point x X, where the stalk at x is considered as an -module. ∈ Fx OX,x We say that satisfies S at a point x X if F k ∈ depth inf k, dim Fy ≥ { Fy} for any point y X such that x y . We say that X satisfies S , if does so. ∈ ∈ { } k OX ′ Remark. In the situation of Definition 2.9, assume that = i∗( ) for a closed ′ ′F F immersion i: X ֒ X and for a coherent ′ -module . Then, satisfies S if → OX F F k and only if ′ does so. In fact, F depth =+ and dim = Fx ∞ Fx −∞ for any x X′, and 6∈ depth = depth ′ and dim = dim ′ Fx Fx Fx Fx GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 13 for any x X′ (cf. [12, 0 , Prop. (16.4.8)]). ∈ IV Remark 2.10. Let A be a Noetherian ring and M a finitely generated A-module. For a positive integer k, we say that M satisfies Sk if the associated coherent sheaf M ∼ on Spec A satisfies S . Then, for X, , and x in Definition 2.9, satisfies S k F F k at x if and only if the -module satisfies S . In fact, by considering Supp OX,x Fx k Fx as a closed subset of Spec and by the canonical morphism Spec X, OX,x OX,x → we can identify Supp with the set of points y Supp such that x y . Fx ∈ F ∈ { } Definition 2.11 (Cohen–Macaulay). Let A be a Noetherian local ring and M a finitely generated A-module. Then, M is said to be Cohen–Macaulay if depth M = dim M unless M = 0 (cf. [12, 0 , Déf. (16.5.1)], [36, 17]). In particular, if IV § dim A = depth A, then A is called a Cohen–Macaulay local ring. Let X be a locally Noetherian scheme and a coherent -module. If the -module F OX OX,x Fx is Cohen–Macaulay for any x X, then is said to be Cohen–Macaulay (cf. [12, ∈ F IV, Déf. (5.7.1)]. If is Cohen–Macaulay, then X is called a Cohen–Macaulay OX scheme. Remark 2.12. For A and M above, it is known that if M is Cohen–Macaulay, then the localization Mp is also Cohen–Macaulay for any prime ideal p of A (cf. [12, 0IV, Cor. (16.5.10)], [36, Th. 17.3]). Hence, M is Cohen–Macaulay if and only if M satisfies S for any k 1. k ≥ Definition 2.13 (S ( ), CM( )). Let X be a locally Noetherian scheme and let k F F be a coherent -module. For an integer k 1, the S -locus S ( ) of is F OX ≥ k k F F defined to be the set of points x X at which satisfies S (cf. Definition 2.9). The ∈ F k Cohen–Macaulay locus CM( ) of is defined to be the set of points x such F F ∈ F that is a Cohen–Macaulay -module. By definition and by Remark 2.12, one Fx OX,x has: CM( )= S ( ). We define S (X) := S ( ) and CM(X) = CM( ), F k≥1 k F k k OX OX and call them the S -locus and the Cohen–Macaulay locus of X, respectively. T k Remark. It is known that S ( ) and CM( ) are open subsets when X is locally a k F F subscheme of a regular scheme (cf. [12, IV, Prop. (6.11.2)(ii)]). In Proposition 4.11 below, we shall prove the openness when X admits a dualizing complex. Remark. For a locally Noetherian scheme X, every generic point of X is contained in the Cohen–Macaulay locus CM(X). For, dim A = depth A = 0 for any Artinian local ring A.

Lemmas 2.14 and 2.15 below are basic properties on the condition Sk. Lemma 2.14. Let X be a locally Noetherian scheme and let be a coherent - G OX module. For a positive integer k, the following conditions are equivalent to each other: (i) The sheaf satisﬁes S . G k (ii) The inequality depth inf k, codim(Z, Supp ) Z G ≥ { G } holds for any closed (resp. irreducible and closed) subset Z Supp . ⊂ G 14 YONGNAMLEEANDNOBORUNAKAYAMA

(iii) The sheaf satisfies S when k 2, and depth k for any closed G k−1 ≥ Z G ≥ (resp. irreducible and closed) subset Z Supp such that codim(Z, Supp ) ⊂ G G k. ≥ (iv) There is a closed subset Z Supp such that depth k and ⊂ G Z G ≥ G|X\Z satisfies Sk. Proof. We may assume that is not zero. The equivalence (i) (ii) follows from G ⇔ Definitions 2.5 and 2.9 and from the equality: dim = codim( x , Supp ) for x Gx { } G ∈ Supp in Property 2.1(1). The equivalence (i) (ii) implies the equivalence: (ii) G ⇔ (iii). We have (i) (iv) by taking a closed subset Z with codim(Z, Supp ) k ⇔ ⇒ G ≥ using the inequality in (ii). It is enough to show: (iv) (i). More precisely, it is ⇒ enough to prove that, in the situation of (iv), the inequality depth inf k, dim Gx ≥ { Gx} holds for any point x X. If x Z, then this holds, since satisfies S . If x ∈ 6∈ G|X\Z k ∈ Z, then dim depth depth k (cf. Property 2.4 and Definition 2.5), Gx ≥ Gx ≥ Z G ≥ and it induces the inequality above. Thus, we are done. Lemma 2.15. Let X be a locally Noetherian scheme and a coherent -module. G OX Then, for any closed subset Z of X, the following hold: (1) One has the inequality depth codim(Z Supp , Supp ). Z G ≤ ∩ G G (2) For an integer k> 0, if satisfies S and if codim(Z Supp , Supp ) k, G k ∩ G G ≥ then depth k. Z G ≥ Proof. The inequality in (1) follows from the inequality depth dim for any Gx ≤ Gx x Supp , since ∈ G codim(Z Supp , Supp ) = inf dim x Z Supp and ∩ G G { Gx | ∈ ∩ G} depth = inf depth x Z Supp Z G { Gx | ∈ ∩ G} when Z Supp = , by Property 2.1 and Definition 2.5. The assertion (2) is ∩ G 6 ∅ derived from the equivalence (i) (ii) of Lemma 2.14. ⇔ For the conditions S1 and S2, we have immediately the following corollary of Lemma 2.14 by considering Property 2.6. Corollary 2.16. Let X be a locally Noetherian scheme and let be a coherent G -module. The following three conditions are equivalent to each other, where j OX :denotes the open immersion X Z ֒ X \ → (i) The sheaf satisfies S (resp. S ). G 1 2 (ii) For any closed subset Z Supp with codim(Z, Supp ) 1 (resp. ⊂ G G ≥ ≥ 2), the restriction homomorphism j ( ) is injective (resp. an G → ∗ G|X\Z isomorphism, and satisfies S ). G 1 (iii) There is a closed subset Z Supp such that satisfies S (resp. ⊂ G G|X\Z 1 S ) and the restriction homomorphism j ( ) is injective (resp. 2 G → ∗ G|X\Z an isomorphism). GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 15

Remark. Let X be a locally Noetherian scheme and a coherent -module. G OX Then, by definition, satisfies S if and only if has no embedded points (cf. G 1 G Remark 2.7). In particular, the following hold when satisfies S : G 1 (1) Every coherent -submodule of satisfies S (cf. Lemma 2.17(2) below). OX G 1 (2) The sheaf om ( , ) satisfies S for any coherent -module . H OX F G 1 OX F (3) Let T be the closed subscheme defined by the annihilator of , i.e., is G OT the image of the natural homomorphism om ( , ). Then, T OX → H OX G G also satisfies S1.

Lemma 2.17. Let X be a locally Noetherian scheme and let be the kernel of a G homomorphism 0 1 of coherent -modules. E →E OX (1) Let Z be a closed subset of X. If depth 0 1, then depth 1. If Z E ≥ Z G ≥ depth 0 2 and depth 1 1, then depth 2. Z E ≥ Z E ≥ Z G ≥ (2) If 0 satisfies S , then satisfies S . E 1 G 1 (3) Assume that Supp Supp 1. If 1 satisfies S and 0 satisfies S , then G ⊂ E E 1 E 2 satisfies S . G 2 Proof. Let be the image of 0 1. Then, we have an exact sequence B E →E 0 0 ( ) 0 ( 0) 0 ( ) 1 ( ) 1 ( 0) → HZ G → HZ E → HZ B → HZ G → HZ E and an injection 0 ( ) 0 ( 1) of local cohomology sheaves with support in HZ B → HZ E Z (cf. [18, Prop. 1.1]). Thus, (1) is derived from Property 2.6. The remaining assertions (2) and (3) are consequences of (1) above and the equivalence: (i) (ii) ⇔ in Lemma 2.14.

n Lemma 2.18. Let P be the n-dimensional projective space Pk over a field k and let be a coherent -module such that satisfies S and that every irreducible G OP G 1 component of Supp has positive dimension. Then, H0(P, (m)) = 0 for any G G m 0, where we write (m)= (m). ≪ G G⊗OP OP Proof. We shall prove by contradiction. Assume that H0(P, ( m)) = 0 for in- G − 6 finitely many m > 0. There is a member D of (k) for some k > 0 such |OP | that D Ass( ) = (cf. Remark 2.7). Thus, the inclusion ( D) in- ∩ G ∅ OP − ⊂ OP duces an injection ( D) := ( D) . Hence, we have an injection G − G ⊗OP OP − → G ( k) ( D) , and we may assume that H0(P, ( m)) = H0(P, ) = 0 for G − ≃ G − → G G − G 6 any m> 0 by replacing with ( l) for some l> 0. Let ξ be a non-zero element G G − of H0(P, ), which corresponds to a non-zero homomorphism . Let T be G OP → G the closed subscheme of P such that is the image of . Then, T is non- OT OP → G empty and is contained in the affine open subset P D, since ξ H0(P, ( D)). \ ∈ G − Therefore, T is a finite set, and T Ass( ). Since satisfies S , every point of T ⊂ G G 1 is an irreducible component of Supp . This contradicts the assumption. G

Definition 2.19 (reflexive sheaf). For a scheme X and an X -module , we ∨ O ∨∨ F write for the dual X -module omOX ( , X ). The double-dual of is F ∨ ∨ O H F O ∨ F F defined as ( ) . The natural composition homomorphism X defines a F ∨∨ F⊗F → O canonical homomorphism c : . Note that c ∨ is always an isomorphism. F F→F F 16 YONGNAMLEEANDNOBORUNAKAYAMA

If is a quasi-coherent -module of finite type and if c is an isomorphism, then F OX F is said to be reflexive. F Remark 2.20. Let π : Y X be a flat morphism of locally Noetherian schemes. → Then, the dual operation ∨ commutes with π∗, i.e., there is a canonical isomorphism

π∗ om ( , ) om (π∗ , ) H OX F OX ≃ H OY F OY for any coherent -module . In particular, if is reﬂexive, then so is π∗ . OX F F F This isomorphism is derived from [12, 0 , (6.7.6)], since every coherent -module I OX has a ﬁnite presentation locally on X.

Lemma 2.21. Let X be a locally Noetherian scheme, Z a closed subset, and a G coherent -module. OX (1) For an integer k = 1 or 2, assume that depth k and that is Z OX ≥ G reflexive. Then, depth k. Z G ≥ (2) For an integer k =1 or 2, assume that X satisfies S and that is reflexive. k G Then, satisfies S . G k (3) Assume that depth 1 and that is reflexive. If depth 2, Z OX ≥ G|X\Z Z G ≥ then is reflexive. G Proof. For the proof of (1), by localizing X, we may assume that there is an exact sequence ∨ 0 for some free -modules and of finite rank. E1 → E0 → G → OX E0 E1 Taking the dual, we have an exact sequence 0 ∨∨ ∨ ∨ (cf. →G≃G → E0 → E1 the proof of [20, Proposition 1.1]). The condition: depth k implies that Z OX ≥ depth ∨ k for i = 0, 1. Thus, depth k by Lemma 2.17(1). This proves Z Ei ≥ Z G ≥ (1). The assertion (2) is a consequence of (1) (cf. Definition 2.9). We shall show (3). ,Let j : X Z ֒ X be the open immersion. Then, j ( ) by Property 2.6 \ → G ≃ ∗ GX\Z since depth 2 by assumption. Hence, we have a splitting of the canonical Z G ≥ homomorphism ∨∨ into the double-dual by the commutative diagram G → G ∨∨ G −−−−→ G ≃

 ≃ ∨∨ j∗( X\Z ) j∗(  X\Z ). G|y −−−−→ G y| Thus, we have an injection ֒ ∨∨ from := ∨∨/ , where Supp Z. The C → G C G G C ⊂ injection corresponds to a homomorphism ∨ , but this is zero, since C ⊗ G → OX depth 1. Therefore, = 0 and is reﬂexive. This proves (3), and we are Z OX ≥ C G done.

Corollary 2.22. Let X be a locally Noetherian scheme, Z a closed subset, and G a coherent -module. Assume that is reflexive and codim(Z,X) 1. Let OX G|X\Z ≥ us consider the following three conditions: (i) satisfies S and codim(Z Supp , Supp ) 2; G 2 ∩ G G ≥ (ii) depth 2; Z G ≥ (iii) is reflexive. G GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 17

Then, (i) (ii) holds always. If depth 1, then (ii) (iii) holds, and if ⇒ Z OX ≥ ⇒ depth 2, then (ii) (iii) holds. If X satisﬁes S and codim(Z,X) 2, Z OX ≥ ⇔ 2 ≥ then these three conditions are equivalent to each other.

Proof. The implication (i) (ii) is shown in Lemma 2.15(2). The next implication ⇒ (ii) (iii) in case depth 1 follows from Lemma 2.21(3), and the converse ⇒ Z OX ≥ implication (iii) (ii) in case depth 2 follows from Lemma 2.21(1). Assume ⇒ Z OX ≥ that X satisfies S and codim(Z,X) 2. Then, depth 2 by Lemma 2.15(2), 2 ≥ Z OX ≥ and we have (ii) (iii) in this case. It remains to prove: (ii) (i). Assume that ⇔ ⇒ depth 2. Then, codim(Z Supp , Supp ) 2 by Lemma 2.15(1). On the Z G ≥ ∩ G G ≥ other hand, the reflexive sheaf satisfies S by Lemma 2.21(2), since X Z G|X\Z 2 \ satisfies S . Thus, satisfies S by the equivalence (i) (iv) of Lemma 2.14, and 2 G 2 ⇔ we are done.

Remark. If X is a locally Noetherian scheme satisfying S1, then the support of a reflexive -module is a union of irreducible components of X. In fact, if is OX G reflexive, then depth 1 for any closed subset Z with codim(Z,X) 1, by Z G ≥ ≥ Lemma 2.21(1), and we have codim(Z Supp , Supp ) 1 by Lemma 2.15(1): ∩ G G ≥ This means that Supp is a union of irreducible components of X. In particular, G if X is irreducible and satisfies S , and if = 0, then Supp = X. However, 1 G 6 G Supp = X in general when X is reducible. For example, let R be a Noetherian ring G 6 with two R-regular elements u and v, and set X := Spec R/uvR and := (R/uR)∼. G Then, we have an isomorphism om ( , ) by the natural exact sequence H OX G OX ≃ G 0 R/uR R/uvR u× R/uvR R/uR 0. → → −−→ → → Thus, is a reflexive -module, but Supp = X when u √vR. G OX G 6 6∈

We have discussed properties S1 and S2 for general coherent sheaves. Finally in Section 2.1, we note the following well-known facts on locally Noetherian schemes satisfying S2.

Fact 2.23. Let X be a locally Noetherian scheme satisfying S2. (1) If X is catenary (cf. Property 2.3), then X is locally equi-dimensional (cf. Deﬁnition 2.2(3)) (cf. [12, IV, Cor. (5.1.5), (5.10.9)]). (2) For any open subset X◦ with codim(X X◦,X) 2 and for any connected \ ≥ component X of X, the intersection X X◦ is connected. This is a α α ∩ consequence of a result of Hartshorne (cf. [12, IV, Th. (5.10.7)], [14, III, Th. 3.6]).

2.2. Relative Sk-conditions. Here, we shall consider the relative Sk-condition for morphisms of locally Noetherian schemes.

Notation 2.24. Let f : Y T be a morphism of schemes. For a point t T , the → ∈ fiber f −1(t) of f over t is defined as Y Spec k(t), and it is denoted by Y . For an ×T t -module , the restriction k(t) to the fiber Y is denoted OY F F ⊗OY OYt ≃F⊗OT t by . F(t) 18 YONGNAMLEEANDNOBORUNAKAYAMA

Remark. The restriction is identified with the inverse image p∗( ) for the F(t) t F projection p : Y Y , and Supp is identified with Y Supp = p−1(Supp ). t t → F(t) t ∩ F t F If f is the identity morphism Y Y , then is a sheaf on Spec k(y) corresponding → F(y) to the vector space k(y) for y Y . Fy ⊗ ∈ Definition 2.25. For a morphism f : Y T of schemes and for an -module → OY , let Fl( /T ) be the set of points y Y such that is a flat -module. F F ∈ Fy OT,f(y) If Y = Fl( /T ), then is said to be flat over T , or f-flat. If S is a subset of F F Fl( /T ), then is said to be flat over T along S, or f-flat along S. F F Fact 2.26. Let f : Y T be a morphism of locally Noetherian schemes and k a → positive integer. For a coherent -module and a coherent -module , the OY F OT G following results are known, where in (2), (3), and (4), we fix an arbitrary point y Y , and set t = f(y): ∈ (1) If f is locally of finite type, then Fl( /T ) is open. F (2) If is flat over and if ( ) is a free -module, then is a free Fy OT,t F(t) y OYt,y Fy -module. In particular, if is flat over T and if is locally free for OY,y F F(t) any t T , then is locally free. ∈ F (3) If is non-zero and flat over , then the following equalities hold: Fy OT,t (II-1) dim( f ∗ ) = dim( ) + dim , F ⊗OY G y F(t) y Gt (II-2) depth( f ∗ ) = depth( ) + depth . F ⊗OY G y F(t) y Gt (4) If is non-zero and flat over and if f ∗ satisfies S at y, then Fy OT,t F⊗OY G k satisfies S at t. G k (5) Assume that is flat over T along the fiber Y over a point t f(Supp ). F t ∈ F If satisfies S and if satisfies S at t, then f ∗ also satisfies F(t) k G k F ⊗OY G Sk at any point of Yt. (6) Assume that f is flat and that every fiber Y satisfies S . Then, f ∗ satisfies t k G S at y if and only if satisfies S at f(y). k G k The assertion (1) is just [12, IV, Th. (11.1.1)]. The assertion (2) is a consequence of Proposition A.1 and Lemma A.5, since = /I for the ideal I = m OYt,y OY,y T,tOY,y and we have TorOY,y ( , /I)=0 and ( ) /I 1 Fy OY,y F(t) y ≃Fy Fy under the assumption of (2). Two equalities (II-1) and (II-2) in (3) follow from [12, IV, Cor. (6.1.2), Prop. (6.3.1)], since ( f ∗ ) and ( ) k(t). F ⊗OY G y ≃Fy ⊗OT,t Gt F(t) y ≃Fy ⊗OT,t The assertions (4) and (5) are shown in [12, IV, Prop. (6.4.1)] by the equalities (II-1) and (II-2), and the assertion (6) is a consequence of (4) and (5) (cf. [12, IV, Cor. (6.4.2)]). Corollary 2.27. Let f : Y T be a flat morphism of locally Noetherian schemes. → Let W be a closed subset of T contained in f(Y ). Then, −1 ∗ codim(f (W ), Y ) = codim(W, T ) and depth −1 f = depth f (W ) G W G for any coherent -module . OT G GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 19

Proof. We may assume that = 0. Then, G 6 codim(f −1(W ), Y ) = inf dim y f −1(W ) , { OY,y | ∈ } ∗ ∗ −1 depth −1 f = inf depth(f ) y f (W ) , f (W ) G { G y | ∈ } by Property 2.1 and Definition 2.5. Thus, we can prove the assertion by applying (II-1) to ( , ) = ( , ) and (II-2) to ( , ) = ( , ), since F G OY OT F G OY G dim = codim(W, T ) and dim =0 OT,t OYt,y for a certain generic point t of W and a generic point y of Yt, and since depth = depth and depth((f ∗ ) ) = depth =0 Gt W G G (t) y OYt,y for a certain point t W Supp and for a generic point y of Y . ∈ ∩ G t Definition 2.28. Let f : Y T be a morphism of locally Noetherian schemes and → a coherent -module. As a relative version of Definition 2.13, for a positive F OY integer k, we define

Sk( /T ) := Fl( /T ) Sk( (t)) and F F ∩ t∈T F [ CM( /T ) := Fl( /T ) CM( (t)), F F ∩ t∈T F and call them the relative S -locus and the[relative Cohen–Macaulay locus of k F over T , respectively. We also write S (Y/T )= S ( /T ) and CM(Y/T ) = CM( /T ), k k OY OY and call them the relative Sk-locus and the relative Cohen–Macaulay locus for f, respectively. The relative Sk-condition and the relative Cohen–Macaulay condition are defined as follows: For a point y Y (resp. a subset S Y ), we say that satisfies relative • ∈ ⊂ F S over T at y (resp. along S) if y S ( /T ) (resp. S S ( /T )). We k ∈ k F ⊂ k F also say that is relatively Cohen–Macaulay over T at y (resp. along S) F if y CM( /T ) (resp. S CM( /T )). ∈ F ⊂ F We say that satisfies relative S over T if Y = S ( /T ). We also say • F k k F that is relatively Cohen–Macaulay over T if Y = CM( /T ). F F Fact 2.29. For f : Y T and in Definition 2.28, assume that f is locally of finite → F type and is flat over T . Then, the following properties are known: F (1) The subset CM( /T ) is open (cf. [12, IV, Th. (12.1.1)(vi)]). F (2) If is locally equi-dimensional (cf. Definition 2.2(3)) for any t T , then F(t) ∈ S ( /T ) is open for any k 1 (cf. [12, IV, Th. (12.1.1)(iv)]). k F ≥ (3) If Y T is flat, then S (Y/T ) is open for any k 1 (cf. [12, IV, → k ≥ Th. (12.1.6)(i)]).

Definition 2.30 (S -morphism and Cohen–Macaulay morphism). Let f : Y T k → be a morphism of locally Noetherian schemes and k a positive integer. The f is called an Sk-morphism (resp. a Cohen–Macaulay morphism) if f is a flat morphism locally of finite type and Y = Sk(Y/T ) (resp. Y = CM(Y/T )). For a subset S of 20 YONGNAMLEEANDNOBORUNAKAYAMA

Y , f is called an Sk-morphism (resp. a Cohen–Macaulay morphism) along S if f : V T is so for an open neighborhood V of S (cf. Fact 2.29(3)). |V → Remark. The Sk-morphisms and the Cohen–Macaulay morphisms defined in [12, IV, Déf. (6.8.1)] are not necessarily locally of finite type. The definition of Cohen– Macaulay morphism in [17, V, Ex. 9.7] coincides with ours. The notion of “CM map” in [7, p. 7] is the same as that of Cohen–Macaulay morphism in our sense for morphisms of locally Noetherian schemes. Lemma 2.31. Suppose that we are given a Cartesian diagram p Y ′ Y −−−−→ f ′ f

′ q  T T −−−−→ of schemes consisting of locally Noetheriany schemes.y Let be a coherent - F OY module, Z a closed subset of Y , k a positive integer, and let t′ T ′ and t T be ∈ ∈ points such that t = q(t′). ′ ′−1 ′ −1 (1) If f is flat, then, for the fibers Yt′ = f (t ) and Yt = f (t), one has −1 ′ ′ codim(p (Z) Y ′ , Y ′ ) = codim(Z Y , Y ), and ∩ t t ∩ t t − ′ ′ depthp 1(Z)∩Y Y ′ = depthZ∩Y Yt . t′ O t t O (2) If is flat over T , then F ∗ depthp−1(Z)∩Y ′ (p )(t′) = depthZ∩Y (t). t′ F t F (3) If is flat over T , then S (p∗ /T ′) p−1S ( /T ). If f is locally of finite F k F ⊂ k F type in addition, then S (p∗ /T ′)= p−1S ( /T ). k F k F (4) If f is locally of finite type and if satisfies relative S over T , then p∗ F k F does so over T ′. ′ (5) If f is an Sk-morphism (resp. Cohen–Macaulay morphism), then so is f . Proof. The assertions (1) and (2) follow from Corollary 2.27 applied to the flat ′ morphism Yt′ Yt and to = Yt or = (t). The first half of (3) follows from → G O G F ′ Definition 2.28 and Fact 2.26(4) applied to Y ′ Yt and to ( , ) = ( Y ′ , (t)). t → F G O t′ F The latter half of (3) follows from Fact 2.26(6), since the fiber p−1(y) over a point ′ ′ y Y is isomorphic to Spec k(y) k k(t ) and since k(y) k k(t ) is Cohen– ∈ t ⊗ (t) ⊗ (t) Macaulay (cf. [12, IV, Lem. (6.7.1.1)]). The assertion (4) is a consequence of (3), and the assertion (5) follows from (3) in the case: = , by Definition 2.30. F OY Lemma 2.32. Let Y T be a morphism of locally Noetherian schemes and let Z → be a closed subset of Y . Let be a coherent -module and k a positive integer. F OY (1) If is flat over T , then F depth inf depth t f(Z) . Z F ≥ { Z∩Yt F(t) | ∈ } (2) If satisfies relative S over T and if F k codim(Z Supp , Supp ) k ∩ F(t) F(t) ≥ for any t T , then depth k. ∈ Z F ≥ GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 21

(3) If Y T is flat and if one of the two conditions below is satisfied, then → depth k: Z OY ≥ (a) depth k for any t T ; Yt∩Z OYt ≥ ∈ (b) Y satisfies S and codim(Y Z, Y ) k for any t T . t k t ∩ t ≥ ∈ Proof. For the first assertion (1), we may assume that Z Supp = . Then, by ∩ F 6 ∅ Definition 2.5, we have the inequality in (1) from the equality (II-2) in Fact 2.26(3) in the case where = , since depth 0 for any t T . The assertion (2) G OT OT,t ≥ ∈ is a consequence of (1) and Lemma 2.15(2) applied to (Y ,Z Y , ). The last t ∩ t F(t) assertion (3) is derived from (1) and (2) in the case where = . F OY The following result gives some relations between the reflexive modules and the relative S -condition. Similar results can be found in [21, 3]. 2 § Lemma 2.33. Let f : Y T be a flat morphism of locally Noetherian schemes, → F a coherent -module, and Z a closed subset of Y . Assume that OY depth 1 Yt∩Z OYt ≥ for any fiber Y = f −1(t). Then, the following hold for the open immersion j : Y t \ :( ) Z ֒ Y and for the restriction homomorphism j → F → ∗ F|Y \Z (1) If is reflexive and if j ( ), then is reflexive. F|Y \Z F ≃ ∗ F|Y \Z F (2) If is reflexive and if depth 2 for any t T , then F Yt∩Z OYt ≥ ∈ F ≃ j ( ). ∗ F|Y \Z (3) If is flat over T and if depth 2 for any t T , then F Yt∩Z F(t) ≥ ∈ F ≃ j ( ). ∗ F|Y \Z (4) If Y satisfies S and codim(Y Z, Y ) 2 for any t T , and if is t 2 t ∩ t ≥ ∈ F reflexive, then j ( ). F ≃ ∗ F|Y \Z (5) If satisfies relative S over T and if codim(Z Supp , Supp ) 2 F 2 ∩ F(t) F(t) ≥ for any t T , then j ( ). ∈ F ≃ ∗ F|Y \Z (6) In the situation of (3) or (5), if is reflexive, then is reflexive; F(t)|Yt\Z F(t) if is reflexive, then is reflexive. F|Y \Z F Proof. Note that j ( ) if and only if depth 2 (cf. Property 2.6). F ≃ ∗ F|Y \Z Z F ≥ We have depth 1 by Lemma 2.32(3). Hence, (1) is a consequence of Z OY ≥ Lemma 2.21(3). In case (2), we have depth 2 by Lemma 2.32(3), and (2) Z OY ≥ is a consequence of Lemma 2.21(1). The assertion (3) follows from Lemma 2.32(1) with k = 2. The assertions (4) and (5) are special cases of (2) and (3), respectively. The first assertion of (6) follows from Corollary 2.22. The second assertion of (6) is derived from (1) and (3).

Remark. The assumption of Lemma 2.33 holds when Y satisfies S and codim(Y t 1 t ∩ Z, Y ) 1 for any t T (cf. Lemma 2.15(2)). t ≥ ∈ Lemma 2.34. In the situation of Lemma 2.31, assume that f is flat, is F|Y \Z locally free, and depth 2 Yt∩Z OYt ≥ 22 YONGNAMLEEANDNOBORUNAKAYAMA for any t T . Then, ∨∨ j ( ) for the open immersion j : Y Z ֒ Y , and ∈ F ≃ ∗ F|Y \Z \ → (p∗ )∨∨ (p∗( ∨∨))∨∨. F ≃ F Moreover, and p∗ are reflexive if is flat over T and F F F depth 2 Yt∩Z F(t) ≥ for any t T . ∈ Proof. Now, depth 2 by Lemma 2.32(3). Hence, depth ∨∨ 2 by Z OY ≥ Z F ≥ Lemma 2.21(1), and this implies the first isomorphism for ∨∨. We have F

′ − ′ depthY ∩p 1(Z) Y ′ 2 t′ O t ≥ by Lemma 2.31(1). Hence, by the previous argument applied to p∗ and p∗( ∨∨), F F we have isomorphisms

∗ ∨∨ ′ ∗ ∗ ∨∨ ∨∨ (p ) j (p ′ −1 ) (p ( )) F ≃ ∗ F|Y \p (Z) ≃ F -for the open immersion j′ : Y ′ p−1(Z) ֒ Y ′. It remains to prove the last as \ → sertion. In this case, is reﬂexive by (1) and (3) of Lemma 2.33. Moreover, by F Lemma 2.31(2), we have

∗ depthY ′ ∩p−1(Z)(p )(t′) 2 t′ F ≥ for any point t′ T ′. Thus, p∗ is reflexive by the same argument as above. ∈ F Lemma 2.35. Let f : Y T be a morphism of locally Noetherian schemes, and → let Z be a closed subset of Y . Assume that f is quasi-flat (cf. [12, IV, (2.3.3)]), i.e., there is a coherent -module such that is flat over T and Supp = Y . OY F F F Then,

(II-3) codim (Z, Y ) codim (Z Y , Y ) y ≥ y ∩ f(y) f(y) for any point y Z. If codim(Z Y , Y ) k for a point t T and for an integer ∈ ∩ t t ≥ ∈ k, then there is an open neighborhood V of Y in Y such that codim(Z V, V ) k. t ∩ ≥ Proof. For the sheaf above, we have Supp = Y for any t T . If z Z Y , F F(t) t ∈ ∈ ∩ t then

(II-4) dim = dim and dim( ) = dim Fz OY,z F(t) z OYt,z by Property 2.1(1), and moreover,

(II-5) dim = dim( ) + dim dim( ) Fz F(t) z OT,t ≥ F(t) z by (II-1), since is ﬂat over T . Thus, we have (II-3) from (II-4) and (II-5) by F Property 2.1(3). The last assertion follows from (II-3) and the lower-semicontinuity of the function x codim (Z, Y ) (cf. [12, 0 , Cor. (14.2.6)]). In fact, the set of 7→ x IV points y Y with codim (Z, Y ) k is an open subset containing Y . ∈ y ≥ t We introduce the following notion (cf. [12, IV, D´ef. (17.10.1)] and [7, p. 6]). GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 23

Definition 2.36 (pure relative dimension). Let f : Y T be a morphism locally → of finite type. The relative dimension of f at y is defined as dimy Yf(y), and it is denoted by dimy f. We say that f has pure relative dimension d if d = dimy f for any y Y . The condition is equivalent to that every non-empty fiber is equi- ∈ dimensional and has dimension equal to d.

Remark 2.37. If a flat morphism f : Y T is locally of finite type and it has pure → relative dimension, then it is an equi-dimensional morphism in the sense of [12, IV, Déf. (13.3.2), (ErrIV, 35)]. Because, a generic point of Y is mapped a generic point of T by (II-1) applied to = and = , and the condition a′′) of [12, IV, F OY G OT Prop. 13.3.1] is satisfied.

Lemma 2.38. Let f : Y T be a flat morphism locally of finite type between → locally Noetherian schemes. For a point y Y and its image t = f(y), assume that ∈ the fiber Y satisfies S at y for some k 2. Let Y ◦ be an open subset of Y with t k ≥ y Y ◦. Then, there exists an open neighborhood U of y in Y such that 6∈ (1) f : U T is an S -morphism having pure relative dimension, and |U → k (2) the inequality ◦ ◦ codim(U ′ Y ,U ′ ) codim (Y Y , Y ) t \ t ≥ y t \ t ′ holds for any t f(U), where U ′ = U Y ′ . ∈ t ∩ t Proof. By Fact 2.29(3), replacing Y with an open neighborhood of y, we may ′ assume that f is an Sk-morphism. For any point y Y and for the fiber Yt′ over ′ ′ ∈ t = f(y ), the local ring ′ ′ is equi-dimensional by Fact 2.23(1), since Y ′ is OYt ,y t catenary satisfying S2. Moreover, the local ring has no embedded primes by the condition S1. Ifan associated prime cycle Γ of Yt′ (cf. [12, IV, Déf. (3.1.1)]) contains ′ y , then Γ corresponds to an associated prime ideal p of ′ ′ , and we have OY ,t ′ ′ dim Γ = dim ′ Γ = dim ′ ′ /p + tr. deg k(y )/k(t ) y OYt ,y ′ by [12, IV, Prop. (5.2.1), Cor. (5.2.3)]. Since p is minimal and Yt′ ,y is equi- O ′ dimensional, it follows that all the associated prime cycles of Yt′ containing y have the same dimension. Thus, by [12, IV, Th. (12.1.1)(ii)], we may assume that f has pure relative dimension, by replacing Y with an open neighborhood of y. Consequently, Y T is an equi-dimensional morphism (cf. Remark 2.37). Then, → the function ′ ◦ Y y codim ′ (Y ′ Y , Y ′ ) ∋ 7→ y f(y ) \ f(y ) is lower semi-continuous by [12, IV, Prop. (13.3.7)]. Hence, we can take an open neighborhood U of y satisfying the inequality in (2). Thus, we are done.

Corollary 2.39. Let f : Y T be an S -morphism of locally Noetherian schemes. → 2 Assume that every ﬁber Yt is connected. (1) If T are connected, then f has pure relative dimension. In particular, f is an equi-dimensional morphism. (2) If f is proper, then the function t codim(Y Z, Y ) is lower semi- 7→ t ∩ t continuous on T for any closed subset Z of Y . 24 YONGNAMLEEANDNOBORUNAKAYAMA

Proof. We may assume that T is connected. We know that every ﬁber Yt is equi- dimensional by the proof of Lemma 2.38, since Yt is connected. Moreover, dim Yt is independent of the choice of t T by Lemma 2.38(1), since T is connected. Hence, ∈ f has pure relative dimension, and (1) has been proved. In the case (2), f(Y )= T , since f(Y ) is open and closed. Let us consider the set F of points y Y such that k ∈ codim (Z Y , Y ) k y ∩ f(y) f(y) ≤ for an integer k. Then, f(F ) is the set of points t T with codim(Y Z, Y ) k. k ∈ t ∩ t ≤ Now, Fk is closed by (1) and by [12, IV, Prop. (13.3.7)]. Since f is proper, f(Fk) is closed. This proves (2), and we are done.

3. Relative S2-condition and flatness We shall study restriction homomorphisms (cf. Definition 3.2 below) of coherent sheaves to open subsets by applying the local criterion of flatness (cf. Section A.1), and give several criteria for the restriction homomorphism on a fiber to be an isomorphism. In Section 3.1, we prove the key proposition (Proposition 3.7) and discuss related properties. Some applications of Proposition 3.7 are given in Sec- tion 3.2: Theorem 3.16 is a criterion for a sheaf to be invertible, which is used in the proof of Theorem 5.10 below. Proposition 3.19 gives infinitesimal and valuative criteria for a reflexive sheaf to satisfy relative S2-condition. Theorem 3.26 on the relative S2 refinement is analogous to the flattening stratification theorem by Mumford in [38, Lect. 8] and to the representability theorem of unramified functors by Murre [40]. Its local version is given as Theorem 3.28. 3.1. Restriction homomorphisms. In Section 3.1, we work under Situation 3.1 below unless otherwise stated: Situation 3.1. We fix a morphism f : Y T of locally Noetherian schemes, a closed → subset Z of Y , and a coherent -module . The complement of Z in Y is written OY F .as U, and j : U ֒ Y stands for the open immersion → Definition 3.2. The restriction morphism of to U is defined as the canonical F homomorphism φ = φ ( ): j ( ). U F F → ∗ F|U Similarly, for a point t T , the restriction homomorphism of to U (or to ∈ F(t) U Y ) is defined as the canonical homomorphism ∩ t φ = φ ( ): j ( ). t U F(t) F(t) → ∗ F(t)|U∩Yt Here, U Y is identical to U Y , and j stands also for the open immersion ∩ t ×Y t . U Y ֒ Y ∩ t → t Remark. The homomorphism φ is an isomorphism along U Y . In particular, φ t ∩ t t is an isomorphism if t f(Z). 6∈ Remark. By Remark 2.8, we see that φ is an injection (resp. isomorphism) along Yt if and only if depth 1 (resp. 2) Fy ≥ ≥ GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 25 for any point y Z such that Y y = . ∈ t ∩ { } 6 ∅ We use the following notation only in Section 3.1. Notation 3.3. For simplicity, we write := j ( ) and := j ( ). F∗ ∗ F|U F(t)∗ ∗ F(t)|U∩Yt When we fix a point t of f(Z), we write A for the local ring and m for the OT,t maximal ideal m , and for an integer n 0, we set T,t ≥ A := A/mn+1,T := Spec A , Y := Y T , n n n n ×T n U = Y U, := , := j ( ). n n ∩ Fn F ⊗OY OYn Fn∗ ∗ Fn|Un In particular, Y = Y , = , = , and Y is a closed subscheme of Y t 0 F(t) F0 F(t)∗ F0∗ n m for any m n. Furthermore, the restriction homomorphisms of and ( ) , ≥ Fn Fn∗ (t) respectively, are written by φ : = j ( ) and ϕ : ( ) = . n Fn →Fn∗ ∗ Fn|Un n Fn∗ (t) Fn∗ ⊗OYn OY0 →F0∗ Remark 3.4. The homomorphism φt in Definition 3.2 equals φ0, and the diagram

φn⊗OY 0 ( ) Fn ⊗OY OY0 −−−−−→ Fn∗ ⊗OY OY0 ≃ ϕn

 φ0    Fy0 −−−−→ Fy0∗ is commutative for any n 0. ≥ Lemma 3.5. Assume that is flat over T . F|U (1) For a point y Z and t = f(y), if φ is injective at y, then is flat over ∈ t Fy . OT,t (2) For a point y Z and t = f(y), if φ is an isomorphism at y, then the ∈ t restriction homomorphism φ : is an isomorphism at y for any n Fn → Fn∗ n 0. ≥ (3) If φ is an isomorphism for any t f(Z), then φ is also an isomorphism. t ∈ Proof. First, we shall prove (3) assuming (1) and (2). Since φt is an isomorphism for any t f(Z), is flat over T by (1), and we have ∈ F depth 2 Yt∩Z F(t) ≥ by (2) (cf. Property 2.6). Then, depth 2 by Lemma 2.32(1), and φ is an Z F ≥ isomorphism (cf. Property 2.6). Next, we shall prove (1) and (2). We may assume that T = Spec A for a local Noetherian ring A in which t = f(y) corresponds to the maximal ideal m of A and that Y = Spec for the given point y (cf. Remark 2.8). We write k = A/m = OY,y k(t) and use Notation 3.3. From the standard exact sequence 0 mn/mn+1 A A 0 → → n → n−1 → of A-modules, by taking tensor products with over A, we have an exact sequence F n n+1 un (III-1) m /m k 0 ⊗ F0 −−→Fn →Fn−1 → 26 YONGNAMLEEANDNOBORUNAKAYAMA of -modules. Here, the left homomorphism u is injective at y for any n 0 if OY n ≥ and only if is flat over by the local criterion of flatness (cf. Proposition A.1). Fy OT,t Now, u is injective on the open subset U , since is flat over T , and u is the n n F|U 0 identity morphism. For each n> 0, there is a natural commutative diagram n n+1 un m /m k ⊗ F0 −−−−→ Fn −−−−→ Fn−1

id ⊗φ0 φn φn−1

n n+1  j∗(un|Un )   0 m /m k j ( ) j (  ) j (  − ) −−−−→ ⊗y ∗ F0|U0 −−−−−−→ ∗ Fyn|Un −−−−→ ∗ Fn−y1|Un 1 of exact sequences. By assumption, φ0 = φt is an injection (resp. isomorphism) at y in case (1) (resp. (2)) (cf. Remark 2.8). Thus, un is injective at y for any n by the diagram. This shows (1). In case (2), by induction on n, we see that φn is an isomorphism at y for any n, by the diagram. Thus, (2) also holds, and we are done.

Applying Lemma 3.5 to = , we have: F OY Corollary 3.6. Suppose that U is flat over T . If the restriction homomorphism φ ( ): j ( ) is injective for a point t f(Z), then f is flat along Y . t OY OYt → ∗ OYt∩U ∈ t If φ ( ) is an isomorphism for any t f(Z), then j ( ). t OY ∈ OY ≃ ∗ OU Proposition 3.7 (key proposition). Suppose that there is an exact sequence 0 0 1 0 →F→E →E → G → of coherent -modules such that OY (i) 0, 1, and are flat over T , and E E G|U (ii) the inequalities depth 0 2 and depth 1 1 Z∩Yt E(t) ≥ Z∩Yt E(t) ≥ hold for any t f(Z). ∈ Then, the following hold: (1) The restriction homomorphism φ: = j ( ) is an isomorphism. F→F∗ ∗ F|U (2) For a fixed point t f(Z) and for any integer n 0, = j ( ) (cf. ∈ ≥ Fn∗ ∗ Fn|Un Notation 3.3) is isomorphic to the kernel ′ of the homomorphism Fn 0 = 0 1 = 1 En E ⊗OY OYn →En E ⊗OY OYn induced by 0 1. In particular, is coherent for any n 0, and E → E Fn∗ ≥ = j ( ) is coherent for any t f(Z). F(t)∗ ∗ F(t)|U∩Yt ∈ (3) For any point y Y and t = f(y), the following conditions are equivalent ∈ to each other, where we use Notation 3.3 in (a′), (b′), and (b′′): (a) φ : is surjective at y; t F(t) →F(t)∗ (b) φt is an isomorphism at y; (c) is flat over ; Gy OT,t (a′) ϕ : ( ) is surjective at y for any n 0; n Fn∗ (t) →F0∗ ≥ (b′) ϕ is an isomorphism at y for any n 0; n ≥ (b′′) φ : is an isomorphism at y for any n 0. n Fn →Fn∗ ≥ GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 27

Note that if (c) is satisfied, then is also flat over . Fy OT,t Proof. By (i), (ii) and by Lemma 2.32(1), we have depth 0 2 and depth 1 Z E ≥ Z E ≥ 1. Thus, depth 2 by Lemma 2.17(1), and we have (1) (cf. Property 2.6). For Z F ≥ each n 0, the exact sequence ≥ 0 ′ 0 1 0 →Fn →En →En → Gn → on Y satisfies the conditions (i) and (ii) for the induced morphism Y T , where n n → n = . Thus, by (1), the restriction homomorphism Gn G ⊗ OYn φ( ′ ): ′ ( ′ ) = j ( ′ ) Fn Fn → Fn ∗ ∗ Fn|Un of ′ is an isomorphism. On the other hand, there is a canonical homomorphism Fn ψ : = ′ . Note that ψ is an isomorphism at a point y Y = Y n Fn F ⊗ OYn →Fn n ∈ t 0 if is flat over . In particular, ψ is an isomorphism on U by the condition Gy OT,t n n (i). Hence, ( ′ ) , and we have an isomorphism ′ , by which φ Fn ∗ ≃ Fn∗ Fn ≃ Fn∗ n is isomorphic to ψn. This proves (2). For the proof of (3), we may assume that y Z. We shall show that there is an exact sequence ∈ (ψ ) (III-2) TorA( , k) ( ) = 0 y ( ′ ) TorA( , k) 0 2 Gy → F(t) y Fy ⊗OY,y OYt,y −−−→ F0 y → 1 Gy → of -modules, where A = and k = k(t): For the image of 0 1, OY,y OT,t B E → E we have two short exact sequences 0 0 0 and 0 1 →F→E → B → →B→E → 0 on Y . Then, the kernel of = 1 = 1 is isomor- G → B0 B ⊗ OY0 → E0 E ⊗ OY0 phic to or OT ( , k), and the kernel of 0 is isomorphic to or OT ( , k) T 1 G F0 → E0 T 1 B ≃ or OT ( , k). Then, we have the exact sequence (III-2) by applying the snake lemma T 2 G to the commutative diagram

0 0 F0 −−−−→ E0 −−−−→ B0 −−−−→

ψ0 =    0 ′ 0 1 −−−−→ Fy0 −−−−→ Ey0 −−−−→ Ey0 of exact sequences. Note that ψ φ by the argument above. We shall prove 0 ≃ 0 (3) using (III-2). If (a) holds, then TorA( , k) = 0 by (III-2), and it implies (c) 1 Gy by the local criterion of flatness (cf. Proposition A.1), since k Gy ⊗ OYt,y ≃ Gy ⊗ is flat over k. If (c) holds, then TorA( , k) = 0 for j = 1 and 2, and it implies j Gy (b) by (III-2). Thus, we have shown the equivalence of the three conditions (a), (b), and (c). By applying the equivalence of three conditions to ′ and Fn ≃ Fn∗ Y T instead of and Y T , we see that (a′) and (b′) are both equivalent n → n F → to that ( ) is flat over for any n 0; This is also equivalent to (c) by Gn y OTn,t ≥ the local criterion of flatness (cf. (i) (iv) in Proposition A.1). If (c) holds, then ⇔ ψ : ′ is an isomorphism as we have noted before, and the isomorphism n Fn → Fn ′ in (2) implies (b′′). Conversely, if (b′′) holds, then ϕ is isomorphic to Fn ≃ Fn∗ n the canonical isomorphism ( ) for any n (cf. Remark 3.4), and it implies Fn (t) ≃F(t) (b′). Thus, we are done. 28 YONGNAMLEEANDNOBORUNAKAYAMA

Remark. The exact sequence (III-2) is obtained as the “edge sequence” of the spectral sequence

Ep,q = or OT ( q( •), k(t)) Ep+q = p+q( • ) 2 T −p H E ⇒ H E(t) of -modules (cf. [12, III, (6.3.2.2)]) arising from the quasi-isomorphism OYt L • • k(t), E(t) ≃qis E ⊗OT where • and • denote the complexes [0 0 1 0] and [0 0 1 E E(t) →E →E → →E(t) →E(t) → 0], respectively.

Remark 3.8. In the situation of Proposition 3.7(2), the canonical homomorphism φ = lim φ : lim lim ∞ n n n Fn → n Fn∗ ←− ←− ←− is an isomorphism, where the projective limit lim is taken in the category of - n OY modules. This is shown as follows. Since ′ ←− , it is enough to show that the Fn ≃Fn∗ homomorphism ψ (V ) := lim H0(V, ψ ): lim H0(V, ) lim H0(V, ′ ) ∞ n n n Fn → n Fn ←− ←− ←− is an isomorphism for any open aﬃne subset V of Y , where we note that the global section functor H0(V, ) commutes with lim. For R = H0(V, ) and R = • OV n R/mn+1R H0(V, ), we have two exact sequences:←− ≃ OYn 0 H0(V, ) H0(V, 0) H0(V, 1), → F → E → E 0 H0(V, ′ ) H0(V, 0) R H0(V, 1) R . → Fn → E ⊗R n → E ⊗R n

Since the mR-adic completion R = lim Rn is flat over R and since lim is left exact, we have an isomorphism ←− ←− b H0(V, ) R Ker(H0(V, 0) R H0(V, 1) R) lim H0(V, ′ ). F ⊗R ≃ E ⊗R → E ⊗R ≃ n Fn ←− Then, ψ∞(V ) isb an isomorphism, since b b lim H0(V, ) lim (H0(V, ) R ) H0(V, ) R. n Fn ≃ n F ⊗R n ≃ F ⊗R ←− ←− Corollary 3.9. In the situation of Proposition 3.7, assume that f is locallyb of finite type. Then, the condition (c) of Proposition 3.7(3) for a point y Y is equivalent ∈ to: (d) there is an open neighborhood V of y in Y such that is flat over T , F|V and φ is an isomorphism on V Y for any t f(V ). t ∩ t ∈ Furthermore, if satisfies S for the point t = f(y) and if ′ is equi- F(t)|U∩Yt 2 F(t ) dimensional and

(III-3) codim(Z Y ′ Supp , Y ′ Supp ) 2 ∩ t ∩ F t ∩ F ≥ for any t′ T , then (d) is equivalent to: ∈ (e) there is an open neighborhood V of y in Y such that satisﬁes relative F|V S over T , i.e., V = S ( /T ). 2 2 F|V GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 29

Proof. For the first assertion, by Proposition 3.7(3), it is enough to show (c) (d) ⇒ assuming that f is locally of finite type and y Z. When (c) holds, is flat over ∈ G|V T for an open neighborhood V of y in Y , by Fact 2.26(1). Thus, is flat over F|V T by Proposition 3.7(i), and moreover, by Proposition 3.7(3) applied to any point in V , we see that φ is an isomorphism on Y V for any t f(V Z). Since φ is t t ∩ ∈ ∩ t an isomorphism for any t f(Z), we have proved: (c) (d). 6∈ ⇒ We shall show (d) (e) in the situation of the second assertion. In this case, if ⇔ φ is an isomorphism, then satisfies S by Corollary 2.16. Hence, we have (d) t F(t) 2 (e) by Fact 2.29(2). Conversely, if (e) holds with V = Y , then ⇒ ′ depth ′ (t ) 2 Yt ∩Z F ≥ ′ for any t f(Z) by Lemma 2.15(2), since (t′) satisfies S2 and the inequality ∈ F ′ (III-3) holds. Hence, φ ′ is an isomorphism for any t f(Z), and (d) holds. Thus, t ∈ we are done. Corollary 3.10. In the situation of Proposition 3.7, for a point t f(Z), assume ∈ that the coherent -module = j ( ) satisfies OYt F(t)∗ ∗ F(t)|Yt∩U (III-4) depth 3. Yt∩Z F(t)∗ ≥ Then, the sheaves and are flat over T along Y , and the restriction homomor- F G t phism φ : is an isomorphism. t F(t) →F(t)∗ Proof. By Proposition 3.7(3), it is enough to prove that φt is an isomorphism. By (III-4), we have R1j ( )= R1j ( )=0 ∗ F(t)|U∩Yt ∗ F0|U0 (cf. Property 2.6). Hence, the exact sequence (III-1) in the proof of Lemma 3.5 induces an exact sequence n n+1 0 m /m k j ( ) j ( ) j ( − ) 0. → ⊗ ∗ F0|U0 → ∗ Fn|Un → ∗ Fn−1|Un 1 → Since = j ( ), the homomorphism ϕ is surjective for any n 0. There- Fn∗ ∗ Fn|Un n ≥ fore, φ is an isomorphism by (b′) (b) of Proposition 3.7(3). t ⇒ Remark 3.11. Corollary 3.10 is similar to a special case of [28, Th. 12], where the sheaf corresponding to above may not have an exact sequence of Proposition 3.7. F However, [28, Th. 12] is not true. Example 3.12 below provides a counterexample.

8 Example 3.12. Let Y be an affine space Ak of dimension 8 over a field k with a 3 coordinate system (y1, y2,..., y8). Let T be a 3-dimensional affine space Ak and let f : Y T be the projection defined by (y1,..., y8) (y1, y2, y3). The fiber −1→ 7→ Y0 = f (0) over the origin 0 = (0, 0, 0) of T is of dimension 5. We define closed subschemes Z and V of Y by Z := y = y = y =0 and { 4 5 6 } V := y + y y + y y = y y = y y = y y =0 . { 1 2 7 3 8 4 − 1 5 − 2 6 − 3 } Then, we can show the following properties: (1) V A4, and V Y = V Z = Y Z A2; ≃ k ∩ 0 ∩ 0 ∩ ≃ (2) codim(Z, Y ) = codim(Z Y , Y ) = 3, and codim(Z V, V ) = 2; ∩ 0 0 ∩ 30 YONGNAMLEEANDNOBORUNAKAYAMA

(3) V Y T is a smooth morphism of relative dimension one, but the fiber \ 0 → V Y of V T over 0 is two-dimensional. ∩ 0 → Let j : U ֒ Y be the open immersion from the complement U := Y Z, and we → \ set := and := . By (1) and (2), we have isomorphisms F OY ⊕ OV F0 F ⊗OY OY0 (III-5) j ( ) j j and ∗ F|U ≃ ∗OU ⊕ ∗OU∩V ≃ OY ⊕ OV (III-6) j ( ) j , ∗ F0|U∩Y0 ≃ ∗OU∩Y0 ≃ OY0 since U V Y = , depth 2, depth 2, and depth 2. ∩ ∩ 0 ∅ Z OY ≥ Z∩V OV ≥ Z∩Y0 OY0 ≥ Thus, we have: (4) = is flat over T by (3); F|U OU ⊕ OU∩V (5) j ( ) is not flat over T by (3) and (III-5); ∗ F|U (6) j ( ) satisfies S by (III-6); ∗ F0|U∩Y0 3 (7) the canonical homomorphism j ( ) j ( ) ∗ F|U ⊗OY OY0 → ∗ F0|U∩Y0 is not an isomorphism by (III-5) and (III-6). Thus, f : Y T , , and U give a counterexample to [28, Th. 12]: The required → F assumptions are satisfied by (2), (4), and (6), but the conclusion is denied by (5) and (7). The kernel of has also an interesting infinitesimal property. Let J OY → OV A = k[y1, y2, y3] be the coordinate ring of T , m = (y1, y2, y3) the maximal ideal at the origin 0 T , and set ∈ A = A/mn+1,T = Spec A , Y = Y T , V = V T , = n n n n ×T n n ×T n Jn J ⊗OY OYn for each n 0 as in Notation 3.3. Then, we can prove: ≥ (III-7) , , = j ( ), and J 6≃ OY J |U 6≃ OU J ≃ J∗ ∗ J |U Jn|U∩Yn ≃ OU∩Yn for any n 0. In fact, the first two of (III-7) are consequences of that the ideal ≥ sheaf of V U is not an invertible -module, and it is derived from codim(V J |U ∩ OU ∩ U,U)=4 > 1. The third isomorphism of (III-7) follows from depth 2 and Z OY ≥ depth 2 (cf. (2)), and the last one from that the kernel of is Z OV ≥ Jn → OYn isomorphic to or OY ( , ), which is supported on V Y Y U. T 1 OV OYn ∩ 0 ⊂ \ Remark 3.13. In the situation of Notation 3.3, not a few people may fail to believe the following wrong assertion: ( ) If φ: j ( ) is an isomorphism, then the morphism ∗ F → ∗ F|U φqcoh : limqcoh limqcoh j (j∗ ) ∞ n Fn → n ∗ Fn ←− ←− induced by φ : = j (j∗ ) is also an isomorphism. n Fn →Fn∗ ∗ Fn Here, limqcoh stands for the projective limit in the category QCoh( ) of quasi- OY coherent←− -modules. We shall show that the ideal sheaf in Example 3.12 OY J provides a counterexample of ( ), and explain why a usual projective limit argument ∗ does not work for the “proof” of ( ). ∗ For simplicity, assume that Y is an affine Noetherian scheme Spec R, and set M = H0(Y, ), which is a finitely generated R-module. We set R = R/mn+1R F n GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 31 and M = M R for integers n 0. Then M ∼ and M ∼. Let n ⊗R n ≥ F ≃ Fn ≃ n R be the mR-adic completion lim R , and let π : Spec R Spec R = Y be the n n → associated morphism of schemes.←− Then, we have isomorphisms b b limqcoh (M R)∼ π (π∗ ). n Fn ≃ ⊗R ≃ ∗ F ←− Note that the projective limit lim in the category Mod( ) of -modules is n Fn b OY OY not quasi-coherent in general. ←− We shall show that the ideal sheaf in Example 3.12 provides a counterexample J of ( ). In this case, the left hand side of φqcoh is isomorphic to π (π∗ ) and the right ∗ ∞ ∗ J hand side is to π π∗ by the last isomorphisms of (III-7). Now, π is faithfully ∗ OY flat if we replace Y = A8 with Spec for the origin 0 = (0, 0,..., 0) Y . Then, OY,0 ∈ φqcoh is never an isomorphism, since . On the other hand, φ: j ( ) ∞ J 6⊂ OY J → ∗ J |U is an isomorphism as the third isomorphism of (III-7). ∗ We remark here on the commutativity of lim with functors j∗ and j . The direct image functor j : QCoh( ) QCoh( ←−) is right adjoint to the restriction ∗ OU → OY functor j∗ : QCoh( ) QCoh( ). Thus, lim commutes with j , and we have OY → OU ∗ an isomorphism ←− α: j limqcoh j∗ ≃ limqcoh j (j∗ ). ∗ n Fn −→ n ∗ Fn ←− ←− On the other hand, j∗ does not have a left adjoint functor. Because, the left adjoint functor j : Mod( ) Mod( ) of j∗ : Mod( ) Mod( ) does not preserve ! OU → OY OY → OU quasi-coherent sheaves. Thus, lim does not commute with j∗ in general, and hence, the canonical morphism ←−

β : j∗ limqcoh limqcoh j∗ n Fn → n Fn ←− ←− in QCoh( ) is not necessarily an isomorphism. OU It is necessary to check the morphism β to be an isomorphism, for the “proof” of ( ) by the projective limit argument. In fact, we can prove: ∗ ( ) When φ is an isomorphism, φqcoh is an isomorphism if and only if β is so. ∗∗ ∞ This is a point to which many people probably do not pay attention.

Proof of ( ). Now, we have a commutative diagram ∗∗ φqcoh π π∗ ≃ limqcoh ∞ limqcoh j j∗( ) ∗ F −−−−→ n Fn −−−−→ n ∗ Fn ←− ←− φˆ ≃ α

  j∗(β) x j j∗(π π∗ ) ≃ j j∗ limqcoh j limqcoh j∗ ∗ y∗ F −−−−→ ∗ yn Fn −−−−→ ∗ n Fn ←− ←− in QCoh( ), where φˆ is the restriction homomorphism of π π∗ . Thus, it suﬃces OY ∗ F to show that if φ is an isomorphism, then φˆ is so. Let us consider an isomorphism γ : π π∗(j j∗ ) ≃ j j∗(π π∗ ) ∗ ∗ F −→ ∗ ∗ F deﬁned as the composite

′ ′′ −1 ∗ ∗ π∗(δ ) ∗ ∗ ∗ ∗ j∗(δ ) ∗ ∗ π∗π (j∗j ( )) π∗ˆj∗(π (j )) j∗πU∗ˆj (π ) j∗j π∗π F −−−−→≃ U F ≃ F −−−−−−→≃ F 32 YONGNAMLEEANDNOBORUNAKAYAMA of canonical isomorphisms; Here ˆj is the open immersion π−1(U) ֒ Spec R and −1 → πU is the restriction of π to π (U), and the morphisms b δ′ : π∗j (j∗ ) ≃ ˆj π∗ (j∗ ) and δ′′ : j∗π (π∗ ) ≃ π ˆj∗(π∗ ) ∗ F −→ ∗ U F ∗ F −→ U∗ F are ﬂat base change isomorphisms (cf. Lemma A.9). Then, we can write φˆ = γ (π π∗(φ)) for the induced morphism ◦ ∗ π π∗(φ): π π∗ π π∗(j j∗ ), ∗ ∗ F → ∗ ∗ F and this shows that if φ is an isomorphism, then φˆ is so. Thus, we are done.

In the rest of Section 3.1, in Lemmas 3.14 and 3.15 below, we shall give sufficient conditions for to admit an exact sequence of Proposition 3.7. F Lemma 3.14. Suppose that f j : U T is flat and ◦ → depth 2 Yt∩Z OYt ≥ for any t f(Z). If is a reflexive -module and if is locally free, then ∈ F OY F|U there exists an exact sequence 0 0 1 0 locally on Y which →F→E → E → G → satisfies the conditions (i) and (ii) of Proposition 3.7.

Proof. The morphism f is ﬂat by Corollary 3.6. Since is coherent, locally on Y , F we have a ﬁnite presentation

⊕m h ⊕n ∨ 0 OY −→OY →F → of the dual -module ∨ = om ( , ). Let be the kernel of the left OY F H OY F OY K homomorphism h. Then, is locally free, since so is . We have an exact K|U F|U sequence ∨ 0 ∨∨ ⊕n h ⊕m →F≃F → OY −−→OY by taking the dual. Let be the cokernel of h∨. Then, is isomorphic to the G G|U locally free sheaf ∨ . Thus, the exact sequence K |U ∨ 0 ⊕n h ⊕m 0 →F→OY −−→OY → G → satisﬁes the conditions (i) and (ii) of Proposition 3.7.

Lemma 3.15. Suppose that f : Y T is a ﬂat morphism and → (III-8) depth 2 Yt∩Z OYt ≥ for any t f(Z). Moreover, suppose that there is a bounded complex ∈ • = [ i i+1 ] E ···→E →E → · · · of locally free -modules of ﬁnite rank satisfying the following four conditions: OY (i) i( •) =0 for any i> 0; H E |Y \Z (ii) 0( •); F ≃ H E (iii) i( • )=0 for any i< 0 and any t T , where • stands for the complex H E(t) ∈ E(t) L [ i i+1 ] • ; ···→E(t) →E(t) → · · · ≃qis E ⊗OY OYt GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 33

i • (iv) the local cohomology group Hy(M ) at the maximal ideal mY,y for the complex M • = τ ≤1 • E(t) y of -modules is zero for any i 1 and any y Z, where t = f(y). OY,y ≤ ∈ Then, i( •)=0 for any i < 0, and admits an exact sequence satisfying the H E F conditions (i) and (ii) of Proposition 3.7. Proof. For an integer k, the truncated complex τ ≥k( •) is expressed as E [ 0 k k+1 k+2 ], ···→ →C →E →E → · · · where k is the cokernel of k−1 k. First, we shall show that • τ ≥0( •) C E →E E ≃qis E and 0 is flat over T . Note that it implies that i( •) =0 for any i< 0. Since • C H E E is bounded, we have an integer k< 0 such that • τ ≥k( •) and k is flat over E ≃qis E C T . Then, by (iii), one has k( • ) Ker( k k+1)=0 H E(t) ≃ C(t) →E(t) for any t T . Hence, k k+1 is injective and k+1 k+1/ k is flat over T by ∈ C →E C ≃E C a version of local criterion of flatness (cf. Corollary A.2). Thus, • τ ≥k+1( •), E ≃qis E and we can increase k by one. Therefore, we can take k = 0, and consequently, • τ ≥0( •), and 0 is flat over T . We write := 0. Then, E ≃qis E C C C • [ 0 1 2 ] E(t) ≃qis ···→ →C(t) →E(t) →E(t) → · · · for any t T , since and i are all flat over T . ∈ C E Second, we shall prove that (III-9) depth 2 Yt∩Z C(t) ≥ for any t f(Z). We define to be the kernel of 1 2 . Then, ∈ Kt E(t) →E(t) depth i 2 and depth 2 Yt∩Z E(t) ≥ Yt∩Z Kt ≥ for i = 0, 1, and for any t f(Z), by (III-8) and by Lemma 2.17(1). In particular, ∈ for any y Z Y , we have the vanishing ∈ ∩ t (III-10) Hi (( ) )=0 y Kt y of the local cohomology group at y for any i 1 (cf. Property 2.6). By construction, ≤ we have a quasi-isomorphism τ ≤1( • ) [ 0 0 ]. E(t) ≃qis ···→ →C(t) → Kt → → · · · In view of the induced exact sequence Hi (M •) Hi (( ) ) Hi (( ) ) ···→ y → y C(t) y → y Kt y → · · · of local cohomology groups, we have Hi (( ) )=0 y C(t) y for any i 1 by (iv) and (III-10). Thus, we have (III-9) (cf. Property 2.6). ≤ Finally, we consider the cokernel of 1. Then, is flat over T by (i). G C→E G|U Therefore, the exact sequence 0 1 0 satisfies the conditions →F→C→E → G → (i) and (ii) of Proposition 3.7. 34 YONGNAMLEEANDNOBORUNAKAYAMA

3.2. Applications of the key proposition. First, we shall prove the following criterion for a sheaf to be invertible.

Theorem 3.16. Let f : Y T be a morphism of locally Noetherian schemes, Z → a closed subset of Y , a coherent -module, and t a point of f(Z). We set F OY :U = Y Z, and write j : U ֒ Y for the open immersion. Assume that \ → (i) depth 1, Z OY ≥ (ii) is ﬂat over T , is invertible, depth 2, and F|U F|U Z F ≥ (iii) the direct image sheaf

= j (( ) ) F(t)∗ ∗ F ⊗OY OYt |U∩Yt (cf. Definition 3.2) is an invertible -module. OYt Assume furthermore that one of the conditions (a) and (b) below is satisfied: (a) depth 3; Z∩Yt OYt ≥ (b) the double-dual [r] of ⊗r is invertible along Y for a positive integer r F F t coprime to the characteristic of the residue field k(t). Then, f is flat along Y , and is invertible along Y . t F t Proof. We may replace Y with its open subset, since the assertions are local on Y . By (ii), U is flat over T . Moreover,

(III-11) depth 2 Z∩Yt OYt ≥ by (iii), since the isomorphism j ( ) implies that depth( ) = F(t)∗ ≃ ∗ F(t)∗|U∩Yt F(t)∗ y depth 2 for any y Z Y . Hence, f : Y T is flat along Y by Corollary 3.6 OYt,y ≥ ∈ ∩ t → t (cf. Property 2.6). Then, is a reflexive -module by Lemma 2.21(3), since we F OY have assumed depth 1 and depth 2 in (i) and (ii). Therefore, by Z OY ≥ Z F ≥ (III-11) and Lemma 3.14, we may assume that admits an exact sequence of F Proposition 3.7. By Fact 2.26(2), we see that is invertible along Y if the two conditions below F t are both satisfied: (1) is flat over T along Y ; F t (2) φ : is an isomorphism. t F(t) →F(t)∗ Here, (1) is a consequence of (2) by Proposition 3.7(3). When (a) holds, we have

depth 3 Yt∩Z F(t)∗ ≥ by (iii), and hence, the condition (2) is satisﬁed by Corollary 3.10. Thus, it remains to prove (2) assuming the condition (b). We use Notation 3.3 for t. By replacing Y with its open subset, we may assume that Y is aﬃne, and there exist isomorphisms

= = and [r] OYt OY0 ≃F(t)∗ F0∗ F ≃ OY in (iii) and in (b), respectively. Note that we have

depth 2 Yn∩Z OYn ≥ GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 35 for any n 0: This follows from (III-11) by Lemma 2.32(3) applied to the flat ≥ morphism Y T . As a consequence, n → n H0(Y , ) H0(U , ) n OYn ≃ n OUn and for any n, and the restriction homomorphism (III-12) H0(U , ) H0(U , ) n OUn → n−1 OUn−1 is surjective for any n> 0, since we have assumed that Y is affine. We set := . It is enough to show that for all n. In fact, if Nn Fn|Un Nn ≃ OUn this is true, then we have an isomorphism = j ( ) j ( ) , Fn∗ ∗ Nn ≃ ∗ OUn ≃ OYn and, as a consequence, the restriction homomorphism ϕ : ( ) is an n Fn∗ (t) → F0∗ isomorphism for any n 0. Hence, in this case, φ : is an isomorphism ≥ t F(t) →F(t)∗ by (b′) (b) of Proposition 3.7(3). ⇒ We shall prove by induction on n. When n = 0, we have the Nn ≃ OUn isomorphism from the isomorphism above. Assume that F0∗ ≃ OY0 Nn−1 ≃ OUn−1 for an integer n > 0. Let be the kernel of . Then, 2 = 0 as an J OYn → OYn−1 J ideal of Yn , and O n n+1 m /m k . J ≃ ⊗ OY0 We have an exact sequence 0 ⋆ ⋆ 1 →J →OYn → OYn−1 → of sheaves on Y = Y with respect to the Zariski topology, where ⋆ stands for the | n| | 0| subsheaf of invertible sections of a sheaf of rings, and where a local section ζ of J is mapped to the invertible section 1 + ζ of . It induces a long exact sequence: OYn 0 1 H0(U , ⋆ ) res H0(U , ⋆ ) H1(U , ) Pic(U ) res Pic(U ), n OUn −−→ n−1 OUn−1 → 0 J → n −−→ n−1 0 1 0 where res and res are restriction homomorphisms to Un−1. Note that res is surjective, since so is (III-12). Hence, the kernel of res1 is a k-vector space isomorphic to H1(U , ). Now, the isomorphism class of in Pic(U ) belongs to the kernel 0 J Nn n by , and its multiple by r is zero by (b), where r is coprime to Nn−1 ≃ OUn−1 char(k). Thus, , and we are done. Nn ≃ OUn We have the following by a direct application of Proposition 3.7. Lemma 3.17. Let f : Y T be a flat morphism of locally Noetherian schemes → and let Z be a closed subset of Y such that depth 2 Yt∩Z OYt ≥ for any fiber Y . Let q : T ′ T be a morphism from another locally Noetherian t → scheme T ′ such that Y ′ = Y T ′ is also locally Noetherian. We write f ′ : Y ′ T ′ ×T → and p: Y ′ Y for the projections. Let 0 0 1 0 be an exact → →F→E → E → G → sequence of coherent -modules such that , 0, 1, and are locally free, OY F|U E E G|U where U = Y Z. Then, is a reflexive -module, and \ F OY ∗ ∨∨ ∗ 0 ∗ 1 ′ ∗ (p ) Ker(p p ) j (p ′ ) F ≃ E → E ≃ ∗ F|U 36 YONGNAMLEEANDNOBORUNAKAYAMA

-for the open immersion j′ : U ′ = p−1(U) ֒ Y ′. Moreover, (p∗ )∨∨ satisfies rela → F tive S over T ′ if and only if p∗ is flat over T ′. 2 G Proof. The exact sequence satisfies the assumptions of Proposition 3.7 for Y T . → Hence, j ( ), i.e., depth 2, by Proposition 3.7(1). Moreover, is F ≃ ∗ F|U Z F ≥ F reflexive by Lemma 2.21(3), since we have depth 2 by Lemma 2.32(3). Let Z OY ≥ ′ be the kernel of p∗ 0 p∗ 1. Then, the exact sequence F E → E 0 ′ p∗ 0 p∗ 1 p∗ 0 →F → E → E → G → on Y ′ satisfies the assumptions of Proposition 3.7 for f ′ : Y ′ T ′, since → ′ − ′ depthY ∩p 1(Z) Y ′ = depthY ∩Z Yt 2 t′ O t t O ≥ ′ ′ ′ ′ ′ ′ for any t T and t = q(t ), by Lemma 2.31(1). Hence, j∗( U ′ ) by ∈ ′ ∗ ′ ∗ ∨∨F ≃ F | Proposition 3.7(1). Since ′ p ′ , we have (p ) by Lemma 2.34. F |U ≃ F|U F ≃ F Furthermore, by Proposition 3.7(3), we see that ′ satisfies relative S over T ′ if F 2 and only if p∗ is flat over T ′. G Here, we introduce the following notion useful to state results in the rest of Section 3.2. Definition 3.18. Let f : Y T be a morphism of locally Noetherian schemes and → let be a coherent -module. We say that is locally free in codimension one F OY F on each fiber of f if there is an open subset U Y such that is locally free ⊂ F|U and codim(Y U, Y ) 2 for any fiber Y = f −1(t). t \ t ≥ t Proposition 3.19 (infinitesimal and valuative criteria). Let f : Y T be a flat → morphism locally of finite type between locally Noetherian schemes and let y Y be ∈ a point such that f satisfies relative S over T at y. Let be a reflexive -module 2 F OY which is locally free in codimension one on each fiber of f. Then, satisfies relative F S2 over T at y if one of the following two conditions (I) and (II) is satisfied, where Y = Y Spec A and = p∗ for the projection p : Y Y : A ×T FA AF A A → (I) Let Spec A T be a morphism defined by a surjective local ring homo- → morphism A to an Artinian local ring A. Then, the double-dual OT,f(y) → ( )∨∨ satisfies relative S over Spec A at the point y = p−1(y). FA 2 A A (II) The local ring is reduced. Let Spec A T be a morphism defined OT,f(y) → by a local ring homomorphism A to a discrete valuation ring A. OT,f(y) → Then, the double-dual ( )∨∨ satisfies relative S over Spec A at any point FA 2 z Y lying over y Y and the closed point m of Spec A. ∈ A ∈ A Proof. We may assume that T = Spec B for the local ring B = and we can OT,f(y) localize Y freely. Thus, we may assume that Y = Spec C for a finitely generated B-algebra C and moreover that Y T is an S -morphism by Fact 2.29(3). By → 2 assumption, there is a closed subset Z Y such that is locally free and ⊂ F|Y \Z codim(Y Z, Y ) 2 for any t T . In particular, t ∩ t ≥ ∈ depth 2 Yt∩Z OYt ≥ for any t T (cf. Lemma 2.15(2)). As in the proof of Lemma 3.14, we have an ∈ exact sequence 0 0 1 0 of coherent -modules such that 0, →F→E →E → G → OY E GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 37

1, and are locally free. By Proposition 3.7(3), we see that satisfies relative E G|U F S over T at y if and only if the stalk is flat over B. Let Spec A T be a 2 Gy → morphism in (I) or (II). Then, Y = Spec C A is Noetherian, since C A A ⊗B ⊗B is a finitely generated A-algebra. By Proposition 3.7(3) and Lemma 3.17, we see also that ( )∨∨ satisfies relative S over Spec A at a point z lying over y and FA 2 m if and only if the stalk is flat over A for the pullback = p∗ , where A GA,z GA AG ( A) . Therefore, the assertions in the cases (I) and (II), respectively, GA,z ≃ Gy ⊗B z follow from the local criterion of flatness (cf. Proposition A.1(iv)) for over B Gy and from the valuative criterion of flatness (cf. [12, IV, Th. (11.8.1)]) for over T G at y.

Definition 3.20 (relative S refinement). Let Y T be an S -morphism of locally 2 → 2 Noetherian schemes and let be a reflexive -module which is locally free in F OY codimension one on each fiber. A morphism S T from a locally Noetherian → scheme S is called a relative S refinement for over T if the following conditions 2 F are satisfied: (i) S T is a monomorphism in the category of schemes (cf. Fact 3.23); → (ii) for any morphism T ′ T of locally Noetherian schemes, and for the pull- → back ′ of to the fiber product Y T ′, the double dual ( ′)∨∨ satisfies F F ×T F relative S over T ′ if and only if T ′ T factors through S T . 2 → → Remark. The fiber product Y T ′ in (ii) is locally Noetherian, since it is locally ×T of finite type over T ′. Thus, we can consider the relative S -condition for ( ′)∨∨ 2 F (cf. Definition 2.28). By (i) and (ii), S T is unique up to unique isomorphism. → Remark 3.21. In the situation of Definition 3.20(ii), we write T ′ for ′, and F ×T F we set ′ ∨∨ ′ ⋆, if ( T T ) satisfies relative S2 over T , F (T ′/T )= F × ( , otherwise, ∅ where ⋆ denotes a one-point set. For any morphism T ′′ T ′ from a locally Noe- → therian scheme T ′′, we can show that if F (T ′/T )= ⋆, then F (T ′′/T )= ⋆. In fact, we have an isomorphism ′′ ∨∨ ′ ∨∨ ′′ ( T ) ( T ) ′ T F ×T ≃ F ×T ×T ′′ by Lemma 2.34, and this sheaf satisfies relative S2 over T by Lemma 2.31(3). Therefore, F is regarded as a functor (LNSch/T )op Set for the category LNSch/T → of locally Noetherian T -schemes, and the relative S2 refinement is a T -scheme representing F .

Remark 3.22. If T ′ = Spec k for a field k, then F (T ′/T )= ⋆, since ( ′)∨∨ satisfies F S by Corollary 2.22. In particular, if the relative S refinement S T exists, then 2 2 → it is bijective.

Fact 3.23. Let h: S T be a morphism locally of ﬁnite type between locally → Noetherian schemes. Then, h is a morphism locally of ﬁnite presentation (cf. [12, IV, 1.4]), and we have the following properties: § 38 YONGNAMLEEANDNOBORUNAKAYAMA

(1) The morphism h is a monomorphism in the category of schemes if and only if h is radicial and unramified, by [12, IV, Prop. (17.2.6)]. (2) If h is an unramified morphism, then it is étale locally a closed immersion, i.e., for any point s S, there exists an open neighborhood V of s such ∈ that the induced morphism V T is written as the composite of a closed → immersion V W and an étale morphism W T (cf. [12, IV, Cor. → → (18.4.7)], [13, I, Cor. 7.8]).

Example 3.24. For a Noetherian scheme T and a finite number of locally closed subschemes S1, S2,..., Sk of T , assume that T is equal to the disjoint union k S as a set; The collection S is called a stratification in [38, Lect. 8]. Then, i=1 i { i} immersions Si T define a morphism h: S T from the scheme-theoretic disjoint F k ⊂ → union S = i=1 Si. This h is a separated surjective monomorphism of finite type and is a local immersion (cf. [12, I, Déf. (4.5.1)]), i.e., a closed immersion Zariski- F locally.

Example 3.25. There is a separated monomorphism h: X Y of finite type of → Noetherian schemes such that X is connected but h is not an immersion. An example is given as follows: For an algebraically closed field k, let D be a reduced 2 effective divisor of degree three on the projective plane Y = Pk having a node P . For the blowing up M Y at P , let X be the proper transform of D in M and → let Q X be one of the two points lying over P . We set X := X Q . Then, X ∈ \{ } is connected, the induced morphism h: X Y is a separated monomorphism of → finite type inducing a bijection X D, and h−1(Y P ) D P . However, h → \{ } ≃ \{ } is not a closed immersion, since X is not isomorphic to D. If D is irreducible, then h is not a local immersion. On the other hand, if D is reducible, then h is a local immersion. In fact, for another node P ′ of D, the inverse image h−1(Y P ′ ) is \{ } isomorphic to a disjoint union of two locally closed subschemes of Y .

The following is analogous to the flattening stratification theorem by Mumford in [38, Lect. 8] or to the representability theorem of unramified functors by Murre [40]: A similar result is stated by Kollár in [29, Th. 2] in the case where f is projective, but where the S2-condition for f, etc., are not assumed.

Theorem 3.26. Let f : Y T be an S -morphism of locally Noetherian schemes. → 2 Let be a reflexive -module which is locally free in codimension one on each F OY fiber (cf. Definition 3.18). If the following condition (i) is satisfied, then there is a relative S refinement for over T as a separated morphism S T locally of 2 F → finite type: (i) satisfies relative S over T for a closed subset Σ Y such that F|Y \Σ 2 ⊂ Σ T is proper. → Furthermore, the morphism S T is a local immersion of finite type if → (ii) f is a projective morphism locally over T .

Proof. For the first assertion, by Fact 3.23(1), it is enough to prove that the functor F in Remark 3.21 is representable by a separated morphism S T locally of finite → GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 39 type. We may replace T freely by an open subset, since S T is unique up to → unique isomorphism and since the second assertion is also local on T . Thus, we assume that T is an affine Noetherian scheme. We set U to be an open subset of Y such that is locally free and codim(Y U, Y ) 2 for any fiber Y . F|U t \ t ≥ t We first consider the case (ii): We may assume that Y is a closed subscheme of PN T for some N > 0. Let be the f-ample invertible -module defined as × A OY the pullback of (1) on PN . Then, we can construct an exact sequence O ′ ′ ( ⊗−l )⊕m ( ⊗−l)⊕m ∨ 0 A → A →F → on Y for positive some integers m, m′, l, and l′, where the kernel of the left homomorphism is locally free on U, since ∨ is so. Taking the dual, we have an exact F sequence 0 0 1 0 of coherent -modules such that 0, 1, →F→E → E → G → OY E E and are locally free (cf. the proof of Lemma 3.14). Let T ′ T be an arbitrary G|U → morphism from another locally Noetherian scheme T ′. Then, F (T ′/T )= ⋆ if and only if T ′ is flat over T ′, by Lemma 3.17. Hence, the functor F is nothing G ×T but the “universal flattening functor” G: (Sch/T )op Set for (cf. Remark 3.27 → G below) restricted to the category LNSch/T . Here,

′ ′ ⋆, if T T is flat over T , G(T ′/T )= G× ( , otherwise, ∅ for any T -scheme T ′. By the Theorem of [38, Lect. 8], it is represented by a separated morphism S T of finite type which is a local immersion. Thus, we → have proved the assertion in the case (ii). In the case (i), we can cover Σ by finitely many open affine subsets Yλ. We may assume that Y = Y , since satisfies relative S over T . By Lemma 3.14, λ F|Y \Σ 2 we may also assume that there exists an exact sequence S 0 0 1 0 → F|Yλ →Eλ →Eλ → Gλ → on each Y such that 0 and 1 are free -modules of finite rank, and that is λ Eλ Eλ OYλ Gλ locally free on U = U Y . Let T ′ T be an arbitrary morphism from a locally λ ∩ λ → Noetherian scheme T ′. By Lemma 3.17, we see that F (T ′/T ) = ⋆ if and only if T ′ is flat over T ′ for any λ. Let G : (Sch/T )op Set be the universal Gλ ×T λ → flattening functor for , which is defined by Gλ ′ ′ ′ ⋆, if λ T T is flat over T ; Gλ(T /T )= G × ( , otherwise. ∅ op ′ Let G: (Sch/T ) Set be the “intersection” functor of all Gλ, i.e., G(T /T ) = ′ → ′ Gλ(T /T ) for any T/ T . By the argument above, F is the restriction of G to LNSch/T . Every functor G satisfies the conditions (F )–(F ) of [40] except (F ), T λ 1 8 3 by the proof of [40, Th. 2]. Hence, the intersection functor G satisfies the same conditions except possibly (F3) and (F8). By [40, Th. 1], we are reduced to check these two conditions for G. Since the two conditions concern only Noetherian schemes, we may take F = G. We write F (T ′) = F (T ′/T ) for simplicity for a morphism T ′ T . → 40 YONGNAMLEEANDNOBORUNAKAYAMA

We shall show that F satisfies (F3) (cf. [40, (F3), p. 244]). Let A be a Noetherian complete local ring with maximal ideal m and let Spec A T be a morphism. A → What we have to prove is the bijectivity of the canonical map F (Spec A) lim F (Spec A/mn ), → n A ←− n or equivalently that F (Spec A)= ⋆ if F (Spec A/mA)= ⋆ for all n> 0. Assume the latter condition. By Corollary 3.9 applied to Y Spec A Spec A for each λ, we λ ×T → have an open neighborhood W of the closed fiber Y Spec A/m in Y Spec A λ λ ×T A λ ×T such that ( Spec A)∨∨ satisfies relative S over Spec A. On the other hand, F×T |Wλ 2 the restriction of ( Spec A)∨∨ to (Y Σ) Spec A also satisfies relative S F ×T \ ×T 2 over Spec A. Then, the union W ((Y Σ) Spec A) equals Y Spec A, since λ ∪ \ ×T ×T the complement of the union is proper over Spec A but its image does not contain S the closed point mA. Therefore, F (Spec A)= ⋆. Next, we shall show that F satisfies (F8) (cf. [40, (F8), p. 246]). Let A be a Noetherian ring containing a unique minimal prime ideal p and let I be a nilpotent ideal of A such that Ip = 0. Note that p = √0. Let Spec A T be a morphism ′ → ′ and assume that F (Spec A/I) = ⋆ but F (Spec Ap/I ) = for any ideal I of Ap ′ ∅ such that I ( Ip. What we have to prove is the existence of an element a A p ∈ \ having the following property: ( ) For any element b A p and for any ideal J of A = A[(ab)−1], if J IA ⋄ ∈ \ ab ⊂ ab and if F (Spec Aab/J)= ⋆, then J = IAab. For each λ, we set B to be an A-algebra such that Spec B Y Spec A over λ λ ≃ λ ×T Spec A and let Mλ be a finitely generated Bλ-module such that the quasi-coherent sheaf M ∼ on Spec B is isomorphic to Spec A. Note that λ λ Gλ ×T M Ap/IAp λ ⊗A is a free Ap/IAp-module, since it is flat over Ap/IAp by Gλ(Spec Ap/IAp)= ⋆ and since Ap is an Artinian local ring. Hence, (M A/I) A = M A /IA λ ⊗A ⊗A a λ ⊗A a a is a free A /IA -module for an element a A p. For each λ, let be the set a a ∈ \ Sλ of ideals J of A such that G (Spec A /J)= ⋆, or equivalently, that M A /J a λ a λ ⊗A a is a flat Aa/J-module. By [12, IV, Cor. (11.4.4)], there exists a unique minimal element I = I in , and λ λ,(a) Sλ ( ) for any A -algebra A′, if M A′ is a flat A′-module, then A′ is an † a λ ⊗A Aa/Iλ-algebra. Note that I is nilpotent, since the nilpotent ideal IA belongs to . We define λ a Sλ I(a) := Iλ,(a) as an ideal of Aa. Then, it has the following property: ( ) For any A -algebra A′, it is an A /I -algebra if and only if M A′ is ‡ P a a (a) λ ⊗A flat over A′ for any λ, i.e., F (Spec A′)= ⋆.

By the assumption of Ip, we have (I(a))p = I(a)Ap = IAp. Thus, there is an element ′ a A p such that I(a)Aaa′ = IAaa′ . Here, Iλ,(aa′) = Iλ,(a)Aaa′ for any λ by the ∈ \ ′ property ( ) of I . Thus, I ′ = I A ′ = IA ′ . Therefore, aa satisﬁes the † λ (aa ) (a) aa aa GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 41 condition ( ) by the property ( ). Thus, we have checked the conditions (F ) and ⋄ ‡ 3 (F8), and the assertion in the case (i) has been proved.

Remark 3.27. The (universal) flattening functor is introduced by Murre in [40], but its origin seems to go back to Grothendieck as the subtitle says. Murre gives a criterion of the representability of the functor in [40, 3, (A)], whose prototype seems § to be [12, IV, Prop. (11.4.5)]. Mumford considers the case of projective morphism in [38, Lect. 8], and proves the representability by using Hilbert polynomials, where the representing scheme is called the “flattening stratification.” He also mentioned that Grothendieck has proved a weaker result by much deeper method. Raynaud [46, Ch. 3] and Raynaud–Gruson [47, Part 1, 4] give further criteria of the repre- § sentability of the universal flattening functor by another method. One of them is used in proving the following “local version” of the existence of relative S2 refinement.

Theorem 3.28. Let f : Y T be an S -morphism of locally Noetherian schemes → 2 and a reflexive -module which is locally free in codimension one on each fiber F OY (cf. Definition 3.18). Assume that T = Spec R for a Henselian local ring R and let o T be the closed point. Then, for any point y of the closed fiber Y = f −1(o), ∈ o there is a closed subscheme S T having the following universal property: Let ⊂ T ′ = Spec R′ T = Spec R be a morphism defined by a local ring homomorphism → R R′ for a Noetherian local ring R′ and let o′ T ′ be the closed point. Let → ∈ f ′ : Y ′ = Y T ′ T ′ and p: Y ′ Y be the induced morphisms. Then, for the ×T → → pullback ′ := p∗ , its double-dual ( ′)∨∨ on Y ′ satisfies relative S over T ′ at F F F 2 any point y′ of Y ′ with p(y′) = y and f ′(y′) = o′ if and only if T ′ T factors → through S.

Proof. Replacing Y with an open neighborhood of y, we may assume that Y is an affine R-scheme of finite type. Then, by the same argument as in the proof of Lemma 3.14, we have an exact sequence 0 0 1 0 of coherent 0 1 →F→E → E → G → Y -modules such that and are locally free and U is also locally free for O E E G| ′ ∨∨ the maximal open subset U such that U is locally free. By Lemma 3.17, ( ) ′ F|′ ∗ F satisfies relative S over T at a point y lying over y if and only if (p ) ′ is flat 2 G y over T ′. Therefore, the universal closed subscheme S T exists by [47, Part 1, Th. ⊂ (4.1.2)] or [46, Ch. 3, Th. 1] applied to . G

4. Grothendieck duality We shall explain the theory of Grothendieck duality with some base change theorems referring to [17], [7], [34], etc. We do not prove the main part of the duality theory but show several consequences. Some of them are useful for studying Q-Gorenstein schemes and Q-Gorenstein morphisms in Sections 6 and 7. Some well-known properties on the dualizing complex are mentioned in Sec- tions 4.1 and 4.2 based on arguments in [17] and [7]. Section 4.1 explains some basic properties and results on a locally Noetherian scheme admitting a dualizing complex, mainly on the codimension function associated with the dualizing complex 42 YONGNAMLEEANDNOBORUNAKAYAMA and on interpretation of Sk-conditions for a coherent sheaf via the dualizing complex. In Section 4.2, we introduce a useful notion of ordinary dualizing complex for locally equi-dimensional locally Noetherian schemes, and study cohomology sheaves of ordinary dualizing complexes. Section 4.3 explains the notion of twisted inverse image and the relative duality theory referring mainly to [17], [7], [34]. Our original base change result for the relative dualizing complex to the ﬁber is proved in Corollary 4.38. In Section 4.4, we explain the relative dualizing sheaf for a Cohen– Macaulay morphism (cf. Deﬁnition 2.30) and its base change property referring to [7], [50], etc.

4.1. Dualizing complex. We shall begin with recalling the notion of dualizing complex, which is introduced in [17, V].

Definition 4.1. A dualizing complex • of a locally Noetherian scheme X is R defined to be a complex of -modules bounded below such that OX it has coherent cohomology and has finite injective dimension, i.e., • • + R ∈ Dcoh(X)fid in the sense of [17], and the natural morphism • R om ( •, •) OX → H OX R R is a quasi-isomorphism (cf. [17, V, Prop. 2.1]).

+ Remark. Every complex in Dcoh(X)fid is quasi-isomorphic to a bounded complex of quasi-coherent injective -modules when X is quasi-compact (cf. [17, II, Prop. OX 7.20]). The derived functor R om of the bi-functor om is considered as a H OX H OX functor D(X)op D(X) ( •, •) R om ( •, •) D(X) × ∋ F G 7→ H OX F G ∈ (cf. [17, I, 6], [53, Th. A(ii)]). § Example. A Noetherian local ring A is said to be Gorenstein if there is a finite injective resolution of A. In particular, is a dualizing complex for X = Spec A. OX There are known several conditions for a local ring A to be Gorenstein (e.g. [17, V, Th. 9.1], [36, Th. 18.1]): For example, A is Gorenstein if and only if A is Cohen– Macaulay and Extn(A/m , A) A/m for the maximal ideal m and n = dim A. A ≃ A A A locally Noetherian scheme Y is said to be Gorenstein if every local ring OY,y is Gorenstein. For a locally Noetherian scheme Y , it is Gorenstein of finite Krull dimension if and only if is a dualizing complex (cf. [17, II, Prop. 7.20]). OY Example (cf. [17, V, Prop. 3.4], [36, Th. 18.6]). For an Artinian local ring A, let I be an injective hull of the residue field A/mA. Then, the associated quasi-coherent sheaf I∼ on Spec A is a dualizing complex.

Remark 4.2 ([17, V, 10]). Let X be a locally Noetherian scheme. If there is § a morphism X Y of finite type to a locally Noetherian scheme Y admitting a → dualizing complex in which the dimensions of fibers are bounded, then X also admits a dualizing complex [17, VI, Cor. 3.5]. In particular, any scheme of finite type over a Noetherian Gorenstein scheme of finite Krull dimension admits a dualizing complex. GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 43

When X is connected, the dualizing complex is unique up to quasi-isomorphism, shift, and up to tensor product with invertible sheaves (cf. [17, V, Th. 3.1], [7, (3.1.30)]).

Fact. For a Noetherian ring A, the aﬃne scheme Spec A admits a dualizing complex if and only if there is a surjection B A from a Gorenstein ring B of ﬁnite Krull → dimension. This is conjectured by Sharp [52, Conj. (4.4)] and has been proved by Kawasaki [25, Cor. 1.4].

We shall explain the notion of codimension function.

Definition 4.3 (cf. [17, V, p. 283]). Let X be a scheme such that every local ring has finite Krull dimension. A function d: X Z is called a codimension OX,x → function if d(x)= d(y) + codim( x , y ) { } { } for any points x and y such that x y . ∈ { } Remark. Let X be a scheme whose local rings all have finite Krull dimension. OX,x If X admits a codimension function, then X is catenary (cf. Property 2.3). In fact, codim( x , z ) = codim( x , y ) + codim( y , z ) { } { } { } { } { } { } holds for any x, y, z X satisfying x y and y z . Moreover, if the ∈ ∈ { } ∈ { } codimension function is bounded, then X has finite Krull dimension.

Lemma 4.4. Let X be a scheme such that every local ring has ﬁnite Krull OX,x dimension, and let d: X Z be a codimension function. Then, → d(y) dim d(x) dim − OX,y ≥ − OX,x holds for any points x, y X with x y . Moreover, the following three conditions ∈ ∈ { } are equivalent to each other: (i) the equality d(y) dim = d(x) dim − OX,y − OX,x holds for any points x, y X with x y ; ∈ ∈ { } (ii) the function X x d(x) dim Z is locally constant; ∋ 7→ − OX,x ∈ (iii) X is locally equi-dimensional (cf. Deﬁnition 2.2(3)).

Proof. The ﬁrst inequality is derived from the well-known inequality dim dim + codim( x , y ) OX,x ≥ OY,y { } { } (cf. Property 2.1(1), [12, 0IV, Prop. (14.2.2)]). To show the equivalence of three conditions (i)–(iii), we may assume that X is connected. Let be the set of generic S points of irreducible components of X and, for a point x X, let (x) be the ∈ S subset consisting of y with x y . Note that is equi-dimensional if and ∈ S ∈ { } OX,x only if (IV-1) codim( x , y ) = codim( x ,X) { } { } { } 44 YONGNAMLEEANDNOBORUNAKAYAMA for any y (x). In fact, a point y (x) corresponds to a minimal prime ideal p ∈ S ∈ S of via the natural morphism Spec X, and (IV-1) is written as OX,x OX,x → dim /p = dim OX,x OX,x (cf. Property 2.1(1)). The implication (ii) (i) is trivial, and (i) (iii) is shown ⇒ ⇒ by the equality dim = codim( x , y ) for any y (x), which holds by OX,x { } { } ∈ S (i). It suﬃces to prove: (iii) (ii). In the situation of (iii), by (IV-1), we have ⇒ d(x) dim = d(y) = d(y) dim for any x X and y (x). This − OX,x − OX,y ∈ ∈ S implies that x d(x) dim is a constant function with value d(y) on y 7→ − OX,x { } for any y , and d(y) = d(y′) for any points y, y′ with y y′ = . ∈ S ∈ S { } ∩ { } 6 ∅ Consequently, x d(x) dim is constant on X, since X is connected. Thus, 7→ − OX,x we are done.

The importance of the codimension function comes from the following:

Fact 4.5. Let X be a locally Noetherian scheme with a dualizing complex •. R Then, we can define a function d: X Z by → 0, for i = d(x); i • i R • ExtOX,x (k(x), x) = H ( HomOX,x (k(x), x)) = 6 R R (k(x), for i = d(x), where k(x) denotes the residue field at x and • denotes the stalk at x (cf. [17, V, Rx Prop. 3.4]). The function d is a bounded codimension function (cf. [17, V, Cor. 7.2]), and we call d the codimension function associated with •. In particular, X is R catenary and has finite Krull dimension.

The following result and Lemma 4.8 below are useful for checking Sk-conditions for coherent sheaves.

Proposition 4.6. Let X be a locally Noetherian scheme admitting a dualizing complex • with codimension function d: X Z. Let be a coherent -module. R → F OX For an integer j, we set

(j) j • j • := Ext ( , ) := (R omO ( , )). G OX F R H H X F R Then, (j) is a coherent -module and G OX (j) j • Ext ( x, ) Gx ≃ OX,x F Rx (j) (j) for the stalk x = ( ) at any point x X. Moreover, the following hold for a G G x ∈ point x X: ∈ (j) (1) If j d(x) < dim x or j d(x) > 0, then x =0. − − F − G(j) (2) For an integer k, depth x k if and only if x =0 for any j > d(x) k. F ≥ G (j) − (3) For an integer k, satisﬁes S at x if and only if y = 0 for any point F k G y X with x y and for any j > d(y) inf k, dim y . ∈ ∈ { } − { F }(j) (4) is a Cohen–Macaulay -module if and only if x = 0 for any Fx OX,x G j = d(x) dim x. 6 − F (i) (5) If x Supp , then x =0 for i = d(x) dim . ∈ F G 6 − Fx GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 45

Proof. The first assertion is derived from: • D+ (X) . The assertions (1) and R ∈ coh fid (2) are essentially proved in [17, V]; (1) is shown in the proof of [17, V, Prop. 3.4], and (2) follows from the local duality theorem [17, V, Cor. 6.3]. The assertion (3) follows from (2) and Definition 2.9. The assertion (4) is a consequence of (1) and (2), since is Cohen–Macaulay if and only if depth = dim unless = 0. Fx Fx Fx Fx The assertion (5) is shown as follows. For the given point x Supp , we can find ∈ F a point y Supp such that y is an irreducible component of Supp containing ∈ F { } (d(Fy)) x and dim x = codim( x , y ). Then, d(x) dim x = d(y). If x = 0, then (d(y)) F { } { } − (j) F G y = 0, since x y . But, in this case, y = 0 for any j Z by (1), i.e., G ∈ { } G ∈ R om ( , •) 0. This is a contradiction, since = 0 and H OX F R y ≃qis Fy 6 R om (R om ( , •), •) F ≃qis H OX H OX F R R (i) by [17, V, Prop. 2.1]. Therefore, x = 0 for i = d(x) dim = d(y). G 6 − Fx Corollary 4.7. Let X be a locally Noetherian scheme admitting a dualizing complex • and let be a coherent -module. R F OX (1) Assume that Supp is connected. Then, is a Cohen–Macaulay - F F OX module if and only if R om ( , •) [ c] H OX F R ≃qis G − for a coherent -module and a constant c Z. In this case, is also OX G ∈ G a Cohen–Macaulay -module and Supp = Supp . OX G F (2) Assume that X is connected. Then, X is Cohen–Macaulay if and only if • [ c] for a coherent -module and a constant c Z. In this R ≃qis L − OX L ∈ case, is also a Cohen–Macaulay -module and Supp = X. L OX L (3) Assume that be a Cohen–Macaulay -module and let S be a closed F OX subscheme of X such that S = Supp as a set. Then, S is locally equi- F dimensional.

Proof. It suﬃces to prove(1) and (3), since (2) is a special case of (1). Let d: X Z → be the codimension function associated with •. First, we shall prove the “if” part R of (1). The quasi-isomorphism in (1) implies that (j) := Extj ( , •) = 0 for G OX F R any j = c and (c). Then, is Cohen–Macaulay and c = d(x) dim for any 6 G ≃ G F − Fx x X by (4) and (5) of Proposition 4.6. Second, we shall prove the remaining part ∈ of (1) and (3). For the proof of (3), we may also assume that Supp is connected. F Suppose that is Cohen–Macaulay. Then, d(x) dim = d(y) dim holds F − Fx − Fy for any points x, y S with x y by (4) and (5) of Proposition 4.6, where ∈ (j) ∈ { } (j) we use the property that x = 0 implies y = 0. As a consequence, c := G G d(x) dim is constant on S = Supp . We have dim = dim for any − Fx F Fx OS,x x S by Property 2.1(1). Thus, S is locally equi-dimensional by Lemma 4.4, and ∈ this proves (3). Furthermore, R om ( , •) [ c] for the cohomology sheaf H OX F R ≃ G − = (c). We have also G G R om ( [ c], •) F ≃ H OX G − R by [17, V, Prop. 2.1]. Thus, Supp = Supp , and is also a Cohen–Macaulay G F G -module by the “if” part of (1). Thus, we are done. OX 46 YONGNAMLEEANDNOBORUNAKAYAMA

Lemma 4.8. Let X, •, , and (j) be as in Proposition 4.6. Then, (j) = 0 R F G G except for finitely many j. For a positive integer k, the following hold: (1) satisfies S at a point x Supp if and only if F k ∈ F codim (Supp (i) Supp (j), Supp ) k + i j x G ∩ G F ≥ − for any i > j; (2) satisfies S if and only if F k codim(Supp (i) Supp (j), Supp ) k + i j G ∩ G F ≥ − for any i > j.

Proof. The first assertion follows from Proposition 4.6(1), since d: X Z is → bounded and dim X < by Fact 4.5. For integers i, j with i > j, we set ∞ Z(i,j) := Supp (i) Supp (j). G ∩ G Note that codim(Z(i,j), Supp )=+ if Z(i,j) = . The assertion (1) is derived F ∞ ∅ from (2) applied to the coherent sheaf ( )∼ on Spec associated with (cf. Fx OX,x Fx Remark 2.10), since codim (Z(i,j), Supp ) = codim(Supp (i) Supp (j), Supp ) x F Gx ∩ Gx Fx (cf. Property 2.1(3)). Hence, it is enough to prove (2). Assume first that satisfies F S . For integers i > j with Z(i,j) = , we can find a generic point x of Z(i,j) such k 6 ∅ that codim(Z(i,j), Supp ) = codim( x , Supp ) = dim . F { } F Fx If dim k, then i = j = d(x) dim by (1) and (3) of Proposition 4.6. This Fx ≤ − Fx is a contradiction, since i > j. Thus, dim > k, and Fx d(x) dim j j. For a point x Supp , we set c(x) := d(x) dim x. By (1) and (5) of Proposition 4.6, we ∈ F (c(x)) − F (i) (i) know that x Supp and x Supp for any ic(x) and 6 dim codim(Z(i,c(x)), Supp ) k + i c(x)= k + i d(x) + dim . Fx ≥ F ≥ − − Fx Hence, i d(x) k and dim > k. Thus, satisfies S by Proposition 4.6(3). ≤ − Fx F k Therefore, (2) has been proved, and we are done.

Corollary 4.9. Let X, •, , and (j) be as in Proposition 4.6. Let k be a R F G positive integer. (1) Assume that Supp is connected and locally equi-dimensional. Then, there F is a positive integer c such that c = d(x) dim for any x X. For the − Fx ∈ integer c, one has Supp (c) = Supp . Moreover, satisﬁes S if and G F F k only if codim(Supp (j), Supp ) k + j c G F ≥ − GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 47

for any j>c. (2) Assume that Supp is connected, equi-dimensional, and equi-codimen- F sional (cf. [12, 0 , Déf. (14.2.1)]). Furthermore, assume that Supp is IV F Noetherian. Let c be the integer in (1). Then, satisfies S if and only if F k dim Supp (j) dim Supp + c j k G ≤ F − − for any j>c. (3) Assume that x =0 and x is equi-dimensional (cf. Definition 2.2). Then, F 6 F (j) satisfies S at x if and only if dim x d(x) j k for any j = F k G ≤ − − 6 d(x) dim . − Fx Proof. (1): For a closed subscheme S with S = Supp , we have the integer c such F that c = d(x) dim = d(x) dim for any x Supp by Lemma 4.4. Then, − OS,x − Fx ∈ F Supp (c) = Supp by Proposition 4.6(5). Assume that satisfies S . Then, G F F k codim(Supp (j), Supp ) = codim(Supp (j) Supp (c), Supp ) k + j c G F G ∩ G F ≥ − for any j>c by Lemma 4.8. Conversely, assume that the inequality in (1) holds (j) for any j>c. If x = 0 for some j>c, then G 6 dim codim(Supp (j), Supp ) k + j c = k + j d(x) + dim Fx ≥ G F ≥ − − Fx (j) as in the proof of Lemma 4.8(2). Hence, x = 0 implies that dim > k and G 6 Fx j d(x) k. This means that satisfies S by Proposition 4.6(3). ≤ − F k (2): The closed subset S = Supp is a bi-equi-dimensional Kolmogorov Noe- F therian space in the sense of EGA (cf. [12, 0IV, Prop. (14.3.3)]), since it is catenary and has finite Krull dimension (cf. Fact 4.5). Then, x dim = codim( x ,S) 7→ OS,x { } is a codimension function on S by [12, 0IV, (14.3.3.2)], and it implies that S is locally equi-dimensional by Lemma 4.4. Moreover, dim Z + codim(Z,S) = dim S for any closed subset Z S by [12, 0 , Cor. (14.3.5)]. Thus, (2) follows from (1). ⊂ IV (3): The closed subset Supp of Spec is equi-codimensional, since x is the Fx OX,x unique closed point of it. Hence, Supp is also a connected bi-equi-dimensional Fx Kolmogorov Noetherian space and it is locally equi-dimensional by the same reason as above. Thus, we can apply (1) to the coherent sheaf ∼ on Spec associated Fx OX,x with . Hence, satisfies S at x (cf. Remark 2.10) if and only if Fx F k codim(Supp (j), Supp ) k + j c(x) Gx Fx ≥ − for any j >c(x), where c(x) := d(x) dim x. Here, the left hand side equals (j) − F dim dim x by [12, 0 , Cor. (14.3.5)]. Therefore, the S -condition at x is Fx − G IV k equivalent to that dim (j) c(x) k j + dim = d(x) k j Gx ≤ − − Fx − − for any j>c(x)= d(x) dim . Thus, we have (3) by Proposition 4.6(1), and we − Fx are done. Definition 4.10 (Gor(X)). The Gorenstein locus Gor(X) of a locally Noetherian scheme X is defined to be the set of points x X such that is Gorenstein. ∈ OX,x 48 YONGNAMLEEANDNOBORUNAKAYAMA

Note that X is Gorenstein if and only if X = Gor(X). The following is a generalization of [12, IV, Prop. (6.11.2)(ii)] (cf. [51, Prop. (3.2)] for Gor(X)).

Proposition 4.11. Let X be a locally Noetherian scheme admitting a dualizing complex locally on X and let be a coherent -module. Then, S ( ) for all F OX k F k 1 and CM( ) are open subsets of X. In particular, CM(X) is open. Moreover, ≥ F Gor(X) is also open.

Proof. Localizing X, we may assume that X is an aﬃne Noetherian scheme with a dualizing complex •. The openness of Gor(X) follows from that of CM(X). In R fact, if X is Cohen–Macaulay, then we may assume that • for a coherent R ≃qis L -module by Corollary 4.7(2), and Gor(X) is the maximal open subset on which OX L is invertible. The openness of CM( ) is derived from Corollary 4.7(1). This L F follows also from the openness of S ( ) for all k 1. In fact, CM( )= S ( ) for k F ≥ F k F k 0, since dim dim X < (cf. Fact 4.5 and Remark 2.12). The openness ≫ F ≤ ∞ of S ( ) is derived from Lemma 4.8(1), since x codim (Z, Supp ) is lower k F 7→ x F semi-continuous for any closed subset Z Supp (cf. Property 2.1(3)). ⊂ F Remark. In the situation of Proposition 4.11, all S ( ) are open if and only if the k F map Supp x codepth := dim depth Z F ∋ 7→ Fx Fx − Fx ∈ ≥0 is upper semi-continuous (cf. [12, IV, Rem. (6.11.4)]).

The following analogy of Fact 2.26(6) for = is known: G OY Fact 4.12 (cf. [36, Th. 23.4], [17, V, Prop. 9.6]). Let Y T be a ﬂat morphism of → locally Noetherian schemes. Then, Y is Gorenstein if and only if T and every ﬁber are Gorenstein.

4.2. Ordinary dualizing complex. We introduce the notion of ordinary dualizing complex • and that of dualizing sheaf as the cohomology sheaf 0( •) for lo- R H R cally Noetherian schemes which are locally equi-dimensional (cf. Deﬁnition 2.2(3)), especially for locally Noetherian schemes satisfying S2. In many articles, the dualizing sheaf is usually deﬁned for a Cohen–Macaulay scheme, and it coincides with the dualizing sheaf in our sense (cf. Remark 4.15 below).

Deﬁnition 4.13. Let X be a locally Noetherian scheme. (1) A dualizing complex • of X is said to be ordinary if the codimension R function d associated with • satisﬁes d(x) = dim for any x X. R OX,x ∈ (2) A coherent sheaf on X is called a dualizing sheaf of X if 0( •) L L ≃ H R for an ordinary dualizing complex • of X. R As a corollary to Lemma 4.4 above, we have:

Lemma 4.14. Let X be a locally Noetherian scheme admitting a dualizing complex. Then, X admits an ordinary dualizing complex if and only if X is locally equi- dimensional. In particular, X admits an ordinary dualizing complex if X satisﬁes S2. GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 49

Proof. We may assume that X is connected. Let • be a dualizing complex of R X with codimension function d: X Z. If it is ordinary, then X is locally equi- → dimensional by Lemma 4.4. Conversely, if X is locally equi-dimensional, then d(x) − dim is a constant c by Lemma 4.4, and hence, the shift •[c] is an ordinary OX,x R dualizing complex. The last assertion follows from Facts 4.5 and 2.23(1). Remark. For a locally Noetherian scheme, the ordinary dualizing complex is unique up to quasi-isomorphism and tensor product with an invertible sheaf (cf. Re- mark 4.2). Similarly, the dualizing sheaf is unique up to isomorphism and tensor product with an invertible sheaf. Remark 4.15. Let X be a locally Noetherian Cohen–Macaulay scheme admitting a dualizing complex. Then, X has an ordinary dualizing complex • which is quasi- R isomorphic to the dualizing sheaf = 0( •). Here, is also a Cohen–Macaulay L H R L -module. These are derived from Proposition 4.6(4) and Corollary 4.7(2). In OX many articles, is called a “dualizing sheaf” for a locally Noetherian Cohen– L Macaulay scheme. Lemma 4.16. Let X be a locally Noetherian scheme admitting an ordinary dualizing complex •. Let Z(i) be the support of the cohomology sheaf i( •) for any R H R i Z. Then, Z(i) = for any i < 0, Z(0) = X, and the following hold for any ∈ ∅ x X: ∈ (1) x Z(i) for any i> dim ; 6∈ OX,x (2) depth = dim sup j x Z(j) ; OX,x OX,x − { | ∈ } (3) for an integer k 1, X satisfies S at x if and only if ≥ k codim (Z(j),X) k + j x ≥ for any j > 0. This is also equivalent to: dim Z(j) dim k j x ≤ OX,x − − for any j > 0; (4) is Cohen–Macaulay if and only if x Z(j) for any j > 0. OX,x 6∈ Proof. Now, d(x) = dim for the codimension function d: X Z associated OX,x → with •, and X is locally equi-dimensional by Lemma 4.4. Thus, applying Proposi- R tion 4.6 to = , we have the assertions except (3). The assertion (3) is obtained F OX by Corollary 4.9(3) (cf. Property 2.1). Remark. Let X be a connected locally Noetherian scheme with a dualizing complex • such that i( •) = 0 for any i < 0 and 0( •) = 0. The sheaf 0( •) is R H R H R 6 H R called the “canonical module” in many articles. But as in Example 4.17 below, the support of the sheaf 0( •) is not always X. This is one of the reasons why H R we do not consider 0( •) as the dualizing sheaf for arbitrary locally Noetherian H R schemes. Example 4.17. Let P be a polynomial ring k[x, y, z] of three variables over a field k. For the ideals I = (x, y) and J = (z) of P , we set A := P/IJ and R• := RHomP (A, P [1]). Then, we have a Noetherian affine scheme X = Spec A and a 50 YONGNAMLEEANDNOBORUNAKAYAMA dualizing complex • on X associated with R• (cf. Example 4.23 below). The R X is a union of a plane Spec P/J and a line Spec P/I in the three-dimensional affine space Spec P A3, where the plane and the line intersect at the origin O ≃ k corresponding to the maximal ideal (x, y, z). Note that the local ring is not OX,O equi-dimensional. We can calculate the cohomology modules of R• as 0, for any i< 0 and i> 1; i • i+1 H (R ) ExtP (A, P ) P/J, for i = 0; ≃ ≃  P/I, for i =1, by the free resolution   g f 0 P P ⊕2 P A 0, → −→ −→ → → where f and g are defined by f(a,b)= xza + yzb and g(c) = (yc, xc) − for any a, b, and c P . Consequently, Supp 0( •) = Spec R/J is a proper subset ∈ H R of X. Lemma 4.18. Let X be a locally Noetherian scheme admitting an ordinary dualizing complex •. We set R (j) := Extj (τ ≤b( •), •) and (j) := Extj (τ ≥b( •), •) G≤b OX R R G≥b OX R R for integers b 0 and j, where τ ≤b and τ ≥b stand for the truncations of a complex ≥ (cf. Notation and conventions, (1)). Then, the following hold: (1) One has: (0) and (i) =0 for any i =0. G≥0 ≃ OX G≥0 6 (2) There exist an exact sequence 0 (−1) (0) (0) (1) 0 → G≤b → G≥b+1 → OX → G≤b → G≥b+1 → and an isomorphism (j) (j+1) G≤b ≃ G≥b+1 for any j = 0, 1 . Moreover, (j) =0 for any j < 0. 6 { − } G≤0 (3) For any integers b 0 and j, one has: ≥ codim(Supp (j),X) j + b for any j Z, • G≥b ≥ ∈ codim(Supp (j),X) j + b +2 for any j =0, and • G≤b ≥ 6 codim(Supp (0),X)=0. • G≤b (4) If X satisfies S for some k 1, then k ≥ (j) =0 for any b> 0 and j < k, and • G≥b (i) =0 for any 0

The exact sequence and the isomorphism in the second assertion (2) are derived from the canonical distinguished triangle τ ≤b( •) • τ ≥b+1( •) τ ≤b( •)[1] . ···→ R → R → R → R → · · · The last vanishing in (2) is expressed as Extj ( , •) = 0 for any j < 0, where OX L R := 0( •), and this is a consequence of Proposition 4.6(1) applied to = with L H R F L the property X = Supp shown in Lemma 4.16. For the remaining assertions (3) L and (4), it is enough to consider only the sheaves (j). In fact, by (2), we have an G≥b injection (i) (i+1) for any i = 0, and an exact sequence (0) (0), G≤b → G≥b+1 6 G≥b+1 → OX → G≤b where codim(Supp (0) ,X) > 0 by the assertion for (0) . Hence, G≥b+1 G≥b+1 codim(Supp (i),X) codim(Supp (i+1),X) G≤b ≥ G≥b+1 for any i = 0 and codim(Supp (0),X) = 0. In order to prove (3) and (4) for (j), 6 G≤b G≥b let us consider the spectral sequence (IV-2) p,q = Extp ( −q(τ ≥b( •)), •) p+q = (p+q) E2 OX H R R ⇒E G≥b of -modules (cf. Remark 4.19 below). Assume that ( p,q) = 0 for a point OX E2 x 6 x X. Then, q b, and ∈ − ≥ (IV-3) dim p dim dim −q( •) = codim (Supp −q( •),X) OX,x ≥ ≥ OX,x − H R x x H R by Proposition 4.6(1), since d(x) = dim for the codimension function d of OX,x •. In particular, p + q dim b. Therefore, if j + b > dim , then R ≤ OX,x − OX,x x Supp (j), since ( p,q) = 0 for any integers p, q with p + q = j. Thus, we 6∈ G≥b E2 x have (3). Assume that X satisfies S . If( p,q) = 0 and q < 0, then p + q k by k E2 x 6 ≥ (IV-3), since codim (Supp −q( •),X) k q x H R ≥ − for any q < 0 by Lemma 4.16(3). Hence, (j) = 0 for any b > 0 and j < k, since G≥b p,q = 0 for any integers p, q with p + q = j. This proves (4), and we are done. E2 Remark 4.19. The spectral sequence (IV-2) is obtained by the same method as follows. Let A be a commutative ring and let M • and N • be complexes of A- modules such that N • is bounded below. We shall construct a spectral sequence Ep,q = Extp (H−q(M •),N •) Ep+q = Extp+q(M •,N •), 2 A ⇒ A p where ExtA denotes the p-th hyper-ext group. Since there is a quasi-isomorphism from N • into a complex of injective A-modules bounded below, we may assume that N • itself is a complex of injective A-modules bounded below. We consider a double complex K•,• defined by Kp,q = Hom (M −p,N q) for p, q Z with the dif- A ∈ ferentials d : Kp,q Kp+1,q and d : Kp,q Kp,q+1, which are induced from the I → II → differentials M −p−1 M −p and N q N q+1, respectively. Then, Extk(M •,N •) → → is isomorphic to the k-th cohomology group of the total complex K• defined by n p,q K = p+q=n K (cf. [17, I, Th. 6.4]). Moreover, we have Q Hq(K•,p) Hom (H−q(M •),N p) I ≃ A 52 YONGNAMLEEANDNOBORUNAKAYAMA for any p and q, since N p is now assumed to be injective. Thus, we have the spectral sequence above as the well-known spectral sequence Hp Hq(K•,•) Hp+q(K•) II I ⇒ associated with the double complex K•,•. Corollary 4.20. Let X be a locally Noetherian scheme admitting an ordinary dualizing complex •. For a point x X and for an integer b 0, the vanishing R ∈ ≥ Hi (τ ≤b( •) )=0 x R x holds for any i 0, where (i) = xti ( , •). For the remaining assertions, we assume that G≤0 E OX L R X satisfies S2 or S3. Note that we have an isomorphism om ( , ) Ext0 ( , •)= (0) H OX L L ≃ OX L R G≤0 and an injection xt 1 ( , ) Ext1 ( , •)= (1) E OX L L → OX L R G≤0 by τ ≤0( •). If X satisfies S , then (0) = (1) = 0 by Lemma 4.18(4), and L≃qis R 2 G≥1 G≥1 hence, (0) by Lemma 4.18(2); thus, om ( , ). If X satisfies S , OX ≃ G≤0 OX ≃ H OX L L 3 then (1) = 0 by Lemma 4.18(4), and consequently, xt 1 ( , ) = 0. G≤0 E OX L L Remark (S2-ification). For a locally Noetherian scheme X admitting an ordinary dualizing complex • and for the dualizing sheaf = 0( •), we consider the R L H R coherent -module := om ( , ). Then, we can show: OX A H OX L L has a structure of -algebra, • A OX GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 53

is an isomorphism on the S -locus S (X) (cf. Definition 2.13), • OX → A 2 2 and satisfies S . • A 2 Therefore, the finite morphism pec X is regarded as the so-called “S - S X A → 2 ification” of X (cf. [12, IV, (5.10.11), Prop. (5.11.1)], [4, Prop. 2], [5, Th. 3.2], [22, Prop. 2.7]). Three properties above are shown as follows: We know that satisfies L S , U := S (X) is an open subset by Proposition 4.11, and that is an 2 2 OX → A isomorphism on U by Proposition 4.21. In particular, j ( ) for the open A ≃ ∗ A|U immersion j : U ֒ X, since it is expressed as → = om ( , ) j ( ) om ( , j ( )). A H OX L L → ∗ A|U ≃ H OX L ∗ L|U Thus, satisfies S by Corollary 2.16, and consequently, j has an - A 2 A ≃ ∗OU OX algebra structure. Corollary 4.22. Let X be a locally Noetherian scheme admitting a dualizing complex •, and set := 0( •). Let X◦ X be an open subset such that R L H R ⊂ ◦ • codim(X X ,X) 1 and ◦ ◦ . \ ≥ R |X ≃qis L|X • ◦ Then, is ordinary and satisfies S2. In particular, if codim(X X ,X) 2, R L ◦ \ ≥ .then j ( ◦ ) for the open immersion j : X ֒ X L≃ ∗ L|X → Proof. It is enough to prove that • is ordinary. In fact, if so, then the dualizing R sheaf satisfies S by Proposition 4.21, and we have the isomorphism j ( ◦ ) L 2 L≃ ∗ L|X by Corollary 2.16 when codim(X X◦,X) 2. Let d: X Z be the codimension • \ ≥ → ◦ function associated with . Then, d(x) = dim X,x for any x X by Proposi- R O ◦ ∈ tion 4.6(5) applied to = ◦ . For a point x X X , we have a generic point F OX ∈ \ y of X such that x y and codim( x , y ) = dim . Then, d(y) = 0, since ∈ { } { } { } OX,x y X◦, and we have ∈ d(x)= d(y) + codim( x , y ) = dim . { } { } OX,x Thus, • is ordinary. R 4.3. Twisted inverse image. We shall explain the twisted inverse image functor, the relative duality theorem, and some base change theorems referring to [17], [7], [34]. Let f : Y T be a morphism of locally Noetherian schemes which is → locally of finite type. In the theory of Grothendieck duality, the “twisted inverse image functor” f ! plays an essential role, which is unfortunately defined only when some suitable conditions are satisfied (cf. [17, III, Th. 8.7], [17, VII, Cor. 3.4], [17, ! Appendix, no. 4], [43], [7], [34]). However, f T has a unique meaning at least ! O locally on Y , where f T is expressed as a complex of Y -modules with coherent O O ! + cohomology which vanish in sufficiently negative degree, i.e., f T Dcoh(Y ). We • ! ! O ∈ write ωY/T := f T whenever f T is defined, and call it the relative dualizing O O • complex for Y/T (or, with respect to f). When T = Spec A, we write ωY/A for • ωY/ Spec A. Example 4.23. For a scheme S, an S-morphism f : Y T of locally Noether- → ian schemes over S is called an S-embeddable morphism if f = p i for a finite ◦ 54 YONGNAMLEEANDNOBORUNAKAYAMA morphism i: Y P T and the second projection p: P T T for a lo- → ×S ×S → cally Noetherian S-scheme P such that P S is a smooth separated morphism → of pure relative dimension (cf. [7, (2.8.1)], [17, III, p. 189]). When S = T , an S- embeddable morphism is called simply an embeddable morphism. There is a theory of f ! : D+ (T ) D+ (Y ) (resp. f ! : D+ (T ) D+ (Y )) for the S-embeddable qcoh → qcoh coh → coh morphisms f : Y T of locally Noetherian S-schemes as in [17, III, Th. 8.7] (cf. → [7, Th. 2.8.1]). For a complex • D+ (T ), if f is separated and smooth of pure G ∈ qcoh relative dimension d (cf. Definition 2.36), then L f !( •)= Ωd [d] Lf ∗( •), G Y/T ⊗OY G and if f is a finite morphism, then f !( •) is defined by G Rf (f !( •)) = R om (f , •). ∗ G H OT ∗OY G In the both cases of f above, f !( •) D+ (Y ) if • D+ (T ). If f = g h for two G ∈ coh G ∈ coh ◦ S-embeddable morphisms h: Y Z and g : Z T , then f ! h! g! as functors → → ≃ ◦ D+ (T ) D+ (Y ) (resp. D+ (T ) D+ (Y )). qcoh → qcoh coh → coh Example 4.24. Let f : Y T be a morphism of finite type between Noetherian → schemes. Then, the dimensions of fibers are bounded. Assume that T admits a dualizing complex • . In this situation, we have the twisted inverse image functor RT f ! : D+ (T ) D+ (Y ) as follows (cf. [17, VI], [7, 3]). For the dualizing complex coh → coh § • of T , we have the corresponding residual complex E( • ) on T (cf. [17, VI, RT RT Prop. 1.1], [7, Lem. 3.2.1]) and the “twisted inverse image” f △(E( • )) on Y as a RT residual complex on Y (cf. [17, VI, Th. 3.1, Cor. 3.5], [7, 3.2]), which corresponds § to a dualizing complex • := f !( • ) := Q(f △(E( • ))) RY RT RT of Y (cf. [17, VI, Prop. 1.1, Remarks in p. 306], [7, 3.3]). Then, one can define § f ! : D+ (T ) D+ (Y ) by coh → coh f !( •)= D (Lf ∗(D ( •))), G Y T G where DY and DT are the dualizing functors defined by: D ( •) := R om ( •, • ) and D ( •) := R om ( •, • ). Y F H OY F RY T G H OT G RT The definition of f ! does not depend on the choice of • (cf. [7, 3.3]), and f ! RT § satisfies expected compatible properties in [17, VII, Cor. 3.4] (cf. [7, Th. 3.3.1]). Moreover, when f is an embeddable morphism, then this f ! is isomorphic to the functor f ! defined in Example 4.23 (cf. [17, VI, Th. 3.1, VII, Cor. 3.4], [7, 3.3]). § The following is shown in [17, V, Cor. 8.4, VI, Prop. 3.4] but with an error concerning (cf. [7, (3.1.25), (3.2.4)]). ± Lemma 4.25. Let f : Y T be a morphism of finite type between Noetherian → schemes such that T admits a dualizing complex • . Let • be the induced du- RT RY alizing complex f !( • ) of Y . Let d : T Z and d : Y Z be the codimension RT T → Y → functions associated with • and • , respectively. Then, RT RY d (y)= d (t) tr. deg k(y)/k(t) Y T − GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 55 for any y Y with t = f(y), where k(t) and k(y) denote the residue fields of ∈ OT,t and , respectively. OY,y Proof. Since the assertion is local on Y , we may assume that Y T is an embed- → dable morphism. Hence, it is enough to prove assuming that f is a finite morphism or a smooth and separated morphism. Assume first that f is finite. Then, Rf R om ( , • ) R om (f , • ) ∗ H OY F RY ≃ H OT ∗F RT for any coherent -module by [17, III, Th. 6.7] (cf. Theorem 4.30 below). OY F Applying this to = for the closed subscheme Z = y with reduced structure, F OZ { } and localizing Y , we have RHom (k(y), ( • ) ) RHom (k(y), ( • ) ). OY,y RY y ≃qis OT,t RT t Since k(y) is a finite-dimensional k(t)-vector space, we have tr. deg k(y)/k(t)=0 and dY (y) = dT (t). Thus, we are done in the case where f is finite. Assume next that f is smooth and separated. We may assume furthermore that f has pure relative dimension d by localizing Y . Then, L • Ωd [d] Lf ∗( • ) RY ≃qis Y/T ⊗OY RT as in Example 4.23, and it implies that i( • ) i+d( • ) , H RY y ≃ H RT t ⊗OT,t OY,y since f is flat. Here, i( • ) = 0 if and only if i+d( • ) = 0, since f is faithfully H RY y 6 H RT t 6 flat. We know that d (t) dim = inf i i( • ) =0 T − OT,t { | H RT t 6 } by (1) and (5) of Proposition 4.6. The similar formula holds also for (Y,y) and • . RY Thus, d (y) dim = d (t) dim d. Y − OY,y T − OT,t − Since f is flat, we have dim = dim + dim OY,y OT,t OYt,y −1 for the fiber Yt = f (t) by (II-1). Furthermore, we have d = dim Y = dim + tr. deg k(y)/k(t), y t OYt,y since Yt is algebraic over k(t) (cf. [12, IV, Cor. (5.2.3)]). Therefore, d (y)= d (t) d + dim dim = d (t) tr. deg k(y)/k(t). Y T − OY,y − OT,t T − Definition 4.26 (canonical dualizing complex). Let X be an algebraic scheme over a field k, i.e., a k-scheme of finite type. We define the canonical dualizing • ! complex ωX/k of X to be the twisted inverse image f (k) for the structure morphism f : X Spec k. → • The dualizing complex ωX/k has the following property related to Serre’s conditions Sk. 56 YONGNAMLEEANDNOBORUNAKAYAMA

Lemma 4.27. Let X be an n-dimensional algebraic scheme over a ﬁeld k. For an integer i, let Z be the support of −i(ω• ). Then, Z = for any i>n, and Z is i H X/k i ∅ n the union of irreducible components of X of dimension n. If X is equi-dimensional, then ω• [ n] is an ordinary dualizing complex, and the following hold for integers X/k − k 1: X satisﬁes S if and only if dim Z i k for any i = n. ≥ k i ≤ − 6 Proof. By Lemma 4.25, d(x)= tr. deg k(x)/k for the codimension function d: X − Z associated with the dualizing complex ω• (cf. Example 4.24). Moreover, → X/k (IV-4) n dim X = dim + tr. deg k(x)/k ≥ x OX,x by [12, IV, Cor. (5.2.3)]. Thus, d(x) dim = dim X n, and −i(ω• ) − OX,x − x ≥− H X/k = 0 for any i>n by Proposition 4.6(1) applied to the case where ( •, ) = R F (ω• , ). Thus, Z = for any i>n. If dim X = n, then −n(ω• ) =0 by X/k OX i ∅ x H X/k x 6 Proposition 4.6(5). If dim X

− dim Xα • ω k := (ω ) X/ |Xα H X/k |Xα for any connected component Xα of X.

Remark. The canonical sheaf ωX/k is a dualizing sheaf of X in the sense of Deﬁni- tion 4.13. In fact, ω• [ dim X ]= ω• [ dim X ] Xα/k − α X/k − α |Xα is an ordinary dualizing complex of the connected component Xα by Lemma 4.27. • In particular, if X is connected and Cohen–Macaulay, then ω ω k[dim X]. X/k ≃qis X/ By Corollary 4.22, we have:

Corollary 4.29. For an algebraic scheme X over k, if it is locally equi-dimensional, then ωX/k satisﬁes S2. For a proper morphism f : Y T of Noetherian schemes, we have the following → general result on the twisted inverse image functor f !, which is derived from [34, Th. 4.1.1]:

Theorem 4.30 (Grothendieck duality for a proper morphism). Let f : Y T be → a proper morphism of Noetherian schemes. Then, there is a triangulated functor f ! : D (T ) D (Y ) which induces D+ (T ) D+ (Y ) and which is right qcoh → qcoh coh → coh GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 57 adjoint to the derived functor Rf : D (Y ) D (T ) in the sense that there ∗ qcoh → qcoh is a functorial isomorphism RHom (Rf ( •), •) RHom ( •,f !( •)) OT ∗ F G ≃qis OY F G for • D (Y ) and • D (T ). F ∈ qcoh G ∈ qcoh Remark. In [34, Th. 4.1.1], the existence of a similar right adjoint f × is proved for a quasi-compact and quasi-separated morphism f : Y T of quasi-compact and → quasi-separated schemes Y and T . When f is proper, it is written as f ! (cf. the paragraph just before [34, Cor. 4.2.2]). By [53, Th. A], the total derived functor RHom of Hom exists for any scheme X as a bi-functor D(X)op D(X) OX OX × → D(Z), and there exists also the total right derived functor Rf : D(Y ) D(T ) of ∗ → the direct image functor f . When f : Y T is a proper morphism of Noetherian ∗ → schemes, we have: Rf (D (Y )) D (T ) by [34, Prop. 3.9.1], • ∗ qcoh ⊂ qcoh Rf (D+ (Y )) D+ (T ) by [17, II, Prop. 2.2], and • ∗ coh ⊂ coh Rf (D− (Y )) D− (T ) by the explanation just before [34, Cor. 4.2.2]. • ∗ coh ⊂ coh The functor f ! is bounded below (cf. [34, Def. 11.1.1]). Thus, f !(D+ (T )) qcoh ⊂ D+ (Y ). The inclusion f !(D+ (T )) D+ (Y ) is proved firstly by reducing to qcoh coh ⊂ coh the case where T is the spectrum of a Noetherian local ring by the base change isomorphism (cf. [34, Cor. 4.4.3]), and secondly by applying [55, Lem. 1]. Remark. When T admits a dualizing complex (or a residual complex), Theorem 4.30 for • D+ (T ) is a consequence of [17, VII, Cor. 3.4]. In [9, Th. 2], Deligne has G ∈ coh proved Theorem 4.30 for • Db (Y ) without assuming the existence of dualizing F ∈ coh complex of T . These results are summarized by Verdier as [55, Th. 1], which is almost the same as Theorem 4.30 in the case where T has finite Krull dimension. Neeman [43] gives a new idea toward the proof of Theorem 4.30 by using Brown representability. He generalizes Theorem 4.30 to the case where Y and T are only quasi-compact and separated schemes but Dqcoh(T ) and Dqcoh(Y ) are replaced with D(QCoh( )) and D(QCoh( )), respectively (cf. [43, Exam. 4.2]). Neeman’s idea OT OY is used in Lipman’s article [34], which contains generalizations of Theorem 4.30 to non-proper and non-Noetherian case. The sheafified form of the duality theorem is as follows (cf. [34, Th. 4.2]): Corollary 4.31. In the situation of Theorem 4.30, there exists a canonical quasi- isomorphism Rf R om ( •,f ! •) R om (Rf •, •) ∗ H OY F G ≃qis H OT ∗F G for any • D (Y ) and • D (T ). F ∈ qcoh G ∈ qcoh As a special case of Theorem 4.30, we have the following, which is called the Serre duality theorem for coherent sheaves. Corollary 4.32. Let X be a projective scheme over a field k. Then, there is a canonical quasi-isomorphism • • • RHom ( ,ω ) RHomk(RΓ(X, ), k) OX F X/k ≃qis F 58 YONGNAMLEEANDNOBORUNAKAYAMA for any • D+ (X). In particular, F ∈ coh i • • i • Ext ( ,ω ) Homk(H (X, ), k) OX F X/k ≃ F for any i, where Exti and Hi stands for the i-th hyper-Ext group and i-th hyper cohomology group, respectively.

Example 4.33. Let f : Y T be a separated morphism of finite type between Noe- → therian schemes. By the Nagata compactification theorem (cf. [41], [42], [35], [8], f is expressed as the composite π j of an open immersion j : Y ֒ Z and a ,([11] ◦ → proper morphism π : Z T . Using the functor π! : D+ (T ) D+ (Z) in Theo- → qcoh → qcoh rem 4.30, we define the twisted inverse image functor f ! : D+ (T ) D+ (Y ) as qcoh → qcoh Lj∗ π!. This is well-defined up to functorial isomorphism, i.e., it is independent of ◦ the choice of factorization f = π j, by [9, Th. 2], [55, Cor. 1]. Deligne [9] defines ◦ a functor Rf : pro Db (Y ) pro Db (T ) and shows in [9, Th. 2] that f ! above ! coh → coh is a right adjoint of Rf!. Fact 4.34. The twisted inverse image functors in Example 4.33 have the following properties. Let f : Y T be a separated morphism of finite type between → Noetherian schemes. (1) Let h: X Y be a separated morphism of finite type from another Noether- → ian scheme X. Then, there is a functorial isomorphism (f h)! h! f !. ◦ ≃ ◦ (2) If f : Y T is a smooth morphism of pure relative dimension d, then ! • → d L ∗ • f ( ) qis Ω [d] Lf ( ). G ≃ Y/T ⊗OY G (3) If T admits a dualizing complex, then f ! is functorially isomorphic to the twisted inverse image functor D Lf ∗ D in Example 4.24. Y ◦ ◦ T (4) For a flat morphism g : T ′ T from a Noetherian scheme T ′, let Y ′ be the → fiber product Y T ′ and let f ′ : Y ′ T ′ and g′ : Y ′ Y be the induced ×T → → morphisms. Then, g′∗ f ! f ′! g∗. ◦ ≃ ◦ The property (1) is derived from the isomorphism R(f h) Rf Rh shown ◦ ! ≃ ! ◦ ! in [9, no. 3]. This is also proved in [34, Th. 4.8.1]. The properties (2) and (3) are proved in [55, Th. 3, Cor. 3] and [34, (4.9.4.2), Prop. 4.10.1]. In order to prove the property (4), we may assume that f is proper, and in this case, this is shown in [34, Cor. 4.4.3] (cf. [55, Th. 2]). As a refinement of the property (1) above, f f ! can be regarded as a pseudo-functor, and Lipman proves in [34, Th.4.8.1] 7→ the uniqueness of f f ! under three conditions corresponding to: 7→ f ! is a right adjoint of Rf when f is proper (Theorem 4.30); • ∗ The property (2) above for étale f; • The property (4) above for proper f and étale g. • Fact 4.35. The following are also known for a flat separated morphism f : Y T → of finite type between Noetherian schemes: (1) The twisted inverse image f ! is an f-perfect complex in D (Y ) (cf. [23, OT coh III, Prop. 4.9], [34, Th. 4.9.4]). For the definition of “f-perfect,” see [23, III, Déf. 4.1] (cf. Remark 4.36 below). Note that a coherent -module OY flat over T is f-perfect. GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 59

(2) For an f-perfect complex •, E D ( •) := R om ( •,f ! ) Y/T E H OY E OT is also f-perfect and the canonical morphism • D (D ( •)) E → Y/T Y/T E is a quasi-isomorphism (cf. [23, III, Cor. 4.9.2]). In particular, (IV-5) R om (f ! ,f ! ) OY → H OY OT OT is a quasi-isomorphism (cf. [34, p. 234]). (3) There is a quasi-isomorphism L L f !( •) Lf ∗( •) f !( • •) F ⊗OY G ≃qis F ⊗OT G for any •, • D+ (T ) with • L • D+ (T ) (cf. [34, Th. 4.9.4]). F G ∈ qcoh F ⊗OT G ∈ qcoh In particular, L (IV-6) f ! Lf ∗( •) f !( •) OT ⊗OY G ≃qis G for any • D+ (T ). Similar results are proved in [17, V, Cor. 8.6], [55, G ∈ qcoh Cor. 2], and [43, Th. 5.4]. Remark 4.36 (cf. [23, III, Prop. 4.4]). Let f : Y T be a morphism of finite type → between Noetherian schemes and let • be an object of D (Y ). Assume that F qcoh f is the composite g i of a closed immersion i: Y P and a smooth separated ◦ → morphism g : P T . Then, • is f-perfect if and only if Ri ( •) is perfect (cf. → F ∗ F [23, I, Déf. 4.7]), i.e., locally on P , it is quasi-isomorphic to a bounded complex of free -modules. OP Lemma 4.37. Let f : Y T be a flat separated morphism of finite type between → Noetherian schemes in which T admits a dualizing complex. Let g : T ′ T be a → finite morphism from another Noetherian scheme T ′. For the fiber product Y ′ = Y T ′, let f : Y ′ T ′ and g′ : Y ′ Y be the projections. Thus, we have a ×T → → Cartesian diagram: g′ Y ′ Y −−−−→ f ′ f

′ g  T T. In this situation, there is a naturaly quasi-isomorphism−−−−→ y ′∗ ! ′ ! Lg (f ) f ′ . OT ≃qis OT Proof. Let DT , DY , DT ′ , and DY ′ , respectively, be the dualizing functors on T , Y , T ′, and Y ′ defined by a dualizing complex on T and their transforms by f !, g!, and (f g′)! (g f ′)! (cf. Fact 4.34(1)) as in Example 4.24. For any • D+ (T ′), ◦ ≃ ◦ G ∈ coh we have ! • ∗ • ∗ • f (Rg ( )) D Lf D (Rg ( )) D Lf Rg (D ′ ( )) ∗ G ≃qis Y ◦ ◦ T ∗ G ≃qis Y ◦ ◦ ∗ T G ′ ′∗ • ′ ′∗ • D Rg Lf (D ′ ( )) Rg D ′ (Lf (D ′ ( ))) ≃qis Y ◦ ∗ ◦ T G ≃qis ∗ ◦ Y T G Rg′ (f ′ !( •)), ≃qis ∗ G 60 YONGNAMLEEANDNOBORUNAKAYAMA where we use the flat base change isomorphism: Lf ∗ Rg Rg′ Lf ′∗ (cf. ◦ ∗ ≃qis ∗ ◦ Proposition A.10), and the duality isomorphisms: DT Rg∗ qis Rg∗ DT ′ and ′ ′ ◦ ′ ≃ ◦ D Rg Rg D ′ for the finite morphisms g and g (cf. Corollary 4.31). On Y ◦ ∗ ≃qis ∗ ◦ Y the other hand, we have L L f !(Rg ( •)) f ! Lf ∗(Rg •) f ! Rg′ (Lf ′∗( •)) ∗ G ≃qis OT ⊗OY ∗G ≃qis OT ⊗OY ∗ G by the quasi-isomorphism (IV-6) in Fact 4.35 and by the flat base change isomor- • phism. Substituting = ′ , we have a quasi-isomorphism G OT ! L ′ ′ ′ ! f Rg ′ Rg (f ′ ). OT ⊗OY ∗OY ≃qis ∗ OT ′∗ ! ′ ! ′ + ′ It is associated with a morphism Lg (f T ) f T in Dcoh(Y ) which induces ′ O →′∗ O! ′ ! a quasi-isomorphism by taking Rg . Hence, Lg (f ) f ′ . ∗ OT ≃qis OT Corollary 4.38 (cf. [44, Prop. 3.3(1)]). Let f : Y T be a flat separated morphism → of finite type between Noetherian schemes. For a point t T , let φt : Spec k(t) T ∈ −1 → be the canonical morphism for the residue field k(t), and let ψt : Yt = f (t) Y be →• the base change of φt by f : Y T . Then, the canonical dualizing complex ω → Yt/k(t) defined in Definition 4.26 is quasi-isomorphic to Lψ∗(f ! ). t OT Proof. Let Spec T be the canonical morphism from the spectrum of the local OT,t → ring . Considering the completion of and the surjection k(t) OT,t OT,t OT,t OT,t → to the residue field, we have a flat morphism b b τ : T ♭ := Spec Spec T OT,t → OT,t → ♭ and a closed immersion ι: Spec k(t) ֒ T ♭. Let Y ♭ be the fiber product Y T →b ×T and let f ♭ : Y ♭ T ♭ and τ ′ : Y ♭ Y be projections, which make a Cartesian → → diagram: ′ Y ♭ τ Y −−−−→ f ♭ f

♭ τ  T T. By Fact 4.34(4), we have a quasi-isomorphismy −−−−→ y ′∗ ! ♭! Lτ (f ) f ♭ . OT ≃qis OT ♭ Hence, we may assume from the beginning that T = T . Then, φt is the closed immersion ι. Now, T admits a dualizing complex, since we have a surjection to from a complete regular local ring by Cohen’s structure theorem. Thus, we OT,t are done by Lemma 4.37. b 4.4. Cohen–Macaulay morphisms and Gorenstein morphisms. The notions of Cohen–Macaulay morphism and Gorenstein morphism are introduced in [12, IV, Déf. (6.8.1)] and [17, V, Ex. 9.7]. By [7, Sect. 3.5] or [50, Th. 2.2.3], one can define the relative dualizing sheaf for a Cohen–Macaulay morphism (cf. Definition 4.43 below), and prove a base change property (cf. Theorem 4.46 below). We shall explain these facts. We have defined the notion of Cohen–Macaulay morphism in Definition 2.30. The notion of Gorenstein morphism is defined as follows. GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 61

Definition 4.39 (Gor(Y/T )). Let Y and T be locally Noetherian schemes and f : Y T a flat morphism locally of finite type. We define → Gor(Y/T ) := Gor(Yt), t∈T and call it the relative Gorenstein locus[ for f. The flat morphism f is called a Gorenstein morphism if Gor(Y/T )= Y .

Remark. The Gorenstein locus Gor(Y/T ) is open. In fact, this is characterized as the maximal open subset of the relative Cohen–Macaulay locus Y ♭ = CM(Y/T ) on which the relative dualizing sheaf ωY ♭/T is invertible (cf. Lemma 4.40 below), where Y ♭ is open by Fact 2.29(1).

The following characterizations of Cohen–Macaulay morphism and Gorenstein morphism are known:

Lemma 4.40 ([17, V, Exer. 9.7], [7, Th. 3.5.1]). Let f : Y T be a flat mor- → phism locally of finite type between locally Noetherian schemes. Then, f is Cohen– Macaulay if and only if, locally on Y , the twisted inverse image f ! is quasi- OT isomorphic to an f-flat coherent -module ω up to shift. Here, f is Gorenstein OY Y/T if and only if ωY/T is invertible. Proof. We may assume that f is a separated morphism of finite type between affine Noetherian schemes by localizing Y and T . Assume first that f ! ω [d] OT ≃qis Y/T for a coherent Y -module ωY/T flat over T and for an integer d. For an arbitrary O • fiber Yt, the dualizing complex ω is quasi-isomorphic to ωY/T OY Yt [d] by Yt/k(t) ⊗ O Corollary 4.38. Hence, Yt is Cohen–Macaulay by Corollary 4.7(2) or Lemma 4.16(4). Conversely, assume that every fiber Yt is Cohen–Macaulay. Then, we may assume that f has pure relative dimension d by Lemma 2.38. We shall show that f ! ω [d] OT ≃qis Y/T for the cohomology sheaf ω := −d(f ! ) and that ω is flat over T . For a Y/T H OT Y/T point t T and the inclusion morphism ψ : Y Y , we have a quasi-isomorphism ∈ t t → ∗ ! (IV-7) Lψ (f ) ω k [d] t OT ≃qis Yt/ (t) ! − for the canonical sheaf ω k by Corollary 4.38. Now, f belongs to D ( ). Yt/ (t) OT coh OY In fact, f ! is f-perfect by Fact 4.35(1). For the stalk (f ! ) at a point y Y , OT OT y ∈ t we have ! L (f ) [ d] k(t) (ω k ) OT y − ⊗OT,t ≃qis Yt/ (t) y by (IV-7). By applying Lemma 4.41 below to (f ! ) [ d] and , we OT y − OT,t → OY,y see that i(f ! ) = 0 for any i = d and −d(f ! ) is a flat -module with H OT y 6 − H OT y OT,t an isomorphism −d ! (f ) k(t) (ω k ) . H OT y ⊗OT,t ≃ Yt/ (t) y Since these hold for arbitrary point y Y , there is a quasi-isomorphism f ! ∈ OT ≃qis ωY/T [d] and ωY/T is flat over T . Therefore, we have proved the first assertion on a characterization of Cohen–Macaulay morphism. For the second assertion, we assume that f is a Cohen–Macaulay morphism. Then, ωY/T is flat over T . Thus, 62 YONGNAMLEEANDNOBORUNAKAYAMA

ω is invertible along a fiber Y if and only if ω is invertible (cf. Y/T t Y/T ⊗OY OYt Fact 2.26(2)). By the isomorphism ω ω k , we see that Y is Y/T ⊗OY OYt ≃ Yt/ (t) t Gorenstein if and only if ωY/T is invertible along Yt. Thus, the second assertion follows, and we are done. The following is used in the proof of Lemma 4.40 above: Lemma 4.41. Let A be a Noetherian local ring with residue field k and let A B → be a local ring homomorphism to another Noetherian local ring B. Let L• be a complex of B-modules such that Hl(L•)=0 for l 0 and Hi(L•) is a finitely ≫ generated B-modules for any i Z, i.e., L• D− (B). Assume that ∈ ∈ coh L Hi(L• k)=0 ⊗A for any i> 0. Then, Hi(L•)=0 for any i> 0, and there exist an isomorphism L H0(L•) k H0(L• k) ⊗A ≃ ⊗A and an exact sequence L TorA(H0(L•), k) H−1(L•) k h H−1(L• k) TorA(H0(L•), k) 0. 2 → ⊗A −→ ⊗A → 1 → Consequently, the following hold: (1) H0(L•) is flat over A if and only if the homomorphism h above is surjective. (2) If Hi(L• L k)=0 for any i = 0, then L• is quasi-isomorphic to a flat ⊗A 6 A-module. Proof. There is a standard spectral sequence L Ep,q = TorA (Hq(L•), k) Ep+q = Hp+q(L• k) 2 −p ⇒ ⊗A p,q (cf. [12, III, (6.3.2.2)]), where E2 = 0 for any p> 0. Let a be an integer such that Hl(L•) = 0 for any l>a. Then, Ep,q = 0 for any q>a, and we have Ea E0,a 2 ≃ 2 and an exact sequence E−2,a E0,a−1 Ea−1 E−1,a 0. 2 → 2 → → 2 → a • 0,a Hence, if a> 0, then H (L )=0 by E2 = 0, and we may decrease a by 1. Thus, we can choose a = 0, and we have the required isomorphism and exact sequence. The assertion (1) is derived from the local criterion of flatness (cf. Proposition A.1), 0 • A 0 • since H (L ) is flat over A if and only if Tor1 (H (L ), k) = 0. The assertion (2) follows from (1) and τ ≤−1(L•) 0, the latter of which is obtained by applying ≃qis the result above to the complex τ ≤−1(L•) instead of L•. Fact 4.42. Let f : Y T be a Cohen–Macaulay morphism having pure relative → dimension d (cf. Definition 2.36). In [7, Sect. 3.5], Conrad defines a sheaf ωf , called the dualizing sheaf for f, on Y such that ω −d((f )! ) f |U ≃ H |U OT for any open subset U Y such that the restriction f : U T factors as a ⊂ |U → closed immersion U ֒ P followed by a smooth separated morphism P T with → → pure relative dimension. Here, the sheaf ωf is obtained by gluing the sheaves −d((f )! ) along natural isomorphisms, where the compatibility of gluing is H |U OT GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 63 checked by explicit calculation of Ext groups. In [50, Th. 2.3.3, 2.3.5], Sastry defines the same sheaf ω by another method: This is obtained by gluing −d((f )! ) f H |V OT for open subsets V Y such that f factors as an open immersion V ֒ V ⊂ |V → followed by a d-proper morphism V T in the sense of [50, Def. 2.2.1]. → Definition 4.43 (relative dualizing sheaf). Let f : Y T be a Cohen–Macaulay → morphism. For any connected component Yα of Y , it is shown in Lemma 2.38 that the restriction morphism f = f : Y T has pure relative dimension. Thus, α |Yα α → one can consider the dualizing sheaf ωfα in Fact 4.42 for fα. We define the relative dualizing sheaf ωY/T of Y over T by ω = ω Y/T |Yα fα for any connected component Yα. The ωY/T is also called the relative dualizing sheaf for f or the relative canonical sheaf of Y over T (cf. Definition 5.3 below).

We sometimes write ωf for ωY/T . By Corollary 4.7(2) and Lemma 4.40, we have: Corollary 4.44. For a Cohen–Macaulay morphism f : Y T , the relative du- → alizing sheaf ωY/T is relatively Cohen–Macaulay over T (cf. Definition 2.28) and Supp ωY/T = Y . By Lemma 2.33(5), we have also: Corollary 4.45. For a Cohen–Macaulay morphism f : Y T , let Y ◦ be an open ◦ → −1 subset of Y such that codim(Yt Y , Yt) 2 for any fiber Yt = f (t). Then, \ ≥ ◦ . ω j (ω ◦ ) for the open immersion j : Y ֒ Y Y/T ≃ ∗ Y /T → The following base change property is known for the relative dualizing sheaves (cf. [7, Th. 3.6.1], [26, Prop. (9)], [50, Th. 2.3.5]): Theorem 4.46. Let f : Y T be a Cohen-Macaulay morphism. For an arbitrary → morphism T ′ T from a locally Noetherian scheme T ′, let Y ′ be the fiber product ′ → ′ ∗ Y T and let p: Y Y be the projection. Then, p (ω ) ω ′ ′ . ×T → Y/T ≃ Y /T Remark. Conrad [7, Th. 3.6.1] and Sastry [50, Th. 2.3.5] prove Theorem 4.46 assuming that f has pure relative dimension, but it is enough for the proof, since the restriction of f to any connected component of Y has pure relative dimension (cf. Lemma 2.38). When f is proper, Theorem 4.46 is shown by Kleiman [26, Prop. (9)(iii)], whose proof uses another version of twisted inverse image functor f !. The proof of [7, Th. 3.6.1] is based on arguments in [17, V], while the proof of [50, Th. 2.3.5] is based on arguments in [9], [55], [26], and [34].

5. Relative canonical sheaves As a generalization of the relative dualizing sheaf for a Cohen–Macaulay morphism, we introduce the notion of relative canonical sheaf for an arbitrary S2- morphism (cf. Definition 2.30). We give some base change properties of the relative canonical sheaf and its “multiple.” These are used for studying Q-Gorenstein morphisms in Section 7. In Section 5.1, we shall study the relative canonical sheaf and 64 YONGNAMLEEANDNOBORUNAKAYAMA the conditions for it to satisfy relative S2. Section 5.2 is devoted to prove Theo- rem 5.10 on a criterion for a certain sheaf related to the relative canonical sheaf to be invertible. This provides sufficient conditions for the base change homomorphism of the relative canonical sheaf to the fiber to be an isomorphism.

5.1. Relative canonical sheaf for an S2-morphism. First of all, we shall give a partial generalization of the notion of canonical sheaf in Deﬁnition 4.28 as follows.

Definition 5.1. Let X be a k-scheme locally of finite type for a field k. Assume that X is locally equi-dimensional, and • codim(X X♭,X) 2 for the Cohen–Macaulay locus X♭ = CM(X). • \ ≥ Note that this assumption is verified when X satisfies S2. For the relative dualizing ♭ ♭ ,sheaf ω ♭ over Spec k in Definition 4.43 and for the open immersion j : X ֒ X X /k → we set ♭ ωX/k := j∗(ωX♭/k) and call it the canonical sheaf of X. Remark. By Corollaries 4.22 and 4.29, we have the following properties in the situation of Definition 5.1: (1) Let U be an arbitrary open subset of X which is of finite type over Spec k. Then, ω k is isomorphic to the canonical sheaf ω k defined in Defini- X/ |U U/ tion 4.28. Thus, the use of the same symbol ωX/k for the canonical sheaf causes no confusion.

(2) The canonical sheaf ωX/k is coherent and satisfies S2. Lemma 5.2. Let X be a scheme locally of finite type over a field k. Assume that

X is Gorenstein in codimension one and satisfies S2. Then, ωX/k is reflexive, and every reflexive -module satisfies S . In particular, the double-dual ω[m] of ω⊗m OX 2 X/k X/k satisfies S for any m Z. 2 ∈ Proof. Let Z be the complement of the Gorenstein locus of X (cf. Definition 4.10). Then, codim(Z,X) 2 and ω k is invertible. Hence, ω k is reflexive by ≥ X/ |X\Z X/ Corollary 2.22, since ω k satisfies S and Supp ω k = X. Every reflexive - X/ 2 X/ OX module satisfies S2 by Lemma 2.21(2). The definition of the canonical sheaf above is partially extended to the relative situation as follows. Definition 5.3 (relative canonical sheaf). Let f : Y T be an S -morphism → 2 of locally Noetherian schemes. Let j : Y ♭ ֒ Y be the open immersion from the ♭ → ♭ relative Cohen–Macaulay locus Y = CM(Y/T ). Note that codim(Yt Y , Yt) 3 −1 \ ≥ for any fiber Yt = f (t), since Yt satisfies S2. In this situation, we define

ωY/T := j∗(ωY ♭/T ) for the relative dualizing sheaf ω ♭ for f ♭ in the sense of Deﬁnition 4.43. We Y /T |Y call ωY/T also the relative canonical sheaf of Y over T . GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 65

Lemma 5.4. Let f : Y T be an S -morphism of locally Noetherian schemes and → 2 let p Y ′ Y −−−−→ ′ f   T′ T be a Cartesian diagram such that yT ′ −−−−→is a locallyy Noetherian scheme ﬂat over T . ∗ Then, ω ′ ′ p (ω ). Y /T ≃ Y/T Proof. Let Y ♭ (resp. Y ′♭) be the relative Cohen–Macaulay locus for f (resp. f ′) and (♭ let j : Y ♭ ֒ Y (resp. j′ : Y ′♭ ֒ Y ′) be the open immersion. Then, Y ′♭ = p−1(Y → → ′ ♭ ′♭ ♭ by Lemma 2.31(3) for = Y , and j is induced from j. Let p : Y Y be the F O ♭∗ → restriction of p. Then, ω ′♭ ′ p (ω ♭ ) by Theorem 4.46. Thus, we have Y /T ≃ Y /T ′ ♭∗ ∗ ∗ ω ′ ′ j (p (ω ♭ )) p (j (ω ♭ )) p ω Y /T ≃ ∗ Y /T ≃ ∗ Y /T ≃ Y/T by the ﬂat base change isomorphism (cf. Lemma A.9) for the Cartesian diagram composed of p, p♭, j, and j′.

Proposition 5.5. Let f : Y T be an S -morphism of locally Noetherian schemes. → 2 Then, the relative canonical sheaf ωY/T deﬁned in Deﬁnition 5.3 is coherent, and moreover, if f is a separated morphism of pure relative dimension d, then

i ! 0, if i< d; (f T ) − H O ≃ (ωY/T , if i = d, − for the twisted inverse image f ! . Let Y ◦ be an open subset of CM(Y/T ) such OT that codim(Y Y ◦, Y ) 2 for any ﬁber Y = f −1(t). For a point t T , let t \ t ≥ t ∈ φ : ω ω k = j (ω ◦ k ) t Y/T ⊗OY OYt → Yt/ (t) t∗ Y ∩Yt/ (t) be the homomorphism induced from the base change isomorphism

(V-1) ω ◦ ◦ ◦ ω ◦ k Y /T ⊗OY OY ∩Yt ≃ Y ∩Yt/ (t) ,cf. Theorem 4.46), where j : Y ◦ Y ֒ Y denotes the open immersion. Then) t ∩ t → t for any point y Y , the following three conditions are equivalent to each other: ∈ (a) The homomorphism φf(y) is surjective at y. (b) The homomorphism φf(y) is an isomorphism at y. (c) There is an open neighborhood U of y in Y such that ω satisﬁes rel- Y/T |U ative S2 over T (cf. Deﬁnition 2.28).

Proof. The coherence of ωY/T and the conditions (a)–(c) are local on Y . Hence, we may assume that f is a separated morphism of pure relative dimension d by Lemma 2.38(1). Then, we have the twisted inverse image f ! with a quasi- OT isomorphism ! (f ) ♭ ω ♭ [d] OT |Y ≃qis Y /T for Y ♭ = CM(Y/T ) by Lemma 4.40, and we have a canonical homomorphism

−d ! ♭ φ: (f ) j (ω ♭ )= ω H OT → ∗ Y /T Y/T 66 YONGNAMLEEANDNOBORUNAKAYAMA

,for the open immersion j♭ : Y ♭ ֒ Y . In order to prove that φ is an isomorphism → since it is a local condition, we may replace Y with an open subset freely. Thus, we may assume that f is the composite p ι of a closed immersion ι: Y ֒ P and a smooth affine • ◦ → morphism p: P T . → By Fact 4.35(1) and Remark 4.36, we know that Rι (f ! ) is perfect. Hence, ∗ OT by localizing Y , we may assume that Rι (f ! ) is quasi-isomorphic to a bounded complex • = [ i • ∗ OT E ··· → E → i+1 ] of free -modules of finite rank. E → · · · OP Then, we have an isomorphism i( •) ι i(f ! ) for any i Z. Moreover, H E ≃ ∗H OT ∈ there exist quasi-isomorphisms L L L • Rι (f ! Lι∗ ) Rι (f ! Lf ∗k(t)) E ⊗OP OPt ≃qis ∗ OT ⊗OY OPt ≃qis ∗ OT ⊗OY ! L • qis Rι∗(f T Y ) qis Rιt∗(ω ) ≃ O ⊗OY O t ≃ Yt/k(t) ,for any t T and for the induced closed immersion ι : Y ֒ P = p−1(t). In fact ∈ t t → t the first quasi-isomorphism is known as the projection formula (cf. [17, II, Prop. 5.6]), the quasi-isomorphisms Lp∗k(t) and Lf ∗k(t) OPt ≃qis ≃qis OYt are derived from the flatness of p and f, and the quasi-isomorphism L f ! ω• OT ⊗OY OYt ≃qis Yt/k(t) is obtained by Corollary 4.38. We shall show that the three data: •[ d],Z := ι(Y Y ◦), := 0( •[ d]) ι −d(f ! ), E − \ F H E − ≃ ∗H OT satisfy the conditions of Lemma 3.15 for the morphism P T . The required → inequality (III-8) of Lemma 3.15 is derived from depth = codim(P Z, P ) = codim(Y Z, P ) codim(Y Z, Y ) 2 Pt∩Z OPt t ∩ t t ∩ t ≥ t ∩ t ≥ (cf. Lemma 2.14). The condition (i) of Lemma 3.15 is derived from (cf. Lemma 4.40):

i • i ! 0, if i = d; ( ) P \Z ι∗( (f T )) P \Z 6 − H E | ≃ H O | ≃ (ι∗ωY ◦/T , if i = d, − and the next condition (ii) has no meaning now. The condition (iii) follows from L i( ) ι ( i(ω• ))=0 H E ⊗OP OPt ≃ t∗ H Yt/k(t) for any i < d (cf. Lemma 4.27). The last condition (iv) of Lemma 3.15 is a − consequence of Corollary 4.20 applied to the ordinary dualizing complex ω• [ d] Yt/k(t) − (cf. Lemma 4.27) and to b = 1, since the complex M • in Lemma 3.15(iv) is quasi-isomorphic to the stalk of • ≤1 • ≤1 • τ (Rι∗ω [ d]) qis Rι∗(τ (ω [ d])), and Yt/k(t) − ≃ Yt/k(t) − dim codim(Z Y , Y ) 2 for any z Z with t = f(z). • OPt,z ≥ ∩ t t ≥ ∈ GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 67

Therefore, all the conditions of Lemma 3.15 are satisfied, and consequently, i( •) ι i(f ! )=0 H E ≃ ∗H OT for any i < d, and we can apply Proposition 3.7 to via Lemma 3.15. Then, − F j ( ) for the open immersion j : P Z ֒ P by Proposition 3.7(1), and it F ≃ ∗ F|P \Z \ → implies that the morphism φ above is an isomorphism. Moreover, the three conditions (a)–(c) are equivalent to each other by Proposition 3.7(3) and Corollary 3.9. Thus, we are done. Proposition 5.6. Let f : Y T be an S -morphism of locally Noetherian schemes → 2 and let j : Y ◦ ֒ Y be the open immersion from an open subset Y ◦ of the relative → Gorenstein locus Gor(Y/T ) for f. Assume that ◦ −1 codim(Yt Y , Yt) 2 for any fiber Yt = f (t). • \ ≥ [m] For an integer m and for the relative canonical sheaf ωY/T , let ωY/T denote the ⊗m double-dual of ωY/T . Then, [m] ⊗m ω j (ω ◦ ) Y/T ≃ ∗ Y /T for any m. In particular, ω is reflexive. For an integer m and a point t T , Y/T ∈ let [m] [m] [m] ⊗m φ : ω O Y ω = jt∗(ω ◦ ) t Y/T ⊗ Y O t → Yt/k(t) Y ∩Yt/k(t) be the homomorphism induced from the base change isomorphism (V-1), where j : Y ◦ Y ֒ Y denotes the open immersion. Then, for any integer m and any t ∩ t → t point y Y , the following three conditions are equivalent to each other: ∈ [m] (a) The homomorphism φf(y) is surjective at y. [m] (b) The homomorphism φf(y) is an isomorphism at y. (c) There is an open neighborhood V of y in Y such that ω[m] satisfies Y/T |V relative S2 over T . Proof. We apply some results in Section 3.1 to the reflexive sheaf = ω[m] and the F Y/T closed subset Z := Y Y ◦. By assumption, is invertible and depth \ F|Y \Z Yt∩Z OYt ≥ 2 (cf. Lemma 2.14(ii)). Thus, we can apply Lemma 3.14, and consequently, we can assume that has an exact sequence of Proposition 3.7, by replacing Y with its F [m] ⊗m open subset. Then, ω j (ω ◦ ) by Proposition 3.7(1). In case m = 1, Y/T ≃ ∗ Y /T [1] we have ω ω j (ω ◦ ) by Corollary 4.45 and Definition 5.3, and as Y/T ≃ Y/T ≃ ∗ Y /T a consequence, ωY/T is reflexive. The equivalence of three conditions (a)–(c) is derived from Proposition 3.7(3) and Corollary 3.9. Corollary 5.7. Let us consider a Cartesian diagram p Y ′ Y −−−−→ f ′ f

′ q  T T −−−−→ of locally Noetherian schemes in whichy f is ay ﬂat morphism locally of ﬁnite type. Then, p−1 CM(Y/T ) = CM(Y ′/T ′) and p−1 Gor(Y/T ) = Gor(Y ′/T ′). Assume that f is an S2-morphism. Then: 68 YONGNAMLEEANDNOBORUNAKAYAMA

∗ (1) If ωY/T satisfies relative S2 over T , then p ωY/T ωY ′/T ′ . −1 ≃ (2) If every fiber Yt = f (t) is Gorenstein in codimension one, then for any m Z, there is a canonical isomorphism ∈ ∗ [m] ∨∨ [m] (p ω ) ω ′ ′ . Y/T ≃ Y /T [m] ∗ [m] [m] Here, if ω satisfies relative S over T , then p ω ω ′ ′ . Y/T 2 Y/T ≃ Y /T

Proof. The equality for CM is derived from Lemma 2.31(3) for = Y . If f is a ∗ F O Cohen–Macaulay morphism, then p ω ω ′ ′ by Theorem 4.46. This implies Y/T ≃ Y /T the equality for Gor by the Remark of Deﬁnition 4.39. Assume that f is an S2- morphism. Then, f ′ is so by Lemma 2.31(5). For open subsets Y ♭ := CM(Y/T ) and Y ′♭ := p−1(Y ♭), we have

′ ′♭ ′ ♭ codim(Y ′ Y , Y ′ ) = codim(Y Y , Y ) 3 t \ t t \ t ≥ for any t′ T ′ and t = q(t) by Lemma 2.31(1) and by the S -condition of Y . If ∈ 2 t ωY/T satisﬁes relative S2 over T , then the canonical base change isomorphism ∗ (V-2) p ω ♭ ω ′♭ ′ Y /T ≃ Y /T in Theorem 4.46 induces an isomorphism

∗ ′ ∗ ′ p ω j (p ω ′♭ ) j ω ′♭ ′ = ω ′ ′ Y/T ≃ ∗ Y/T |Y ≃ ∗ Y /T Y /T = (for the open immersion j′ : Y ′♭ ֒ Y ′, by Lemma 2.32(2) applied to ( ,Z → F (p∗ω , Y ′ Y ′♭). This proves (1). In the situation of (2), codim(Y Y ◦, Y ) 2 Y/T \ t \ t ≥ for any t T , where Y ◦ = Gor(Y/T ). In particular, ∈ depth ◦ 2 Yt\Y F ⊗OY OYt ≥ for any coherent -module satisfying relative S over T , by Lemma 2.15(2). OY F 2 Thus, (2) is a consequence of Lemma 2.34 via the isomorphism (V-2).

Proposition 5.8. Let f : Y T be an S -morphism of locally Noetherian schemes. → 2 Then, om (ω ,ω ) H OY Y/T Y/T ≃ OY for the relative canonical sheaf ωY/T in the sense of Definition 5.3. If every fiber satisfies S3, then xt1 (ω ,ω )=0. E OY Y/T Y/T Proof. Let j : Y ♭ ֒ Y be the open immersion from the relative Cohen–Macaulay → locus Y ♭ = CM(Y/T ). Now, we have a quasi-isomorphism

♭ R omO (ω ♭ ,ω ♭ ) OY ≃ H Y ♭ Y /T Y /T by (IV-5) in Fact 4.35(2). This induces another quasi-isomorphism

R omOY (ωY/T , Rj∗(ω ♭ )) qis Rj∗ R omO (ω ♭ ,ω ♭ ) Rj∗ ♭ H Y /T ≃ H Y ♭ Y /T Y /T ≃ OY and the spectral sequence

p,q p q p+q p+q = xt (ω , R j∗(ω ♭ )) = R j∗ ♭ . E2 E OY Y/T Y /T ⇒E OY GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 69

0,0 0 1,0 1 , ֒ Since ω = j (ω ♭ ), the isomorphism and the injection Y/T ∗ Y /T E2 ≃ E E2 → E respectively, correspond to an isomorphism omOY (ωY/T ,ωY/T ) j∗ Y ♭ and an 1 1 H ≃ O injection xt (ω ,ω ) ֒ R j ♭ . Therefore, it suffices to prove that E OY Y/T Y/T → ∗OY (1) Y j∗ Y ♭ , and O ≃ O 1 (2) if every fiber satisfies S , then R j ♭ = 0. 3 ∗OY Here, (1) (resp. (1) with the conclusion of (2)) is equivalent to: depth 2 Z OY ≥ (resp. 3) for Z := Y Y ♭ (cf. Property 2.6). If a fiber Y satisfies S , then ≥ \ t k codim(Z Y , Y ) > k, and depth k by Lemma 2.15(2). Hence, we have ∩ t t Z∩Yt OYt ≥ depth 2 (resp. 3) by Lemma 2.32(3) when every fiber Y satisfies S (resp. Z OY ≥ ≥ t 2 S3). Thus, we are done.

5.2. Some base change theorems for the relative canonical sheaf. For an S -morphism f : Y T of locally Noetherian schemes and for a ﬁber Y = f −1(t), 2 → t let ♭ φt(ωY/T ): ωY/T OY Yt ωYt/k(t) = j (ω ♭ k ) ⊗ O → ∗ Yt / (t) be the canonical homomorphism induced from the base change isomorphism

♭ ♭ ♭ ωY /T O ♭ ω k ⊗ Y OYt ≃ Yt / (t) (cf. Theorem 4.46), where Y ♭ = CM(Y/T ), Y ♭ = Y ♭ Y , and j♭ is the open im- t ∩ t mersion Y ♭ ֒ Y . The homomorphism φ (ω ) is not necessarily an isomorphism → t Y/T (e.g. Fact 7.7 below). We shall give a suﬃcient condition for φt(ωY/T ) to be an isomorphism in Theorem 5.10 below.

Lemma 5.9. Let f : Y T be a Cohen–Macaulay morphism of locally Noetherian → schemes. Let be a coherent -module ﬂat over T with an isomorphism L OY (V-3) ω k L⊗OY OYt ≃ Yt/ (t) for the ﬁber Y = f −1(t) over a given point t T . Then, for the sheaf := t ∈ M om ( ,ω ), the canonical homomorphism ω is an isomorphism H OY L Y/T L⊗M → Y/T along Y , and is an invertible sheaf along Y with an isomorphism t M t M⊗OY OYt ≃ . OYt

Proof. Since the assertions are local on Yt, we may assume that (1) f has pure relative dimension d (cf. Lemma 2.38), and f is the composite p ι of a closed immersion ι: Y ֒ P and a smooth affine (2) ◦ → morphism p: P T of pure relative dimension e. → Then, f ! ω [d] and ω is flat over T by Lemma 4.40. The complex OT ≃ Y/T Y/T R om ( ,f ! ) is f-perfect by Fact 4.35(2), and there is a quasi-isomorphism H OY L OT Rι R om ( ,f ! ) R om (ι ,ω [e]) ∗ H OY L OT ≃qis H OP ∗L P/T by Corollary 4.31, where p! = ω [e] by (2) above. Localizing Y , by Re- OT P/T mark 4.36, we may assume furthermore that (3) Rι R om ( ,f ! ) is quasi-isomorphic to a bounded complex • = ∗ H OY L OT E [ i i+1 ] of free -modules of finite rank. ···→E →E → · · · OP 70 YONGNAMLEEANDNOBORUNAKAYAMA

Note that we have an isomorphism −d( •) ι om ( ,ω ) ι . H E ≃ ∗ H OY L Y/T ≃ ∗M ֒ For the closed immersion ι: Y ֒ P and the induced closed immersion ιt : Yt −1 → → Pt = p (t), we have quasi-isomorphisms

• L L L P qis R omO ((ι∗ ) P ,ω [e] P ) E ⊗OP O t ≃ H Pt L ⊗OP O t P/T ⊗OP O t qis R omO (ιt∗( O Y ),ω k [e]) ≃ H Pt L⊗ Y O t Pt/ (t) by [23, I, Prop. 7.1.2], since is ﬂat over T , ι is perfect (cf. Fact 4.35(1) and L ∗L Remark 4.36), and since P T is smooth. From the isomorphism (V-3) and the → base change isomorphism

φ (ω ): ω ω k t Y/T Y/T ⊗OY OYt ≃ Yt/ (t) (cf. Theorem 4.46), by duality for ιt (cf. Corollary 4.31), we have quasi-isomorphisms

• L • L k(t) qis P qis Rιt∗ R omO ( O Y ,ω k [d]) E ⊗OT ≃ E ⊗OP O t ≃ H Yt L⊗ Y O t Yt/ (t) qis Rιt∗ R omO (ω k ,ω k [d]) qis ιt∗ Y [d], ≃ H Yt Yt/ (t) Yt/ (t) ≃ O t where the last quasi-isomorphism follows from that ωYt/k(t)[d] is a dualizing complex of Yt. Then, by Lemma 4.41, we see that (4) •[ d] is quasi-isomorphic to −d( •) ι along Y , E − H E ≃ ∗M t (5) ι is flat over T along Y , and ∗M t (6) there is an isomorphism L ι −d( • k(t)) ι . ∗M⊗OP OPt ≃ H E ⊗OT ≃ t∗OYt Hence, is flat over T along Y with an isomorphism by (5) M t M⊗OY OYt ≃ OYt and (6). As a consequence, is an invertible -module along Y by Fact 2.26(2). M OY t Now, we have a quasi-isomorphism R om ( ,ω ) H OY L Y/T ≃qis M along Yt by (3) and (4). By the duality quasi-isomorphism R om (R om ( ,ω ),ω ) L≃qis H OY H OY L Y/T Y/T (cf. Fact 4.35(2)), we have an isomorphism om ( ,ω ) ω −1 L ≃ H OY M Y/T ≃ Y/T ⊗OY M along Y , since is invertible along Y . Thus, we are done. t M t Theorem 5.10. For an S -morphism f : Y T of locally Noetherian schemes, let 2 → be a coherent -module and set := om ( ,ω ). For an open subset L OY M H OY L Y/T U of Y and for the fiber Y = f −1(t) over a given point t T , assume that t ∈ (i) codim(Y U, Y ) 2, t \ t ≥ (ii) is flat over T with an isomorphism ω k , and L L⊗OY OYt ≃ Yt/ (t) (iii) one of the following two conditions is satisfied: (a) Y satisfies S and codim(Y U, Y ) 3; t 3 t \ t ≥ GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 71

(b) there is a positive integer r coprime to the characteristic of k(t) such that [r] = ( ⊗r)∨∨ and ω[r] = (ω⊗r )∨∨ are invertible -module L L Y/T Y/T OY along Yt. Then, is an invertible -module along Y with an isomorphism M OY t M⊗OY OYt ≃ , and the canonical homomorphism ω is an isomorphism along OYt L⊗OY M → Y/T Yt. Moreover, the “base change homomorphism”

φ (ω ): ω ω k t Y/T Y/T ⊗OY OYt → Yt/ (t) is an isomorphism.

Proof. Since the assertions are local on Yt, we may replace Y with an open subset freely. Let Y ♭ be the relative Cohen–Macaulay locus CM(Y/T ), which is an open subset by Fact 2.29(1). Then, codim(Y Y ♭, Y ) 3 (resp. 4 in the case (a)), t \ t ≥ ≥ since Y satisﬁes S (resp. S ). We set U ♭ := U Y ♭. Then, t 2 3 ∩ (V-4) codim(Y U ♭, Y ) = codim((Y U) (Y Y ♭), Y ) 2 (resp. 3). t \ t t \ ∪ t \ t ≥ ≥ By Lemma 5.9 applied to the Cohen–Macaulay morphism U ♭ T , there is an → isomorphism

(1) ♭ O ♭ ♭ , M|U ⊗ U♭ OU ∩Yt ≃ OU ∩Yt and there is an open neighborhood U ′ of U ♭ Y in U ♭ such that ∩ t (2) ′ is an invertible sheaf, and M|U (3) the canonical homomorphism ω is an isomorphism on U ′. L⊗OY M → Y/T We set Z = Y U ′. Then, codim(Y Z, Y ) = codim(Y U ♭, Y ) 2 by (V-4). Since \ t ∩ t t \ t ≥ f is an S -morphism, by Lemma 2.38, we may assume that codim(Y ′ Z, Y ′ ) 2 2 t ∩ t ≥ for any t′ T by replacing Y with an open subset. Then, depth 2 by ∈ Z OY ≥ Lemma 2.32(3), and ′ ω j (ω ) j (ω ′ ) Y/T ≃ ∗ U/T ≃ ∗ U /T ′ ′ ,for the open immersion j : U ֒ Y by Corollary 4.45. In particular, depthZ 2 ′ → M≥ i.e., j ( ′ ), by the isomorphism M≃ ∗ M|U ′ ′ om ( ,ω ) om ( , j (ω ′ )) j om ′ ( ′ ,ω ′ ). H OY L Y/T ≃ H OY L ∗ U /T ≃ ∗ H OU L|U U /T

By (ii), satisﬁes relative S2 over T along Yt, since ωYt/k(t) satisﬁes S2 by Corol- L ′ lary 4.29. Hence, we have also an isomorphism j ( ′ ) by Lemma 2.33(5). L≃ ∗ L|U We shall show that is invertible along Y by applying Theorem 3.16 to Y T , M t → the closed subset Z = Y U ′, and to the sheaf as . By the previous argument, \ M F we have checked the conditions (i) and (ii) of Theorem 3.16. The condition (iii) is derived from (1): In fact, we have

′ ′ (V-5) = j (( ) ′ ) j ( ′ ) , M(t)∗ ∗ M⊗OY OYt |U ∩Yt ≃ ∗ OU ∩Yt ≃ OYt since we have depth 2 by the S -condition on Y and by codim(Y Yt∩Z OYt ≥ 2 t t ∩ Z, Y ) 2 (cf. Lemma 2.14). Similarly, in the case of (a) above, we have the t ≥ condition (a) of Theorem 3.16 by the S -condition on Y and by codim(Y Z, Y )= 3 t t ∩ t 72 YONGNAMLEEANDNOBORUNAKAYAMA codim(Y U ♭, Y ) 3 (cf. (V-4)). In the case of (b) above, [r] is an invertible t \ t ≥ M -module along Y . In fact, the restriction homomorphisms OY t [r] ′ [r] [r] ′ [r] j ( ′ ) and ω j (ω ′ ) M → ∗ M |U Y/T → ∗ U /T are isomorphisms by Lemma 2.33(4), since [r] and ω[r] are reﬂexive, and the M Y/T isomorphism [r] [r] [r] ′ ′ ′ ω ′ L |U ⊗OU M |U ≃ U /T obtained by (2) and (3) induces an isomorphism

[r] ′ [r] ′ [r] [r] j ( ′ ) j om ′ ( ′ ,ω ′ ) M ≃ ∗ M |U ≃ ∗ H OU L |U U /T [r] ′ [r] [r] [r] om ( , j (ω ′ )) om ( ,ω ). ≃ H OY L ∗ U /T ≃ H OY L Y/T Thus, the condition (b) of Theorem 3.16 is also satisﬁed in the case of (b). Hence, we can apply Theorem 3.16, and as a result, we see that is an invertible sheaf M along Yt. Then, we have an isomorphism by (V-5), and the canonical M⊗OY OYt ≃ OYt homomorphism ω is an isomorphism along Y by (3): In fact, it is L⊗OY M → Y/T t expressed as the composite

′ ′ ′ j ( ′ ) j ( ′ ′ ) j (ω ′ ) ω , L⊗OY M≃ ∗ L|U ⊗OY M → ∗ L|U ⊗ M|U ≃ ∗ U /T ≃ Y/T where the middle arrow is an isomorphism along Yt by the projection formula, since is invertible along Yt. In particular, ωY/T OY Yt satisﬁes S2, and as a M ⊗ O consequence, φt(ωY/T ) is an isomorphism by (V-4). Thus, we are done.

6. Q-Gorenstein schemes A normal algebraic variety defined over a field is said to be Q-Gorenstein if some positive multiple of the canonical divisor is Cartier. We shall generalize the notion of Q-Gorenstein to locally Noetherian schemes. In Section 6.1, the notion of Q- Gorenstein scheme is defined and its basic properties are given. In Section 6.2, we consider the case of affine cones over polarized projective schemes over a field, and determine when it is a Q-Gorenstein scheme.

6.1. Basic properties of Q-Gorenstein schemes.

Deﬁnition 6.1 (Q-Gorenstein scheme). Let X be a locally Noetherian scheme admitting a dualizing complex locally on X and assume that X is Gorenstein in codimension one, i.e., codim(X X◦) 2 for the Gorenstein locus X◦ = Gor(X) \ ≥ (cf. Deﬁnition 4.10). (1) The scheme X is said to be quasi-Gorenstein (or 1-Gorenstein) at a point P if there exist an open neighborhood U of P and a dualizing complex • R of U such that 0( •) is invertible at P . If X is quasi-Gorenstein at every H R point, then X is said to be quasi-Gorenstein (or 1-Gorenstein). (2) The scheme X is said to be Q-Gorenstein at P if there exist an open neighborhood U of P , a dualizing complex • of U, and an integer r > 0 R GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 73

such that = 0( •) is invertible on the Gorenstein locus U ◦ = U X◦ L H R ∩ and ⊗r j ◦ ∗ L |U is invertible at P , where j : U ◦ ֒ U denotes the open immersion. If X is → Q-Gorenstein at every point, then X is said to be Q-Gorenstein.

Definition 6.2 (Gorenstein index). For a Q-Gorenstein scheme X, the Gorenstein index of X at P X is defined to be the smallest positive integer r satisfying the ∈ condition (2) of Definition 6.1 for an open neighborhood of P . The least common multiple of Gorenstein indices of X at all the points is called the Gorenstein index of X, which might be + . ∞ Remark. The conditions (1) and (2) of Definition 6.1 do not depend on the choice of • by the essential uniqueness of the dualizing complex (cf. Remark 4.2). R Lemma 6.3. (1) A quasi-Gorenstein (1-Gorenstein) scheme is nothing but a Q-Gorenstein scheme of Gorenstein index one.

(2) Every Q-Gorenstein scheme satisfies S2. Proof. (1): Let X be a locally Noetherian scheme admitting a dualizing complex • R such that it is Gorenstein in codimension one and that := 0( •) is invertible on ◦ L H R the Gorenstein locus X . Then, satisfies S and j ( ◦ ) is an isomorphism L 2 L → ∗ L|X by Corollary 4.22. Hence, X is Q-Gorenstein with Gorenstein index one if and only if is invertible, equivalently, X is quasi-Gorenstein. L (2): We may assume that X admits a dualizing complex • such that = 0 • ◦ R L ( ) is invertible on the Gorenstein locus X , since the condition S2 is local. H R ⊗r Then, := j ( ◦ ) is invertible for some r by Definition 6.1(2). Hence, Mr ∗ L |X Mr satisfies S2 by Corollary 2.16. Therefore, X satisfies S2. Lemma 6.4. Let X be a locally Noetherian scheme admitting a dualizing complex •. For the cohomology sheaf := 0( •) and for an open subset U with R L H R codim(X U,X) 2, assume that is invertible and • . Then, the \ ≥ L|U R |U ≃qis L|U following hold: (1) If X satisfies S , then • is an ordinary dualizing complex of X and the 1 R dualizing sheaf is a reflexive -module satisfying S . L OX 2 (2) If X satisfies S , then the double-dual [m] of ⊗m satisfies S for any 2 L L 2 integer m, and in particular, [m] j ( ⊗m ) L ≃ ∗ L |U .for the open immersion j : U ֒ X → (3) The scheme X is Q-Gorenstein if and only if X satisfies S2 and, locally on X, there is a positive integer r such that [r] is invertible. L Proof. (1): This follows from Corollary 4.22 with Lemmas 2.14 and 2.21(3). (2): Since depth 2 by the S -condition, we have the isomorphism X\U OX ≥ 2 [m] j ( ⊗m ) by Lemma 2.21(1). Hence, [m] satisfies S by Corollary 2.16, L ≃ ∗ L |U L 2 since is invertible. L|U 74 YONGNAMLEEANDNOBORUNAKAYAMA

(3): This is a consequence of (2) above and Lemma 6.3(2) by the uniqueness of dualizing complex explained in Remark 4.2. Example 6.5. Let X be a k-scheme locally of finite type for a field k. Assume that ◦ ◦ X satisfies S2 and codim(X X ,X) 2 for the Gorenstein locus X = Gor(X). \ ≥ [m] Let ωX/k be the canonical sheaf defined in Definition 5.1 and let ωX/k denote the double-dual of ω⊗m for any m Z (cf. Proposition 5.6). Then, X is Q-Gorenstein X/k ∈ [r] at a point x if and only if ωX/k is invertible at x for some r> 0. Example 6.6. Let X be a normal algebraic k-variety for a field k, i.e., a normal integral separated scheme of finite type over k. Then, X is Q-Gorenstein if and only if the multiple rKX of the canonical divisor KX is Cartier for some r > 0. ◦ In fact, X satisfies S , ω ◦ k ◦ (K ) for the Gorenstein locus X = Gor(X), 2 X / ≃ OX X where codim(X X◦,X) 2, and hence ω[m] (mK ) for any m Z. \ ≥ X/k ≃ OX X ∈ Lemma 6.7. Let X be a locally Noetherian scheme and let π : Y X be a smooth → surjective morphism. Then, for any integer k 1, Y satisfies S if and only if X ≥ k satisfies Sk. In particular, Y is Cohen–Macaulay if and only if X is so. Moreover, Y is Gorenstein if and only if X is so. Assume that X admits a dualizing complex locally on X. Then, Y is quasi-Gorenstein (resp. Q-Gorenstein of index r) if and only if X is so. Proof. The first assertion follows from Fact 2.26(6). In particular, we have the equivalence for the Cohen–Macaulay property (cf. Remark 2.12). The Gorenstein case follows from Fact 4.12. It remains to prove the case of Q-Gorenstein property, since “quasi-Gorenstein” is nothing but “Q-Gorenstein of index one” (cf. Lemma 6.3(1)). Since the Q-Gorenstein property is local and it implies S2, we may assume that X has a dualizing complex • , • RX X and Y are affine schemes satisfying S , and • 2 π = p λ for an étale morphism λ: Y X Ad and the first projection • ◦ → × p: X Ad X for the “d-dimensional affine space” Ad = Spec Z[x ,..., x ] × → 1 d for some integer d 0 (cf. [12, IV, Cor. (17.11.4)]). ≥ In particular, π has pure relative dimension d. We may assume also that • is an RX ordinary dualizing complex by Lemma 4.14. We set to be the dualizing sheaf LX 0( • ). H RX By Examples 4.23 and 4.24, we see that • := π!( • ) is a dualizing complex RY RX of Y , and we have an isomorphism d ∗ ω Ω λ (ω d ) Y/X ≃ Y/X ≃ X×A /X ≃ OY for the relative dualizing sheaf ω . Thus, π!( ) [d], and Y/X OX ≃qis OY L • π!( ) Lπ∗( • ) Lπ∗( • )[d] RY ≃qis OX ⊗OY RX ≃qis RX (cf. Example 4.23, Fact 4.34(2)). Since Y satisfies S , the shift • [ d] is an 2 RY − ordinary dualizing complex on Y by the proof of Lemma 4.14. Here, the associated dualizing sheaf := 0( • [ d]) is isomorphic to π∗( ). Since π is faithfully LY H RY − LX GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 75

ﬂat, we see that is invertible if and only if is so (cf. Lemma A.7). For an LY LX integer m, let [m] (resp. [m]) be the double-dual of ⊗m (resp. ⊗m). Then, LX LY LX LY [m] π∗( [m]) for any m Z by Remark 2.20. Hence, for a given integer r, [r] is LY ≃ LX ∈ LY invertible if and only if [r] is invertible by the same argument as above. Therefore, LX by Lemma 6.4(3), Y is Q-Gorenstein of index r if and only if X is so. Thus, we are done.

Remark 6.8. By Lemma 6.7, we see that the Q-Gorenstein property is local even in the ´etale topology. More precisely, for an ´etale morphism X′ X, for a point → P X, and for a point P ′ X′ lying over P , X is Q-Gorenstein of index r at P if ∈ ∈ and only if X′ is so at P ′.

6.2. Affine cones of polarized projective schemes over a field. For an affine cone over a projective scheme over a field k, we shall determine when it is Cohen– Macaulay, Gorenstein, Q-Gorenstein, etc., under suitable conditions. We fix a field k which is not necessarily algebraically closed.

Definition 6.9 (affine cone). A polarized projective scheme over k is a pair (S, ) A consisting of a projective scheme S over k and an ample invertible sheaf on S. A The polarized projective scheme (S, ) is said to be connected if S is connected. For A a connected polarized projective scheme (S, ), the affine cone of (S, ) defined to A A be Spec R for the graded k-algebra R = R(S, ) := H0(S, ⊗m). A m≥0 A We denote the affine cone by Cone(S,M). Note that the closed subscheme of A Cone(S, ) = Spec R defined by the ideal A 0 ⊗m R+ = H (S, ) m>0 A of R is isomorphic to Spec H0(S, M), and the support of the closed subscheme is a OS point, since the finite-dimensional k-algebra H0(S, ) is an Artinian local ring by OS the connectedness of S. The point is called the vertex of Cone(S, ). A Remark. The k-algebra R(S, ) above is finitely generated, since S is projective A and is ample. Moreover, S Proj R(S, ). In some articles, the affine cone of A ′≃ A ′ ′ (S, ) is defined to be Spec R for the graded subring R of R such that Rn = Rn A ′ for any n> 0 and R0 = k. Similar results to the following are well-known on the structure of affine cones (cf. [12, II, Prop. (8.6.2), (8.8.2)]).

Lemma 6.10. For a connected polarized projective scheme (S, ) over k, let X be A the aﬃne cone Cone(S, ). Let π : Y S be the geometric line bundle associated A → with , i.e., Y = V( ) = pec , where = ⊗m. Let E be the zero- A A S S R R m≥0 A section of π corresponding to the projection to the component of degree R →L OS zero. Then, E is a relative Cartier divisor over S (cf. [12, IV, D´ef. (21.15.2)]) with an isomorphism ( E) π∗ . Moreover, there exists a projective k-morphism OY − ≃ A µ: Y X such that → 76 YONGNAMLEEANDNOBORUNAKAYAMA

(1) µ is an isomorphism, OX → ∗OY (2) π∗ is µ-ample, A (3) µ−1(P )= E as a closed subset of Y for the vertex P of X, and (4) µ induces an isomorphism Y E X P . \ ≃ \ Proof. For an open subset U = Spec B of S with an isomorphism ε: , we A|U ≃ OU have an isomorphism ϕ: π−1(U) Spec B[t] for the polynomial B-algebra B[t] of ≃ one variable such that ϕ induces an isomorphism

0 −1 0 ⊗m m H (π (U), Y )= H (U, ) B[t]= Bt O m≥0 A ≃ m≥0 M M of graded B-algebras. Then, E −1 is a Cartier divisor corresponding to div(t) |π (U) on Spec B[t], which is relatively Cartier over Spec B (cf. [12, IV, (21.15.3.3)]). Thus, E is a relative Cartier divisor over S, since such open subsets U cover S. The exact sequence 0 ( E) 0 induces an isomorphism → OY − → OY → OE → ⊗m π∗ Y ( E) O ( 1) O − ≃ m≥1 A ≃A⊗ S R − M of graded -modules, where ( 1) denotes the twisted graded module. In partic- R ∗ R − ular, Y ( E) π . O − ≃ A 0 ⊗m ⊗m The canonical homomorphisms H (S, ) k induce a graded A ⊗ OS → A homomorphism Φ: R k of graded -algebras, where R := R(S, ). The ⊗ OS → R OS A cokernel of Φ is a finitely generated -module, since ⊗m is generated by global OS A sections for m 0. Hence, is a finitely generated R k -module. Therefore, ≫ R ⊗ OS Φ defines a finite morphism

ν : Y = pec pec (R k ) X k S S S R → S S ⊗ OS ≃ ×Spec over S. Let p : X k S X and p : X k S S be the first and 1 ×Spec → 2 ×Spec → second projections. Then, µ := p ν : Y X is a projective morphism, since 1 ◦ → S is projective over k. Here, µ , since H0(Y, ) H0(S, ) R. OX ≃ ∗OY OY ≃ R ≃ Moreover π∗ is µ-ample, since p∗ is relatively ample over X and π∗ is the A 2A A pullback by the finite morphism ν. Thus, µ satisfies the conditions (1) and (2). Since the projection defining E induces the projection R = H0(S, ) R → OS R → H0(S, ) to the component of degree zero, the scheme-theoretic image µ(E) is OS the zero-dimensional closed subscheme Spec H0(S, ) of X defined by the ideal OS R = H0(S, ⊗m) of R. Hence, the image µ(E) is set-theoretically the + m>0 A vertex P . We shall show that the morphism L µ′ : Y ′ := Y µ−1(P ) X′ := X P \ → \ induced by µ is an isomorphism. Since µ is proper, so is µ′. Moreover, the structure ′ ∗ ′ sheaf Y ′ is µ -ample, since π Y ( E) is µ-ample by (2). Hence, µ is a O ′ A ≃ O − ′ finite morphism. Thus, µ is an isomorphism by (1), since ′ µ ′ . Asa OX ≃ ∗OY consequence, (4) is derived from (3), and it remains to prove (3) for µ and P . For a global section f of ⊗m for some m > 0, we set V (f) to be the closed A subscheme Spec(R/fR) of X = Spec R by regarding f as a homogeneous element GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 77 of R of degree m. We also set a closed subscheme W (f) of S to be the “zero- subscheme” of f, i.e., it is defined by the exact sequence ⊗f ⊗−m 0. A −−→OS → OW (f) → The condition (3) is derived from the following ( ) for any f and for any affine open ∗ subsets U = Spec B with an isomorphism ε: : A|U ≃ OU ( ) µ−1V (f) π−1(U) = (π−1W (f) E) π−1(U) as a subset of π−1(U). ∗ ∩ ∪ ∩ In fact, if ( ) holds for all U and f, then µ−1V (f)= π−1W (f) E for any f, and ∗ ∪ we have µ−1(P ) = E by V (f) = P and W (f) = . Here, V (f) = P f f ∅ f and W (f) = hold, since all of such f R generate the ideal R and since f ∅ T ∈T T+ is ample. We shall prove ( ) as follows. Let ϕ: π−1(U) Spec B[t] be the A T ∗ ≃ isomorphism above defined by ε. We set b = ε⊗m(f ) H0(U, )= B |U ∈ OU for the induced isomorphism ε⊗m : ⊗m . Then, W (f) U = Spec B/bB, A |U ≃ OU ∩ and ϕ induces isomorphisms µ−1V (f) π−1(U) Spec B[t]/(btm) and E π−1(U) ∩ ≃ ∩ Spec B[t]/(t). This implies ( ), and we are done. ≃ ∗ Corollary 6.11. In the situation of Lemma 6.10, for an integer k 1, S satis- ≥ fies S if and only if X P satisfies S . Moreover, S is Cohen–Macaulay (resp. k \ k Gorenstein, resp. quasi-Gorenstein, resp. Q-Gorenstein of Gorenstein index r) if and only if X P is so. \ Proof. This is a consequence of Lemmas 6.7 and 6.10, since X P Y E is smooth \ ≃ \ and surjective over S.

The following result is essentially well-known (cf. [37, Prop. 1.7], [44, Lem. 4.3]).

Proposition 6.12. Let X be the affine cone of a connected polarized projective scheme (S, ) over k and let P be the vertex of X. For a coherent -module , A OS G we set = µ (π∗ ) for the morphisms µ: Y X and π : Y S in Lemma 6.10 F ∗ G → → for the geometric line bundle Y = V ( ) over S. We define also := j ( ) S A F ∗ F|X\P for the open immersion j : X P ֒ X, and for simplicity, we define \ → e Hi( (m)) := Hi(S, ⊗m) G G⊗OS A for m Z and i 0. Then, the following hold: ∈ ≥ (0) If = , then . G OS F ≃ OX .The inequality depth 1 holds; Equivalently, ֒ is injective (1) FP ≥ F → F (2) The inequality depth 2 holds if and only if H0( (m)) = 0 for any FP ≥ G m< 0. This condition is also equivalent to that .e F ≃ F (3) The quasi-coherent -module is coherent if and only if H0( (m)) = OX F G 0 for m 0. In particular, is coherent if satisfiese S and every ≪ F G 1 irreducible component of Supp ehas positive dimension. G (4) Assume that is coherent. Then,e for an integer k 3, depth k F ≥ FP ≥ holds if and only if Hi( (m)) =0 for any m Z and 0

(6) The satisfies S if and only if satisfies S and H0( (m)) =0 for any F 2 G 2 G m< 0. (7) Assume that is coherent. Then, for an integer k 3, satisfies S if and F ≥ F k only if satisfies S and Hi( (m))=0 for any m Z and 0

0 H0 (X, ′) H0(X, ′) H0(X P, ′) H1 (X, ′) 0 → P F → F → \ F → P F → and isomorphisms Hi(X P, ′) Hi+1(X, ′) for all i 1 (cf. [18, Prop. 2.2]). \ F ≃ P F ≥ Hence, if ′ is a coherent -module, then depth ′ k if and only if F OX FP ≥ (i) H0(X, ′) H0(X P, ′) is injective, when k = 1, F → \ F (ii) H0(X, ′) H0(X P, ′) is an isomorphism, when k = 2, and F → \ F (iii) H0(X, ′) H0(X P, ′) is an isomorphism and Hi(X P, ′) = 0 for F → \ F \ F any 0

H0(X, ) H0(Y, π∗ ) H0( (m)), and F ≃ G ≃ m≥0 G Hi(X P, ) Hi(Y E, π∗ ) M Hi( (m)) \ F ≃ \ G ≃ m∈Z G M for any i 0, where the homomorphism H0(Y, π∗ ) H0(Y E, π∗ ) is an injection ≥ G → \ G and is the identity on each component H0( (m)) of degree m 0. We have (1), G ≥ (2), and (4) by considering the conditions (i)–(iii) above. Moreover, (3) holds, since is coherent if and only if / is a finite-dimensional k-vector space, and since F FP FP we have an isomorphism e e 0 P / P H ( (m)) F F ≃ m<0 G M by the argument above: Thise implies the first half of (3), and the second half follows from Lemma 2.18. For an integer k > 0, satisfies S if and only if satisfies S by [12, IV, F|X\P k G k Cor. (6.4.2)], since Y E X P . Thus, the assertion (5) (resp. (6), resp. (7)) \ ≃ \ follows from (1) (resp. (2), resp. (4)) by the equivalence: (i) (iv) in Lemma 2.14 ⇔ applied to Z = P . The last assertion (8) is a consequence of (7), since dim = FP dim Supp + 1. G e GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 79

Proposition 6.13. Let (S, ) be a connected polarized projective scheme over k A and let X be the aﬃne cone Cone(S, ). Let π : Y S and µ: Y X be the A → → morphisms in Lemma 6.10. Assume that X satisﬁes S2 and n := dim S > 0. Then,

(0) S and Y also satisfy S2, and the schemes S, Y , and X are all equi- dimensional.

Let ωX/k (resp. ωY/k, resp. ωS/k) be the canonical sheaf of X (resp. Y , resp. S) in [r] [r] the sense of Deﬁnition 4.28, and let ωX/k (resp. ωS/k) denote the double-dual of ⊗r ⊗r ωX/k (resp. ωS/k) for an integer r. (1) There exist isomorphisms ∗ (VI-1) ω k π (ω k ) and Y/ ≃ S/ ⊗OS A (VI-2) ω[r] π∗(ω[r] ⊗r) Y/k ≃ S/k ⊗OS A [r] for any integer r. Moreover, ωX/k is isomorphic to the double-dual of [r] µ∗(ωY/k) for any integer r. (2) For any integer r and for any integer k 3, ≥ depth(ω[r] ) k X/k P ≥ holds for the vertex P of X if and only if Hi(S,ω[r] ⊗m)=0 S/k ⊗OS A for any m Z and any 0

Proof. The assertion (0) is a consequence of Proposition 6.12(6) for = S, • • • G O Lemma 6.7, and Fact 2.23(1). Let ωS/k (resp. ωY/k, resp. ωX/k) be the canonical dualizing complex of S (resp. Y , resp. X) in the sense of Definition 4.26. Note that ω• [ n] (resp. ω• [ n 1], resp. ω• [ n 1]) is an ordinary dualizing S/k − Y/k − − X/k − − complex by Lemma 4.27 for n = dim S. Then, L L ω• Ω1 [1] Lπ∗(ω• ) Lπ∗( ω• )[1], Y/k ≃qis Y/S ⊗OY S/k ≃qis A⊗OS S/k since π is separated and smooth (cf. Example 4.23) and since there is an isomorphism Ω1 π∗ (cf. [12, IV, Cor. 16.4.9]). Thus, we have the isomorphism Y/S ≃ A (VI-1). By taking double-dual of tensor powers of both sides of (VI-1), we have the isomorphism (VI-2) for any integer r by Remark 2.20. Since X satisfies S2, any 80 YONGNAMLEEANDNOBORUNAKAYAMA reflexive -module satisfies S by Corollary 2.22, and moreover, depth 2, OX F 2 P F ≥ since codim(P,X) = dim X = n +1 2. Thus, we have isomorphisms ≥ ω[r] j (ω[r] ) j (µ (ω[r] ) )) (µ (ω[r] ))∨∨ X/k ≃ ∗ X\P/k ≃ ∗ ∗ Y/k |X\P ≃ ∗ Y/k .(for any integer r and for the open immersion j : X P ֒ X. This proves (1 \ → By (1), we see that (2) is a consequence of (4) and (7) of Proposition 6.12 applied to the case: = ω[r] ⊗r, where ω[r] . It remains to prove the equivalence G S/k ⊗ A F ≃ X/k of the conditions (i)–(iii) of (3). Since (i) (ii) is trivial, it is enough to prove (ii) ⇒ (iii) and (iii) (i). e ⇒ ⇒ Proof of (iii) (i): Assume that ω[r] ⊗l for some r > 0 and l Z. Since ⇒ S/k ≃ A ∈ ( E) π∗ for the zero-section E of Lemma 6.10, we have OY − ≃ A ω[r] ((r + l)E) π∗(ω[r] ⊗r ⊗−(r+l)) Y/k ⊗OY OY ≃ S/k ⊗ A ⊗OS A ≃ OY from the isomorphism in (1). By taking µ , we have: ω[r] π . ∗ X/k ≃ ∗OY ≃ OX Proof of (ii) (iii): Assume that ω[r] is invertible. Then, ω[r] is invertible on ⇒ X/k Y/k Y E, since Y E X P . Moreover, ω[r] is invertible by (VI-2), since Y E S \ \ ≃ \ S/k \ → [r] is faithfully flat (cf. Lemma A.7). Thus, ωY/k is also invertible again by (VI-2). There is an injection ( [φ: ω[r] ( bE) ֒ µ∗(ω[r Y/k ⊗OY OY − → X/k for some integer b such that the cokernel of φ is supported on E. In fact, for any integer b, we have a canonical homomorphism

( ((µ (ω[r] ( bE)) ֒ j (µ (ω[r] ( bE ∗ Y/k ⊗OY OY − → ∗ ∗ Y/k ⊗OY OY − |X\P j (µ (ω[r] ) ) ω[r] ≃ ∗ ∗ Y/k |X\P ≃ X/k whose cokernel is supported on P , and if b is sufficiently large, then µ∗µ (ω[r] ( bE)) ω[r] ( bE) ∗ Y/k ⊗OY OY − → Y/k ⊗OY OY − is surjective, since ( E) π∗ is relatively ample over X. Thus, OY − ≃ A µ∗µ (ω[r] ( bE)) µ∗(ω[r] ) ∗ Y/k ⊗OY OY − → X/k ∗ [r] induces the injection φ, since the invertible sheaf µ (ωX/k) does not contain non-zero coherent -submodule whose support is contained in E, by the S -condition on Y . OY 1 Let b be a minimal integer with an injection φ above. Then, φ is an isomorphism. This is shown as follows. The homomorphism φ : (ω[r] ( bE)) µ∗(ω[r] ) |E Y/k ⊗OY OY − ⊗OY OE → X/k ⊗OY OE is not zero by the minimality of b. Here, φ corresponds to a non-zero homomor- |E phism ω[r] ⊗(r+b) S/k ⊗OS A → OS by the isomorphism π : E S and by (VI-2). In particular, there is an non- |E ≃ empty open subset U S such that φ is an isomorphism on π−1(U). On the other ⊂ hand, since φ is an injection between invertible sheaves, there is an effective Cartier GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 81 divisor D on Y such that the cokernel of φ is isomorphic to π∗(ω[r] ) OD ⊗OY X/k and that Supp D E. Then, D is a relative Cartier divisor over S, since every ⊂ fiber of π is A1 (cf. [12, IV, (21.15.3.3)]). Thus, π : D S is a flat and finite |D → morphism. If D = 0, then π(D)= S by the connectedness of S, and it contradicts 6 Supp D π−1(U)= . Thus, D = 0, and consequently, φ is an isomorphism. ∩ ∅ Therefore, we have an isomorphism ω[r] ⊗(r+b) S/k ⊗OS A ≃ OS corresponding to the isomorphism φ , and the condition (iii) is satisfied for l = |E (r + b). Thus, we have proved the equivalence of (i)–(iii), and we are done. − Corollary 6.14. Let X be the affine cone of a connected polarized scheme (S, ) A over k. Assume that n = dim S > 0 and H0(S, ⊗m)=0 for any m < 0. Then, A the following hold: (1) The scheme X is Gorenstein if and only if S is Gorenstein, • Hi(S, ⊗m)=0 for any 0 0 and l. A Proof. The assertion (1) follows from (2) and Proposition 6.12(8). The “only if” parts of (2) and (3) are shown as follows. Assume that X is Q-Gorenstein of Gorenstein index r. Note that X is quasi-Gorenstein if and only if r = 1 by Lemma 6.3(1). Then, S is Q-Gorenstein by Corollary 6.11. Moreover, ω[r] ⊗l S/k ≃ A for some l Z by the implication (ii) (iii) of Proposition 6.13(3). Thus, the ∈ ⇒ “only if” parts are proved. The “if” parts of (2) and (3) are shown as follows. Assume that S is Q-Gorenstein. Then, X P is Q-Gorenstein by Corollary 6.11. In \ particular, codim(X X◦,X) 2 for the Gorenstein locus X◦ = Gor(X). Moreover, \ ≥ X satisfies S by Proposition 6.12(6), since S satisfies S and H0(S, ⊗m)=0 for 2 2 A any m < 0 by assumption. If ω[r] ⊗l for integers r > 0 and l, then ω[r] S/k ≃ A X/k is invertible by the implication (iii) (ii) of Proposition 6.13(3). Thus, X is Q- ⇒ Gorenstein. This proves the “if” part of (3). The “if” part of (2) follows also from the argument above by setting r = 1. Thus, we are done. Corollary 6.15. Let X be the affine cone of a connected polarized scheme (S, ) A over k. Assume that S is Cohen–Macaulay, n := dim S > 0, and i ⊗m i ⊗m H (S, ) = H (S,ω k )=0 A S/ ⊗ A for any i> 0 and m> 0. Then, the following hold:

(1) The aﬃne cone X satisﬁes S2. In particular, S is reduced (resp. normal) if and only if X is so. (2) The following conditions are equivalent to each other for an integer k 3: ≥ (a) depth k; OX,P ≥ 82 YONGNAMLEEANDNOBORUNAKAYAMA

(b) X satisﬁes Sk; (c) Hi(S, )=0 for any 0 0 and l. ⊗l (6) When S is Gorenstein, X is Gorenstein if and only if ω k for some S/ ≃ A l Z and if Hi(S, )=0 for any 0

i ⊗m n−i ⊗−m ∨ H (S, ) H (S,ω k ) A ≃ S/ ⊗OS A for any integers m and i, and by assumption, this is zero either if m> 0 and i> 0 or if m < 0 and i

7. Q-Gorenstein morphisms Section 7 introduces the notion of “Q-Gorenstein morphism” and its weak forms: “naively Q-Gorenstein morphism” and “virtually Q-Gorenstein morphism.” We inspect relations between these three notions, and prove basic properties and several theorems on Q-Gorenstein morphisms. In Sections 7.1 and 7.2, we define the notions of Q-Gorenstein morphism, naively Q-Gorenstein morphism, and virtually Q-Gorenstein morphism, and we discuss their properties giving some criteria for a morphism to be Q-Gorenstein. A Q- Gorenstein morphism is always naively and virtually Q-Gorenstein. In Section 7.1, we provide a new example of naively Q-Gorenstein morphisms which are not Q- Gorenstein, by Lemma 7.8 and Example 7.9, and discuss the relative Gorenstein index for a naively Q-Gorenstein morphism in Proposition 7.11. Theorem 7.18 in Section 7.2 shows that a virtually Q-Gorenstein morphism is a Q-Gorenstein morphism under some mild conditions. In Section 7.3, several basic properties including base change of Q-Gorenstein morphisms and of their variants are discussed. Finally, in Section 7.4, we shall prove notable theorems. We prove three criteria for a morphism to be Q-Gorenstein: an infinitesimal criterion (Theorem 7.25), a valuative criterion (Theorem 7.29), and a criterion by S3-conditions on fibers (The- orem 7.30). Moreover, we prove the existence theorem of Q-Gorenstein refinement (Theorem 7.32) and its variants (Theorems 7.34 and 7.35). GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 83

7.1. Q-Gorenstein morphisms and naively Q-Gorenstein morphisms.

Definition 7.1. Let f : Y T be an S -morphism of locally Noetherian schemes → 2 such that every fiber is Q-Gorenstein. Let ωY/T denote the relative canonical sheaf [m] ⊗m in the sense of Definition 5.3 and let ωY/T denote the double-dual of ωY/T for m Z. ∈ (1) The morphism f is said to be naively Q-Gorenstein at a point y Y if [r] ∈ ωY/T is invertible at y for some integer r> 0. If f is naively Q-Gorenstein at every point of Y , then it is called a naively Q-Gorenstein morphism. (2) If ω[m] satisfies relative S over T (cf. Definition 2.28) for any m Z, then Y/T 2 ∈ f is called a Q-Gorenstein morphism. If f : U T is a Q-Gorenstein |U → morphism for an open neighborhood U of a point y Y , then f is said to ∈ be Q-Gorenstein at y.

Remark 7.2. For an S -morphism f : Y T of locally Noetherian schemes, if every 2 → fiber is Gorenstein in codimension one and if ω is an invertible -module, then Y/T OY f is a Q-Gorenstein morphism. In fact, ω[m] ω⊗m satisfies relative S over T Y/T ≃ Y/T 2 for any m Z and every fiber Y = f −1(t) is Q-Gorenstein of Gorenstein index ∈ t one, since ω ω k (cf. Proposition 5.5). Y/T ⊗OY OYt ≃ Yt/ (t) The Q-Gorenstein morphisms and the naively Q-Gorenstein morphisms are characterized as follows. Lemma 7.3. Let Y and T be locally Noetherian schemes and f : Y T a flat → morphism locally of finite type. Let j : Y ◦ ֒ Y be the open immersion from an → open subset Y ◦ of the relative Gorenstein locus Gor(Y/T ). For a point y Y , the ∈ fibers Y = f −1(t) and Y ◦ = Y ◦ Y over t = f(y), and for a positive integer r, let t t ∩ t us consider the following conditions: ◦ (i) The fiber Yt satisfies S2 at y and codimy(Yt Y , Yt) 2. ⊗r \ ≥ (ii) The direct image sheaf j∗(ωY ◦/T ) is invertible at y. (iii) The fiber Yt is Q-Gorenstein at y, and r is divisible by the Gorenstein index of Yt at y. (iv) For any 0 < k r, the base change homomorphism ≤ [k] ⊗k [k] ⊗k φt : j∗(ω ◦ ) OY Yt ω k = j∗(ω ◦ k ) Y /T ⊗ O → Yt/ (t) Yt / (t) induced from the base change isomorphism ω ◦ Y ω ◦ k (cf. Y /T ⊗ O t ≃ Yt / (t) Proposition 5.6) is surjective at y. (v) There is an open neighborhood U of y such that f U : U T is a naively [r] | → Q-Gorenstein morphism and ωU/T is invertible. (vi) There is an open neighborhood U of y such that f U : U T is a Q- [r] | → Gorenstein morphism and ωU/T is invertible. Then, one has the following equivalences and implication on these conditions: (i) + (ii) (v); • ⇔ (i) + (ii) (iii); • ⇒ (iii) + (iv) (vi). • ⇔ 84 YONGNAMLEEANDNOBORUNAKAYAMA

⊗r Proof. First, we shall prove: (i) + (ii) (iii). We set := j (ω ◦ ). Then, ⇒ Mr ∗ Y /T is invertible at y by (ii), and Mr ⊗OY OYt [r] ◦ ⊗r r OY Yt j∗ (( r OY Yt ) Y ) j∗(ω ◦ k ) ω k M ⊗ O → M ⊗ O | ≃ Yt / (t) ≃ Yt/ (t) is an isomorphism at y by (i). In particular, ω[r] is invertible at y. Thus, (iii) Yt/k(t) holds (cf. Definitions 6.1(2) and 6.2). Second, we shall prove (v) (i) + (ii) and (vi) (iii) + (iv). We may assume ⇒ ⇒ that f is naively Q-Gorenstein. Since every fiber Yt is a Q-Gorenstein scheme, we have (i) (cf. Definition 6.1). Moreover,

[k] ⊗k ω j (ω ◦ ) Y/T ≃ ∗ Y /T for any k Z by Proposition 5.6. Hence, (ii) is also satisfied, since ω[r] is invertible ∈ Y/T for an integer r> 0. If f is Q-Gorenstein, then ω[k] is flat over T and ω[k] Y/T Y/T ⊗OY OYt satisfies the S -condition for any t T (cf. Definition 7.1(2)); thus, φ[k] is an 2 ∈ t isomorphism for any t T and k Z, and in particular, (iii) and (iv) are satisfied. ∈ ∈ Finally, we shall prove (i) + (ii) (v) and (iii) + (iv) (vi). Assume that (i) ⇒ ⇒ holds. By Lemma 2.38, there is an open neighborhood U of y such that f U : U T ◦ | → is an S2-morphism having pure relative dimension and codim(Ut′ Y ,Ut′ ) 2 for ′ \ ≥ any t f(U), where U ′ = U Y ′ ; Thus, ∈ t ∩ t [k] ⊗k ω j (ω ◦ ) U/T ≃ ∗ U∩Y /T [r] for any k Z by Lemma 2.33(4). Therefore, if (ii) also holds, then ωU ′/T is ∈ ′ ′ invertible for an open neighborhood U of y in U, and f ′ : U T is a naively Q- |U → Gorenstein morphism. This proves (i) + (ii) (v). Next, assume that (iii) and (iv) ⇒ hold. Note that (iii) implies (i). Thus, we have the same open neighborhood U of ′ ′ y as above. For any integer 0 < k r, there is an open neighborhood Uk of y in U [k] ≤ above such that ω ′ satisfies relative S2 over T , by (iv) and by Proposition 5.6. Uk/T [r] In particular, ωY/T is invertible at y by Fact 2.26(2). In fact, it is flat over T at y [r] and its restriction to the fiber Yt is invertible at y. Then, ωY/T is invertible on an ′′ ′ ′′ ′ open neighborhood Ur of y in Ur. We set U to be the intersection of Uk for all ′′ [l] 0

:Remark. For f : Y T and j : Y ◦ ֒ Y in Lemma 7.3, we have → → (1) The set of points y Y satisfying the condition (i) of Lemma 7.3 is open. ∈ (2) If every ﬁber of f satisﬁes S2 and is Gorenstein in codimension one, then ◦ j ◦ and codim(Y Y , Y ) 2. Here, if Y is connected in addition, OY ≃ ∗OY \ ≥ then f has pure relative dimension. (3) The set of points y Y at which f is naively Q-Gorenstein is open. ∈ GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 85

(4) The set of points y Y at which f is Q-Gorenstein is open. ∈ In fact, the property (1) is mentioned in the proof of Lemma 7.3, and the property (2) is derived from Lemmas 2.33(4), 2.35, and 2.38. The properties (3) and (4) are deduced from Deﬁnition 7.1.

An S2-morphism of locally Noetherian schemes is not necessarily naively Q- Gorenstein even if every ﬁber is Q-Gorenstein. The following example is well known.

Example 7.4. Let S = P( (4)) be the Hirzebruch surface of degree 4 over an O ⊕ O algebraically closed field k. By contracting of the unique ( 4)-curve Γ, we have a − birational morphism S X to the weighted projective plane X = P(1, 1, 4). Note → that X is Q-Gorenstein and its Gorenstein index is two. Let η be the extension 1 class in Ext 1 ( (4), P1 ) of an exact sequence P O O (VII-1) 0 P1 (2) (2) (4) 0 → O → O ⊕ O → O → 1 1 1 1 on P . We set T = A = Spec k[t] and P = P k T , and let p: P P and × → q : P T be projections. Let us consider the element η of Ext1 (p∗ (4), ) → P P O OP corresponding to η t by the isomorphism ⊗ 1 ∗ 1 0 Ext (p (4), ) Ext 1 ( (4), P1 ) k H (T, ), P O OP ≃ P O O ⊗ OT and let (VII-2) 0 p∗ (4) 0 → OP → E → O → be an exact sequence on P whose extension class is η . Let π : V P be the P → P1-bundle associated with , and let f : Y T be the T -scheme defined as E → Proj q (Sym( )) for the symmetric -algebra Sym( ). Then, Y is a normal T ∗ E OP E projective variety and there is a birational morphism µ: V Y over T such that → the induced morphism µ : V Y of fibers over t T is described as follows: t t → t ∈ µ0 is isomorphic to the contraction morphism S X of Γ, and • 1 → 1 µ is isomorphic to the identity morphism of P k P . • t k(t) × (t) k(t) In fact, µ∗ for the tautological invertible sheaf on V associated with and L≃ H L E for the f-ample tautological invertible sheaf on Y associated with the graded H algebra q (Sym ). In particular, f is a flat projective morphism whose fibers are ∗ E all Q-Gorenstein. However, f is not naively Q-Gorenstein. For, if the canonical 2 2 divisor KY is Q-Cartier, then K is constant for t T , but we have K = 9 and Yt ∈ Y0 K2 = 8 for t = 0. Yt 6 Lemma 7.5. In the situation of Example 7.4, let Spec A T = A1 be the closed 2 ⊂ immersion defined by k[t] A = k[t]/(t ). Then, for the base change fA : YA → [2] → Spec A of f : Y T by the closed immersion, the reflexive sheaf ω on YA → YA/A is not invertible. Moreover, ω[2] does not satisfy relative S over Spec A, and YA/A 2 Y Spec A is not a Q-Gorenstein morphism. A → Proof. The last assertion follows from the previous one by Fact 2.26(2) and Propo- sition 5.6, since ω[2] is invertible. Let g : V Spec A be the base change of X/k A A → g := q π : V T , and let µ : V Y and π : V P := P Spec A ◦ → A A → A A A → A ×Spec T ≃ 86 YONGNAMLEEANDNOBORUNAKAYAMA

1 PA be the induced morphisms from the morphisms µ and π over T , respectively. Note that the further base change by Spec k Spec A produces the contraction → morphism S X and the ruling S P1 from µ and π , respectively. Assume → → A A that ω[2] is invertible. Then, YA/A := ω⊗2 µ∗ (ω[2] )−1 M VA/A ⊗ A YA/A is invertible on VA. We have canonical homomorphisms ⊗2 [2] ∗ ⊗2 ⊗2 (µA)∗ω ω and µ (µA)∗ω ω . VA/A → YA/A A VA/A → VA/A The left one is obtained by taking double-dual. The right one is surjective, since ∗ ∗ ⊗−2 ω ω π p P1 (2) VA/A ≃ V/T ⊗OV OVA ≃ O ⊗OV L ⊗OV OVA ∗ and since µ . Therefore, there is a homomorphism V which is an L ≃ H M → O A isomorphism outside Γ. Since V Spec k S, we have depth 1 by A ×Spec A ≃ Γ M ≥ Lemma 2.32(3), and it implies that V is injective and O S S M → O A M⊗ VA O → O is also injective. Therefore, the closed subscheme D of VA defined by the ideal sheaf is an effective Cartier divisor, and it is flat over Spec A by the local M criterion of flatness (cf. Proposition A.1(ii)). Moreover, D Spec k Γ by ×Spec A ≃ the isomorphism ω⊗2 (Γ) µ∗(ω[2] ). S/k ⊗OS OS ≃ 0 X/k Hence, the composite D V P is a finite morphism, and the corresponding ⊂ A → A ring homomorphism π is an isomorphism, since its base change by OPA → A∗OD A k is isomorphic to the isomorphism P1 and since π is flat over → O → OΓ A∗OD A. Therefore, D is a section of π : V P . Then, the pullback of the exact A A → A sequence (VII-2) to PA is split by the surjection π ( ) π ( ) π . E⊗OP OPA ≃ A∗ L⊗OV OVA → A∗ L⊗OV OD ≃ A∗OD ≃ OPA This means that the morphism Spec A T factors through Spec k T ; This is a → ⊂ contradiction. Therefore, ω[2] is not invertible. YA/A Remark 7.6. Let f : Y T be an S -morphism of locally Noetherian schemes → 2 whose fibers are all Gorenstein in codimension one. The Kollár condition for f −1 along a fiber Yt = f (t) is a condition that the base change homomorphism [m] [m] [m] φ : ω O Y ω t Y/T ⊗ Y O t → Yt/k(t) is an isomorphism for any m Z. By Proposition 5.6, we see that the Kollár ∈ [m] condition is equivalent to that ωY/T satisfies relative S2 over T along Yt for any m Z. Therefore, when Y is Q-Gorenstein, the Kollár condition for f is satisfied ∈ t along Yt if and only if f is Q-Gorenstein along Yt. The Kollár condition has been considered for deformations of Q-Gorenstein algebraic varieties of characteristic zero in [27, 2.1.2], [21, 2, Property K], etc. § Fact 7.7. Some naively Q-Gorenstein morphisms are not Q-Gorenstein. Kollár gives an example of a naively Q-Gorenstein morphism which is not Q-Gorenstein in the positive characteristic case (cf. [16, 14.7], [31, Exam. 7.6]). See Remark 7.28 below for a detail. Patakfalvi has constructed an example of characteristic zero in GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 87

[44, Th. 1.2] using some example of projective cones (cf. [44, Prop. 5.4]): This is a projective flat morphism B of normal algebraic varieties over a field k of H → characteristic zero such that B is an open subset of P1, • k a closed fiber has a unique singular point, but other fibers are all smooth • H0 of dimension 3, ≥ ω[r] is invertible for some r> 0, but • H/B

ω H ω k. H/B ⊗O OH0 6≃ H0/ Recently, Altmann and Kollár [3] have constructed several examples of natively Q-Gorenstein morphisms which are not Q-Gorenstein as infinitesimal deformations of two-dimensional cyclic quotient singularities. We can construct another example by the following lemma, which is inspired by Patakfalvi’s work [44]. Lemma 7.8. Let S be a non-singular projective variety of dimension 2 over ≥ an algebraically closed field k of characteristic zero, and let be an invertible - L OS module of order l > 1, i.e., l is the smallest positive integer such that ⊗l . L ≃ OS Assume that H1(S, )=0, H1(S, ) = 0, and that K is ample. For an integer OS L 6 S r 2, we set ≥ := (rK ) −1 = ω⊗r −1, A OS S ⊗ L S/k ⊗ L and let X be the affine cone Cone(S, ) with a vertex P . Then, A (1) X is a normal Q-Gorenstein variety with one isolated singularity P of Gorenstein index lr, and the following hold for any non-constant function f : X T := A1: → k (2) f is a naively Q-Gorenstein morphism along the fiber F = f −1(f(P )); (3) ω[r] ω[r] does not satisfy relative S over T at P . In particular, f is X/T ≃ X/k 2 not Q-Gorenstein at P . Proof. (1): The affine cone X is Q-Gorenstein by Corollary 6.14(3). Here, X P \ is a non-singular variety by Lemma 6.10(4). Therefore, X is a normal variety. We [lr] [m] have ωX/k X by Proposition 6.13(3). If ωX/k is invertible for some m> 0, then ≃′ O ⊗m ⊗l ′ ′ ωS/k for some integer l by Proposition 6.13(3), but it implies that m = l r, ≃ A′ and ⊗l . Hence, the Gorenstein index of X is lr. L ≃ OS (2): For any i> 0 and m> 0, we have i ⊗m i ⊗m H (S, ) = H (S,ω k )=0 A S/ ⊗OS A by the Kodaira vanishing theorem, since ⊗m ω−1 ⊗m−1 ω⊗r−1 −1 A ⊗OS S/k ≃ A ⊗OS S/k ⊗ L is ample. Then, we can apply Corollary 6.15(2). As a consequence, X satisfies S , since H1(S, ) = 0. Now, f is a flat morphism, since X is irreducible and 3 OS dominates T . Hence, F satisfies S2 by the equality depth = depth depth = depth 1 OF,x OX,x − OT,f(x) OX,x − 88 YONGNAMLEEANDNOBORUNAKAYAMA for any closed point x F (cf. (II-2) in Fact 2.26). Thus, f is an S -morphism ∈ 2 along F , and f is a naively Q-Gorenstein morphism along F (cf. Definition 7.1(1)), since ω[lr] ω[lr] is invertible by (1). X/T ≃ X/k (3): By assumption, we have H1(S,ω[r] −1) H1(S, ) =0. S/k ⊗ A ≃ L 6 [r] [r] Then, depth(ωX/k)P = 2 by Proposition 6.13(2). Since ωX/k is flat over T , we have depth(ω[r] ) = depth(ω[r] ) depth =1 X/k ⊗OX OF P X/k P − OT,f(P ) by (II-2) in Fact 2.26. This implies that ω[r] ω[r] does not satisfy relative S X/k ≃ X/T 2 over T at P . Therefore, f is not Q-Gorenstein at P (cf. Definition 7.1(2)). We have the following example of non-singular projective varieties S with invertible -module of order l = 2 in Lemma 7.8: OS L Example 7.9. Let V be an abelian variety of dimension d 3 and let ι: V V be ≥ → the involution defined by ι(v)= v with respect to the group structure on V . Let − W be the quotient variety V/ ι . Then, W is a normal projective variety with only h i isolated singular points, and (VII-3) H1(W, )=0, OW since it is isomorphic to the invariant part of H1(V, ) by the induced action of OV ι, which is just the multiplication map by 1. The quotient morphism π : V W − → is a double-cover étale outside the singular locus of W , and we have isomorphisms [2] π ω k and ω . In particular, ∗OV ≃ OW ⊕ W/ W/k ≃ OW 1 1 ⊕d (VII-4) H (W, ω k) H (V, ) k W/ ≃ OV ≃ by (VII-3). We can take a smooth ample divisor S on W away from the singular locus of W . Then, dim S = d 1 2. By the Kodaira vanishing theorem applied − ≥ to the ample divisor π∗S on V , we have Hi(V, π∗ ( S))= 0 for any 0

Deﬁnition 7.10 (relative Gorenstein index). For a naively Q-Gorenstein morphism f : Y T and for a point y Y , the relative Gorenstein index of f at y → ∈ [r] is the smallest positive integer r such that ωY/T is invertible at y. The least common multiple of relative Gorenstein indices at all the points is called the relative Gorenstein index of f, which might be + . ∞ Proposition 7.11. Let f : Y T be a naively Q-Gorenstein morphism. For → a point y Y , let m be the relative Gorenstein index of f at y and let r be the ∈ −1 Gorenstein index of Yt = f (t) at y, where t = f(y). Then, m = r in the following three cases: (i) f is Q-Gorenstein at y; (ii) Yt is Gorenstein in codimension two and satisﬁes S3 at y; (iii) m is coprime to the characteristic of k(t).

Proof. Note that m is divisible by r. In fact, the base change homomorphism [m] [m] ω O Y ω Y/T ⊗ Y O t → Yt/k(t) is an isomorphism at y, since the left hand side is invertible at y and since Yt satisfies S . We set := ω[r] . It is enough to prove that is invertible at 2 M Y/T M y. Let Z be the complement of the relative Gorenstein locus Gor(Y/T ) and let j : Y Z ֒ Y be the open immersion. Note that codim(Z Y , Y ) 2 ( 3 in \ → ∩ t t ≥ ≥ case (ii)) and codim(Z, Y ) 2. If f is Q-Gorenstein, then satisfies relative S ≥ M 2 over T ; in particular, [r] OY Yt j∗ ( OY Yt ) Y \Z ω M⊗ O ≃ M⊗ O | t ≃ Yt/k(t) and hence, is invertible at y by Fact 2.26(2). Thus, it is enough to consider the M cases (ii) and (iii). By replacing Y with an open neighborhood of y, we may assume the following: (1) depth 2 (cf. Lemma 2.32(3)); Z OY ≥ (2) Y \Z is invertible and depthZ 2 (cf. Proposition 5.6); M| [r] M≥ (3) j∗( OY Yt Y \Z ) ω is invertible; M⊗ O | t ≃ Yt/k(t) (4) one of the following holds: (a) depth 3; Z∩Yt OYt ≥ (b) [m/r] ω[m] is invertible, where m/r is coprime to the character- M ≃ Y/T istic of k(t). Then, is invertible by Theorem 3.16, and we are done. M Remark 7.12. A special case of Proposition 7.11 for naively Q-Gorenstein morphisms is stated in [30, Lem. 3.16], where T is the spectrum of a complete Noether- ian local C-algebra and the closed fiber Yt is a normal complex algebraic surface. However, the proof of [30, Lem. 3.16] has two problems. We explain them using the notation there, where (X S, 0 S) corresponds to (Y T,t T ) in our → ∈ → ∈ situation, and 0 is the closed point of S. The central fiber X0 is only a germ of complex algebraic surface in [30, 3], but here, for simplicity, we consider X as a § 0 usual algebraic surface and hence consider X S as a morphism of finite type. → 90 YONGNAMLEEANDNOBORUNAKAYAMA

The authors of [30] write X0 for Gor(X/S) and write Y 0 X0 for the cyclic étale → cover associated with an isomorphism ω[m] . They want to prove that m is X/S ≃ OX equal to the Gorenstein index r of the fiber X of X S over 0. 0 → The first problem is in the proof in the case where S = Spec A is Artinian. This [m] is minor and is caused by omitting an explanation of the isomorphism ωX/S X . 0 0≃ O In this situation, they assert that it is enough to prove the fiber Y0 of Y S 0 → over 0 to be connected. However, Y0 is connected even if r = m. In fact, for [r] [m] 6 isomorphisms u: ω X and v : ω X , we have an invertible element X0/C ≃ O 0 X/S ≃ O θ of such that OX0 v = θu⊗m/r |X0 as an isomorphism ω[m] . Here, we can take v so that θ can not X/S ⊗OX OX0 ≃ OX0 have k-th root in for any integer k dividing r. Then, Y 0 is connected for such OX0 0 v. Of course, this problem is resolved by replacing the isomorphism v with v(θ˜)−1 for a function θ˜ which is a lift of θ . ∈ OX ∈ OX0 The second problem is in the reduction to the Artinian case. They set An = A/mn, S = Spec A , and X0 = X0 S , for n 1 and for the maximal ideal m n n n ×S n ≥ of A, and they obtain an isomorphism ⊗r Φn : ω 0 X0 Xn/Sn ≃ O n for any n by applying the assertion: m = r, to the Artinian case. However, just ⊗r after the isomorphism Φ , they deduce an isomorphism ω 0 0 without n X /S ≃ OX mentioning any reason. This is thought of as a lack of the proof, by Remark 3.13. For, their isomorphism above induces isomorphisms

[r] ⊗r ω Xn j∗(ω 0 ) X/S ⊗ O ≃ Xn/Sn for all n, while we always have an isomorphism

[r] ⊗r ω j (ω 0 ), X/S ≃ ∗ X /S .where j : X0 ֒ X denotes the open immersion → 7.2. Virtually Q-Gorenstein morphisms.

Definition 7.13. Let f : Y T be a morphism locally of finite type between → locally Noetherian schemes. For a given point y Y and the image o = f(y), the ∈ morphism f is said to be virtually Q-Gorenstein at y if f is flat at y, • the fiber Y = f −1(o) is Q-Gorenstein at y, • o and if there exist an open neighborhood U of y in Y and a reflexive -module OU L satisfying the following conditions:

(i) ω k , where U = U Y ; L⊗OU OUo ≃ Uo/ (o) o ∩ o (ii) for any integer m, the double-dual [m] of ⊗m satisﬁes relative S over T L L 2 at y. If f is virtually Q-Gorenstein at every point of Y , then it is called a virtually Q-Gorenstein morphism. GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 91

Remark 7.14. If the morphism f above is virtually Q-Gorenstein at y, then there exist an open neighborhood U of y in Y and a reflexive -module such that OU L (1) f : U T is an S -morphism of pure relative dimension, |U → 2 (2) every non-empty fiber U = U Y of f is Gorenstein in codimension one, t ∩ t |U i.e., codim(U Y ◦,U ) 2 for any t f(U), where Y ◦ = Gor(Y/T ), t \ t ≥ ∈ (3) ω k , L⊗OU OUo ≃ Uo/ (o) (4) ◦ is invertible, L|U∩Y (5) [r] is invertible for some integer r> 0, and L (6) [m] satisfies relative S over T for any integer m. L 2 In fact, we have an open neighborhood U satisfying (1) and (2) by Lemma 2.38. By shrinking U and by Fact 2.26(2), we may assume the existence of satisfying L (3), (4), and (5), where r is a multiple the Gorenstein index of Yo at y. Then, for any point t f(U), the coherent sheaf [m] = [m] is locally equi- ∈ L(t) L ⊗ OUt dimensional by Fact 2.23(1), since Supp [m] = U, Supp [m] = U , and since L L(t) t U is catenary satisfying S . Hence, the relative S -locus S ( [m]/T ) is an open t 2 2 2 L subset of U by Fact 2.29(2), and now, y S ( [m]/T ) for any m Z. We have ∈ 2 L ∈ S ( [m+r]/T ) = S ( [m]/T ) for any m by [m+r] [r] [m], and hence the 2 L 2 L L ≃ L ⊗ L intersection of S ( [m]/T ) for all m is still an open neighborhood of y. Thus, we 2 L can also assume (6). As a consequence of (1)–(6), we see that

(7) Uo = U Yo is Q-Gorenstein, and [m] ∩ [m] (8) O U ω for any m Z. L ⊗ U O o ≃ Uo/k(o) ∈ In fact, [m] satisﬁes S by (6) and its depth along U Y ◦ is 2 by (1) L ⊗OU OUo 2 o \ ≥ and (2) (cf. Lemma 2.15(2)); this implies (8). The condition (7) follows from (5) and (8).

Remark. The set of points y T at which f is virtually Q-Gorenstein, is not open ∈ in general. Even if a morphism f : Y T is virtually Q-Gorenstein at any point → of a fiber Y , the other fibers Y are not necessarily Q-Gorenstein even if t T is o t ∈ sufficiently close to the point o. The following gives such an example.

Example 7.15. Let X be a non-singular projective variety over an algebraically closed field k of characteristic zero such that the dualizing sheaf ωX/k is ample, 1 1 H (X, ) = 0, and H (X,ω k) = 0. Then, n := dim X 3. As an example OX 6 X/ ≥ of X, we can take the product C S of a non-singular projective curve C of × genus 2 and a non-singular projective surface S such that ω k is ample and ≥ S/ H1(S, ) = H2(S, ) = 0. Let us take a positive-dimensional nonsingular affine OS OS subvariety T = Spec A of the Picard scheme Pic0(X) which contains the origin 0 of 0 Pic (X). Then, there is an invertible sheaf on X := X k T such that N A ×Spec is algebraically equivalent to zero for any t T , and • N(t) ∈ if and only if t = 0, • N(t) ≃ OXt where Xt = X Spec k Spec k(t) and (t) = OX Xt (cf. Notation 2.24). We × N N ⊗ A O define a Z≥0-graded A-algebra R = m≥0 Rm by 0 ∗ ⊗m Rm := H (XA,L(p (ωX/k) OX ) ) ⊗ A N 92 YONGNAMLEEANDNOBORUNAKAYAMA for the projection p: X X, and let f : Y := Spec R T = Spec A be the A → → induced affine morphism. We shall prove the following by replacing T with a suitable open neighborhood of 0: (1) f is a flat morphism; (2) for any t T , the fiber Y = f −1(t) is isomorphic to the affine cone of the ∈ t polarized scheme (X ,ω k ); t Xt/ (t) ⊗ N(t) (3) the set of points t T such that Y is Q-Gorenstein, is a countable set; ∈ t (4) f is virtually Q-Gorenstein at any point of the fiber Y0. t t For the proof, we consider a graded k(t)-algebra R = m≥0 Rm defined by t 0 ⊗m R = H (Xt, (ω k O L) ). m Xt/ (t) ⊗ Xt N(t) Then, Spec Rt is the affine cone associated with (X ,ω ). On the other t Xt ⊗ N(t) hand, Y = Spec(R k(t)), and we have a natural homomorphism t ⊗A ϕt : R k(t) Rt ⊗A → ∗ t of graded k(t)-algebras, since (p ωX/k) OX Xt ωX /k(t). Let ϕ be the ⊗ A O ≃ t m homomorphism R k(t) Rt of m-th graded piece of ϕt. Note that m ⊗A → m 1 ⊗m H (Xt, (ω k O ) )=0 Xt/ (t) ⊗ Xt N(t) for any m 2 by the Kodaira vanishing theorem, since ω k is ample and is ≥ X/ N(t) algebraically equivalent to zero. Moreover, there is an open neighborhood U of 0 in T such that 1 H (Xt,ω k O )=0 Xt/ (t) ⊗ Xt N(t) for any t U by the upper semi-continuity theorem (cf. [12, III, Th. (7.7.5) I], [39, ∈ 1 5, Cor., p. 50]), since we have assumed that H (X,ω k) = 0. We may replace T § X/ with U. Then, ϕt is an isomorphism for any m 1 and for any t T by [12, III, m ≥ ∈ Th. (7.7.5) II] (cf. [39, 5, Cor. 3, p. 53]). Since ϕt is obviously an isomorphism, ϕt § 0 is an isomorphism and Y Spec Rt for any t T . Moreover R is a flat A-module t ≃ ∈ m for any m 0 by [12, III, Cor. (7.5.5)] (cf. [19, III, Th. 12.11]), and it implies that ≥ Y = Spec R is flat over T . This proves (1) and (2). By Corollary 6.15(5), Y is Q-Gorenstein if and only if ⊗r for some t N(t) ≃ OXt r> 0. For an integer r> 0, let F be the kernel of the r-th power map Pic0(X) r → Pic0(X) which sends an invertible sheaf to ⊗r. Then, F is a finite set, and L L r F T is just the set of points t T such that ⊗r . Thus, Y is Q-Gorenstein r ∩ ∈ N(t) ≃ OXt t if and only if t is contained in the countable set F T . This proves (3). r>0 r ∩ Note that ω k by Proposition 6.13(3). Hence, f : Y T is virtually Y0/ ≃ OY0 S → Q-Gorenstein at any point of Y , since plays the role of in Definition 7.13. 0 OY L This proves (4).

Lemma 7.16. Let f : Y T be a flat morphism locally of finite type between → −1 locally Noetherian schemes and let o T be a point such that Yo = f (o) is Q- ∈ [r] Gorenstein. For a given isomorphism u: ω Y for a positive integer r, Yo/k(o) → O o we set r−1 = ω[i] R i=0 Yo/k(o) M GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 93 to be the Z/rZ-graded -algebra defined by the isomorphism u. Then, the follow- OYo ing two conditions are equivalent to each other: (1) Locally on Y , there exists a Z/rZ-graded coherent -algebra ∼ flat over OY R T with an isomorphism

∼ R ⊗OY OYo ≃ R as a Z/rZ-graded -algebra. OYo (2) The morphism f is virtually Q-Gorenstein along Yo.

Proof. We write X = Y and k = k(o) for short. First, we shall show (1) (2). o ⇒ We may assume that ∼ is defined on Y . Thus, there exist coherent -modules R OY for 0 i r 1 such that Li ≤ ≤ − r−1 ∼ = i R i=0 L as a Z/rZ-graded -algebra. Hence, Mare all flat over T , and moreover, OY Li ω[i] for any 0 i r 1, • Li ⊗OY OX ≃ X/k ≤ ≤ − the multiplication map ⊗i restricts to the canonical homomorphism • L1 → Li ω⊗i ω[i] for any 1 i r 1, and X/k → X/k ≤ ≤ − the multiplication map ⊗r induces the isomorphism u: ω[r] • L1 → OY X/k → . OX We shall show that [r] and [i] for any 1 i r 1 along X = Y . L1 ≃ OY Li ≃ L1 ≤ ≤ − o Now, satisfies relative S over T along X for any 0 i r 1, since ω[i] Li 2 ≤ ≤ − X/k satisfies S2 (cf. Lemma 5.2). Thus, there is a closed subset Z of Y such that Gor(X) X Z, • ⊂ \ is invertible for any 0 i r 1 (cf. Fact 2.26(2)), • Li|Y \Z ≤ ≤ − the multiplication maps ⊗i and ⊗r are isomorphisms on • L1 → Li L1 → OY Y Z. \ By replacing Y with its open subset, we may assume that codim(Y Z, Y ) 2 for t ∩ t ≥ any t T by Lemma 2.38, since codim(Y Z, Y ) codim(X Gor(X),X) 2 and ∈ o ∩ o ≥ \ ≥ may assume that satisfies relative S over T for all i (cf. Fact 2.29(2)). Then, Li 2 for any m 1 and any 1 i r 1, we have ≥ ≤ ≤ − [m] j ( ⊗m ) and j ( ) L1 ≃ ∗ L1 |Y \Z Li ≃ ∗ Li|Y \Z .for the open immersion j : Y Z ֒ Y by (4) and (5) of Lemma 2.33, respectively \ → This argument shows that [i] and [r] along X = Y . Li ≃ L1 OY ≃ L1 o As a consequence, satisfies the conditions in Definition 7.13 for any point of L1 Y , and we have proved (1) (2). o ⇒ Next, we shall show: (2) (1). We may assume the existence of a reflexive ⇒ -module which satisfies the conditions of Remark 7.14 for U = Y and for the OY L fiber Yo = X. By replacing Y with an open neighborhood of an arbitrary point of Y , we may assume that there is an isomorphism u∼ : [r] which restricts o L → OY to the composite of the isomorphism [r] ω[r] and the isomorphism L ⊗OY OX ≃ X/k 94 YONGNAMLEEANDNOBORUNAKAYAMA u: ω[r] . Then, u∼ defines a Z/rZ-graded -algebra X/k → OX OY r−1 ∼ = [i], R i=0 L which satisfies the condition (1). Thus,M we are done. Remark 7.17. The Q-Gorenstein deformation in the sense of Hacking [15, Def. 3.1] is considered as a virtually Q-Gorenstein deformation by Lemma 7.16. Hack- ing’s notion is generalized to the notion of Kollár family of Q-line bundles by Abramovich–Hassett (cf. [1, Def. 5.2.1]). This is related to the notion of virtually Q-Gorenstein morphism as follows. Let f : Y T be an S -morphism between → 2 Noetherian schemes such that every fiber is connected, reduced, and Q-Gorenstein. Let be a reflexive -module. Then, satisfies the conditions (i) and (ii) of L OY L Definition 7.13 for U = Y and for any y Y , if and only if (Y T, ) is a Kollár ∈ → L family of Q-line bundles with ω k for all t T . However, in their L ⊗ OYt ≃ Yt/ (t) ∈ study of Kollár families (Y T, ) for = ω , every fiber and every ω[m] are → L L Y/T Y/T assumed to be Cohen–Macaulay (cf. [1, Rem. 5.3.9, 5.3.10]). A Q-Gorenstein morphism is always virtually Q-Gorenstein. The following theorem shows conversely that a virtually Q-Gorenstein morphism is a Q-Gorenstein morphism under some mild conditions. In particular, we see that a virtually Q- Gorenstein morphism is Q-Gorenstein if it is a Cohen–Macaulay morphism. Theorem 7.18. Let Y and T be locally Noetherian schemes and f : Y T a flat → morphism locally of finite type. For a point t T , assume that f is virtually Q- −1 ∈ Gorenstein at any point of the fiber Yt = f (t) and that one of the following two conditions is satisfied:

(a) Yt satisfies S3; (b) there is a positive integer r coprime to the characteristic of k(t) such that [r] ωY/T is invertible along Yt. Then, f is Q-Gorenstein along Yt. Proof. Since the assertion is local, by Remark 7.14, we may assume that f is an S -morphism and there is a reflexive -module satisfying the following two 2 OY L conditions: (1) [m] = ( ⊗m)∨∨ satisfies relative S over T for any integer m; L L 2 (2) there is an isomorphism ω k . L⊗OY OYt ≃ Yt/ (t) We can prove the following for := om ( ,ω ) applying Theorem 5.10: M H OY L Y/T (3) is an invertible -module along Y ; M OY t (4) ω −1 along Y . L≃ Y/T ⊗OY M t In fact, the condition (ii) of Theorem 5.10 holds by (1) and (2) above, and the condition (i) of Theorem 5.10 holds for U = CM(Y/T ) (resp. U = Gor(Y/T )) in case (a) (resp. (b)). The remaining condition (iii) of Theorem 5.10 is checked as follows. In case (a), the condition (iii)(a) of Theorem 5.10 is satisfied for U above. In case (b), [r] is invertible along Y by (1) and (2), since L t [r] [r] [r] O Y ω ω O Y L ⊗ Y O t ≃ Yt/k(t) ≃ Y/T ⊗ Y O t GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 95 is invertible (cf. Fact 2.26(2)); Thus, the condition (iii)(b) of Theorem 5.10 is satisfied in this case. Therefore, we can apply Theorem 5.10 and obtain (3) and (4). As a consequence, we have an isomorphism

ω[m] [m] ⊗m Y/T ≃ L ⊗OY M for any m Z along Y . Therefore, ω[m] satisfies relative S over T along Y by ∈ t Y/T 2 t (1), and hence f : Y T is Q-Gorenstein along Y . → t Corollary 7.19. Let Y and T be locally Noetherian schemes and f : Y T a flat → −1 morphism locally of finite type. For a point t T , assume that the fiber Yt = f (t) [r] ∈ is quasi-Gorenstein. If ωY/T is invertible for a positive integer r coprime to the characteristic of k(t), then f is Q-Gorenstein along Yt.

Proof. The morphism f is virtually Q-Gorenstein at any point of Y , since plays t OY the role of in Deﬁnition 7.13. Thus, we are done by Theorem 7.18 in the case L (b).

7.3. Basic properties of Q-Gorenstein morphism. We shall prove some basic properties of Q-Gorenstein morphism and its variants. The following is a criterion for a morphism to be naively Q-Gorenstein.

Lemma 7.20. Let f : Y T be an S -morphism of locally Noetherian schemes. → 2 Assume that T is Q-Gorenstein and that every ﬁber of f is Gorenstein in codimension one. Then, f is a naively Q-Gorenstein morphism if and only if Y is Q-Gorenstein.

Proof. Since the Q-Gorenstein properties are local, we may assume that T and Y are affine and that f is of finite type with pure relative dimension (cf. Lemma 2.38). Since the Q-Gorenstein scheme T satisfies S2 (cf. Lemma 6.3(2)), we may assume the following (cf. Lemma 6.4): T admits an ordinary dualizing complex • (cf. Lemma 4.14) with the • R dualizing sheaf ω := 0( •); T H R the double-dual ω[m] of ω⊗m satisfies S for any integer m; • T T 2 ω[r] is invertible for a positive integer r. • T For the Gorenstein locus T ◦ := Gor(T ) and the relative Gorenstein locus Y ◦ := Gor(Y/T ), we set U := f −1(T ◦) and U ◦ := U Y ◦. Then, codim(Y U, Y ) 2 by ∩ \ ≥ (II-1) in Fact 2.26 and Property 2.1(3), since f is flat and codim(T T ◦,T ) 2. \ ≥ Hence, codim(Y U ◦, Y ) 2 by codim(Y Y ◦, Y ) 2, since f is an S -morphism \ ≥ \ ≥ 2 (cf. Lemma 2.35). The twisted inverse image • := f !( •) is a dualizing complex RY R of Y (cf. Example 4.24) with a quasi-isomorphism

L • f ! Lf ∗( •) RY ≃qis OT ⊗OY R by (IV-6) in Fact 4.35, where

! ω ◦ [d] f ◦ Y /T ≃qis OT |Y 96 YONGNAMLEEANDNOBORUNAKAYAMA for the relative dimension d of f. Note that Y satisﬁes S2 by Fact 2.26(6). Thus, • [ d] is an ordinary dualizing complex of Y , and ω := −d( • ) is a dualizing RY − Y H RY sheaf of Y . In particular, U ◦ is a Gorenstein scheme with the dualizing sheaf

−d • ∗ ω ◦ = ( ) ◦ ω ◦ ◦ ◦ (f ◦ ) (ω ◦ ). Y |U H RY |U ≃ Y /T |U ⊗OU |U T By Lemma 6.4, we have an isomorphism (VII-6) ω[m] ω[m] f ∗(ω[m]) Y ≃ Y/T ⊗OY T for any integer m. For a point y Y , Y is Q-Gorenstein at y if and only if ω[m] ∈ Y is invertible at y for some m > 0. On the other hand, f is naively Q-Gorenstein [m] [r] at y if and only if ωY/T is invertible at y for some m > 0. Since ωT is invertible, the isomorphism (VII-6) implies that Y is Q-Gorenstein if and only if f is naively Q-Gorenstein.

The following is a criterion for a morphism to be Q-Gorenstein.

Proposition 7.21. Let f : Y T be a flat morphism locally of finite type between → locally Noetherian schemes. For a point t T , assume that the fiber Y = f −1(t) is ∈ t a Q-Gorenstein scheme. If there exist coherent -modules for m 1 such OY Mm ≥ that [m] ⊗m m O Y ω and m Y ◦ ω ◦ , M ⊗ Y O t ≃ Yt/k(t) M | ≃ Y /T where Y ◦ is the relative Gorenstein locus Gor(Y/T ), then f is a Q-Gorenstein morphism along Yt. Proof. We set = . Then, = satisfies S along Y for M0 OY Mm,(t) Mm ⊗OY OYt 2 t any m 0. For the complement Z = Y Y ◦, we have codim(Z Y , Y ) 2, since ≥ \ ∩ t t ≥ Yt is Q-Gorenstein. Hence, m is flat over T along Yt by Lemma 3.5(1), since ⊗m M ◦ ω ◦ is flat over T and Mm|Y ≃ Y /T depth 2 Z∩Yt Mm,(t) ≥ (cf. Lemma 2.15(2)). As a consequence, satisfies relative S over T along Y Mm 2 t for any m 0. In particular, f is an S -morphism along Y by considering the ≥ 2 t case m = 0. By replacing Y with an open neighborhood of Yt, we may assume ′ that f is an S -morphism and that codim(Z Y ′ , Y ′ ) 2 for any t f(Y ), by 2 ∩ t t ≥ ∈ Lemma 2.38. ◦ Now, Supp m = Y , since it contains the dense open subset Y . Hence, M ′ Supp ′ = Y ′ for any t T , and it is locally equi-dimensional by Fact 2.23(1). Mm,(t ) t ∈ Thus, U := S ( ) is open by Fact 2.29(2), and m 2 Mm depth 2 Z∩Um Mm|Um ≥ , by Lemma 2.32(1). It implies that, for the open immersion j : Y ◦ ֒ Y → ⊗m [m] j ( ◦ ) j (ω ◦ )= ω Mm → ∗ Mm|Y ≃ ∗ Y /T Y/T [m] is an isomorphism along Yt. As a consequence, ωY/T satisfies relative S2 over T along Y for any m 0. Therefore, f is a Q-Gorenstein morphism along Y . t ≥ t GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 97

We have the following base change properties for Q-Gorenstein morphisms and for their variants.

Proposition 7.22. Let f : Y T be a ﬂat morphism locally of ﬁnite type between → locally Noetherian schemes and let p Y ′ Y −−−−→ f ′ f

′ q  T T y −−−−→ y be a Cartesian diagram of schemes such that T ′ is also locally Noetherian. ′ [r] (1) If f is a naively Q-Gorenstein morphism, then so is f . Here, if ωY/T is [r] ∗ [r] invertible, then ω ′ ′ p (ω ). Y /T ≃ Y/T (2) In case q : T ′ T is a flat and surjective morphism, if f ′ is naively Q- → Gorenstein, then so is f. (3) If every fiber of f is Q-Gorenstein, then every fiber of f ′ is so. The converse holds if q is surjective. (4) If f is virtually Q-Gorenstein at a point y Y , then f ′ is so at any point ∈ of p−1(y). ′ [m] ∗ [m] (5) If f is Q-Gorenstein, then f is so and ω ′ ′ p (ω ) for any m Z. Y /T ≃ Y/T ∈ (6) In case q : T ′ T is a flat and surjective morphism, if f ′ is Q-Gorenstein, → then so is f.

Proof. Note that Y ′◦ = p−1(Y ◦) for Y ′◦ := Gor(Y ′/T ′) (cf. Corollary 5.7) and that

◦ ′ ′◦ ′ (VII-7) codim(Y Y , Y ) = codim(Y ′ Y , Y ) t \ t t \ for any t′ T ′ and t = q(t′) (cf. Lemma 2.31(1)). ∈ ′ (1): The base change f is an S2-morphism by Lemma 2.31(5), and we have an [r] ∗ [r] ′ isomorphism ω ′ ′ p (ω ) by Corollary 5.7(2). In particular, f is a naively Y /T ≃ Y/T Q-Gorenstein morphism. (2): The morphism f is an S2-morphism by Lemma 2.31(3) applied to = ′ F Y , since p: Y Y is surjective. Moreover, every fiber of f is Gorenstein in O → ∗ [m] codimension one by (VII-7). Now, p (ωY/T ) is reflexive for any m by Remark 2.20, ∗ [m] [m] ∗ [r] since p is flat. Hence, p (ω ) ω ′ ′ for any m by Corollary 5.7(2). If p (ω ) Y/T ≃ Y /T Y/T [r] is invertible, then so is ωY/T , since p is fully faithful (cf. Lemma A.7). Therefore, f is naively Q-Gorenstein. (3): This is obtained by applying (1) and (2) to the case where T = Spec k(t) and T ′ = Spec k(t′) and by Lemma 7.20. (4): We may assume that the conditions of Remark 7.14 are satisfied for U = Y , a certain reflexive -module , and for o = f(y). Then, the conditions (1) and OY L (2) of Remark 7.14 imply

depth ◦ 2 Yt\Y OYt ≥ ∗ [m] for any t f(Y ), by Lemma 2.15(2). Hence, p ( ) is a reﬂexive ′ -module and ∈ L OY (p∗ )[m] p∗( [m]) for any m, by Lemma 2.34 applied to Z = Y Y ◦ and to [m]. L ≃ L \ L 98 YONGNAMLEEANDNOBORUNAKAYAMA

Here, (p∗ )[m] satisfies relative S over T ′ by Remark 7.14(6) and Lemma 2.31(4). L 2 Furthermore, for any point t′ T ′ and t = q(t′), we have isomorphisms ∈ ∗ ′ ′ ′ ′ ′ p O ′ Y ( OY Yt ) k(t) k(t ) ωY /k(t) k(t) k(t ) ωY /k(t ), L⊗ Y O t′ ≃ L⊗ O ⊗ ≃ t ⊗ ≃ t′ by applying Lemma 5.4 to Spec k(t′) Spec k(t). Therefore, f ′ is virtually Q- → Gorenstein at any point of p−1(y), since p∗ plays the role of in Definition 7.13. ′ L L [m] (5): By (1), f is an S2-morphism whose fibers are all Q-Gorenstein. If ωY/T ∗ [m] ′ satisfies relative S2 over T , then p ωY/T does so over T by Lemma 2.31(4), and ∗ [m] [m] ′ p ω ω ′ ′ by Corollary 5.7(2). Therefore, f is Q-Gorenstein (cf. Defini- Y/T ≃ Y /T tion 7.1(2)). (6): By (2) above, f is naively Q-Gorenstein. By Lemma 7.3, it is enough to prove that the base change homomorphism [m] [m] [m] φ : ω O Y ω t Y/T ⊗ Y O t → Yt/k(t) is an isomorphism for any m Z and any point t T . For any point t′ q−1(t), ∈ ∈ ∈ the base change morphism [m] [m] [m] ′ ′ ′ ′ ′ ′ φt : ωY /T OY ′ Y ′ ωY /k(t ) ⊗ O t → t′ ′ [m] is an isomorphism, since f is Q-Gorenstein. Now, φt′ is isomorphic to the homo- [m] ∗ ′ ′ morphism pt′ (φt ) for the morphism pt : Yt′ Yt induced from p, since we have [m] ∗ [m] → an isomorphism ω ′ ′ p ω as in the proof of (2). Since p ′ is faithfully flat, Y /T ≃ Y/T t φ[m] is an isomorphism for any m Z and t T . Therefore, f is Q-Gorenstein. t ∈ ∈ We have the following properties for compositions of Q-Gorenstein morphisms and of their variants. Proposition 7.23. Let f : Y T and g : X Y be flat morphisms of locally → → Noetherian schemes. (1) If f and g are naively Q-Gorenstein, then f g is so, and ◦ ω[r] ω[r] g∗(ω[r] ) X/T ≃ X/Y ⊗OX Y/T [r] [r] for an integer r> 0 such that ωX/Y and ωY/T are invertible. (2) Assume that g is a Q-Gorenstein morphism. If f is virtually Q-Gorenstein at a point y, then f g virtually Q-Gorenstein at any point of g−1(y). ◦ (3) If f and g are Q-Gorenstein morphisms, then f g is so, and ◦ ω[m] ω[m] g∗(ω[m] ) X/T ≃ X/Y ⊗OX Y/T for any integer m. Proof. (1): Every fiber of the composite f g is Q-Gorenstein by Lemma 7.20 and ◦ by Proposition 7.22(1). In particular, f g is an S -morphism. For the relative ◦ 2 Gorenstein loci Y ◦ := Gor(Y/T ) and X◦ := Gor(X/Y ), let V be the intersection X◦ g−1(Y ◦). Then, V Gor(X/T ) and codim(X V,X ) 2 for any fiber ∩ ⊂ t \ t ≥ X = (f g)−1(t) of f g. We set t ◦ ◦ := ω[r] g∗(ω[r] ) Mr X/Y ⊗ Y/T GROTHENDIECK DUALITY AND Q-GORENSTEINMORPHISMS 99 for an integer r> 0 such that ω[r] and ω[r] are invertible. Then, ω⊗r X/Y Y/T Mr|V ≃ V/T and j (ω⊗r )= ω[r] Mr ≃ ∗ V/T Y/T for the open immersion j : V ֒ X, since f g is an S -morphism. Thus, f g is → ◦ 2 ◦ naively Q-Gorenstein. (2): We may assume that the conditions of Remark 7.14 are satisfied for U = Y , a certain reflexive -module , and for o = f(y). We set OY L := ω[m] g∗( [m]) Nm X/Y ⊗OX L for an integer m. This is flat over T , since [m] is so over T and ω[m] is so L X/Y over Y . Let go = g Xo : Xo Yo be the induced Q-Gorenstein morphism (cf. | → −1 Proposition 7.22(5)). Then, Xo = g (Yo) is Q-Gorenstein by Remark 7.14(7) and Lemma 7.20, and we have isomorphisms [m] ∗ [m] m O X (ω O X ) O g (ω ) N ⊗ X O o ≃ X/Y ⊗ X O o ⊗ Xo o Yo/k(o) [m] ∗ [m] [m] ω O g (ω ) ω , ≃ Xo/Yo ⊗ Xo o Yo/k(o) ≃ Xo/k(o) where the first isomorphism is derived from Remark 7.14(8) and the last one from (VII-6) in the proof of Lemma 7.20. In particular, satisfies relative S over T Nm 2 along X . Then, for := , we have isomorphisms o N N1 j ( ) j ω⊗m (g∗ )⊗m j ( ⊗m )= [m] Nm ≃ ∗ Nm|V ≃ ∗ V/Y ⊗OV L |V ≃ ∗ N |V N along Xo by Lemma 2.33(5), where j : V ֒ X is the open immersion in the proof of [m] → (1). Hence, satisfies relative S2 over T along Xo for any m, and OX Xo N −N⊗1 O ≃ ω k . Therefore, f g is virtually Q-Gorenstein at any point of g (y), since Xo/ (o) ◦ N plays the role of in Definition 7.13. L (3): We can apply the argument in the proof of (2) by setting = ω . Then, L Y/T [m] j ( ) j ω⊗m (g∗ω⊗m ) j (ω⊗m )= ω[m] N ≃ ∗ Nm|V ≃ ∗ V/Y ⊗OV Y/T |V ≃ ∗ V/T X/T along X . Hence, ω[m] satisfies relative S over T for any m. Consequently, f g o X/T 2 ◦ is Q-Gorenstein with an isomorphism ω[m] ω[m] g∗(ω[m] ) for any m Z. X/T ≃ X/Y ⊗OX Y/T ∈ Thus, we are done. Corollary 7.24. Let Y and T be locally Noetherian schemes and f : Y T a flat → morphism locally of finite type. Let g : X Y be a smooth separated surjective → morphism from a locally Noetherian scheme X. Then, f is Q-Gorenstein if and only if f g : X Y T is so. ◦ → → Proof. For the relative Gorenstein loci Y ◦ := Gor(Y/T ) and X◦ := Gor(X/T ), we have X◦ = g−1(Y ◦) by Lemma 6.7. Let g◦ : X◦ Y ◦ be the induced smooth → morphism. Then, ◦∗ (VII-8) ω ◦ ω ◦ ◦ ◦ g (ω ◦ ) X /T ≃ X /Y ⊗OX Y /T for the relative canonical sheaves ωY ◦/T , ωX◦/T , and ωX◦/Y ◦ (cf. (1) and (2) of Fact 4.34). For a point t T , let g : X Y be the smooth morphism induced on ∈ t t → t the fibers Y = f −1(t) and X = (f g)−1(t). t t ◦ 100 YONGNAMLEEANDNOBORUNAKAYAMA

By Proposition 7.23(2), it is enough to prove the “if” part. Assume that f g is ◦ Q-Gorenstein. Then, every fiber Yt is Q-Gorenstein by Lemma 6.7. In particular, Y satisfies S and codim(Y Y ◦, Y ) 2. Hence, by Lemma 2.33(4), t 2 t \ t ≥ [m] ⊗m ω j (ω ◦ ) Y/T ≃ ∗ Y /T for any m Z, where j : Y ◦ ֒ Y is the open immersion. For the open immersion ∈ → j : X◦ ֒ X, we have an isomorphism X → ∗ [m] ∗ ⊗m ◦∗ ⊗m g (ω ) g (j (ω ◦ )) j (g (ω ◦ )) Y/T ≃ ∗ Y /T ≃ X∗ Y /T by the flat base change isomorphism (cf. Lemma A.9). Thus,

[m] ⊗m ∗ ⊗m ◦∗ ⊗m ⊗m ∗ [m] ω j (ω ◦ ) j (j (ω ) ◦ g (ω ◦ ◦ )) ω g (ω ) X/T ≃ X∗ X /T ≃ X∗ X X/Y ⊗OX Y /T ≃ X/Y ⊗OY Y/T for any m Z by (VII-8). In particular, ω[m] is ﬂat over T , since g is faithfully ∈ Y/T ﬂat (cf. Lemma A.6). Moreover,

∗ [m] ⊗−m [m] ⊗−m [m] g (ω O Y ) ω O (ω O X ) ω O ω t Y/T ⊗ Y O t ≃ Xt/Yt ⊗ Xt X/T ⊗ X O t ≃ Xt/Yt ⊗ Xt Xt/k(t) satisfies S for any t T (cf. Lemma 6.4(2)). As a consequence, ω[m] 2 ∈ Y/T ⊗OY OYt [m] satisfies S2 by Fact 2.26(6). Therefore, ωY/T satisfies relative S2 over T for any m, and Y T is a Q-Gorenstein morphism. → Remark. Considering an étale morphism g in Corollary 7.24, we see that, for a given flat morphism f : Y T locally of finite type between locally Noetherian → schemes, the Q-Gorenstein condition at a point of Y is not only Zariski local but also étale local (cf. Remark 6.8).

7.4. Theorems on Q-Gorenstein morphisms. First of all, we shall prove some inﬁnitesimal criteria for a morphism to be Q-Gorenstein or naively Q-Gorenstein.

Theorem 7.25 (infinitesimal criterion). Let f : Y T be a flat morphism locally → of finite type between locally Noetherian schemes. For a point y Y and its image −1 ∈ t = f(y), assume that the fiber Yt = f (t) is Q-Gorenstein at y. Then, for a positive integer m, the following two conditions (i) and (ii) are equivalent to each other: [m] (i) The sheaf ωY/T satisfies relative S2 over T at y. (ii) Let A be a surjective local ring homomorphism to an Artinian local OT,t → ring A and let YA = Y T Spec A Spec A be the base change of f by the × → [m] associated morphism Spec A T . Then, ω satisfies relative S2 over → YA/A A at the point y Y lying over y. A ∈ A Moreover, the following two conditions (iii) and (iv) are also equivalent to each other: [m] (iii) The sheaf ωY/T is invertible at y. [m] (iv) For the same morphism YA Spec A in (ii), ω is invertible at yA. → YA/A GROTHENDIECK DUALITY AND Q-GORENSTEIN MORPHISMS 101

Proof. Since the fiber Yt satisfies S2 at y, by localizing Y , we may assume that f is an S2-morphism (cf. Fact 2.29(3)). Moreover, we may assume that every fiber of f is Gorenstein in codimension one by Lemma 2.38(2), since

codim (Y U, Y ) 2 y t \ t ≥ for the relative Gorenstein locus U = Gor(Y/T ). Let p : Y Y be the projec- A A → tion for a morphism Spec A T in (ii). Then, by Corollary 5.7(2), we have an → isomorphism (p∗ ω[m] )∨∨ ω[m] , A Y/T ≃ YA/A and moreover the base change homomorphism

p∗ ω[m] ω[m] A Y/T → YA/A is an isomorphism at y when (i) holds. In particular, we have (i) (ii), and the A ⇒ converse (ii) (i) is a consequence of Proposition 3.19 in the case (I) applied to ⇒ = ω[m] . If (i) or (ii) holds, then the base change homomorphisms F Y/T [m] [m] [m] [m] ω O Y ω and ω O Y ω Y/T ⊗ Y O t → Yt/k(t) YA/A ⊗ YA O t → Yt/k(t) are isomorphisms at y or yA, again by Corollary 5.7(2). Hence, by Fact 2.26(2), we have equivalences (iii) (i) + (v) and (iv) (ii) + (v) for the following condition: ⇔ ⇔ (v) The sheaf ω[m] is invertible at y. Yt/k(t) Thus, we have (iii) (iv). ⇔ By the equivalence (i) (ii) in Theorem 7.25 for all m Z and for any y Y , ⇔ ∈ ∈ t we have the following inﬁnitesimal criterion for a morphism to be Q-Gorenstein:

Corollary 7.26 (infinitesimal criterion). Let f : Y T be a flat morphism locally → of finite type between locally Noetherian schemes. Then, for a given point t T , −1 ∈ the morphism f is Q-Gorenstein along the fiber Yt = f (t) if and only if the base change f : Y = Y Spec A Spec A is Q-Gorenstein for any morphism A A ×T → Spec A T defined by a surjective local ring homomorphism A to any → OT,t → Artinian local ring A.

Remark. For an Artinian local ring A, a flat morphism Y Spec A of finite A → type is not necessarily a Q-Gorenstein morphism even if YA is Q-Gorenstein and A is Gorenstein. For example, let us consider a naively Q-Gorenstein morphism f : Y T = Spec R for a discrete valuation ring R such that f is not Q-Gorenstein → −1 along the closed fiber Yo = f (o), where o is the closed point of T corresponding to the maximal ideal mR. See Fact 7.7 or Lemma 7.8 and Example 7.9 for such an example of f. Let Spec A T be a closed immersion for a local Artinian ring → A. Then, A is Gorenstein, and the base change f : Y = Y Spec A Spec A A A ×T → of f is a naively Q-Gorenstein morphism by Proposition 7.22(1). Hence, YA is Q- Gorenstein by Lemma 7.20. However, fA is not a Q-Gorenstein morphism for some A by Corollary 7.26. 102 YONGNAMLEEANDNOBORUNAKAYAMA

By the equivalence (iii) (iv) in Theorem 7.25 for any y Y , we have the ⇔ ∈ t following version of an inﬁnitesimal criterion for naively Q-Gorenstein morphisms with bounded relative Gorenstein index:

Corollary 7.27 (infinitesimal criterion). Let f : Y T be a flat morphism locally → of finite type between locally Noetherian schemes. For a point t T and a positive ∈ integer m, the following two conditions are equivalent to each other:

(i) The morphism f is naively Q-Gorenstein along Yt and the relative Goren- stein index of f along Yt is a divisor of m. [m] (ii) The sheaf ω is invertible for the base change fA : YA = Y T Spec A YA/A × → Spec A by any morphism Spec A T defined by a surjective local ring → homomorphism A to any Artinian local ring A. OT,t → Remark 7.28. The infinitesimal criterion does not hold for naively Q-Gorenstein morphisms without boundedness conditions for the relative Gorenstein index: Let f : Y T be a flat morphism of finite type of Noetherian schemes such that → n+1 T = Spec R for a discrete valuation ring R. For an integer n, we set Rn := R/mR , T = Spec R , and let Y = Y T T be the base change of f by the closed n n n ×T n → n immersion T T . Assume that the special fiber Y is a Q-Gorenstein scheme and n ⊂ 0 the residue field k = R/m has characteristic p> 0. Then, Y T is naively Q- R n → n Gorenstein for any n 0 by an argument of Kollár in [16, 14.7] or [31, Exam. 7.6]. ≥ But, there is an example of f : Y T above such that f is not naively Q-Gorenstein → (cf. Example 7.4). Let r be the relative Gorenstein index of Y T ; this equals n n → n the Gorenstein index of Y . Then, r is not bounded by Corollary 7.27 if f n { n}n≥0 is not naively Q-Gorenstein.

By a similar argument in the proof of Theorem 7.25 and by Proposition 3.19 in the case (II) instead of (I), we have the following valuative criterion for a morphism to be Q-Gorenstein.

Theorem 7.29 (valuative criterion). Let f : Y T be a flat morphism locally of → finite type between locally Noetherian schemes. Assume that T is reduced. Then, f is a Q-Gorenstein morphism if and only if the base change f : Y = Y Spec R R R ×T → Spec R is a Q-Gorenstein morphism for any discrete valuation ring R and for any morphism Spec R T . → Proof. It is enough to check the ‘if’ part by Proposition 7.22(5). Then, every fiber Yt is Q-Gorenstein, since we can consider R as the localization at the prime ideal (x) of the polynomial ring k(t)[x] for the residue field k(t) and consider the morphism Spec R T defined by the composite k(t) R. Therefore, it → OT,t → ⊂ is enough to prove that ω[m] satisfies relative S over T for any m Z. For the Y/T 2 ∈ base change morphism f : Y Spec R and the projection p: Y Y , we have R R → R → an isomorphism (p∗ω[m] )∨∨ ω[m] Y/T ≃ YR/R for any m by Corollary 5.7(2). Therefore, the assertion is a consequence of Propo- sition 3.19 in the case (II). GROTHENDIECK DUALITY AND Q-GORENSTEIN MORPHISMS 103

The following theorem gives a criterion for a morphism to be Q-Gorenstein only by conditions on fibers. Theorem 7.30. Let Y and T be locally Noetherian schemes and f : Y T be → a flat morphism locally of finite type. For a point t T , if the following three ∈ −1 conditions are all satisfied, then f is Q-Gorenstein along the fiber Yt = f (t):

(i) Yt is Q-Gorenstein; (ii) Yt is Gorenstein in codimension two; [m] (iii) ω satisfies S3 for any m Z. Yt/k(t) ∈ Proof. As in the first part of the proof of Theorem 7.25, by (i) and (ii), we may assume that Y T an S -morphism and its fibers are all Gorenstein in codimension → 2 one. Then, it is enough to prove that = ω[m] satisfies relative S over T along F Y/T 2 Y for any m. Since is reflexive, we can apply Proposition 3.7 and its corollaries t F to the morphism Y T and the closed subset Z = Y Gor(Y/T ). Then, (ii) and → \ (iii) imply the inequality (III-4) of Corollary 3.10. Thus, satisfies relative S over F 2 T along Yt by Corollaries 3.9 and 3.10. Definition 7.31 (Q-Gorenstein refinement). Let f : Y T be an S -morphism → 2 of locally Noetherian schemes such that every fiber is Q-Gorenstein. A morphism S T from a locally Noetherian scheme S is called a Q-Gorenstein refinement of → f if the following two conditions are satisfied: (i) S T is a monomorphism in the category of schemes; → (ii) for any morphism T ′ T from a locally Noetherian scheme T ′, the base → change Y ′ T ′ T ′ is a Q-Gorenstein morphism if and only if T ′ T ×T → → factors through S T . → Remark. If the Q-Gorenstein refinement S T exists, then it is bijective, since → every fiber is Q-Gorenstein. Theorem 7.32. Let f : Y T be an S -morphism of locally Noetherian schemes → 2 whose fibers are all Q-Gorenstein. Assume that (i) Y Σ T is a Q-Gorenstein morphism for a closed subset Σ proper over \ → T , and −1 (ii) the Gorenstein indices of all the fibers Yt = f (t) are bounded above. Then, f admits a Q-Gorenstein refinement as a separated morphism S T lo- → cally of finite type. Furthermore, S T is a local immersion of finite type if the → assumption (i) is replaced with (iii) f is a projective morphism locally on T . Proof. By (ii), we have a positive integer m such that ω[m] is invertible for any Yt/k(t) t T . Let S T be the relative S2 refinement for the reflexive Y -module ∈ → m O = ω[i] . F i=1 Y/T It exists as a separated morphism S MT locally of finite type by Theorem 3.26, → since is locally free for the relative Gorenstein locus U = Gor(Y/T ) in which F|U codim(Y U, Y ) 2 for any t T , and since satisfies relative S over T t \ t ≥ ∈ F|Y \Σ 2 104 YONGNAMLEEANDNOBORUNAKAYAMA by (i). Moreover, S T is a local immersion of finite type in case (iii), also by → Theorem 3.26. It is enough to show that S T is the Q-Gorenstein refinement. → Let T ′ T be a morphism from a locally Noetherian scheme T ′. Then, T ′ T → [i] ′ → factors through S T if and only if ω ′ ′ satisfies relative S over T for the base → Y /T 2 change Y ′ = Y T ′ T ′ for any 0 i m, since we have an isomorphism ×T → ≤ ≤ [i] ∗ [i] ∨∨ ω ′ ′ (p ω ) Y /T ≃ Y/T by Corollary 5.7(2) for the projection p: Y ′ Y . This condition is also equivalent → to that Y ′ T ′ is Q-Gorenstein. Therefore, S T is the relative Q-Gorenstein → → refinement. Example 7.33. Let f : Y T be the morphism in Example 7.4. Then, the Q- → Gorenstein refinement S T of f is just the disjoint union of T 0 and the → \{ } closed point 0 . This is shown as follows. Since f is smooth over T 0 and since { } \{ } K2 = 9 and K2 = 8 for t = 0, we see that S (T 0 ) Spec A over T for a Y0 Yt 6 ≃ \{ } ⊔ closed immersion Spec A T defined by a surjection k[t] k[t]/(tn) = A. On → → the other hand, if n 2, the base change f : Y Spec A is not a Q-Gorenstein ≥ A A → morphism by Lemma 7.5. Therefore, A = k. The following theorem is a local version of Theorem 7.32: Theorem 7.34. Let f : Y T be a flat morphism locally of finite type from a → locally Noetherian scheme Y such that T = Spec R for a Noetherian Henselian local ring R. For the closed point o T and for a point y of the closed fiber Y = f −1(o), ∈ o assume that Y is Q-Gorenstein at y. Then, there is a closed subscheme S T o ⊂ having the following universal property: Let T ′ = Spec R′ T be a morphism → defined by a local ring homomorphism R R′ to a Noetherian local ring R′. Then, → T ′ T factors through S if and only if the base change Y ′ = Y T ′ T ′ is a → ×T → Q-Gorenstein morphism at any point y′ Y ′ lying over y and the closed point of ∈ T ′. Proof. As in the first part of the proof of Theorem 7.25, we may assume that f is an S2-morphism and every fiber is Gorenstein in codimension one. For the Gorenstein index m of Yo at y, we consider the reflexive sheaf m = ω[i] F i=1 Y/T on Y . This is locally free in codimensionM one on each fiber. Let S T be the ⊂ universal subscheme in Theorem 3.28 for . We shall show that S satisfies the F required condition. Let T ′ = Spec R T be the morphism above. Then, we have → an isomorphism ∗ [i] ∨∨ [i] (p ω ) ω ′ ′ Y/T ≃ Y /T for any i Z by Corollary 5.7(2). Thus, T ′ T factors through S if and only if [i] ∈ ′ → ′ ωY ′/T ′ satisfies relative S2 over T at any point y lying over y and the closed point ′ ′ [m] ′ o of T , for any 1 i m. In this case, ω ′ ′ is invertible at y by Fact 2.26(2), ≤ ≤ Y /T since ω[m] is so at y and since the canonical morphism Yo/k(o) [m] [m] [m] ′ ′ ′ ′ ′ ′ ωY /T OY ′ Y ′ ωY /k(o ) ωY /k(o) OYo Y ′ ⊗ O o → o′ ≃ o ⊗ O o GROTHENDIECK DUALITY AND Q-GORENSTEIN MORPHISMS 105 is an isomorphism at y′ (cf. Corollary 5.7(2)). Therefore, the latter condition at ′ [i] ′ y is equivalent to that ω ′ ′ satisfies relative S over T at y for any i Z; Y /T 2 ∈ This means that Y ′ T ′ is a Q-Gorenstein morphism at y′. Thus, S satisfies the → required condition.

The following theorem is similar to Theorem 7.32, and it links a projective S2- morphism Gorenstein in codimension one in each ﬁber, to a naively Q-Gorenstein morphism by a speciﬁc base change.

Theorem 7.35. Let f : Y T be a projective S -morphism of locally Noetherian → 2 schemes such that every ﬁber is Gorenstein in codimension one. Then, for any positive integer r > 0, there exists a separated monomorphism S T from a r → locally Noetherian scheme Sr satisfying the following conditions: (i) The morphism S T is a local immersion of ﬁnite type. r → (ii) Let T ′ T be a morphism from a locally Noetherian scheme T ′. Then, → it factors through S T if and only if Y T ′ T ′ is a naively Q- r → ×T → Gorenstein morphism whose relative Gorenstein index is a divisor of r.

Proof. By Theorem 3.26, there is a relative S refinement S T for the reflexive 2 → -module = ω[r] . In fact, is locally free for the relative Gorenstein locus OY F Y/T F|U U = Gor(Y/T ) and codim(Y U, Y ) 2 for any t T by assumption. Here, t \ t ≥ ∈ S T is a separated monomorphism and a local immersion of finite type, since f → is a projective morphism. By the universal property of relative S2 refinement and by Corollary 5.7(2), we see that, for a morphism T ′ T from a locally Noetherian ′ [r→] ′ scheme T , it factors through S T if and only if ω ′ ′ satisfies relative S over T → Y /T 2 ′ ′ ′ [r] for the base change morphism Y = Y T T . Note that ω ′ ′ is invertible if ×T → Y /T and only if Y ′ T ′ is a naively Q-Gorenstein morphism whose relative Gorenstein → index is a divisor of r. Let Br be the set of points P YS := Y T S such that [r] ∈ × ω is not invertible at P . Then, Br is a closed subset of YS. Let Sr S be the YS/S ⊂ complement of the image of Br in S. Then, Sr is an open subset. For the morphism ′ [r] ′ T T above, if ω ′ ′ is invertible, then T T factors through S T , and for → Y /T → → the induced morphism h: Y ′ Y lying over T ′ S, we have an isomorphism → S → [r] ∗ [r] ω ′ ′ h (ω ) Y /T ≃ YS /S by Corollary 5.7(2). This implies that h(Y ′) B = and that the image of T ′ S ∩ r ∅ → is contained in the open subset S . Therefore, the composite S S T is the r r ⊂ → required morphism.

Remark. When f : Y T is a projective morphism, similar results to Theo- → rems 7.32 and 7.35 are found in [29, Cor. 24, 25].

Appendix A. Some basic properties in scheme theory For readers’ convenience, we collect here famous results on the local criterion of ﬂatness and the base change isomorphisms. 106 YONGNAMLEEANDNOBORUNAKAYAMA

A.1. Local criterion of flatness. Here, we summarize results related to the “local criterion of flatness.” It is usually considered as Proposition A.1 below. But, the subsequent Corollaries A.2, A.3, A.4 are also useful in the scheme theory. For the detail, the reader is referred to [13, IV, 5], [6, III, 5], [12, 0 , 10.2], [2, V, 3], § § III § § [36, 22], etc. We also mention a “local criterion of freeness” as Lemma A.5, and § explain two more results on flatness and local freeness for sheaves on schemes.

Proposition A.1 (local criterion of flatness). For a ring A, an ideal I of A, and for an A-module M, assume that (1) I is nilpotent, or (2) A is Noetherian and M is I-adically ideally separated, i.e., a M is ⊗A separated for the I-adic topology for all ideals a of A. Then, the following four conditions are equivalent to each other: (i) M is flat over A; A (ii) M/IM is flat over A/I and Tor1 (M, A/I)=0; (iii) M/IM is flat over A/I and the canonical homomorphism

M/IM Ik/Ik+1 IkM/Ik+1M ⊗A/I → is an isomorphism for any k 0; ≥ (iv) M/IkM is flat over A/Ik for any k 1. ≥ Remark. The proof is found in [13, IV, Cor. 5.5, Th. 5.6], [6, III, 5.2, Th. 1], [12, § 0III, (10.2.1)], [2, V, Th. (3.2)], [36, Th. 22.3]. The condition (2) is satisfied, for example, when there is a ring homomorphism A B of Noetherian rings such → that M is originally a finitely generated B-module and that IB is contained in the Jacobson radical rad(B) of B (cf. [6, III, 5.4, Prop. 2], [12, 0 , (10.2.2)], [36, § III p. 174]).

Corollary A.2. Let A B be a local ring homomorphism of Noetherian local → rings and let u: M N be a homomorphism of B-modules such that M and N → are finitely generated B-modules and that N is flat over A. Then, the following two conditions are equivalent to each other: (i) u is injective and the cokernel of u is flat over A; (ii) u k: M k N k is injective for the residue field k of A. ⊗A ⊗A → ⊗A

The proof is given in [13, IV, Cor. 5.7], [12, 0III, (10.2.4)], [2, VII, Lem. (4.1)], [36, Th. 22.5].

Corollary A.3 (cf. [12, 0 , Prop. (15.1.16)], [36, Cor. to Th. 22.5]). Let A B IV → be a local ring homomorphism of Noetherian local rings and let M be a finitely generated B-module. Let k be the residue field of A and let x¯ denote the image of x B in B k. For elements x , ..., x in the maximal ideal m , the following ∈ ⊗A 1 n B two conditions are equivalent to each other: n (i) (x1,...,xn) is an M-regular sequence and M/ i=1 xiM is flat over A; (ii) (¯x ,..., x¯ ) is an M k-regular sequence and M is flat over A. 1 n ⊗A P GROTHENDIECK DUALITY AND Q-GORENSTEIN MORPHISMS 107

Corollary A.4. Let A B and B C be local ring homomorphisms of Noether- → → ian local rings and let k be the residue field of A. Assume that B is flat over A. Then, for a finitely generated C-module M, the following conditions are equivalent to each other: (i) M is flat over B; (ii) M is flat over A and M k is flat over B k. ⊗A ⊗A The proof is given in [13, IV, Cor. 5.9], [6, III, 5.4, Prop. 3], [12, 0 , (10.2.5)], § III [2, V, Prop. (3.4)]. Next, we shall give the “local criterion of freeness” as Lemma A.5 below, which is similar to Proposition A.1. This result is well known (cf. [13, IV, Prop. 4.1], [6, II, 3.2, Prop. 5], [12, 0 , (10.1.2)]), but is not usually called the “local criterion § III of freeness” in articles.

Lemma A.5 (local criterion of freeness). Let A be a ring, I an ideal of A, and M an A-module such that I is nilpotent or • A is Noetherian, I rad(A), and M is a finitely generated A-module. • ⊂ Then, the following conditions are equivalent to each other: (i) M is a free A-module; A (ii) M/IM is a free A/I-module and Tor1 (M, A/I) = 0; (iii) M/IM is a free A/I-module and the canonical homomorphism M/IM Ik/Ik+1 IkM/Ik+1M ⊗A/I → is an isomorphism for any k 0. ≥ Remark. Applying Lemma A.5 to the case where A is a Noetherian local ring and I is the maximal ideal, we have the equivalence of flatness and freeness for finitely generated A-modules (cf. [13, IV, Cor. 4.3], [12, 0III, (10.1.3)]). On the other hand, the equivalence of flatness and freeness can be proved by other methods (cf. [36, Th. 7.10], [2, Lem. 5.8]), and using the equivalence, we obtain Lemma A.5 for the same local ring (A, I) and for a finitely generated A-module M, as a corollary of Proposition A.1.

Remark. The equivalence explained above implies the following well-known fact: For a locally Noetherian scheme X, a coherent flat -module is nothing but a OX locally free -module of finite rank. OX The following is proved immediately from the definitions of flatness and faithful flatness (cf. [6, I, 3, no. 2, Prop. 4]): § Lemma A.6. Let f : X Y and g : Y Z be morphisms of schemes such that f → → is faithfully flat, i.e., flat and surjective. Then, for an -module , it is flat over OY G Z if and only if f ∗ is flat over Z. G As a corollary in the case where Y = Z, we have the following descent property of locally freeness by the relation with flat coherent sheaves. 108 YONGNAMLEEANDNOBORUNAKAYAMA

Lemma A.7. Let f : X Y be a ﬂat surjective morphism of locally Noetherian → schemes. For a coherent -module , it is locally free if and only if f ∗ is so. OY G G The authors could not ﬁnd a good reference for Lemma A.7. For example, we have a weaker result as a part of [13, VIII, Prop. 1.10], where f is assumed additionally to be quasi-compact; However, the quasi-compactness is related to the other part. A.2. Base change isomorphisms. Let us consider a Cartesian diagram

g′ X′ X −−−−→ f ′ f

′ g  S S of schemes, i.e., X′ X S′. Then,y −−−−→ for anyy quasi-coherent -module , one ≃ ×S OX F has a functorial canonical homomorphism θ( ): g∗(f ) f ′ (g′∗ ) F ∗F → ∗ F of ′ -modules, and more generally, a functorial canonical homomorphism OS θi( ): g∗(Rif ) Rif ′ (g′∗ ) F ∗F → ∗ F for each i 0. We have the following assertions on θ( ) and θi( ). ≥ F F Lemma A.8 (aﬃne base change). If f is an aﬃne morphism, then θ( ) is an F isomorphism.

Lemma A.9 (flat base change). Assume that g is flat and that f is quasi-compact and quasi-separated. Then, θi( ) is an isomorphism for any i. F A proof of Lemma A.8 is given [12, II, Cor. (1.5.2)], and a proof of Lemma A.9 is given in [12, III, Prop. (1.4.15)] (cf. [12, IV, (1.7.21)]). Here, the morphism f : X S is said to be “quasi-separated” if the diagonal morphism X X X → → ×S is quasi-compact (cf. [12, IV, Déf. (1.2.1)]). We have also the following generalization of Lemma A.9 to the case of complexes by [17, II, Prop. 5.12], [23, IV, Prop. 3.1.0], and [34, Prop. 3.9.5].

• Proposition A.10. In the situation of Lemma A.9, let be a complex of X - + F O modules in Dqcoh(X). Then, there is a functorial quasi-isomorphism Lg∗(Rf ( •)) Rf ′ (Lg′∗( •)). ∗ F → ∗ F References

[1] D. Abramovich and B. Hassett, Stable varieties with a twist, Classification of algebraic varieties (eds. C. Faber, et al.), pp. 1–38, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011. [2] A. Altman and K. Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Math. 146, Springer-Verlag, 1970. [3] K. Altmann and J. Kollár, The dualizing sheaf on first order deformations of toric surface singularities, preprint, arXiv:1601.17805v2, 2016. [4] Y. Aoyama, On the depth and the projective dimension of the canonical module, Japan. J. Math. 6 (1980), 61–66. GROTHENDIECK DUALITY AND Q-GORENSTEIN MORPHISMS 109

[5] Y. Aoyama, Some basic results on canonical modules, J. Math. Kyoto Univ. 23 (1983), 85–94. [6] N. Bourbaki, Elements of Mathematics, Commutative Algebra, Chapters 1-7, Springer 1989; original publication: Eléments´ de mathématique, Algèbre Commutative, Ch. 1, 2 (1961), Ch. 3, 4 (1961), Ch. 5, 6 (1964), Ch. 7 (1965), Hermann. [7] B. Conrad, Grothendieck Duality and Base Change, Lecture Notes in Math. 1750, Springer- Verlag, 2000. [8] B. Conrad, Deligne’s notes on Nagata compactifications, J. Ramanujan Math. Soc. 22 (2007), 205–257. [9] P. Deligne, Cohomology àsupport propre et construction du foncteur f !, Appendix to Residues and Duality, pp. 404–421, Lecture Notes in Math. 20, Springer-Verlag, 1966, [10] P. Deligne, Cohomologie àsupports propres (avec un appendice de B. Saint-Donat), Théorie des Toposes et Cohomologie Etales´ des Schémas, Tome 3 (SGA4 III), pp. 250–480 (Exposé XVII), Lecture Notes in Math. 305, Springer-Verlag, 1973. [11] P. Deligne, Le théorème de plongement de Nagata, Kyoto J. Math. 50 (2010), 661–670. [12] A. Grothendieck, Eléments´ de géométrie algébrique (rédigés avec la collaboration de J. Dieudonné), Publ. Math. I.H.E.S.,´ 4 (1960), 8 (1961), 11 (1961), 17 (1963), 20 (1964), 24 (1965), 28 (1966), 32 (1967). [13] A. Grothendieck, et al., Revêtements Etales´ et Groupe Fondamental (SGA1), Lecture Notes in Math. 224, Springer-Verlag, 1971; A new updated edition: Documents Math. 3, Soc. Math. France, 2003. [14] A. Grothendieck, et al., Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA2), Adv. Stud. Pure Math. Vol. 2, North-Holland and Masson & Cie, 1968; A new updated edition: Documents in Math. 4, Soc. Math. France, 2005. [15] P. Hacking, Compact moduli of plane curves, Duke Math. J. 124 (2004), 213–257. [16] C. D. Hacon and S. J. Kovács, Classification of higher dimensional algebraic varieties, Oberwolfach Seminars, 41, Birkhäuser, 2010. [17] R. Hartshorne, Residues and Duality, Lecture Notes in Math. 20, Springer-Verlag, 1966. [18] R. Hartshorne, Local Cohomology, Lecture Notes in Math. 41, Springer-Verlag, 1967. [19] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer-Verlag, 1977. [20] R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), 121–176. [21] B. Hassett and S. J. Kovács, Reflexive pull-backs and base extension, J. Alg. Geom. 13 (2004), 233–247. [22] M. Hochster and C. Huneke, Indecomposable canonical modules and connectedness, Com- mutative Algebra: Syzygies, Multiplicities, and Birational Algebra (eds. C. Huneke et. al.), pp. 197–208, Comtemp. Math. 159, 1994. [23] L. Illusie, Exposés I–IV, Théorie des Intersections et Théorème de Riemann-Roch (SGA6), pp. 78–296, Lecture Notes in Math. 225, Springer-Verlag, 1971. [24] Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985 (T. Oda ed.), pp. 283–360, Adv. Stud. Pure Math., 10, Kinokuniya and North-Holland, 1987. [25] T. Kawasaki, On arithmetic Macaulayfication of Noetherian rings, Trans. Amer. Math. Soc. 354 (2002), 123–149. [26] S. L. Kleiman, Relative duality for quasi-coherent sheaves, Compo. Math. 41 (1980), 39–60. [27] J. Kollár, Projectivity of complete moduli, J. Diff. Geom. 32 (1990), 235–268. [28] J. Kollár, Flatness criteria, J. of Algebra 175 (1995), 715–727. [29] J. Kollár, Hulls and husks, preprint, arXiv:0805.0576v4, 2008. [30] J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), 299–338. [31] S. J. Kovács, Young person’s guide to moduli of higher dimensional varieties, Algebraic Geometry, Seattle 2005 (eds. D. Abramovich, et. al.), pp. 711–743, Proc. Sympos. Pure in Math., 80, Part 2, Amer. Math. Soc., 2009. 110 YONGNAMLEEANDNOBORUNAKAYAMA

[32] Y. Lee and N. Nakayama, Simply connected surfaces of general type in positive characteristic via deformation theory, Proc. London Math. Soc. (3) 106 (2013), 225–286. 2 [33] Y. Lee and J. Park, A simply connected surface of general type with pg = 0 and K = 2, Invent. Math. 170 (2007), 483–505. [34] J. Lipman, Notes on derived functors and Grothendieck duality, Part I of Foundations of Grothendieck duality for diagrams of schemes, pp. 1–250, Lecture Notes in Math. 1960, Springer, 2009. [35] W. Lütkebohmert, On compactifications of schemes, Manuscripta Math. 80 (1993), 95–111. [36] H. Matsumura, Commutative Ring Theory, translated from the Japanese by M. Reid, Cam- bridge Stud. in Adv. Math., 8, Cambridge Univ. Press, 1989. [37] S. Mori, On affine cones associated with polarized varieties, Japan. J. Math. 1 (1975), 301– 309. [38] D. Mumford, Lectures on curves on surfaces, Ann. Math. Studies, 59, Princeton Univ. Press, 1966. [39] D. Mumford, Abelian Varieties, Tata Studies in Math., Oxford Univ. Press, 1970. [40] J. P. Murre, Representation of unramified functors. Applications, Séminaire N. Bourbaki, 1964–1966, exp. no 294, pp. 243–261. [41] M. Nagata, Imbedding of an abstract variety in a complete variety, J. Math. Kyoto Univ. 2 (1962), 1–10. [42] M. Nagata, A generalization of the imbedding problem of an abstract variety in a complete variety, J. Math. Kyoto Univ. 3 (1963), 89–102. [43] A. Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc., 9 (1996), 205–236. [44] Z. Patakfalvi, Base change behavior of the relative canonical sheaf related to higher dimensional moduli, Algebra and number theory 7 (2013), 353–378. [45] H. C. Pinkham, Deformations of algebraic varieties with Gm action, Astérisque 20, Soc. Math. France, 1974. [46] M. Raynaud, Flat modules in algebraic geometry, Compos. Math. 24 (1972), 11–31. [47] M. Raynaud and L. Gruson, Critères de platitude et de projectivité, Invent. Math. 13 (1971), 1–89. [48] M. Reid, Canonical 3-folds, Journées de Géometrie Algébrique d’Angers (ed. A. Beauville), pp. 273–310, Sijthoff and Noordhoff, 1980. [49] M. Reid, Minimal models of canonical 3-folds, Algebraic Varieties and Analytic Varieties (ed. S. Iitaka), pp. 131–180, Adv. Stud. in Pure Math. 1, Kinokuniya and North-Holland, 1983. [50] P. Sastry, Base change and Grothendieck duality for Cohen–Macaulay maps, Compo. Math. 140 (2004), 729–777. [51] R. Y. Sharp, A commutative Noetherian ring which possesses a dualizing complex is accept- able, Math. Proc. Camb. Phil. Soc. 82 (1977), 197–213. [52] R. Y. Sharp, Necessary conditions for the existence of dualizing complexes in commutative algebra, Séminaire d’Algèbre Paul Dubreil, Proceedings, Paris 1977–78 (31ème Année), pp. 213–219, Lect. Notes in Math., 740, Springer, 1979. [53] N. Spaltenstein, Resolutions of unbounded complexes, Composit. Math. 65 (1988), 121–154. [54] N. Tziolas, Q-Gorenstein deformations of non-normal surfaces, Amer. J. Math., 131 (2009), 171–193. [55] J.-L. Verdier, Base change for twisted inverse image of coherent sheaves, Algebraic Geometry, papers presented at the Bombay Colloquium (1968), pp. 393–408, Tata Inst. Fund. Res., Oxford Univ. Press, 1969.

Department of Mathematical Sciences Korea Advanced Institute of Science and Technology GROTHENDIECK DUALITY AND Q-GORENSTEIN MORPHISMS 111

291, Daehak-ro, Yuseong-gu, Daejeon 34141 South Korea E-mail address: [email protected]

Research Institute for Mathematical Sciences Kyoto University, Kyoto 606-8502 Japan E-mail address: [email protected]