Duality and Flat Base Change on Formal Schemes

Home , Derived category, Formal scheme, Noetherian scheme, Residue field

Leovigildo Alonso Tarr´ıo, Ana Jerem´ıas L´opez, and Joseph Lipman

Abstract. We give several related versions of global Grothendieck Duality for unbounded complexes on noetherian formal schemes. The proofs, based on a non-trivial adaptation of Deligne’s method for the special case of ordinary schemes, are reasonably self-contained, modulo the Special Adjoint Functor Theorem. An alternative approach, inspired by Neeman and based on recent results about “Brown Representability,” is indicated as well. A section on applications and examples illustrates how our results synthesize a number of different duality-related topics (local duality, formal duality, residue theorems, dualizing complexes,. . .). A flat-base-change theorem for pseudo-proper maps leads in particular to sheafified versions of duality for bounded-below complexes with quasi-coherent homology. Thanks to Greenlees-May duality, the results take a specially nice form for proper maps and bounded-below complexes with coherent homology.

Contents 1. Preliminaries and main theorems. 4 2. Applications and examples. 10 3. Direct limits of coherent sheaves on formal schemes. 31 4. Global Grothendieck Duality. 43 5. Torsion sheaves. 47 6. Duality for torsion sheaves. 59 7. Flat base change. 71 8. Consequences of the ﬂat base change isomorphism. 86 References 90

First two authors partially supported by Xunta de Galicia research project XUGA20701A96 and Spain’s DGES grant PB97-0530. They also thank the Mathematics Department of Purdue University for its hospitality, help and support. . Third author partially supported by the National Security Agency.

c 1999 American Mathematical Society 3 4 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

1. Preliminaries and main theorems. First we need some notation and terminology. Let X be a ringed space, i.e., a topological space together with a sheaf of commutative rings . Let (X) be the OX A category of X -modules, and qc(X) (resp. c(X), resp. ~c(X)) the full subcategory of (XO) whose objects areA the quasi-coherenA t (resp.Acoherent, resp. lim’s of A 1 −−→ coherent) X -modules. Let K(X) be the homotopy category of (X)-complexes, and let DO(X) be the corresponding derived category, obtainedAfrom K(X) by adjoining an inverse for every quasi-isomorphism (= homotopy class of maps of complexes inducing homology isomorphisms).

For any full subcategory ...(X) of (X), denote by D...(X) the full subcategory of D(X) whose objectsAare those complexesA whose homology sheaves all lie + − in ...(X), and by D...(X) (resp. D...(X)) the full subcategory of D...(X) whose A m objects are those complexes D...(X) such that the homology H ( ) vanishes for all m 0 (resp. m 0).F ∈ F The full subcategory (X) of (X) is plump if it contains 0 and for every ex- A... A act sequence 1 2 3 4 in (X) with 1, 2, 3 and 4 in (X), Mis →in M (X→)Mtoo.→IfM →(XM) is plumpA then itMis abMelian,Mand hasMa A... M A... A... derived category D( ...(X)). For example, c(X) is plump [GD, p. 113, (5.3.5)]. If X is a locally noetherianA formal scheme,2 thenA (X) (X) (Corollary 3.1.5)— A~c ⊂ Aqc with equality when X is an ordinary scheme, i.e., when X has discrete topology [GD, p. 319, (6.9.9)]—and both of these are plump subOcategories of (X), see Proposition 3.2.2. A

Let K1, K2 be triangulated categories with respective translation functors T1 , T2 [H1, p. 20]. A (covariant) ∆-functor is a pair (F, Θ) consisting of an additive ∼ functor F : K1 K2 together with an isomorphism of functors Θ : F T1 T2F → u v w −→ such that for every triangle A B C T A in K , the diagram −→ −→ −→ 1 1 ◦ F A F u F B F v F C Θ F w T F A −−−→ −−−→ −−−−→ 2 is a triangle in K2. Explicit reference to Θ is often suppressed—but one should keep it in mind. (For example, if ...(X) (X) is plump, then each of D...(X) A ⊂ A and D...(X) carries a unique triangulation for which the translation is the restriction of that on D(X) and such that inclusion into D(X) together with Θ:=identity is a ∆-functor; in other words, they are all triangulated subcategories of D(X). See e.g., Proposition 3.2.4 for the usefulness of this remark.) Compositions of ∆-functors, and morphisms between ∆-functors, are defined in the natural way.3 A ∆-functor (G, Ψ): K2 K1 is a right ∆-adjoint of (F, Θ) if G is a right adjoint of F and the resulting→functorial map F G 1 (or equivalently, 1 GF ) is a morphism of ∆-functors. → → We use R to denote right-derived functors, constructed e.g., via K-injective resolutions (which exist for all (X)-complexes [Sp, p. 138, Thm. 4.5]).4 For a A 1 “lim−→” always denotes a direct limit over a small ordered index set in which any two elements have an upper bound. More general direct limits will be referred to as colimits. 2Basic properties of formal schemes can be found in [GD, Chap. 1, §10]. 3See also [De, §0, §1] for the multivariate case, where signs come into play—and ∆-functors are called “exact functors.” 4A complex F in an abelian category A is K-injective if for each exact A-complex G the • abelian-group complex HomA(G, F ) is again exact. In particular, any bounded-below complex of injectives is K-injective. If every A-complex E admits a K-injective resolution E → I(E) (i.e., DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 5 map f : X Y of ringed spaces (i.e., a continuous map f : X Y together with → ∗ → ∗ a ring-homomorphism Y f∗ X ), Lf denotes the left-derived functor of f , constructed via K-flat Oresolutions→ O [Sp, p. 147, 6.7]. Each derived functor in this paper comes equipped, implicitly, with a Θ making it into a ∆-functor (modulo obvious modifications for contravariance), cf. [L4, Example (2.2.4)].5 Conscientious readers may verify that such morphisms between derived functors as occur in this paper are in fact morphisms of ∆-functors.

1.1. Our ﬁrst main result, global Grothendieck Duality for a map f : X Y → of quasi-compact formal schemes with X noetherian, is that, D( ~c(X)) being the derived category of (X) and j : D( (X)) D(X) being the naturalA functor, the A~c A~c → ∆-functor Rf∗◦ j has a right ∆-adjoint. A more elaborate—but readily shown equivalent—statement is:

Theorem 1. Let f : X Y be a map of quasi-compact formal schemes, with → X noetherian, and let j : D( ~c(X)) D(X) be the natural functor. Then there × A → exists a ∆-functor f : D(Y) D ( ~c(X)) together with a morphism of ∆-functors × → A τ : Rf∗ jf 1 such that for all D( ~c(X)) and D(Y), the composed map (in the derive→d category of abelianGgr∈oupsA) F ∈

• × natural • × RHom ( , f ) RHom (Rf∗ , Rf∗f ) A~c(X) G F −−−−→ A(Y) G F via τ RHom• (Rf , ) −−−−→ A(Y) ∗ G F is an isomorphism.

× Here we think of the ~c(X)-complexes and f as objects in both D( ~c(X)) X A G F A and D( ). But as far as we know, the natural map HomD(A~c(X)) HomD(X) need not always be an isomorphism. It is when X is properly algebraic,→ i.e., the J-adic completion of a proper B-scheme with B a noetherian ring and J a B-ideal: then j induces an equivalence of categories D( ~c(X)) D~c(X), see Corollary 3.3.4. So for properly algebraic X, we can replace AD( (X→)) in Theorem 1 by D (X), and A~c ~c let be any (X)-complex with ~c(X)-homology. GWe proveATheorem 1 (= TheoremA 4.1) in 4, adapting the argument of Deligne § in [H1, Appendix] (see also [De, 1.1.12]) to the category ~c(X), which presents itself as an appropriate generalization§ to formal schemes ofAthe category of quasi- coherent sheaves on an ordinary noetherian scheme. For this adaptation what is needed, mainly, is the plumpness of ~c(X) in (X), a non-obvious fact mentioned above. In addition, we need some factsAon “boundedness”A of certain derived functors in order to extend the argument to unbounded complexes. (See section 3.4, which makes use of techniques from [Sp].)6 a quasi-isomorphism into a K-injective complex I(E)), then every additive functor Γ: A → A0 (A0 abelian) has a right-derived functor RΓ: D(A) → D(A0) which satisfies RΓ(E) = Γ(I(E)). R • • For example, HomA(E1, E2) = HomA(E1, I(E2)). 5We do not know, for instance, whether Lf ∗—which is defined only up to isomorphism—can always be chosen so as to commute with translation, i.e., so that Θ = Identity will do. 6A ∆-functor φ is bounded above if there is an integer b such that for any n and any complex E such that HiE = 0 for all i ≤ n it holds that Hj(φE) = 0 for all j < n + b. Bounded below and bounded (above and below) are defined analogously. Boundedness (way-outness) is what makes the very useful “way-out Lemma” [H1, p. 68, 7.1] applicable. 6 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

In Deligne’s approach the “Special Adjoint Functor Theorem” is used to get right adjoints for certain functors on qc(X), and then these right adjoints are applied to injective resolutions of complexes.A . . There is now a neater approach to duality on a quasi-compact separated ordinary scheme X, due to Neeman [N1], in which “Brown Representability” shows directly that a ∆-functor F on D( qc(X)) has a right adjoint if and only if F commutes with coproducts. Both approacA hes need a small set of category-generators: coherent sheaves for (X) in Deligne’s, and Aqc perfect complexes for D( qc(X)) in Neeman’s. Lack of knowledge about perfect complexes over formal schemesA discouraged us from pursuing Neeman’s strategy. Recently however (after this paper was essentially written), Franke showed in [Fe] that Brown Representability holds for the derived category of an arbitrary Grothen- 7 dieck category . Consequently Theorem 1 also follows from the fact that ~c(X) is a GrothendiecAk category (straightforward to see once we know it—by plump-A ness in (X)—to be abelian) together with the fact that Rf∗ ◦ j commutes with coproductsA (Proposition 3.5.2).

1.2. Two other, probably more useful, generalizations—from ordinary schemes to formal schemes—of global Grothendieck Duality are stated below in Theorem 2 and treated in detail in 6. To describe them, and related results, we need some preliminaries about torsion§ functors. 1.2.1. Once again let (X, ) be a ringed space. For any -ideal , set OX OX J Γ := lim om ( / n, ) ( (X)), J −−→ OX X M n>0 H O J M M ∈ A and regard Γ as a subfunctor of the identity functor on -modules. If J OX N ⊂ M then ΓJ = ΓJ ; and it follows formally that the functor ΓJ is idempotent (Γ Γ N= Γ M) and∩ N left exact [St, p. 138, Proposition 1.7]. J J M J M Set (X) := Γ ( (X)), the full subcategory of (X) whose objects are the AJ J A A -torsion sheaves, i.e., the X -modules such that ΓJ = . Since ΓJ is Jan idempotent subfunctor ofOthe identityMfunctor, thereforeMit isMright-adjoint to the inclusion i = iJ : J (X) , (X). Moreover, J (X) is closed under (X)- colimits: if F is any functorA in→toA (X) such thatAiF has a colimit A(X), AJ M ∈ A then, since i and ΓJ are adjoint, the corresponding functorial map from iF to the constant functor with value factors via a functorial map from iF to the constant M functor with value ΓJ , and from the deﬁnition of colimits it follows that the monomorphism Γ ,M has a right inverse, so that it is an isomorphism, and J M → M thus J (X). In particular, if the domain of a functor G into J (X) is a smallMcategory∈ A , then iG does have a colimit, which is also a colimit ofAG; and so J (X) has small colimits, i.e., it is small-cocomplete. A Submodules and quotient modules of -torsion sheaves are -torsion sheaves. J J If is ﬁnitely-generated (locally) and if are X -modules such that andJ / are -torsion sheaves then N is⊂aM -torsionO sheaf too; and henceN (XM) isNplumpJin (X).8 In this case, Mthe stalkJof Γ at x X is AJ A J M ∈ (Γ ) = lim Hom ( / n, ). J x −−→ OX,x X,x x x M n>0 O J M

7So does the closely-related existence of K-injective resolutions for all A-complexes. (See also [AJS, §5].) 8 Thus the subcategory AJ (X) is a hereditary torsion class in A(X), in the sense of Dickson, see [St, pp. 139–141]. DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 7

Let X be a locally noetherian scheme and Z X a closed subset, the support ⊂ 0 of X / for some quasi-coherent X -ideal . The functor ΓZ := ΓJ does not depOendJon the quasi-coherent ideal O determiningJ Z. It is a subfunctor of the left- J exact functor ΓZ which associates to each X -module its subsheaf of sections O 0 M supported in Z. If is quasi-coherent, then ΓZ ( ) = ΓZ ( ). M M M 0 More generally, for any complex Dqc(X), the D(X)-map RΓZ RΓZ 0 E ∈ E → E induced by the inclusion ΓZ , ΓZ is an isomorphism [AJL, p. 25, Corollary (3.2.4)]; so for such we usually iden→tify RΓ 0 and RΓ . E Z E Z E Set Z (X) := J (X), the plump subcategory of (X) whose objects are the Z-torsionA sheaves,A that is, the -modules such Athat Γ 0 = ; and set OX M Z M M qcZ (X):= qc(X) Z (X), the plump subcategory of (X) whose objects are Athe quasi-coherenA t ∩ A-modules supported in Z. A OX 0 For a locally noetherian formal scheme X with ideal of definition J, set ΓX := ΓJ , a left-exact functor depending only on the sheaf of topological rings X , not on 0 O the choice of J—for (X), ΓX is the submodule whose sections are those of annihilatedM ∈loAcally by anMop⊂enMideal. Say that is a torsion sheaf 0 M M if Γ = . Let (X) := J(X) be the plump subcategory of (X) whose XM M At A A objects are all the torsion sheaves; and set qct(X) := qc(X) t(X), the full (in fact plump, see Corollary 5.1.3) subcategoryA of (XA) whose∩ obA jects are the quasi-coherent torsion sheaves. It holds that (X) A (X), see Corollary 5.1.4. Aqct ⊂ A~c If X is an ordinary locally noetherian scheme (i.e., J = 0), then t(X) = (X) and (X) = (X) = (X). A A Aqct Aqc A~c 1.2.2. For any map f : X Y of locally noetherian formal schemes there are → ideals of definition I Y and J X such that I X J [GD, p. 416, (10.6.10)]; and correspondingly⊂thereO is a map⊂ Oof ordinary scOhemes⊂ (= formal schemes having (0) as ideal of definition) (X, X/J) (Y, Y/I) [GD, p. 410, (10.5.6)]. We say that f is separated (resp. affine,Oresp. pseudo→ O-proper, resp. pseudo-finite, resp. of pseudo-finite type) if for some—and hence any—such I, J the corresponding scheme- map is separated (resp. affine, resp. proper, resp. finite, resp. of finite type), see [GD, 10.15–10.16, p. 444 ff.], keeping in mind [GD, p. 416, (10.6.10)(ii)].9 Any affine map§§ is separated. Any pseudo-proper map is separated and of pseudo-finite type. The map f is pseudo-finite it is pseudo-proper and affine it is pseudo- proper and has finite fibers [EGA⇔, p. 136, (4.4.2)]. ⇔ We say that f is adic if for some—and hence any—ideal of definition I Y , ⊂ O I X is an ideal of definition of X [GD, p. 436, (10.12.1)]. We say that f is proper (resp.O finite, resp. of finite type) if f is pseudo-proper (resp. pseudo-finite, resp. of pseudo-finite type) and adic, see [EGA, p. 119, (3.4.1)], [EGA, p. 148, (4.8.11)] and [GD, p. 440, (10.13.3)].

9In [Y, Definition 1.14], pseudo-finite-type maps are called “maps of formally finite type.” The proof of Prop. 1.4 in [Y] (with A0 = A) yields the following characterization of pseudo-finite-type maps of affine formal schemes (cf. [GD, p. 439, Prop. (10.13.1)]): The map f : Spf(B) → Spf(A) corresponding to a continuous homomorphism h: A → B of noetherian adic rings is of pseudo- finite type ⇔ for any ideal of definition I of A, there exists an A-algebra of finite type A0, an A0-ideal I0 ⊃ IA0, and an A-algebra homomorphism A0 → B inducing an adic surjective map c c A0 B where A0 is the I0-adic completion of A0. 8 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

1.2.3. Here is our second main result, Torsion Duality for formal schemes. (See Theorem 6.1 and Corollary 6.1.4 for more elaborate statements.) In the as- 0 −1 sertion, Dqc(X):= RΓX (Dqct(X)) is the least ∆-subcategory of D(X) containing 0 −1 both Dqce(X) and RΓX (0) (Deﬁnition 5.2.9, Remarks 5.2.10, (1) and (2)). For example, when X is an ordinary scheme then Dqc(X) = Dqc(X). e Theorem 2. Let f : X Y be a map of noetherian formal schemes. Assume either that f is separated or→that X has ﬁnite Krull dimension, or else restrict to bounded-below complexes. (a) The restriction of Rf : D(X) D(Y) takes D (X) to D (Y), and it ∗ → qct qct has a right ∆-adjoint f × : D(Y) D (X). t → qct (b) The restriction of Rf RΓ 0 takes D (X) to D (Y) D (Y), and it has ∗ X qc qct ⊂ qc a right ∆-adjoint f # : D(Y) D (X). e e → qc e Remarks 1.2.4. (1) The “homology localization” functor

Λ ( ):= R om•(RΓ 0 , ) X − H XOX − 0 −1 0 −1 is right-adjoint to RΓX, and ΛX (0) = RΓX (0) (Remarks 6.3.1). The ∆-functors # × f and ft are connected thus (Corollaries 6.1.4 and 6.1.5(a)):

# × × 0 # f = ΛXft , ft = RΓXf .

0 −1 (2) In the footnote on page 70 it is indicated that RΓX (0) admits a “Bousfield 0 colocalization” in D(X), with associated “cohomology colocalization” functor RΓX ; and in Remark 6.3.1(3), Theorem 2 is interpreted as duality with coefficients in the corresponding quotient D (X)/RΓ 0 −1(0) = D (X)/(D (X) RΓ 0 −1(0)). qc X ∼ qc qc ∩ X (3) The proof of Theoreme 2 is similar to that of Theorem 1, at least when the formal scheme X is separated (i.e., the unique formal-scheme map X Spec(Z) is separated) or finite-dimensional, in which case there is an equivalence→of categories D( qct(X)) Dqct(X) (Proposition 5.3.1). (As mentioned before, we know the correspA onding→result with “~c ” in place of “qct” only for properly algebraic formal schemes.) In addition, replacing separatedness of X by separatedness of f takes a technical pasting argument. 0 (4) For an ordinary scheme X (having (0) as ideal of definition), ΓX is just the identity functor of (X), and Dqct(X) = Dqc(X). In this case, Theorems 1 and 2 both reduce to theAusual global (non-sheafified) version of Grothendieck Duality. In 2 we will describe how Theorem 2 generalizes and ties together various strands in the§ literature on local, formal, and global duality. In particular, the behavior of Theorem 2 vis-à-vis variable f gives compatibility of local and global duality, at least on an abstract level—i.e., without the involvement of differentials, residues, etc. (See Corollary 6.1.6.)

1.3. As in the classic paper [V] of Verdier, the culminating results devolve × # from ﬂat-base-change isomorphisms, established here for the functors ft and f of × ! Theorem 2, with f pseudo-proper—in which case we denote ft by f . DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 9

Theorem 3. Let X, Y and U be noetherian formal schemes, let f : X Y be a pseudo-proper map, and let u: U Y be flat, so that in the natural diagr→am → v X Y U =:V X × −−−−→ g  f   Uy yY −−−u−→ the formal scheme V is noetherian, g is pseudo-proper, and v is flat (Proposi- tion 7.1). Then for all D+ (Y) := D (Y) D+(Y) the base-change map β of Defi- F ∈ qc qc ∩ F nition 7.3 is an isomorphisme e 0 ∗ ! ∼ ! 0 ∗ ! ∗ βF : RΓV v f g RΓUu = g u . F −→ F 6.1.5(b)∼ F In particular, if u is adic then we have a functorial isomorphism v∗f ! ∼ g!u∗ . F −→ F 0 This theorem is proved in 7 (Theorem 7.4). The functor RΓV has a right § # adjoint ΛV, see (15). Theorem 3 leads quickly to the corresponding result for f (see Theorem 8.1 and Corollary 8.3.3): Theorem 4. Under the preceding conditions, let β# : v∗f # g#u∗ ( D+ (Y)) F F → F F ∈ qc be the map adjoint to the natural composition e 0 ∗ # ∼ ! ∗ ∗ Rg∗RΓV v f Rg∗g u u . F Thm.−→3 F → F # Then the map ΛV(βF ) is an isomorphism # ∗ # ∼ # ∗ # ∗ ΛV(βF ): ΛV v f ΛV g u = g u . F −→ F 6.1.5(a)∼ F + # Moreover, if u is an open immersion, or if Dc (Y), then βF itself is an isomorphism. F ∈ The special case of Theorems 3 and 4 when u is an open immersion is equivalent to what may be properly referred to as Grothendieck Duality (unqualified by the prefix “global”), namely the following sheafified version of Theorem 2 (see Theorem 8.2): Theorem 5. Let X and Y be noetherian formal schemes and let f : X Y be a pseudo-proper map. Then the following natural compositions are isomorphisms:→ Rf R om• ( , f # ) R om• (Rf RΓ 0 , Rf RΓ 0 f # ) ∗ H X G F → H Y ∗ X G ∗ X F R om• (Rf RΓ 0 , ) ( D (X), D+ (Y)); → H Y ∗ X G F G ∈ qc F ∈ qc e e Rf R om• ( , f ! ) R om• (Rf , Rf f ! ) R om• (Rf , ) ∗ H X G F → H Y ∗G ∗ F → H Y ∗G F ( D (X), D+ (Y)). G ∈ qct F ∈ qc + # + + e Finally, if f is proper and Dc (Y), then f : Dc (Y) Dc (X) is right- + +F ∈ 0 → adjoint to Rf∗ : Dc (X) Dc (Y), and RΓX in Theorem 5 can be deleted, see Theorem 8.4. → 10 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

In this—and several other results about complexes with coherent homology—an essential ingredient is Proposition 6.2.1, deduced here from Greenlees-May duality for ordinary aﬃne schemes, see [AJL]: Let X be a locally noetherian formal scheme, and let D(X). Then for all D (X) the natural map RΓ 0 induces an isomorphismE ∈ F ∈ c XE → E R om•( , ) ∼ R om•(RΓ 0 , ). H E F −→ H XE F In closing this introductory section, we wish to express our appreciation for illuminating interchanges with Amnon Neeman and Amnon Yekutieli.

2. Applications and examples. Again, Theorem 2 generalizes global Grothendieck Duality on ordinary schemes. This section illustrates further how Theorem 2 provides a common home for a number of different duality-related results (local duality, formal duality, residue theorems, dualizing complexes. . .). For a quick example, see Remark 2.3.8. Section 2.1 reviews several forms of local duality. In section 2.2 we sheafify these results, and connect them to Theorem 2. In particular, Proposition 2.1.6 is an abstract version of the Local Duality theorem of [HuK¨ , p. 73, Theorem 3.4]; and Theorem 2.2.3 (Pseudo-finite Duality) globalizes it to formal schemes. Section 2.3 relates Theorems 1 and 2 to the central “Residue Theorems” in [L1] and [HuS¨ ] (but does not subsume those results). Section 2.4 indicates how both the Formal Duality theorem of [H2, p. 48, Propo- sition (5.2)] and the Local-Global Duality theorem in [L3, p. 188] can be deduced from Theorem 2. Section 2.5, building on work of Yekutieli [Y, 5], treats dualizing complexes on formal schemes, and their associated dualizing functors.§ For a pseudo-proper map f, the functor f # of Theorem 2 lifts dualizing complexes to dualizing complexes (Proposition 2.5.11). For any map f : X Y of noetherian formal schemes, there is natural isomorphism → R om• (Lf ∗ , f # ) ∼ f #R om• ( , ) ( D−(Y), D+(Y)), H X F G −→ H Y F G F ∈ c G ∈ (Proposition 2.5.13). For pseudo-proper f, if Y has a dualizing complex , so that f # is a dualizing complex on X, and if Y := R om•( , ) and RX are the correspR onding dualizing functors, one deducesD a naturalH isomorphism− R (wDell-known for ordinary schemes) f # = XLf ∗ Y ( D+(Y)), E ∼ D D E E ∈ c see Proposition 2.5.12. × There are corresponding results for ft as well.

2.1. (Local Duality.) All rings are commutative, unless otherwise specified. Let ϕ: R S be a ring homomorphism with S noetherian, let J be an S- → ideal, and let ΓJ be the functor taking any S-module to its submodule of elements which are annihilated by some power of J. Let E and E0 be complexes in D(S), the derived category of S-modules, and let F D(R). With denoting derived ∈ ⊗= tensor product in D(S) (defined via K-flat resolutions [Sp, p. 147, Proposition 6.5]), there is a natural isomorphism E RΓ E0 ∼ RΓ (E E0), see e.g., [AJL, p. 20, ⊗= J −→ J ⊗= DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 11

• 0 op Corollary(3.1.2)]. Also, viewing RHomR(E , F ) as a functor from D(S) D(R) to D(S), one has a canonical D(S)-isomorphism × RHom• (E E0, F ) ∼ RHom• (E, RHom• (E0, F )), R ⊗= −→ S R see [Sp, p. 147; 6.6]. Thus, with ϕ# : D(R) D(S) the functor given by J → ϕ#( ):= RHom• (RΓ S, ) = RHom• (RΓ S, RHom• (S, )), J − R J − ∼ S J R − there is a composed isomorphism RHom• (E, ϕ#F ) ∼ RHom• (E RΓ S, F ) ∼ RHom• (RΓ E, F ). S J −→ R ⊗= J −→ R J Application of homology H 0 yields the (rather trivial) local duality isomorphism (2.1.1) Hom (E, ϕ#F ) ∼ Hom (RΓ E, F ). D(S) J −→ D(R) J # “Non-trivial” versions of (2.1.1) include more information about ϕJ . For example, Greenlees-May duality [AJL, p. 4, (0.3)aff] gives a canonical isomorphism # • (2.1.2) ϕJ F ∼= LΛJ RHomR(S, F ), where ΛJ is the J-adic completion functor, and L denotes “left-derived.” In particular, if R is noetherian, S is a finite R-module, and F Dc(R) (i.e., each homology module of F is finitely generated), then as in [AJL, p.∈6, Proposition (0.4.1)], (2.1.3) ϕ#F = RHom• (S, F ) Sˆ (Sˆ = J-adic completion of S). J R ⊗S More particularly, for S = R and ϕ = id (the identity map) we get id# F = F Rˆ (F D (R)). J ⊗R ∈ c Hence, classical local duality [H1, p. 278 (modulo Matlis dualization)] is just (2.1.1) when R is local, ϕ = id, J is the maximal ideal of R, and F is a normalized dualizing complex—so that, as in Corollary 5.2.3, and by [H1, p. 276, Proposition 6.1],

HomD(R)(RΓJ E, F ) = HomD(R)(RΓJ E, RΓJ F ) = HomD(R)(RΓJ E, I) where I is an R-injective hull of the residue ﬁeld R/J. (See also Lemma 2.5.7.)

For another example, let S = R[[t]] where t := (t1, . . . , td) is a sequence of variables, and set J := tS. The standard calculation (via Koszul complexes) gives an isomorphism RΓJ S ∼= ν[ d] where ν is the free R-submodule of the localiza- − n1 nd tion St1...td generated by those monomials t1 . . . td with all exponents ni < 0, the S-module structure being induced by that of S /S ν. The relative canonical t1...td ⊃ module ωR[[t]]/R := HomR(ν, R) is a free, rank one, S-module. There result, for ﬁnitely-generated R-modules F , functorial isomorphisms (2.1.4) ϕ# F = Hom (ν[ d], F ) = ω [d] F = R[[t]] F [d]; tR[[t]] ∼ R − ∼ R[[t]]/R ⊗R ∼ ⊗R and when R is noetherian, the usual way-out argument [H1, p. 69, (ii)] yields the same for any F D+(R). ∈ c Next, we give a commutative-algebra analogue of Theorem 2 in 1, in the form of a “torsion” variant of the duality isomorphism (2.1.1). Proposition§ 2.2.1 will clarify the relation between the algebraic and formal-scheme contexts. With ϕ: R S and J an S-ideal as before, let J (S) be the category of J-torsion S-modules,→ i.e., S-modules M such that A M = Γ M := m M J nm = 0 for some n > 0 . J { ∈ | } 12 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

The derived category of (S) is equivalent to the full subcategory D (S) of D(S) AJ J with objects those S-complexes E whose homology lies in J (S), (or equivalently, such that the natural map RΓ E E is an isomorphism),Aand the functor RΓ is J → J right-adjoint to the inclusion DJ(S) , D(S) (cf. Proposition 5.2.1 and its proof). × → Hence the functor ϕ : D(R) D (S) deﬁned by J → J ϕ×( ):= RΓ RHom• (S, ) = RΓ S RHom• (S, ) J − J R − ∼ J ⊗= R − is right-adjoint to the natural composition DJ(S) , D(S) D(R): in fact, for E D (S) and F D(R) there are natural isomorphisms→ → ∈ J ∈ (2.1.5) RHom• (E, ϕ×F ) ∼ RHom• (E, RHom• (S, F )) ∼ RHom• (E, F ). S J −→ S R −→ R × Here is another interpretation of ϕJ F . For S-modules A and R-modules B set

HomR,J (A, B):= ΓJ HomR(A, B), the S-module of R-homomorphisms α vanishing on J nA for some n (depending on α), i.e., continuous when A is J-adically topologized and B is discrete. If E is • a K-ﬂat S-complex and F is a K-injective R-complex, then HomR(E, F ) is a K- injective S-complex; and it follows for all E D(S) and F D(R) that ∈ ∈ • • RHomR,J (E, F ) ∼= RΓJ RHomR(E, F ). Thus, × • ϕJ F = RHomR,J (S, F ). A torsion version of local duality is the isomorphism, derived from (2.1.5): Hom (E, RHom• (S, F )) ∼ Hom (E, F ) (E D (S), F D(R)). DJ(S) R,J −→ D(R) ∈ J ∈ × # Apropos of Remark 1.2.4(1), the functors ϕJ and ϕJ are related by • ∼ # ∼ × LΛJ RHomR(S, F ) = ϕJF = LΛJ ϕJ F, (2.1.2) • × ∼ # RΓJ RHomR(S, F ) = ϕJ F = RΓJ ϕJF.

The ﬁrst relation is the case E = RΓJ S of (2.1.5), followed by Greenlees-May duality. The second results, e.g., from the sequence of natural isomorphisms, holding for G ∈ DJ(S), E ∈ D(S), and F ∈ D(R): • ∼ • HomD(S)(G, RΓJ RHomR(E, F )) = HomD(S)(G, RHomR(E, F ))

=∼ HomD (RΓ S ⊗S G ⊗S E, F ) (R) J = = ∼ • = HomD(S)(G, RHomR(RΓJ E, F )) ∼ • = HomD(S)(G, RΓJ RHomR(RΓJ E, F )), which entail that the natural map is an isomorphism • ∼ • RΓJ RHomR(E, F ) −→ RΓJ RHomR(RΓJ E, F ).

Local Duality theorems are often formulated, as in (c) of the following, in terms • • of modules and local cohomology (HJ := H RΓJ ) rather than derived categories. Proposition 2.1.6. Let ϕ : R S be a homomorphism of noetherian rings, → let J be an S-ideal, and suppose that there exists a sequence u = (u1, . . . , ud) in J such that S/uS is R-ﬁnite. Then for any R-ﬁnite module F : (a) Hnϕ#F = 0 for all n < d, so that there is a natural D(S)-map J − h: (H−dϕ#F )[d] ϕ#F. J → J DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 13

(b) If τ : RΓ ϕ#F F corresponds in (2.1.1) to the identity map of ϕ#F, 10 F J J → J and = d (F ) is the composed map R Rϕ,J RΓ (h) τ RΓ (H−dϕ#F )[d] J RΓ ϕ#F F F, J J −−−−→ J J −−→ −d # then (H ϕ F, ) represents the functor HomD (RΓ E[d], F ) of S-modules E. J R (R) J (c) If J √uS then there is a bifunctorial isomorphism (with E, F as before): ⊂ Hom (E, H−dϕ#F ) ∼ Hom (HdE, F ). S J −→ R J Proof. If ϕˆ is the obvious map from R to the u-adic completion Sˆ of S, then in D(S), ϕ#F = ϕˆ#F since RΓ S = RΓ Sˆ. In proving (a), therefore, we may J J J J ψ χ assume that S is u-adically complete, so that ϕ factors as R R[[t]] S with → → t = (t1, . . . , td) a sequence of indeterminates and S finite over R[[t]]. (ψ is the nat- # # ◦ # ural map, and χ(ti) = ui .) In view of the easily-verified relation ϕJ = χJ ψtR[[t]], (2.1.3) and (2.1.4) yield (a). Then (b) results from the natural isomorphisms −d # ∼ # ∼ HomS(E, H ϕJ F ) HomD(S)(E[d], ϕJ F ) HomD(R)(RΓJ E[d], F ). via−→h (2.1.1)−→ i i Finally, (c) follows from (b) because HJ E = HuSE = 0 for all i > d (as one sees i from the usual calculation of HuSE via Koszul complexes), so that the natural map is an isomorphism Hom (RΓ E[d], F ) ∼ Hom (HdE, F ). D(R) J −→ R J d 2.1.7. Under the hypotheses of Proposition 2.1.6(c), the functor HomR(HJ E, R) of S-modules E is representable. Under suitable extra conditions (for example, Sˆ a generic local complete intersection over R[[t]], Hubl¨ and Kunz represent this functor by a canonical pair described explicitly via differential forms, residues, and certain trace maps [HuK¨ , p. 73, Theorem 3.4]. For example, with S = R[[t]], J = tS, and d ν as in (2.1.4), the S-homomorphism from the module ΩS/R of universally finite d-forms to the relative canonical module ωR[[t]]/R = HombR(ν, R) sending the form −1 −1 dt1 . . . dtd to the R-homomorphism ν R which takes the monomial t1 . . . td to 1 n1 nd → and all other monomials t1 . . . td to 0, is clearly an isomorphism; and the resulting isomorphism Ωd [d] ∼ ϕ#R does not depend on the d-element sequence t S/R −→ J generating J—it correspb onds under (2.1.1) to the residue map RΓ Ωd [d] = Hd Ωd R J S/R J S/R → b b (see, e.g., [L2, 2.7]). Thus Hom (HdE, R) is represented by Ωd together with the § R J S/R residue map. The general case reduces to this one via tracesbof differential forms.

2.2. (Formal sheaﬁﬁcation of Local Duality). For f : X Y as in Theorem 2 × → in 1, there is a right ∆-adjoint ft for the functor Rf∗ : Dqct(X) Dqct(Y). Fur- thermore,§ with j : D( (X)) D (X) the canonical functor, we ha→ve A~c → ~c 0 0 Rf∗RΓXjD( ~c(X)) Rf∗RΓX Dqc(X) Rf∗Dqct(X) Dqct(Y) D~c(Y). A (3.1.5)⊂ (5.2.1)⊂ (5.2.6)⊂ (3.1.7)⊂ It results from (15) and Proposition 3.2.3 that Rf RΓ 0 j : D( (X)) D (Y) has ∗ X A~c → ~c the right ∆-adjoint RQ f # := RQ R om•(RΓ 0 , f ×). X X H XOX t

10 R • τF may be thought of as “evaluation at 1”: HomR,J (S, F ) → F . 14 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

If, moreover, X is properly algebraic (Definition 3.3.3)—in particular, if X is affine—then j is an equivalence of categories (Corollary 3.3.4), and so the functor 0 Rf∗RΓX : D~c(X) D~c(Y) has a right ∆-adjoint. For affine f, →these results are closely related to the Local Duality isomorphisms (2.1.5) and (2.1.1). Recall that an adic ring is a pair (R, I) with R a ring and I an R-ideal such that with respect to the I-adic topology R is Hausdorff and complete. The topology on R having been specified, the corresponding affine formal scheme is denoted Spf(R).

Proposition 2.2.1. Let ϕ: (R, I ) (S, J ) be a continuous homomorphism f of noetherian adic rings, and let X := Spf→(S) Spf(R) =: Y be the corresponding (aﬃne) formal-scheme map. Let κ : X X →:= Spec(S), κ : Y Y := Spec(R) X → Y → be the completion maps, and let ∼ = ∼S denote the standard exact functor from S-modules to quasi-coherent -modules. Then: OX (a) The restriction of Rf∗ takes Dqct(X) to Dqct(Y), and this restricted functor has a right adjoint f × : D (Y) D (X) given by t qct → qct ∼ ∼ f × := κ∗ (ϕ×RΓ(Y, )) = κ∗ (RHom• (S, RΓ(Y, ))) ( D (Y)). t F X J F X R,J F F ∈ qct 0 (b) The restriction of Rf∗ RΓX takes D~c(X) to D~c(Y), and this restricted functor has a right adjoint f # : D (Y) D (X) given by ~c ~c → ~c f # := κ∗ (ϕ#RΓ(Y, ))∼ = κ∗ (RHom• (RΓ S, RΓ(Y, )))∼ ( D (Y)). ~c F X J F X R J F F ∈ ~c (c) There are natural isomorphisms RΓ(X, f × ) ∼ ϕ×RΓ(Y, ) ( D (Y)), t F −→ J F F ∈ qct RΓ(X, f # ) ∼ ϕ# RΓ(Y, ) ( D (Y)). ~c F −→ J F F ∈ ~c Proof. The functor ∼ induces an equivalence of categories D(S) D (X), → qc with quasi-inverse RΓX := RΓ(X, ) ([BN, p. 225, Thm. 5.1], [AJL, p. 12, Propo- sition (1.3)]); and Proposition 3.3.1−below implies that κ∗ : D (X) D (X) is an X qc → ~c equivalence, with quasi-inverse (RΓ κ )∼ = (RΓ )∼. 11 X X∗− X− It follows that the functor taking G D(S) to κ∗ G is an equivalence, with ∈ X quasi-inverse RΓ : D~c(X) D(S), and similarly for Y eand R. Moreover, there X → is an induced equivalence between DJ(S) and Dqct(X) (see Proposition 5.2.4). In particular, (c) follows from (a) and (b). Corresponding to (2.1.5) and (2.1.1) there are then functorial isomorphisms

× ∼ ∗ ∼R Hom X ( , f ) Hom Y (κ (RΓ ) , ) ( D (X), D (Y)), D( ) E t F −→ D( ) Y XE F E ∈ qct F ∈ qct # ∼ ∗ ∼R Hom X ( , f ) Hom Y (κ (RΓ RΓ ) , ) ( D (X), D (Y)); D( ) E ~c F −→ D( ) Y J XE F E ∈ ~c F ∈ ~c and it remains to demonstrate functorial isomorphisms

κ∗ (RΓ )∼R ∼ Rf ( D (X)), Y XE −→ ∗E E ∈ qct κ∗ (RΓ RΓ )∼R ∼ Rf RΓ 0 ( D (X)), Y J XE −→ ∗ XE E ∈ ~c the ﬁrst a special case of the second.

11 In checking this note that κX∗ has an exact left adjoint, hence preserves K-injectivity. DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 15

To prove the second, let E := RΓ , let Z := Spec(S/J ) X, and let XE ⊂ f0 : X Y be the scheme-map corresponding to ϕ. The desired isomorphism comes from→ the sequence of natural isomorphisms

Rf RΓ 0 = Rf RΓ 0 κ∗ E ∗ XE ∼ ∗ X X ∗ e ∼= Rf∗κXRΓZ E (Proposition 5.2.4(c)) ∗ e ∼= κYRf0∗RΓZ E (Corollary 5.2.7) ∗ e ∼ ∼= κYRf0∗(RΓJ E) ([AJL, p. 9, (0.4.5)]) ∗ ∼R ∼= κY(RΓJ E) . (The last isomorphism—well-known for bounded-below E—can be checked via the equivalences RΓX and RΓY , which satisfy RΓY Rf0∗ ∼= RΓX (see [Sp, pp. 142–143, 5.15(b) and 5.17]).

Theorem 2.2.3 below globalizes Proposition 2.1.6. But first some preparatory remarks are needed. Recall from 1.2.2 that a map f : X Y of noetherian formal schemes is pseudo-finite if it is pseudo-proper and has finite→ fibers, or equivalently, if f is pseudo-proper and affine. Such an f corresponds locally to a homomorphism ϕ: (R, I ) (S, J ) of noetherian adic rings such that ϕ(I) J and S/J is a finite R-module.→This ϕ can be extended to a homomorphism from⊂ a power series ring R[[t]]:= R[[t1, t2, . . . , te]] such that the images of the variables ti together with ϕ(I ) generate J, and thereby S becomes a finite R[[t]]-module. Pseudo-finiteness is preserved under arbitrary (noetherian) base change. We say that a pseudo-finite map f : X Y of noetherian formal schemes has relative dimension d if each y Y has→an affine neighborhood U such that ≤ ∈ −1 the map ϕU : R S of adic rings corresponding to f U U has a continuous extension R[[→t , . . . , t ]] S making S into a finite R[[t→, . . . , t ]]-module, or 1 d → 1 d equivalently, there is a topologically nilpotent sequence u = (u1, . . . , ud) in S (i.e., n limn→∞ ui = 0 (1 i d)) such that S/uS is finitely generated as an R-module. The relative dimension≤ ≤dim f is defined to be the least among the integers d such that f has relative dimension d. For any pseudo-proper map≤ f : X Y of noetherian formal schemes, we have the functor f # : D(Y) D(X) of Corollary→ 6.1.4, commuting with open base change on Y (Theorem 4). The→ next Lemma is a special case of Proposition 8.3.2.

Lemma 2.2.2. For a pseudo-finite map f : X Y of noetherian formal schemes and for any D+(Y), it holds that f # D+→(X). F ∈ c F ∈ c Proof. Since f # commutes with open base change, the question is local, so we may assume that f corresponds to ϕ: (R, I ) (S, J ) as above. Moreover, # # # → the isomorphism (gf) ∼= f g in Corollary 6.1.4 allows us to assume that either S = R[[t1, . . . , td]] and ϕ is the natural map or S is a finite R-module and J = IS. In either case f is obtained by completing a proper map f0 : X Spec(R) along a −1 → closed subscheme Z f0 Spec(R/I ). (In the first case, take X to be the projective Pd ⊂ space R Spec(R[t1, . . . , td]), and Z := Spec(R[t1, . . . , td]/(I, t1, . . . , td)).) The conclusion⊃is given then by Corollary 6.2.3. 16 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

Theorem 2.2.3 (Pseudo-finite Duality). Let f : X Y be a pseudo-finite map → of noetherian formal schemes, and let be a coherent Y-module. Then: (a) Hnf # = 0 for all n < dim fF. O (b) If dimF f d and X is−covered by affine open subsets with d-generated defining ideals, then≤ with f 0 := f Γ 0 and, for i Z and J a defining ideal of X, X∗ ∗ X ∈ i 0 i 0 i 0 i • n 12 R f := H Rf = H Rf RΓ = lim H Rf R om ( X/J , ), X∗ X∗ ∗ X −−→ ∗ n H O − there is, for quasi-coherent X-modules , a functorial isomorphism O E −d # ∼ d 0 f omX( , H f ) omY(R f , ). ∗H E F −→ H X∗E F (Here H−df # is coherent (Lemma 2.2.2), and by (a), vanishes unless d = dim f.) F Proof. Since f # commutes with open base change we may assume that Y is affine and that f corresponds to a map ϕ: (R, I ) (S, J ) as in Proposition 2.1.6. Then there is an isomorphism of functors → ∼ jRQ f # = κ∗ (ϕ#RΓ(Y, )) , X ∼ X J − 0 both of these functors being right-adjoint to Rf∗RΓX : D~c(X) D~c(Y) (Proposi- → # + tion 2.2.1(b) and remarks about right adjoints preceding it). Since f Dc (X) (Lemma 2.2.2), therefore, by Corollary 3.3.4, the natural map is an isomorphismF ∈ jRQ f # ∼ f # ; and so, since κ∗ is exact, Proposition 2.1.6 gives (a). X F −→ F X Next, consider the presheaf map associating to each open U Y the natural composition (with V:= f −1U): ⊂ −d # ∼ # ∼ 0 HomV( , H f ) HomD(V)( [d], f ) HomD(U)(Rf∗ RΓX [d], ) E F b−y→(a) E F 6−.→1.4 E F d 0 HomU(R f , ). −→ X∗E F To prove (b) by showing that the resulting sheaf map −d # d 0 f omX( , H f ) omY(R f , ) ∗H E F → H X∗E F i 0 is an isomorphism, it suffices to show that R fX∗ = 0 for all i > d, a local problem for which we can (and do) assume that f correspEonds to ϕ: R S as above. Now RΓ 0 D (X) (Proposition 5.2.1), so Proposition 5.2.4→ for X := Spec(S) XE ∈ qct and Z := Spec(S/J) gives RΓ 0 = κ∗ with := κ RΓ 0 D+ (X). Since X XE ∼ XE0 E0 X∗ XE ∈ qcZ has, locally, a d-generated defining ideal, we can represent RΓ 0 locally by a lim of X −−→ E i 0 Koszul complexes on d elements [AJL, p. 18, Lemma 3.1.1], whence H RΓX = 0 i E for all i > d, and so, κX∗ being exact, H 0 = 0. Since the map f0 := Spec(ϕ) i E is affine, it follows that H Rf0∗ 0 = 0, whereupon, κY being flat, Corollary 5.2.7 yields E Rif 0 = HiRf κ∗ = Hiκ∗ Rf = κ∗ HiRf = 0 (i > d), X∗E ∼ ∗ XE0 ∼ Y 0∗E0 ∼ Y 0∗E0 as desired. (Alternatively, use Lemmas 3.4.2 and 5.1.4.)

12 X The equalities hold because being noetherian, any lim−→ of ﬂasque sheaves (for example, Jn lim−→ Hom(OX/ , E) with E an injective OX-module) is f∗-acyclic, and lim−→ commutes with f∗. (For an additive functor φ: A(X) → A(Y), an A(X)-complex F is φ-acyclic if the natural map φF → RφF is a D(Y)-isomorphism. Using a standard spectral sequence, or otherwise (cf. [L4, (2.7.2)]), one sees that any bounded-below complex of φ-acyclic OX-modules is φ-acyclic. DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 17

2.3. Our results provide a framework for “Residue Theorems” such as those appearing in [L1, pp. 87–88] and [HuS¨ , pp. 750-752] (central theorems in those papers): roughly speaking, Theorems 1 and 2 in section 1 include both local and global duality, and Corollary 6.1.6 expresses the compatibility between these dualities. But the dualizing objects we deal with are determined only up to isomorphism. The Residue Theorems run deeper in that they include a canonical realization of dualizing data, via differential forms. (See the above remarks on the Hubl-Kunz¨ treatment of local duality.) This extra dimension belongs properly to a theory of the “Fundamental Class” of a morphism, a canonical map from relative differential forms to the relative dualizing complex, which will be pursued in a separate paper. 2.3.1. Let us be more explicit, starting with some remarks about “Grothendieck Duality with supports” for a map f : X Y of noetherian separated schemes with respective closed subschemes W →Y and Z f −1W . Via the natural ⊂ ⊂ equivalence of categories D( qc(X)) Dqc(X) (see 3.3), we regard the func- × A → § tor f : D(Y ) D( ~c(X)) = D( qc(X)) of Theorem 1 as being right-adjoint to → A 13 A 0 Rf∗ : Dqc(X) D(Y ). The functor RΓZ can be regarded as being right-adjoint to the inclusion→ D (X) , D(X) (cf. Proposition 5.2.1(c)); and its restriction Z → to Dqc(X) agrees naturally with that of RΓZ , both restrictions being right-adjoint to the inclusion D (X) , D (X). Similar statements hold for W Y . Since qcZ → qc ⊂ Rf∗(DqcZ (X)) DW (Y ) (cf. proof of Proposition 5.2.6), we find that the functors × ⊂ × 0 RΓZ f and RΓZ f RΓW are both right-adjoint to Rf∗ : DqcZ (X) D(Y ), so are isomorphic. We define the local integral (a generalized residue map,→cf. [HuK¨ , 4]) § ρ( ): Rf RΓ f × RΓ 0 ( D(Y )) G ∗ Z G → W G G ∈ to be the natural composition Rf RΓ f × ∼ Rf RΓ f ×RΓ 0 Rf f ×RΓ 0 RΓ 0 . ∗ Z G −→ ∗ Z W G → ∗ W G → W G 0 ∼ Noting that for DW (Y ) there is a canonical isomorphism RΓW (proof similar to thatFof∈Proposition 5.2.1(a)), we have then: F −→ F

Proposition 2.3.2 (Duality with supports). For DqcZ (X), DW (Y ), the natural composition E ∈ F ∈ Hom ( , RΓ f × ) Hom (Rf , Rf RΓ f × ) DqcZ (X) E Z F −−−→ DW (Y ) ∗E ∗ Z F HomDW (Y )(Rf∗ , ) −ρ−(−F→) E F is an isomorphism.

× This follows from adjointness of Rf∗ and f , via the natural diagram Rf RΓ f × Rf f × ∗ Z G −−−−→ ∗ G ρ(G)  ( D(Y )),   G ∈ RΓy0 y W G −−−−→ G whose commutativity is a cheap version of the Residue Theorem [HuS¨ , pp. 750- 752]. Again, however, to be worthy of the name a Residue Theorem should in- volve canonical realizations of dualizing objects. For instance, when V is a proper

13For ordinary schemes, this functor f × is well-known, and usually denoted f ! when f is proper. When f is an open immersion, the functors f × and f !(= f ∗) need not agree. 18 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN d-dimensional variety over a field k and v V is a closed point, taking X = V , Z = v , W = Y = Spec(k), = k, and setting∈ ω := H−df ×k, we get an - { } G V OV,v module ωV,v (commonly called “canonical”, though defined only up to isomorphism) together with the k-linear map induced by ρ(k): Hd(ω ) k, v V,v → a map whose truly-canonical realization via differentials and residues is indicated in [L1, p. 86, (9.5)]. 2.3.3. With preceding notation, consider the completion diagram κ X =: X X X /Z −−−−→ ˆ f f   y y Y/W =: Y Y −−−κY−→ Duality with supports can be regarded more intrinsically—via fˆrather than f— as a special case of the Torsion-Duality Theorem 6.1 (= Theorem 2 of 1) for fˆ: ∼ ∗ § First of all, the local integral ρ is completely determined by κY(ρ): for D(Y), 0 ∗ 0 G ∈ the natural map RΓW κY∗κYRΓW is an isomorphism (Proposition 5.2.4); and the same holds for RfGR→Γ f × κ Gκ∗ Rf RΓ f × since as above, ∗ Z G → Y∗ Y ∗ Z G Rf RΓ f × Rf (D (X)) D (Y ) ∗ Z G ∈ ∗ qcZ ⊂ W ∗ ∗ —and so ρ = κY∗κY(ρ). Furthermore, κY(ρ) is determined by the “trace” map ˆ ˆ× τt : Rf∗ft 1, as per the following natural commutative diagram, whose rows are isomorphisms:→ ∗ × ˆ ∗ × ∗ ˆ ˆ× ∗ ˆ ˆ× 0 ∗ κYRf∗ RΓZ f G −−−→ Rf∗κXRΓZ f κY∗κYG −−−→ Rf∗ft κYG −−−→ Rf∗ft RΓY κYG ? 5f.2.7 6f.1.6 6.1f.5(b) ? ∗ ? ?τ κY(ρ)y y t ∗ 0 0 ∗ κYRΓW G ←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− RΓY κYG 5f.2.4 × × ∗ (To see that the natural map RΓZ f RΓZ f κY∗κY is an isomorphism, re- × G → × 0 G place RΓZ f by the isomorphic functor RΓZ f RΓW and apply Proposition 5.2.4.) Finally, we have isomorphisms (for D (X), D (Y )), E ∈ qcZ F ∈ W × ∼ ∗ ∗ × ∗ Hom ( , RΓ f ) Hom X (κ , κ RΓ f κ κ ) (5.2.4) D(X) E Z F −→ D( ) XE X Z Y∗ YF ∼ ∗ ˆ× ∗ Hom X (κ , f κ ) (6.1.6) −→ D( ) XE t YF ∼ ∗ ∗ Hom Y (Rfˆκ , κ ) (6.1) −→ D( ) ∗ XE YF ∼ ∗ ∗ Hom Y (κ Rf , κ ) (5.2.7) −→ D( ) Y ∗E YF ∼ Hom (Rf , ) (5.2.4), −→ D(Y ) ∗E F whose composition can be checked, via the preceding diagram, to be the same as the isomorphism of Proposition 2.3.2. 2.3.4. Proposition 2.3.6 expresses some homological consequences of the fore- going dualities, and furnishes a general context for [L1, pp. 87–88, Theorem (10.2)]. For any noetherian formal scheme X, D(X), and n Z, set E ∈ ∈ H0n( ):= HnRΓ(X, RΓ 0 ). X E XE DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 19

κ For instance, if X = X/Z X is the completion of a noetherian scheme X along a closed Z X, then for−→ D(X), Proposition 5.2.4 yields natural isomorphisms ⊂ F ∈ RΓ(X, RΓ 0 κ∗ ) = RΓ(X, κ RΓ 0 κ∗ ) X F ∗ X F = RΓ(X, κ κ∗RΓ 0 ) = RΓ(X, RΓ 0 ), ∼ ∗ Z F ∼ ZF and so if D (X), then with H• the usual cohomology with supports in Z, F ∈ qc Z H0n(κ∗ ) = Hn ( ). X F ∼ Z F Let J X be an ideal of definition. Writing ΓX for the functor Γ(X, ), we have a functorial⊂ O map − 0 0 γ( ): R(Γ ◦ Γ ) RΓ ◦ RΓ ( D(X)), E X X E → X X E E ∈ which is an isomorphism when is bounded-below, since for any injective X- E i O module , lim of the flasque modules om( X/J , ) is Γ -acyclic. Whenever −−→i X γ( ) is anI isomorphism, the induced homologyH Omaps areI isomorphisms E n i ∼ 0n lim Ext ( X/J , ) H ( ). −−→ X i O E −→ E 0 If Dqc(X), then RΓX Dqct(X) (Proposition 5.2.1). For any map g : X E Y∈ satisfying the hypothesesE ∈ of Theorem 6.1, for D(Y), and with →0 G ∈ R:= H (Y, Y), there are natural maps O 0 × ∼ 0 × 0 Hom X (RΓ , g ) Hom X (RΓ , g RΓ ) (6.1.5(b)) D( ) XE t G −→ D( ) XE t YG ∼ 0 0 (6) Hom Y (Rg RΓ , RΓ ) −→ D( ) ∗ XE YG Hom (H0n , H0n ) −→ R X E Y G where the last map arises via the functor HnRΓ(Y, ) (n Z). ˆ − ∈ ∗ In particular, if g = f in the completion situation of 2.3.3, and if := κX 0 , ∗ § E E = κY 0 ( 0 Dqc(X), 0 Dqc(Y )), then preceding considerations show that Gthis compG osedE ∈map operatesG via∈ Duality with Supports for f (Proposition 2.3.2), i.e., it can be identified with the natural composition × ∼ HomD(X)(RΓZ 0, RΓZ f 0) HomD(Y )(Rf∗ RΓZ 0, RΓW 0) E G 2−.→3.2 E G n n Hom 0 (H , H ). −→ H (Y,OY ) Z E0 W G0 2.3.5. Next, let R be a complete noetherian local ring topologized as usual by its maximal ideal I, let (S, J) be a noetherian adic ring, let ϕ: (R, I) (S, J) be a continuous homomorphism, and let → f Y:= Spf(S) Spf(R) =: V −→ be the corresponding formal-scheme map. As before, g : X Y is a map as in Theorem 6.1, and we set h:= fg. Since the underlying space of→V is a single point, at which the stalk of V is just R, therefore the categories of V-modules and of O O R-modules are identical, and accordingly, for any D(X) we can identify Rh∗ with RΓ(X, ) D(R). E ∈ E E ∈ Let K be an injective R-module, and the corresponding injective V-module. There exist integers r, s such that H i(f #K) = 0 for all i < r (resp. HOi(h# ) = 0 K −r # − −s # K for all i < s) (Corollary 6.1.4). Set ωY := H (f ) (resp. ωX := H (h )). − K K 20 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

Proposition 2.3.6. In the preceding situation ωX represents—via (6)—the 0s 00 # functor HomS(HX , HY (f )) of quasi-coherent X-modules . If ωY is the only E # K O E 0s 0r non-zero homology of f , this functor is isomorphic to Hom (H , H ωY). K S XE Y Proof. There are natural maps

0r 00 h 00 # ∼ H (ωY) = H (ωY[r]) H (f ) Hom (S, K) Y Y −−→ Y K −→ R,J where the last isomorphism results from Proposition 2.2.1(a), in view of the identity 0 # × RΓYf = ft (Corollary 6.1.5(a)) and the natural isomorphisms ∗ ∼ ∗ ∼ ∼ + RΓ(Y, κYG) RΓ(Y, κY∗κYG) RΓ(Y, G) G (G D (S)), −→ 5−.→2.4 −→ ∈ J e e e for G := RHom• (S, RΓ(V, )). (In fact RΓ(Y, κ∗ G) = G for any G D(S), see R,J K Y ∼ ∈ Corollary 3.3.2 and the beginning of 3.3.) In caseeωY is the only non-vanishing homology of f # , then h is an isomorphism§ too. The assertionsK follow from the (easily checked) commutativity, for any quasi- coherent X-module , of the diagram O E # # 6.1.5(a) 0 × # HomX( , ωX) == Hom X ( [s], g f ) Hom X (RΓ [s], g f ) E D( ) E K −−−−→ D( ) XE t K ' (6)   y 0 00 y 00 # Hom V (Rh RΓ [s], ) Hom (H ( [s]), H (f )) D( ) ∗ XE K S X E Y K '  Hom (H 0s , K) Hom (H0s , Homy (S, K)) R XE −−−−→ S XE R,J 2.3.7. Now let us fit [L1, pp. 87–88, Theoremf (10.2)] into the preceding setup. The cited Theorem has both local and global components. The first deals with maps ϕ: R S of local domains essentially of finite type over a perfect field k, with residue→fields finite over k. To each such ring T one associates the canonical module ωT of “regular” k-differentials of degree dim T . Under mild restrictions on ϕ, the assertion is that the functor dimS dimR HomR(H G, H ωR) (m:= maximal ideal) mSˆ mR of Sˆ-modules G is represented by the completion ωS together with a canonical map, the relative residue c dimS dimS dimR ρϕ : Hm ωS = Hm ωS Hm ωR. Sˆ S → R This may be viewed as a consequencec of concrete local duality over k ( 2.1.7). The global aspect concerns a proper map of irreducible k-varieties §g : V W of respective dimensions s and r with all fibers over codimension 1 points →of W having dimension s r, a closed point w W, the fiber E := g−1(w), and the − ∈ completion V := V/E. The assertion is that the functor b 0s r HomR(H , HmR ωR) (R:= W,w) Vb G O of coherent -modules is represented by the completion ωV along E of the OVb G canonical sheaf ωV of regular differentials, together with a canonicalc map 0s s r θ : H ωV = HE ωV HmR ωR . Vb → Moreover, the local and globalcrepresentations are compatible in the sense that if v E is any closed point and ϕ : R S := is the canonical map, then the ∈ v → OV,v DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 21 residue ρ := ρ factors as the natural map Hs ω Hs ω followed by θ. This v ϕv mS S → E V compatibility determines θ uniquely if the ρv (v E) are given [L1, p. 95, (10.6)]; and of course conversely. ∈ Basically, all this—without the explicit description of the ω’s and the maps ρv via differentials and residues—is contained in Proposition 2.3.6, as follows. In the completion situation of 2.3.3, take X and Y to be finite-type separated schemes over an artinian local ring§ R, of respective pure dimensions s and r, let W = w with w a closed point of Y , write g in place of f, and assume that Z g{−1W} is proper over R (which is so, e.g., if g is proper and Z is closed). Let⊂K be an injective hull of the residue field of R, and let be the corresponding injective sheaf on Spec(R) = Spf(R). With f : Y Spec(KR) the canonical map, and h = fg, define the dualizing sheaves → ω := H−sh! , ω := H−rf ! , X K Y K where h! is the Grothendieck duality functor (compatible with open immersions, and equal to h× when h is proper), and similarly for f !. It is well-known (for example via a local factorization of h as smooth ◦ finite) that h! has coherent homology, vanishing in all degrees < s; and similarly f ! has coherenK t homology, vanishing in all degrees < r. − K Let − fˆ: Y:= Spf(\) Spf(R) =: V OW,w → be the completion of f. We may assume, after compactifying f and g—which does not affect fˆ or gˆ (see [Lu¨]), that f and g are proper maps. Then Corollary 6.2.3 # ∗ ! shows that hˆ = κ h , and so κX being flat, we see that K X K ∗ (7) κX ωX = ωX ∗ where ωX is as in Proposition 2.3.6; and similarly κYωY = ωY . Once again, some form of the theory of the Fundamental Class will enable us to represent ωX by means of regular differential forms; and then both the local and global components of the cited Theorem (10.2) become special cases of Propo- sition 2.3.6 (modulo some technicalities [L1, p. 89, Lemma (10.3)] which allow a # weakening of the condition that ωY be the only non-vanishing homology of fˆ ). As for the local-global compatibility, consider quite generally a pair of mapsK X q X p Y 1 −→ −→ of noetherian formal schemes. In the above situation, for instance, we could take p to be gˆ, X1 to be the completion of X at a closed point v Z, and q to be the natural map. Theorem 2 gives us the adjunction ∈

Rp∗ Dqct(X) Dqct(Y). −−−×→ ←p−t− ∼ The natural isomorphism R(pq)∗ Rp∗Rq∗ gives rise then to an adjoint isomor- × × ∼ × −→ × phism qt pt (pq)t ; and for Dqct(Y) the natural map R(pq)∗(pq)t factors as −→ E ∈ E → E R(pq) (pq)× ∼ Rp Rq q×p× Rp p× . ∗ t E −→ ∗ ∗ t t E → ∗ t E → E This factorization contains the compatibility between the above maps θ and ρv , as one sees by interpreting them as homological derivatives of maps of the type Rp p× (with := RΓ 0fˆ# ). Details are left to the reader. ∗ t E → E E Y K 22 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

Remark 2.3.8. In the preceding situation, suppose further that Y = Spec(R) (with R artinian) and f = identity, so that h = g : X Y is a ﬁnite-type sepa- → rated map, X being of pure dimension s, and κX : X X is the completion of X along a closed subset Z proper over Y . Again, K is →an injective R-module, is −s ! K the corresponding Y -module, and ωX := H g is a “dualizing sheaf ” on X. Now Proposition 2.3.6O is just the instance i = s Kof the canonical isomorphisms, 0• • 0 for D (X), i Z (and with H := H RΓ(X, RΓ ), see 2.3.4, and gˆ:= g ◦ κX): E ∈ qc ∈ X X § # ∼ 0 ∼ 0i 0i HomD(X)( [i], gˆ ) HomD(Y)(Rgˆ∗RΓX [i], ) HomR(HX , K) =:(HX )ˇ. E K Thm.−→2 E K −→ E E

! If X is Cohen-Macaulay then all the homology of g other than ωX vanishes, so # ∗ ! K∗ all the homology of gˆ ∼= κXg other than ωX = κXωX vanishes (see (7)), and the preceding composedK isomorphismK becomes

s−i ∼ 0i Ext ( , ωX) (H )ˇ. X E −→ X E In particular, when Z = X (so that X = X) this is the usual duality isomorphism

Exts−i( , ω ) ∼ Hi(X, )ˇ. X E X −→ E If X is Gorenstein and is a locally free X-module of ﬁnite rank, then ω is F O X invertible; and taking := omX( , ωX) = ˇ ωX we get the isomorphism E H F F ⊗ s−i ∼ 0i H (X, ) (H ( ˇ ωX))ˇ, F −→ X F ⊗ which generalizes the Formal Duality theorem [H2, p. 48, Proposition (5.2)].

2.4. Both [H2, p. 48; Proposition (5.2)] (Formal Duality) and the Theorem in [L3, p. 188] (Local-Global Duality) are contained in Proposition 2.4.1, see [AJL, 5.3]. § Let R be a noetherian ring, discretely topologized, and set

Y := Spec(R) = Spf(R) =: Y.

Let g : X Y be a ﬁnite-type separated map, let Z X be proper over Y , let → ⊂ κ: X = X/Z X be the completion of X along Z, and set gˆ:= g ◦ κ: X Y. Assume that→ R has a residual complex [H1, p. 304]. Then the corresp→ onding quasi-coherent -complex := is a dualizingR complex, and := g! is a OY RY R RX RY dualizing complex on X [V, p. 396,eCorollary 3]. For any Dc(X) set F ∈ 0 := R om• ( , ) D (X), F H X F RX ∈ c so that = 00 = R om• ( 0, ). F ∼ F H X F RX

Proposition 2.4.1. In the preceding situation, with ΓZ ( ) := Γ(X, ΓZ ( )) there is a functorial isomorphism − −

RΓ(X, κ∗ ) = RHom• (RΓ 0, ) ( D (X)). F ∼ R Z F R F ∈ c

Proof. Replacing g by a compactiﬁcation ([Lu¨]) doesn’t aﬀect X or RΓZ , so assume that g is proper. Then Corollary 6.2.3 gives an isomorphism κ∗ = gˆ# . RX ∼ RY DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 23

Now just compose the chain of functorial isomorphisms RΓ(X, κ∗ ) = RΓ(X, κ∗R om• ( 0, )) (see above) F ∼ H X F RX = RΓ(X, R om• (κ∗ 0, κ∗ )) (Lemma 2.4.2) ∼ H X F RX = RHom• (κ∗ 0, gˆ# )) (see above) ∼ X F RY = RHom• (Rgˆ RΓ 0 κ∗ 0, ) (Theorem 2) ∼ Y ∗ X F RY = RHom• (Rg RΓ 0, ) (Proposition 5.2.4) ∼ Y ∗ Z F RY • ^ 0 = RHom (RΓ , ) [AJL, footnote, 5.3] ∼ Y Z F RY § • 0 = RHom (RΓZ , ) [AJL, p. 9, (0.4.4)]. ∼ R F R Lemma 2.4.2. Let X be a locally noetherian scheme, and let κ: X X be its → completion along some closed subset Z. Then for Dqc(X) of finite injective dimension and for D (X), the natural map is anG ∈isomorphism F ∈ c κ∗R om• ( , ) ∼ R om• (κ∗ , κ∗ ). H X F G −→ H X F G Proof. By [H1, p. 134, Proposition 7.20] we may assume that is a bounded G complex of quasi-coherent injective X -modules, vanishing, say, in all degrees > n. When is bounded-above theO(well-known) assertion is proved by localizing to the affineFcase and applying [H1, p. 68, Proposition 7.1] to reduce to the trivial m Z case = X (0 < m ). To do the same for unbounded we must first show, for fixedF O, that the con∈ travariant functor R om• (κ∗ , κ∗ F) is bounded-above. G H X F G In fact we will show that if H i = 0 for all i < i then for all j > n i , F 0 − 0 Hj R om• (κ∗ , κ∗ ) = Hjκ R om• (κ∗ , κ∗ ) = Hj R om• ( , κ κ∗ ) = 0. H X F G ∗ H X F G H X F ∗ G The homology in question is the sheaf associated to the presheaf which assigns Hom ( [ j], (κ κ∗ ) ) = Hom ( [ j], RQ (κ κ∗ ) ) D(U) F|U − ∗ G |U D(U) F|U − U ∗ G |U to each affine open subset U = Spec(A) in X. (Here we abuse notation by omit- ting jU in front of RQU , see beginning of 3.3). −1 § Let U:= κ U, and Aˆ:= Γ(U, X), so that κ U factors naturally as O | κ U = Spf(Aˆ) 1 U := Spec(Aˆ) k Spec(A) = U. −−→ 1 −→

The functors RQU k∗ and k∗RQU , both right-adjoint to the natural composition ∗ 1 D (U) k D (U ) , D(U ), are isomorphic; so there are natural isomorphisms qc −→ qc 1 → 1 ∗ ∗ ∗ ∼ ∗ ∗ ∼ ∗ RQ (κ∗κ ) = RQ k∗κ1∗κ1k ( ) k∗RQU κ1∗κ1k ( ) k∗k ( ) U G |U U G|U −→ 1 G|U 3−.→3.1 G|U and the presheaf becomes U Hom ( [ j], k k∗( )). 7→ D(U) F|U − ∗ G|U The equivalence of categories Dqc(U) ∼= D( qc(U)) = D(A) indicated at the beginning of 3.3 yields an isomorphism A § Hom ( [ j], k k∗( )) ∼ Hom (F [ j], G Aˆ) D(U) F|U − ∗ G|U −→ D(A) − ⊗A i where F is a complex of A-modules with H F = 0 for i < i0 , and both G and G AAˆ are complexes of injective A-modules vanishing in all degrees > n (the latter ⊗since Aˆ is A-ﬂat). Hence the presheaf vanishes, and the conclusion follows. 24 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

2.5. (Dualizing complexes.) Let X be a noetherian formal scheme, and write 0 D for D(X), etc. The derived functor Γ := RΓX : D D (see Section 1.2.1) has a • 0 → right adjoint Λ = ΛX := R om (RΓX X, ). This adjunction is given by (15), a natural isomorphism of whicHh we’ll needO the−sheafified form, proved similarly: (8) R om•( , Λ ) = R om•(Γ , ). H M R ∼ H M R There are natural maps Γ 1 Λ inducing isomorphisms ΛΓ ∼ Λ ∼ ΛΛ, Γ Γ ∼ Γ ∼ Γ Λ (Remark→ 6.3.1→ (1)). Proposition 6.2.1, a form−→of Greenlees-−→ May −dualit→ y,−sho→ws that Λ(D ) D . (Recall that the objects of the ∆-subcategory c ⊂ c Dc D are the complexes whose homology sheaves are all coherent.) ⊂ ∗ Let Dc be the essential image of Γ Dc , i.e., the full subcategory of D such that ∗ Γ | ∗ Dc ∼= with Dc. Proposition 5.2.1 shows that Dc Dqct. It folloE ∈ ws from⇔ Ethe precedingF Fparagraph∈ that ⊂ D∗ Γ ∼ and Λ D , E ∈ c ⇐⇒ E −→ E E ∈ c D ∼ Λ and Γ D∗. F ∈ c ⇐⇒ F −→ F F ∈ c ∗ (In particular, Dc is a ∆-subcategory of D.) Moreover Γ and Λ are quasi-inverse ∗ equivalences between the categories Dc and Dc . Definition 2.5.1. A complex is a c-dualizing complex on X if R + (i) Dc (X). R ∈ ∼ • (ii) The natural map is an isomorphism X R om ( , ). (iii) There is an integer b such that for everyO coheren−→ tHtorsionRsheafR and for every i > b, xti( , ):= Hi R om•( , ) = 0. M E M R H M R A complex is a t-dualizing complex on X if R+ (i) Dt (X). R ∈ ∼ • (ii) The natural map is an isomorphism X R om ( , ). (iii) There is an integer b such that for everyO coheren−→ tHtorsionRsheafR and for every i > b, xti( , ):= Hi R om•( , ) = 0. M E M R H M •R (iv) For some ideal of definition J of X, R om ( X/J, ) Dc(X). H O• R ∈ (Equivalently—by simple arguments—R om ( , ) Dc(X) for every coherent torsion sheaf .) H M R ∈ M Remarks. (1) On an ordinary scheme, (iii) signifies finite injective dimension [H1, p. 83, Definition, and p. 134, (iii)c], so both c-dualizing and t-dualizing mean the same as what is called “dualizing” in [H1, p. 258, Definition]. (For the extension to arbitrary noetherian formal schemes, see (4) below.) (2) By (i) and (iv), Proposition 5.2.1(a), and Corollary 5.1.3, any t-dualizing complex is in D+ (X); and then (iii) implies that is isomorphic in D to a R qct R bounded complex of qct-injectives. To see this, beginA by imitating the proof of [H1, p. 80, (iii) (i)], using [Y, ⇒ Theorem 4.8] and Lemma 2.5.6 below, to reduce to showing that if qct(X) 1 N ∈ A is such that xt ( , ) = 0 for every coherent torsion sheaf then is qct- injective. E M N M N A For the last assertion, suppose first that X is affine. Lemma 5.1.4 implies that om( , ) (X); and then Ext1( , ) = 0, by the natural exact sequence H M N ∈ A~c M N 0 = H1(X, om( , )) Ext1( , ) H0(X, xt1( , )). (3.1.8) H M N → M N → E M N DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 25

Since coherent torsion sheaves generate (X) (Corollary 5.1.3, Lemma 5.1.4), a Aqct standard argument using Zorn’s Lemma shows that is indeed qct-injective. In the general case, let U X be any affine openNsubset. For Aany coherent tor- ⊂ sion U-module 0, Proposition 5.1.1 and Lemma 5.1.4 imply there is a coherent O M 1 torsion X-module restricting on U to , whence xt ( , U) = 0. By O M M0 E U M0 N | the affine case, then, U is (U)-injective, hence (U)-injective [Y, Propo- N | Aqct At sition 4.2]. Finally, as in [H1, p. 131, Lemma 7.16], using [Y, Lemma 4.1],14 one concludes that is (X)-injective, hence (X)-injective. N At Aqct (3) With (2) in mind, one finds that what is called here “t-dualizing complex” is what Yekutieli calls in [Y, 5] “dualizing complex.” § (4) A c-dualizing complex has finite injective dimension: there is an inte- R ger n0 such that for any i > n0 and any X-module , HomD( , [i]) = 0. To see this, note first that O E E R

HomD( , [i]) = HomD( , ΛΓ [i]) = HomD(Γ , Γ [i]). E R ∼ E R ∼ E R Lemma 2.5.3(b) below and (2) above show that Γ is isomorphic to a bounded complex of -injectives. The complex Γ —obtainedR by applying the functor Γ 0 Aqct E X to an injective resolution of —consists of torsion X-modules, and so as in [Y, Corollary 4.3] (see also the proE of of Lemma 2.5.6 beloO w, with Proposition 5.1.2 in place of Proposition 3.1.1), the natural map i • i • H (Hom (Γ , Γ )) H (RHom (Γ , Γ )) = HomD(Γ , Γ [i]) E R → E R E R is an isomorphism. Since Γ vanishes in degrees < 0, the asserted result holds for i E any n0 such that H (Γ ) = 0 for i > n0 . R + − (5) For a complex Dc Dc , conditions (ii) and (iii) in Definition 2.5.1 hold iff they hold stalkwiseR ∈for x ∩X, with an integer b independent of x. (The idea is that such an is locally resolv∈able by a bounded-above complex of finite-rank R • F • locally free X-modules, as is in (iii), and om ( , ) ∼= R om ( , ). . . .) Proceeding Oas in the proofs ofM[H1], PropositionH 8.2,Fp.R288, andH CorollaryF R 7.2, p. 283, one concludes that is c-dualizing iff X has finite Krull dimension and R x is a dualizing complex for the category of X,x-modules for every x X. (It is Renough that the latter hold for all closed pointsO x X.) ∈ ∈ Examples 2.5.2. (1) If is c-(or t-)dualizing then so is [n] for any R R ⊗ L invertible X-module and n Z. The converse also holds, see Proposition 2.5.4. O ∈ (2) (Cf. [Y, Example 5.12].) If X is an ordinary scheme and κ: X X is its → completion along some closed subscheme Z, then for any dualizing X -complex , ∗ ∗ ∗ O R κ is c-dualizing on X, and Γ κ = κ RΓZ (see Proposition 5.2.4(c)) is a R ∗ R ∼ R t-dualizing complex lying in Dc (X). Proof. For κ∗ , conditions (i) and (ii) in the definition of c-dualizing follow easily from the sameR for (because of Lemma 2.4.2). So does (iii), after we reduce to the case X affine, whereR Proposition 3.1.1 allows us to write = κ∗ with M M0 0 (X). (Recall from remark (1) above that has finite injective dimension.) MThe ∈lastA assertion is given by Lemma 2.5.3(b). R (3) If X = Spf(A) where A is a complete local noetherian ring topologized by its maximal ideal m—so that (X) is just the category of A-modules—then a A c-dualizing X-complex is an A-dualizing complex in the usual sense; and by (2) O 14where one may assume that X and X have the same underlying space 26 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

(via [H1, p. 276, 6.1]), or directly from Definition 2.5.1, the injective hull of A/m ∗ is a t-dualizing complex lying in Dc (X). (4) It is clear from Definition 2.5.1 and remark (4) above that X is c-dualizing O iff X has finite injective dimension over itself. By remark (5), X is c-dualizing iff O O X is finite dimensional and X is Gorenstein for all x X [H1, p. 295, Definition]. O ,x ∈ (5) For instance, if the finite-dimensional noetherian formal scheme Y is regular (i.e., the local rings Y,y (y Y) are all regular), and is a coherent Y-ideal, defining a closed formal subscO heme∈ i: X , Y [GD, p. 441,(10.14.2)],I thenOby remark (3), R om•(i , ) is c-dualizing on→X. So Lemma 2.5.3 gives that H ∗OX OY • 0 0 • ∗ R om ( Y/ , RΓY Y) = RΓX R om (i∗ X, Y) Dc (X) H O I O 5.2.10(4)∼ H O O ∈ is t-dualizing on X. (This is also shown in [Y, Proposition 5.11, Theorem 5.14].)

Lemma 2.5.3. (a) If D∗ is t-dualizing then Λ is c-dualizing. R ∈ c R (b) If is c-dualizing then Γ is t-dualizing, and lies in D∗. R R c ∗ + + Proof. (a) If Dc then of course Λ Dc . Also, Λ(D ) D because RΓ 0 is given locallyR ∈by a ﬁnite complex R• ∈, see proof of Proposition⊂ 5.2.1(a). XOX K∞ For condition (ii), note that if D+ (X) then Γ = (Proposition 5.2.1), R ∈ qct R ∼ R then use the natural isomorphisms (see (8): R om•(Λ , Λ ) = R om•(Γ Λ , ) = R om•(Γ , ) = R om•( , ). H R R ∼ H R R ∼ H R R ∼ H R R For (iii) note that Γ = (Proposition 5.2.1), then use (8). M ∼ M (b) Proposition 5.2.1 makes clear that if D+ then Γ D+ D∗. R ∈ c R ∈ qct ∩ c For (ii) use the isomorphisms (the second holding because D ): R ∈ c R om•(Γ , Γ ) = R om•(Γ , ) = R om•( , ). H R R 5.2.3∼ H R R 6.2.1∼ H R R For (iii) use the isomorphism R om•( , Γ ) = R om•( , ). For (iv), H M R 5.2.3∼ H M R note that when = X/J (J any ideal of deﬁnition) this isomorphism gives M O • • R om ( X/J, Γ ) = R om ( X/J, ) Dc , H O R ∼ H O R 3.2.4∈

The essential uniqueness of t-(resp. c-)dualizing complexes is expressed by: Proposition 2.5.4. (a) (Yekutieli) If is t-dualizing then a complex 0 is t- R R dualizing iff there is an invertible sheaf and an integer n such that 0 = [n]. L R ∼ R⊗L (b) If is c-dualizing then a complex 0 is c-dualizing iff there is an invertible R R sheaf and an integer n such that 0 = [n]. L R ∼ R ⊗ L Proof. Part (a) is proved in [Y, Theorem 5.6]. Now for a fixed invertible sheaf there is a natural isomorphism of functors L (9) Λ( ) ∼ Λ ( D), F ⊗ L −→ F ⊗ L F ∈ as one deduces, e.g., from a readily-established natural isomorphism between the respective right adjoints Γ −1 ∼ Γ ( −1) ( D). E ⊗ L ←− E ⊗ L E ∈ DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 27

Part (b) results, because Γ 0 and Γ are t-dualizing (Lemma 2.5.3), so that by (a) (and taking := Γ [n] Rin (9)) weRhave isomorphisms F R 0 = Λ(Γ 0) = Λ(Γ [n]) ( invertible, n Z) R ∼ R ∼ R ⊗ L L ∈ = (ΛΓ ) [n] = [n]. ∼ R ⊗ L ∼ R ⊗ L Corollary 2.5.5. If X is locally embeddable in a regular ﬁnite-dimensional ∗ formal scheme then any t-dualizing complex on X lies in Dc .

Proof. Whether a t-dualizing complex satisfies Λ Dc is a local question, so we may assume that X is a closed subschemeRof a finite-dimensionalR ∈ regular formal scheme, and then Example 2.5.2(5) shows that some—hence by Proposition 2.5.4, ∗ any—t-dualizing complex lies in Dc . Lemma 2.5.6. Let X be a locally noetherian formal scheme, let be a bounded complex of (X)-injectives, say i = 0 for all i > n, and letI D+(X), Aqct I F ∈ say H`( ) = 0 for all ` < m. Suppose there exists an open cover (X ) of X by F − α completions of ordinary noetherian schemes Xα along closed subsets, with completion maps κα : Xα Xα , such that for each α the restriction of to Xα is D-isomorphic to κ∗ F →for some F D(X ). Then F α α α ∈ α xti( , ):= HiR om• ( , ) = 0 for all i > m + n. E F I H X F I Moreover, if X has finite Krull dimension d then Exti( , ):= HiRHom• ( , ) = 0 for all i > m + n + d. F I X F I Remarks. In the published version of this paper (Contemporary Math. 244) Lemma 2.5.6 stated: Let D~c and let be a bounded-below complex of qct- injectives. Then the canonicF al∈ map is a D-isomorphismI A om•( , ) ∼ R om•( , ). H F I −→ H F I Suresh Nayak pointed out that the proof given applies only to -complexes, not, A~c as asserted, to arbitrary D~c. (Cf. [Y, Corollary 4.3].) Lemma 2.5.6 is used four times in 2.5, so theseF ∈four places need to be revisited. (There are no other references to Lemma§ 2.5.6 in the paper.) First, in Remark (2) on p. 24, the reference to Lemma 2.5.6 is not necessary: the cited theorem 4.8 in [Y] (see also Proposition 5.3.1 below) shows that the 0• t-dualizing complex is D-isomorphic to a bounded-below complex of qct- injectives; and then Rone can proceed as indicated to show that for Xsome nAthe 0• 0• (bounded) truncation σ≤n is qct-injective and D-isomorphic to . (To follow the details, it helps to keepX in mindA 5.1.3 and 5.1.4 below.) X Since, by Remark (2), any t-dualizing complex is D-isomorphic to a bounded complex of qct-injectives, in view of Propositions 3.3.1 and 5.1.2 one finds that the remainingA three references to Lemma 2.5.6 can be replaced by references to the present Lemma 2.5.6. For the reference in the proof of 2.5.7(b) this is clear. The same is true for Remark (4) on p. 25, but i > n0 at the end should be i > n0 + d, where, by Remark (5), the Krull dimension d of X is finite. Finally, for the reference in the proof of 2.5.12, one can note, via 5.1.4 and 5.1.2, that D∗ D D . c ⊂ qct ⊂ ~c Proof of 2.5.6. By the proof of [Y, Proposition 4.2], qct-injectives are just direct sums of sheaves of the form (x) (x X), where Afor any open U X, J ∈ ⊂ Γ(U, (x)) is a fixed injective hull of the residue field of X,x if x U, and van- ishesJotherwise. Hence the restriction of an (X)-injectivO e to an∈open V X Aqct ⊂ 28 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN is qct(V)-injective; and so the first assertion is local. Thus to prove it one may assumeA that X itself is a completion, with completion map κ: X X, and that → in D(X), = κ∗F for some F D(X). As κ∗, being exact, commutes with the F ∼ ∈ truncation functor σ≥−m , there are D-isomorphisms (the first as in [H1, p. 70]): ∗ ∗ = σ≥ = σ≥ κ F = κ σ≥ F ; F ∼ −m F ∼ −m ∼ −m so one can replace F by σ F and assume further that F ` = 0 for all ` < m. ≥−m − From the above description of -injectives, one sees that κ is a bounded Aqct ∗I complex of X -injectives, vanishing in degree > n. Since κ∗ is exact, therefore for all i > m +On, κ HiR om• ( , ) = Hiκ R om• (κ∗F, ) ∗ H X F I ∼ ∗ H X I = HiR om• (F, κ ) [Sp, p. 147, 6.7(2)] ∼ H X ∗ I = Hi om• (F, κ ) = 0, ∼ H X ∗ I and hence HiR om• ( , ) = 0. H X F I If X has Krull dimension d, and Γ:= Γ(X, ) is the global-section functor, then by a well-known theorem of Grothendieck the −restriction of the derived functor RΓ to the category of abelian sheaves has cohomological dimension d; and so since • ≤ RHom = RΓR om• [L4, Exercise 2.5.10(b)], the second assertion follows from X ∼ H X [L4, Remark 1.11.2(iv)]. Proposition 2.5.8 below brings out the basic property of the dualizing functors associated with dualizing complexes. (For illustration, one might keep in mind the special case of Example 2.5.2(3).)

Lemma 2.5.7. Let be a c-dualizing complex on X, let t be the t-dualizing complex := Γ , andRfor any D set R Rt R E ∈ := R om•( , ), := R om•( , ). DE H E R Dt E H E Rt (a) There are functorial isomorphisms Λ = Λ = Λ = = Γ = Γ. Dt ∼ D ∼ D ∼ D ∼ D ∼ Dt (b) For all D , D and there is a natural isomorphism = Γ . F ∈ c DF ∈ c Dt F ∼ DF Proof. (a) For any D, Proposition 6.2.1 gives the isomorphism E ∈ = R om•( , ) = R om•(Γ , ) = Γ . DE H E R ∼ H E R D E In particular, Λ = ΓΛ = Γ . Thus = Γ = Λ. D E ∼ D E ∼ D E D ∼ D ∼ D Furthermore, using that the natural map Γ X Γ is an isomorphism = (localize, and see [AJL, p. 20, Corollary (3.1.2)])Owe⊗getE natural→ E isomorphisms • • ∼ • • R om (Γ X , om ( , )) om (Γ , ) = R om (Γ , Γ ) H O H E R −→ H E R 5.2.3∼ H E R • • = R om (Γ X , om ( , Γ )), ∼ H O H E R giving Λ = Γ = Γ = Λ . D ∼ D ∼ Dt ∼ Dt (b) Given remark (2) following Deﬁnition 2.5.1, Lemma 2.5.6 implies that the functor := R om•( , ) is bounded on D . The same holds for = Γ Dt H − Rt ~c D Dt (see (a)), because Γ (D~c) Dqct D~c (Lemma 5.1.4), and Γ is bounded. (Γ is ⊂ ⊂ • given locally by tensoring with a bounded ﬂat complex ∞ , see proof of Proposi- tion 5.2.1(a)). K • Arguing as in Proposition 3.2.4, we see that t := R om ( , t) Dqct, so that Γ ∼ (Proposition 5.2.1(a)); andD FsimilarlyH, F RD .∈Further- Dt F −→ Dt F DF ∈ c more, the argument in Remark 5.2.10(4) gives an isomorphism Γ = Γ . Dt F ∼ DF DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 29

Proposition 2.5.8. With notation as in Lemma 2.5.7 we have, for , D: E F ∈ (a) D∗ D and the natural map is an isomorphism ∼ . E ∈ c ⇐⇒ Dt E ∈ c E −→ DtDt E (b) D D and the natural map is an isomorphism ∼ . F ∈ c ⇐⇒ DF ∈ c F −→ DDF (c) D D∗ and the natural map is an isomorphism ∼ . F ∈ c ⇐⇒ Dt F ∈ c F −→ DtDt F ∼ Remark. The isomorphism F −→ DtDt F is a formal version of “Aﬃne Duality, ” see [AJL, §5.2].

Proof. For D , Lemma 2.5.7(b) gives D , so = Γ D∗. F ∈ c DF ∈ c DtF ∼ DF ∈ c Moreover, from the isomorphism tΓ = of Lemma 2.5.7(a) it follows that ∗ D F ∼ DF t(Dc ) Dc. The = implications in (a), (b) and (c) result, as do the ﬁrst parts Dof the =⊂ implications.⇐ Establishing⇒ the isomorphisms ∼ ∼ is a local problem, so DDF ←− F −→ DtDt F we may assume X aﬃne. Since the functors and t are bounded on D~c (see proof of Lemma 2.5.7(b)), and both take D into DD , thereforeD the functors and c ~c DD DtDt are bounded on Dc , and so [H1, p. 68, 7.1] (dualized) reduces the problem to the tautological case = X (cf. [H1, p. 258, Proposition 2.1].) F O For assertion (a) one may assume that = Γ ( Dc), so that there is a composed isomorphism (which one checks toEbe theFnaturalF ∈ map):

= Γ = Γ = t = t tΓ = t t . E F ∼ DDF 2.5.7(b)∼ D DF 2.5.7(a)∼ D D F D D E

Corollary 2.5.9. With the preceding notation, (a) The functor induces an involutive anti-equivalence of D with itself. D c (b) The functor induces quasi-inverse anti-equivalences between D and D∗. Dt c c Lemma 2.5.10. Let J be an ideal of definition of X. Then a complex D R ∈ c (resp. Dqct) is c-dualizing (resp. t-dualizing) iff for every n > 0 the complex R• ∈ n n R om ( X/J , ) is dualizing on the scheme X := (X, X/J ). H O R n O Proof. Remark (1) after Definition 2.5.1 makes it straightforward to see that • n if is either c- or t-dualizing on X then R om ( X/J , ) is dualizing on Xn. RFor the converse, to begin with, CorollaryH 5.2.3O gives R • n • n R om ( X/J , ) = R om ( X/J , Γ ), H O R H O R and it follows from Lemma 2.5.3 that it suffices to consider the t-dualizing case. So • n suppose that Dqct and that for all n, R om ( X/J , ) is dualizing on Xn. R ∈ H O R ∼ • Taking ˜ = in the proof of [Y, Theorem 5.6], one gets X R om ( , ). R R O −→ H R R It remains to check condition (iii) in Definition 2.5.1. We may assume to be • n R K-injective, so that := om ( X/J , ) is K-injective on X for all n. Then, Rn H O R n since Γ 0 = RΓ 0 = (Proposition 5.2.1(a)), XR ∼ XR ∼ R Hi HiΓ 0 Hi lim lim Hi (i Z). = X = −−→ n = −−→ n R ∼ R ∼ n R ∼ n R ∈ For each n, is quasi-isomorphic to a residual complex, which is an injective Rn Xn-complex vanishing in degrees outside a certain finite interval I := [a, b] ([H1, pp.O 304–306]). If m n, the same holds—with the same I—for the complex m≤ i m = omXn ( X/J , n). It follows that H = 0 for i / I. In particular, R ∼ +H O R R ∈ D . R ∈ qct 30 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

So now we we may assume that is a bounded-below complex of qct-injectives [Y, Theorem 4.8]. Then for any coherenR t torsion sheaf , the homologyA of M R om• ( , ) om• ( , ) om• ( , lim ) lim om• ( , ) X = X = X −−→ n = −−→ X n H M R 2.5.6∼ H M R ∼ H M n R ∼ n H M R vanishes in all degrees > b, as required by (iii). Proposition 2.5.11. Let f : X Y be a pseudo-proper map of noetherian formal schemes. → (a) If is a t-dualizing complex on Y, then f × is t-dualizing on X. R t R (b) If is a c-dualizing complex on Y, then f # is c-dualizing on X. R R Proof. (a) Let J be a defining ideal of X, and let I be a defining ideal of Y such j i that I X J. Let XJ := (X, X/J) , X and YI := (Y, Y/I) , Y be the resulting O ⊂ O → × O →• closed immersions. Example 6.1.3(4) shows that i = R om ( Y/I, ), which t R ∼ H O R is a dualizing complex on YI. Pseudo-properness of f means the map fIJ : XJ YI induced by f is proper, so as in [V, p. 396, Corollary 3] (hypotheses about →finite × Krull dimension being unnecessary here for the existence of ft , etc.), • × × × × × R om ( X/J, f ) = j f = (fIJ) i H O t R ∼ t t R ∼ t t R is a dualizing complex on XJ. The assertion is given then by Lemma 2.5.10. # (b) By Proposition 8.3.2, f Dc(X). By Corollary 6.1.5, Lemma 2.5.3(b), and the just-proved assertion (a),R ∈ RΓ 0 f # = f × = f ×RΓ 0 , X R ∼ t R ∼ t YR # 0 # is t-dualizing on X. So by Lemma 2.5.3(a), f = ΛXRΓ f is c-dualizing. R ∼ X R The following proposition generalizes [H1, p. 291, 8.5] (see also [H1, middle of p. 384] and [V, p. 396, Corollary 3]). Proposition 2.5.12. Let f : X Y be a pseudo-proper map of noetherian → formal schemes. Suppose that Y has a c-dualizing complex c, or equivalently (by ∗ R # Lemma 2.5.3 ), a t-dualizing complex t Dc (Y), so that f c is c-dualizing × R ∈ R (resp. f is t-dualizing) on X (Proposition 2.5.11). Define dualizing functors t Rt Y( ):= R om• ( , ), Y( ):= R om• ( , ), Dt − H Y − Rt Dc − H Y − Rc X( ):= R om• ( , f × ), X( ):= R om• ( , f # ). Dt − H X − t Rt Dc − H X − Rc Then there are natural isomorphisms f × = XLf ∗ Y , ( D∗(Y) D+(Y)), t E ∼ Dt Dt E E ∈ c ∩ f # = XLf ∗ Y ( D+(Y)). E ∼ Dc Dc E E ∈ c ∗ + + Y Proof. When Dc (Y) D (Y) (resp. Dc (Y)) set := t (resp. Y E ∈ ∩ E ∈ F D E− := c ). In either case, Dc(Y) (Proposition 2.5.8), and also D (Y)— inF theDfirstE case by remark F(2)∈following Definition 2.5.1 and LemmaF2.5.6,∈ in the second case by remark (1) following Definition 2.5.1. So, by Proposition 2.5.8, we need to find natural isomorphisms f × Y = XLf ∗ , t Dt F ∼ Dt F f # Y = XLf ∗ . Dc F ∼ Dc F Such isomorphisms are given by the next result—a generalization of [H1, p. 194, 8.8(7)]—for := (resp. ). G Rt Rc DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 31

Proposition 2.5.13. Let f : X Y be a map of noetherian formal schemes. Then for D−(Y) and D+(Y→) there are natural isomorphisms F ∈ c G ∈ R om• (Lf ∗ , f × ) ∼ f ×R om• ( , ), H X F t G −→ t H Y F G R om• (Lf ∗ , f # ) ∼ f #R om• ( , ). H X F G −→ H Y F G # × Proof. The second isomorphism follows from the ﬁrst, since f = Λft and since there are natural isomorphisms

• ∗ × • 0 • ∗ × ΛR omX(Lf , ft ) = R omX(RΓX X , R omX(Lf , ft )) H F G H • 0 O H ∗ × F G = R omX(RΓX X Lf , f )) ∼ H O ⊗= F t G = R om• (Lf ∗ , R om• (RΓ 0 , f × )) ∼ H X F H X XOX t G = R om• (Lf ∗ , f # ). H X F G For fixed the source and target of the first isomorphism in Proposition 2.5.13 F + are functors from D (Y) to Dqct(X) (see Proposition 3.2.4), right adjoint, respec- ∗ tively, to the functors Rf∗( Lf ) and Rf∗ ( Dqct(X)). The functorial E ⊗= F E ⊗= F E ∈ “projection” map ∗ Rf∗ Rf∗( Lf ), E ⊗= F → E ⊗= F is, by definition, adjoint to the natural composition

∗ ∗ ∗ ∗ Lf (Rf∗ ) Lf Rf∗ Lf Lf ; E ⊗= F → E ⊗= F → E ⊗= F and it will suﬃce to show that this projection map is an isomorphism. For this, the standard strategy is to localize to where Y is aﬃne, then use boundedness of some functors, and compatibilities with direct sums, to reduce to the trivial case = Y. Details appear, e.g., in [L4, pp. 123–125, Proposition 3.9.4], F O modulo the following substitutions: use D~c in place of Dqc, and for boundedness and direct sums use Lemma 5.1.4 and Propositions 3.4.3(b) and 3.5.2 below.

3. Direct limits of coherent sheaves on formal schemes. In this section we establish, for a locally noetherian formal scheme X, properties- of (X) needed in 4 to adapt Deligne’s proof of global Grothendieck Duality to A~c § the formal context. The basic result, Proposition 3.2.2, is that ~c(X) is plump (see opening remarks in 1), hence abelian, and so (being closed underA lim ) cocomplete, § −−→ i.e., it has arbitrary small colimits. This enables us to speak about D( (X)), and A~c to apply standard adjoint functor theorems to colimit-preserving functors on ~c(X). (See e.g., Proposition 3.2.3, Grothendieck Duality for the identity map of XA). The preliminary paragraph 3.1 sets up an equivalence of categories which allows us to reduce local questions about the (globally deﬁned) category ~c(X) to corresponding questions about quasi-coherent sheaves on ordinary noetherianA schemes. Paragraph 3.3 extends this equivalence to derived categories. As one immediate application, Corollary 3.3.4 asserts that the natural functor D( ~c(X)) D~c(X) is an equivalence of categories when X is properly algebraic, i.e., theAJ-adic→completion of a proper B-scheme with B a noetherian ring and J a B-ideal. This will yield a stronger version of Grothendieck Duality on such formal schemes—for D~c(X) rather than D( ~c(X)), see Corollary 4.1.1. We do not know whether such global results hold overAarbitrary noetherian formal schemes. 32 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

Paragraph 3.4 establishes boundedness for some derived functors, a condition which allows us to apply them freely to unbounded complexes, as illustrated, e.g., in Paragraph 3.5. 3.1. For X a noetherian ordinary scheme, (X) = (X) [GD, p. 319, 6.9.9]. A~c Aqc The inclusion jX : qc(X) (X) has a right adjoint QX : (X) qc(X), the “quasi-coherator,”A necessarily→ A left exact [I, p. 187, Lemme 3.2].A (See→ PropA osition 3.2.3 and Corollary 5.1.5 for generalizations to formal schemes.) Proposition 3.1.1. Let A be a noetherian adic ring with ideal of deﬁnition I, let f : X Spec(A) be a proper map, set Z := f −1Spec(A/I), and let 0 → 0 κ: X = X X /Z → ∗ be the formal completion of X along Z. Let Q:= QX be as above. Then κ induces equivalences of categories from (X) to (X) and from (X) to (X), both Aqc A~c Ac Ac with quasi-inverse Qκ∗.

Proof. For any quasi-coherent X -module the canonical maps are isomorphisms O G (3.1.2) Hi(X, ) ∼ Hi(X, κ κ∗ ) = Hi(X, κ∗ ) (i 0). G −→ ∗ G G ≥ ∗ (The equality holds because κ∗ transforms any ﬂasque resolution of κ into one of κ κ∗ .) G ∗ G For, if ( λ) is the family of coherent submodules of , ordered by inclusion, then X and GX being noetherian, one checks that (3.1.2) isGthe composition of the sequence of natural isomorphisms Hi(X, ) ∼ Hi(X, lim ) [GD, p. 319, (6.9.9)] −−→ λ G −→ λ G ∼ lim Hi(X, ) −−→ λ −→ λ G ∼ lim Hi(X, κ∗ ) [EGA, p. 125, (4.1.7)] −−→ λ −→ λ G ∼ Hi(X, lim κ∗ ) −−→ λ −→ λ G ∼ Hi(X, κ∗lim ) ∼ Hi(X, κ∗ ). −−→ λ −→ λ G −→ G Next, for any and in (X) the natural map is an isomorphism G H Aqc ∼ ∗ ∗ (3.1.3) Hom ( , ) HomX(κ , κ ) X G H −→ G H For, with as above, (3.1.3) factors as the sequence of natural isomorphisms Gλ Hom ( , ) ∼ lim Hom ( , ) X ←−− X λ G H −→ λ G H ∼ lim H0(X, om ( , )) ←−− X λ −→ λ H G H ∼ lim H0(X, κ∗ om ( , )) (see (3.1.2)) ←−− X λ −→ λ H G H ∼ 0 ∗ ∗ lim H (X, omX(κ , κ )) ←−− λ −→ λ H G H ∼ ∗ ∗ lim HomX(κ , κ ) ←−− λ −→ λ G H ∼ ∗ ∗ ∼ ∗ ∗ HomX(lim κ , κ ) HomX(κ , κ ). −−→ λ −→ λ G H −→ G H Finally, we show the equivalence of the following conditions, for (X): ∗ F ∈ A (1) The functorial map α( ): κ Qκ∗ (adjoint to the canonical map Qκ κ F ) is an isomorphism.F → F ∗F → ∗F DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 33

∗ ∼ (2) There exists an isomorphism κ with qc(X). (3) (X). G −→ F G ∈ A F ∈ A~c Clearly (1) (2); and (2) (3) because lim κ∗ ∼ κ∗ ( as before). −−→λ λ λ Since κ∗ comm⇒ utes with lim⇒ and induces an equivG −alence→ Gof categoriesG from −−→ c(X) to c(X) [EGA, p. 150, (5.1.6)], we see that (3) (2). A A ∗ ⇒ For qc(X), let β( ): Qκ∗κ be the canonical map (the unique G ∈ A G G ∗→ ∗G ∗ one whose composition with Qκ∗κ κ∗κ is the canonical map κ∗κ ). Then for any (X) we have theG →naturalG commutative diagramG → G H ∈ Aqc via β Hom( , ) Hom( , Qκ κ∗ ) H G −−−−→ H ∗ G ' '   Hom(κ∗y , κ∗ ) Hom( y, κ κ∗ ) H G −−−−→ H ∗ G where the left vertical arrow is an isomorphismf by (3.1.3), the right one is an isomorphism because Q is right-adjoint to qc(X) , (X), and the bottom arrow A ∗→ A is an isomorphism because κ∗ is right-adjoint to κ ; so “via β” is an isomorphism for all , whence β( ) is an isomorphism. The implication (2) (1) follows now H G ⇒ from the easily checked fact that α(κ∗ )◦ κ∗β( ) is the identity map of κ∗ . G G G We see also that Qκ∗( c(X)) c(X), since by [EGA, p. 150, (5.1.6)] every A ∗ ⊂ A c(X) is isomorphic to κ for some c(X), and β( ) is an isomorphism. F ∈ A G ∗ G ∈ A G Thus we have the functors κ : qc(X) ~c(X) and Qκ∗ : ~c(X) qc(X), both of which preserve coherence, andA the functorial→ A isomorphismsA → A α( ): κ∗Qκ ∼ ( (X)); β( ): ∼ Qκ κ∗ ( (X)). F ∗F −→ F F ∈ A~c G G −→ ∗ G G ∈ Aqc Proposition 3.1.1 results.

Since κ∗ is right-exact, we deduce: Corollary 3.1.4. For any aﬃne noetherian formal scheme X, (X) iﬀ F ∈ A~c is a cokernel of a map of free X-modules (i.e., direct sums of copies of X). F O O Corollary 3.1.5. For a locally noetherian formal scheme X, (X) (X), A~c ⊂ Aqc i.e., any lim of coherent X-modules is quasi-coherent. −−→ O Proof. Being local, the assertion follows from Corollary 3.1.4.

Corollary 3.1.6 (cf. [Y, 3.4, 3.5]). For a locally noetherian formal scheme X let and be quasi-coherent X-modules. Then: F G O (a) The kernel, cokernel, and image of any X-homomorphism are quasi-coherent. O F → G (b) is coherent iff is locally finitely generated. (c) IfF is coherent andF is a sub- or quotient module of then is coherent. (d) If F is coherent thenG om( , ) is quasi-coherent; andFif alsoG is coherent then om(F , ) is coherent. H(For aFgenerG alization, see Proposition 3.2.4G .) H F G Proof. The questions being local, we may assume X = Spf(A) (A noetherian adic), and, by Corollary 3.1.4, that and are in (X). Then, κ∗ being exact, F G A~c Proposition 3.1.1 with X := Spec(A) and f0 := identity reduces the problem to noting that the corresponding statements about coherent and quasi-coherent sheaves on X are true. (These statements are in [GD, p. 217, Cor. (2.2.2) and p. 228, 34 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

(2.7.1)]. Observe also that if F and G are X -modules with F coherent then § ∗ ∗ ∗ O omX(κ F, κ G) = κ om (F, G).) H ∼ H X Corollary 3.1.7. For a locally noetherian formal scheme X, any (X) F ∈ A~c is the lim of its coherent X-submodules. −−→ O Proof. Note that by Corollary 3.1.6(a) and (b) the sum of any two coherent submodules of is again coherent. By deﬁnition, = lim with coherent, F F −−→µ Fµ Fµ and from Corollary 3.1.6(a) and (b) it follows that the canonical image of µ is a coherent submodule of , whence the conclusion. F F Corollary 3.1.8. For any aﬃne noetherian formal scheme X, any ~c(X) and any i > 0, F ∈ A Hi(X, ) = 0. F Proof. Taking f in Proposition 3.1.1 to be the identity map, we have = κ∗ 0 F ∼ G with quasi-coherent; and so by (3.1.2), Hi(X, ) = Hi(Spec(A), ) = 0. G F ∼ G

3.2. Proposition 3.1.1 will now be used to show, for locally noetherian formal schemes X, that ~c(X) (X) is plump, and that this inclusion has a right adjoint, extending to derivAed categories.⊂ A Lemma 3.2.1. Let X be a noetherian formal scheme, let (X), and let F ∈ Ac ( α , γαβ : β α)α,β∈Ω be a directed system in c(X). Then for every q 0 the naturG al mapG is→anG isomorphism A ≥ lim Extq( , ) ∼ Extq( , lim ). −−→ α −−→ α α F G −→ F α G Proof. For an X-module , let E( ) denote the usual spectral sequence O M M Epq( ):= Hp(X, xtq( , )) Extp+q( , ). 2 M E F M ⇒ F M It suﬃces that the natural map of spectral sequences be an isomorphism lim E( ) ∼ E(lim ) (lim := lim ), −−→ α −−→ α −−→ −−→ G −→ G α pq and for that we need only check out the E2 terms, i.e., show that the natural maps lim Hp(X, xtq( , )) Hp(X, lim xtq( , )) Hp(X, xtq( , lim )) −−→ E F Gα → −−→ E F Gα → E F −−→ Gα are isomorphisms. The ﬁrst one is, because X is noetherian. So we need only show that the natural map is an isomorphism lim xtq( , ) ∼ xtq( , lim ). −−→ E F Gα −→ E F −−→ Gα For this localized question we may assume that X = Spf(A) with A a noetherian adic ring. By Proposition 3.1.1 (with f0 the identity map of X := Spec(A)) there is a coherent X -module F and a directed system (Gα , gαβ : Gβ Gα)α,β∈Ω of co- O ∗ ∗ → ∗ herent X -modules such that = κ F, α = κ Gα , and γα,β = κ gα,β. Then the well-knoOwn natural isomorphismsF (see [EGAG , (Chapter 0), p. 61, Prop. (12.3.5)]— or the proof of Corollary 3.3.2 below) lim xtq ( , ) ∼ lim κ∗ xtq (F, G ) ∼ κ∗lim xtq (F, G ) −−→ E X F Gα −→ −−→ E X α −→ −−→ E X α ∼ κ∗ xtq (F, lim G ) ∼ xtq (κ∗F, κ∗lim G ) ∼ xtq ( , lim ) −→ E X −−→ α −→ E X −−→ α −→ E X F −−→ Gα give the desired conclusion. DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 35

Proposition 3.2.2. Let X be a locally noetherian formal scheme. If

F1 → F2 → F → F3 → F4 is an exact sequence of X-modules and if , , and are all in (X) O F1 F2 F3 F4 Aqc (resp. ~c(X)) then qc(X) (resp. ~c(X)). Thus qc(X) and ~c(X) are plump— hence Aabelian—subcFate∈goriesA of (X)A, and both D A(X) and its Asubcategory D (X) A qc ~c are triangulated subcategories of D(X). Furthermore, ~c(X) is closed under arbitrary small (X)-colimits. A A Proof. Part of the case is covered by Corollary 3.1.6(a), and all of it by Aqc [Y, Proposition 3.5]. At any rate, since every quasi-coherent X-module is locally O in ~c qc (see Corollaries 3.1.4 and 3.1.5), it suffices to treat the ~c case. ALet⊂ usA first show that the kernel of an map A K A~c ψ : lim = = lim ( , (X)) −−→β Hβ H → G −−→α Gα Gα Hβ ∈ Ac is itself in (X). It will suffice to do so for the kernel of the composition A~c Kβ ψ ψ : natural , β Hβ −−−−−→ H −→ G since = lim . K −−→β Kβ By the case q = 0 of Corollary 3.2.1, there is an α such that ψβ factors as ψ βα natural ; Hβ −−−→ Gα −−−−−→ G 0 and then with 0 (α > α) the (coherent) kernel of the composed map Kβα ψβα natural 0 Hβ −−−→ Gα −−−−−→ Gα we have = lim 0 0 (X). Kβ −−→α Kβα ∈ A~c Similarly, we find that coker(ψ) (X). Being closed under small direct sums, ∈ A~c then, ~c(X) is closed under arbitrary small (X)-colimits [M1, Corollary 2, p. 109]. ConsiderationA of the exact sequence A 0 coker( ) ker( ) 0 −→ F1 → F2 −→ F −→ F3 → F4 −→ now reduces the original question to where = = 0. Since is the lim of 1 4 3 −−→ its coherent submodules (Corollary 3.1.7) andF isFthe lim of theF inverse images F −−→ of those submodules, we need only show that each such inverse image is in ~c(X). Thus we may assume coherent (and = lim with coherent). A F3 F2 −−→α Gα Gα The exact sequence 0 0 represents an element → F2 → F → F3 → η Ext1( , ) = Ext1( , lim ); ∈ F3 F2 F3 −−→α Gα and by Corollary 3.2.1, there is an α such that η is the natural image of an element 1 ηα Ext ( 3, α), represented by an exact sequence 0 α α 3 0. Then∈ isFcoherenG t, and by [M2, p. 66, Lemma 1.4], we→havGe an→isomorphismF → F → Fα ∼ . F −→ F2 ⊕Gα Fα Thus is the cokernel of a map in (X), and so as above, (X). F A~c F ∈ A~c Proposition 3.2.3. On a locally noetherian formal scheme X, the inclusion functor j : (X) (X) has a right adjoint QX : (X) (X); and RQ is X A~c → A A → A~c X right-adjoint to the natural functor D( ~c(X)) D(X). In particular, if κ: X X A∗ → ∗ → is as in Proposition 3.1.1 then QX ∼= κ QX κ∗ and RQX ∼= κ RQX κ∗ . 36 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

Proof. Since ~c(X) has a small family of (coherent) generators, and is closed under arbitrary smallA (X)-colimits, the existence of Q follows from the Special A X Adjoint Functor Theorem ([F, p. 90] or [M1, p. 126, Corollary]).15 In an abelian category , a complex J is, by definition, K-injective if for each A • exact -complex G, the complex HomA(G, J) is exact too. Since jX is exact, it folloAws that its right adjoint Q transforms K-injective (X)-complexes into X A K-injective ~c(X)-complexes, whence the derived functor RQX is right-adjoint to the natural Afunctor D( (X)) D(X) (see [Sp, p. 129, Proposition 1.5(b)]). A~c → The next assertion is a corollary of Proposition 3.1.1: any ~c(X) is iso- ∗ M ∈ A morphic to κ for some qc(X), and then for any (X) there are natural isomorphismsG G ∈ A N ∈ A ∗ HomX(j , ) = HomX(j κ , ) XM N ∼ X G N = Hom (j , κ ) = Hom ( , Q κ ) ∼ X X G ∗ N ∼ Aqc(X) G X ∗ N ∗ ∗ ∗ = Hom X (κ , κ Q κ ) = Hom X ( , κ Q κ ). ∼ A~c( ) G X ∗ N ∼ A~c( ) M X ∗ N ∗ Moreover, since κ∗ has an exact left adjoint (viz. κ ), therefore, as above, κ∗ transforms K-injective (X)-complexes into K-injective (X)-complexes, and it follows A ∗ A at once that RQX ∼= κ RQX κ∗. Let X be a locally noetherian formal scheme. A property P of sheaves of modules is local if it is defined on (U) for arbitrary open subsets U of X, and is such that for any (U) andAany open covering (U ) of U, P( ) holds iff E ∈ A α E P( Uα) holds for all α. E|For example, coherence and quasi-coherence are both local properties—to which by Proposition 3.2.2, the following Proposition applies. Proposition 3.2.4. Let X be a locally noetherian formal scheme, and let P be a local property of sheaves of modules. Suppose further that for all open U X the ⊂ full subcategory P(U) of (U) whose objects are all the (U) for which P( ) A A E ∈ A + E holds is a plump subcategory of (U). Then for all D−(X) and D (X), A F ∈ c G ∈ P it holds that R om•( , ) D+(X). H F G ∈ P Proof. Plumpness implies that DP(X) is a triangulated subcategory of D(X), as is D (X), so [H1, p. 68, Prop. 7.1] gives a “way-out” reduction to where and c F G are X-modules. The question being local on X, we may assume X affine and replaceO by a quasi-isomorphic bounded-above complex • of finite-rank free F • F• • • X-modules, see [GD, p. 427, (10.10.2)]. Then R om ( , ) = om ( , ), andO the conclusion follows easily. H F G H F G 3.3. Proposition 3.2.3 applies in particular to any noetherian scheme X. When equivalence of categories j X is separated, jX induces an X : D( qc(X)) ∼= Dqc(X), with quasi-inverse RQ . (See [H1, p. 133, Corollary 7.19]Afor bounded-below X |Dqc(X) complexes, and [BN, p. 230, Corollary 5.5] or [AJL, p. 12, Proposition (1.3)] for the general case.) We do not know if such an equivalence, with “~c ” in place of “qc,” always holds for separated noetherian formal schemes. The next result will at least take care of the “properly algebraic” case, see Corollary 3.3.4.

15 It follows that A~c(X) is closed under all A(X)-colimits (not necessarily small): if F is any functor into A~c(X) and F ∈ A(X) is a colimit of jX ◦ F, then QXF is a colimit of F, and the ∼ natural map is an isomorphism F −→ jXQXF. (Proof: exercise, given in dual form in [F, p. 80].) DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 37

Proposition 3.3.1. In Proposition 3.1.1, the functor κ∗ : D(X) D(X) in- → duces equivalences from Dqc(X) to D~c(X) and from Dc(X) to Dc(X), both with quasi-inverse RQκ∗ (where RQ stands for jX ◦ RQX ). ∗ ∗ Proof. Since κ is exact, Proposition 3.1.1 implies that κ (Dqc(X)) D~c(X) and κ∗(D (X)) D (X). So it will be enough to show that: ⊂ c ⊂ c (1) If D (X) then the functorial D(X)-map κ∗RQκ adjoint to the F ∈ ~c ∗F → F natural map RQκ∗ κ∗ is an isomorphism. F → F ∼ ∗ (2) If Dqc(X) then the natural map RQκ∗κ is an isomorphism. (3) If G ∈ D (X) then RQκ D (X).G −→ G F ∈ c ∗F ∈ c Since D~c(X) is triangulated (Proposition 3.2.2), we can use way-out reasoning [H1, p. 68, Proposition 7.1 and p. 73, Proposition 7.3] to reduce to where or is a single sheaf. (For bounded-below complexes we just need the obvious factsF thatG ∗ κ and the restriction of RQκ∗ to D~c(X) are both bounded-below (= way-out right) functors. For unbounded complexes, we need those functors to be bounded-above ∗ as well, which is clear for the exact functor κ , and will be shown for RQκ∗ D~c(X) in Proposition 3.4.4 below.) | ∗ Any ~c(X) is isomorphic to κ for some qc(X); and one checks F ∈ A ∗ ∗G ∗ G ∈∗A that the natural composed map κ κ RQκ∗κ κ is the identity, whence X G → G → G (2) (1). Moreover, if c( ) then ∼= Qκ∗ c(X), whence (2) (3). ⇒Now a map ϕ : F ∈ Ain D+ (X) isGan isomorphismF ∈ A iﬀ ⇒ G1 → G2 qc ( ): the induced map HomD(X)( [ n], 1) HomD(X)( [ n], 2) ∗ is an E − G → E − G isomorphism for every (X) and every n Z. E ∈ Ac ∈ (For, if is the vertex of a triangle with base ϕ, then ( ) says that for all , n, Hom V( [ n], V) = 0; but if ϕ is not an isomorphism, ∗i.e., V has non-vanishingE D(X) E − homology, say H n( ) = 0 and Hi( ) = 0 for all i < n, then the inclusion into H n( ) of any coherent non-zeroV 6 submoduleV gives a non-zero map [ n] V.) So for (2)V it’s enough to check that the naturalEcomposition E − → Hom ( [ n], ) Hom ( [ n], RQκ κ∗ ) D(X) E − G −→ D(X) E − ∗ G ∼ ∗ ∼ ∗ ∗ Hom ( [ n], κ κ ) Hom X (κ [ n], κ ) −→ D(X) E − ∗ G −→ D( ) E − G n ∼ n ∗ ∗ is the isomorphism ExtX ( , ) ExtX(κ , κ ) in the following consequence of (3.1.2): E G −→ E G

Corollary 3.3.2. With κ: X X as in Proposition 3.1.1 and Dqc(X), the natural map RΓ(X, ) RΓ(→X, κ∗ ) is an isomorphism. In particular,L ∈ for − +L → L n n ∗ ∗ Dc (X) and Dqc(X) the natural map ExtX ( , ) ExtX(κ , κ ) is an isomorphism.E ∈ G ∈ E G → E G

Proof. After “way-out” reduction to the case where qc(X) (the RΓ’s are bounded, by Corollary 3.4.3(a) below), the first assertionL ∈ isAgiven by (3.1.2). To get the second assertion, take := R om• ( , ) (which is in D+ (X), [H1, L H X E G qc p. 92, Proposition 3.3]), so that κ∗ = R om• (κ∗ , κ∗ ) (as one sees easily after L ∼ H X E G way-out reduction to where and are X -modules, and further reduction to where X is affine, so that hasE a resolutionG Oby finite-rank free modules. . .). E Definition 3.3.3. A formal scheme X is said to be properly algebraic if there exist a noetherian ring B, a B-ideal J, a proper B-scheme X, and an isomorphism from X to the J-adic completion of X. 38 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

Corollary 3.3.4. On a properly algebraic formal scheme X the natural functor j : D( (X)) D (X) is an equivalence of categories with quasi-inverse RQ ; X A~c → ~c X and therefore j ◦ RQ is right-adjoint to the inclusion D (X) , D(X). X X ~c → Proof. If X is properly algebraic, then with A:= J-adic completion of B and I := JA, it holds that X is the I-adic completion of X B A, and so we may assume the hypotheses and conclusions of Proposition 3.1.1. ⊗We have also, as above, the equivalence of categories jX : D( qc(X)) Dqc(X); and so the assertion follows from Propositions 3.3.1 and 3.2.3.A → Proposition 3.3.5. For a map g : Z X of locally noetherian formal schemes, → Lg∗(D (X)) D (Z). ~c ⊂ qc If X is properly algebraic, then Lg∗(D (X)) D (Z). ~c ⊂ ~c Proof. The first assertion, being local on X, follows from the second. Assum- ing X properly algebraic we may, as in the proof of Corollary 3.3.4, place ourselves in the situation of Proposition 3.1.1, so that any D~c(X) is, by Corollary 3.3.4 ∗ G ∈ and Proposition 3.1.1, isomorphic to κ for some Dqc(X). By [AJL, p. 10, Proposition (1.1)]), is isomorphic to aElim of bounded-abE ∈ ove quasi-coherent flat E −−→ complexes (see the very end of the proof of ibid.); and therefore = κ∗ is iso- ∗ G ∼ ∗ E morphic to a K-flat complex of ~c(X)-objects. Since Lg agrees with g on K-flat complexes, and g∗( (X)) A(Z), we are done. A~c ⊂ A~c Remarks 3.3.6. (1) Let X be a properly algebraic formal scheme (necessarily 0 0 noetherian) with ideal of definition I, and set I := H (X, I) A:= H (X, X). Then A is a noetherian I-adic ring, and X is Spf(A)-isomorphic⊂to the I-adic Ocompletion of a proper A-scheme. Hence X is proper over Spf(A), via the canonical map given by [GD, p. 407, (10.4.6)]. Indeed, with B, J and X as in Definition 3.3.3, [EGA, p. 125, Theorem (4.1.7)] implies that the topological ring 0 n 0 n A = lim H (X, X/I X) = lim H (X, /I ) ←−− ←−− X X n>0 O O n>0 O O 0 is the J-adic completion of the noetherian B-algebra A0 := H (X, X ), and that the J-adic and I-adic topologies on A are the same; and then XOis the I-adic completion of X A0 A. (2) It follows⊗that a quasi-compact formal scheme X is properly algebraic iff so is each of its connected components. (3) While (1) provides a less relaxed characterization of properly algebraic formal schemes than Definition 3.3.3, Corollary 3.3.8 below provides a more relaxed one.

Lemma 3.3.7. Let X be a locally noetherian scheme, 1 2 quasi-coherent -ideals, Z the support of / , and X the completion XI ⊂(iI= 1, 2). Suppose OX i OX Ii i /Zi that X is an ideal of deﬁnition of X . Then X is a union of connected I1O 2 2 2 components of X1 (with the induced formal-subscheme structure).

Proof. We need only show that Z2 is open in Z1. Locally we have a noetherian ring A and A-ideals I J equal to their own radicals such that with Aˆ ⊂ the J-adic completion, J nAˆ IAˆ for some n > 0; and we want the natural map ⊂ DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 39

2 A/I A/I A/J to be flat. (For then with L := J/I, L/L = Tor1 (A/J, A/J) = 0, whence (1 `)L = (0) for some ` L, whence ` = `2 and L = `(A/I), so that − ∈ A/I ∼= L (A/J) and Spec(A/J) , Spec(A/I) is open.) So it ×suffices that the localization→ (A/I ) (A/J ) = A/J by the multi- 1+J → 1+J plicatively closed set 1 + J be an isomorphism, i.e., that its kernel J(A/I )1+J be nilpotent (hence (0), since A/I is reduced.) But this is so because the natural map A Aˆ is faithfully flat, and therefore J nA IA . 1+J → 1+J ⊂ 1+J Corollary 3.3.8. Let A be a noetherian ring, let I be an A-ideal, and let Aˆ be the I-adic completion of A. Let f0 : X Spec(A) be a separated finite-type → −1 scheme-map, let Z be a closed subscheme of f0 (Spec(A/I)), let X = X/Z be the completion of X along Z, and let f : X Spf(Aˆ) be the formal-scheme map → induced by f0 : X:= X X /Z −−−−→ f f0   Spfy(Aˆ) Spec(y A) −−−−→ If f is proper (see 1.2.2) then X is properly algebraic. § Proof. Consider a compactification of f0 (see [Lu¨, Theorem 3.2]):

¯ X , X f0 Spec(A). op→en prop−→er Since f is proper, therefore Z is proper over Spec(A), hence closed in X. Thus we may replace f0 by f¯0 , i.e., we may assume f0 proper. Since f, being proper, is adic, −1 Lemma 3.3.7, with Z2 := Z and Z1 := f0 (Spec(A/I)), shows that X is a union of connected components of the properly algebraic formal scheme X/Z1 . Conclude by Remark 3.3.6(2).

3.4. To deal with unbounded complexes we need the following boundedness results on certain derived functors. (See, e.g., Propositions 3.5.1 and 3.5.3 below.) 3.4.1. Refer to 1.2.2 for the definitions of separated, resp. affine, maps. § A formal scheme X is separated if the natural map fX : X Spec(Z) is sep- → arated, i.e., for some—hence any—ideal of definition J, the scheme (X, X/J) is separated. For example, any locally noetherian affine formal scheme is separated.O A locally noetherian formal scheme X is affine if and only if the map fX is affine, i.e., for some—hence any—ideal of definition J, the scheme (X, X/J) is affine. Hence the intersection V V0 of any two affine open subsets of a separatedO locally noetherian formal scheme∩ Y is again affine. In other words, the inclusion V , Y is an affine map. More generally, if f : X Y is a map of locally noetherian formal→ schemes, if Y is separated, and if V and V0→are affine open subsets of Y and X respectively, then f −1V V0 is affine [GD, p. 282, (5.8.10)]. ∩ Lemma 3.4.2. If g : X Y is an affine map of locally noetherian formal → i schemes, then every ~c(X) is g∗-acyclic, i.e., R g∗ = 0 for all i > 0. M ∈ A Mi More generally, if D~c(X) and e Z are such that H ( ) = 0 for all i e, then Hi(Rg ) = 0Gfor∈ all i e. ∈ G ≥ ∗G ≥ 40 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

i i −1 Proof. R g∗ is the sheaf associated to the presheaf U H (g (U), ), (U open in Y) [EGAM , Chap. 0, (12.2.1)]. If U is affine then so is7→g−1(U) X,Mand Corollary 3.1.8 gives Hi(g−1(U), ) = 0 for all i > 0. ⊂ Now consider in K(X) a quasi-isomorphismM I where I is a “special” G → ≥e inverse limit of injective resolutions I−e of the truncations (see (13)), so that Hi(Rg ) is the sheaf associated to the presheaf U HiG(Γ(g−1U, I)), see [Sp, ∗G 7→ p. 134, 3.13]. If C−e is the kernel of the split surjection I−e I1−e then C−e[e] e → is an injective resolution of H ( ) ~c(X), and so for any affine open U Y i −1 G ∈ A ⊂ and i > e, H (Γ(g U, C−e)) = 0. Applying [Sp, p. 126, Lemma], one finds then that i −1 i −1 for i e the natural map H (Γ(g U, I)) H (Γ(g U, I−e)) is an isomorphism. ≥ i → ≥e X Consequently if H ( ) = 0 for all i e (whence I−e ∼= = 0 in D( )) then Hi(Γ(g−1U, I)) = 0. G ≥ G Proposition 3.4.3. Let X be a noetherian formal scheme. Then: (a) The functor RΓ(X, ) is bounded-above on D~c(X). In other words, there − i is an integer e 0 such that if D~c(X) and H ( ) = 0 for all i i0 then i ≥ G ∈ G ≥ H (RΓ(X, )) = 0 for all i i0 + e. (b) For− any formal-scheme≥ map f : X Y with Y quasi-compact, the func- → tor Rf∗ is bounded-above on D~c(X), i.e., there is an integer e 0 such that if D (X) and Hi( ) = 0 for all i i then Hi(Rf ) = 0 for al≥l i i + e. G ∈ ~c G ≥ 0 ∗G ≥ 0 Proof. Let us prove (b). (The proof of (a) is the same, mutatis mutandis.) Suppose first that X is separated, see 3.4.1. Since Y has a finite affine open cover § and Rf∗ commutes with open base change, we may assume that Y itself is affine. Let n(X) be the least positive integer n such that there exists a finite affine open n cover X = i=1Xi , and let us show by induction on n(X) that e:= n(X) 1 will do. The case∪ n(X) = 1 is covered by Lemma 3.4.2. So assume that n:=−n(X) 2, n n ≥ let X = i=1Xi be an affine open cover, and let u1 : X1 , X, u2 : i=2 Xi , X, n ∪ → ∪ → u3 : i=2 (X1 Xi) , X be the respective inclusion maps. Note that X1 Xi is affine∪ because∩X is separated.→ So by the inductive hypothesis, the assertion∩ holds for the maps fi := f ◦ ui (i = 1, 2, 3). Now apply the ∆-functor Rf∗ to the “Mayer-Vietoris” triangle Ru u∗ Ru u∗ Ru u∗ +1 G −→ 1∗ 1G ⊕ 2∗ 2G −→ 3∗ 3G −→ (derived from the standard exact sequence 0 u u∗ u u∗ u u∗ 0 → E → 1∗ 1E ⊕ 2∗ 2E → 3∗ 3E → where is a K-injective resolution) to get the D(Y)-triangle G → E Rf Rf u∗ Rf u∗ Rf u∗ +1 ∗G −→ 1∗ 1G ⊕ 2∗ 2G −→ 3∗ 3G −→ whose associated long exact homology sequence yields the assertion for f. The general case can now be disposed of with a similar Mayer-Vietoris induction on the least number of separated open subsets needed to cover X. Proposition 3.4.4. Let X be a separated noetherian scheme, let Z X be a closed subscheme, and let κ : X = X X be the completion map.⊂ Then X /Z → the functor RQX κ∗ is bounded-above on D~c(X).

Proof. Set κ:= κX. Let n(X) be the least number of affine open subschemes needed to cover X. When X is affine, QX is the sheafification of the global section ∗ functor, and since κ∗ is exact and, being right adjoint to the exact functor κ , DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 41 preserves K-injectivity, we find that for any D(X), RQ κ is the sheafifica- F ∈ X ∗F tion of the complex RΓ(X, κ∗ ) = RΓ(X, ). Thus Proposition 3.4.3(a) yields the desired result for n(X) = 1. F F Proceed by induction when n(X) > 1, using a “Mayer-Vietoris” argument as in the proof of Proposition 3.4.3. The enabling points are that if v : V , X is an open immersion with n(V ) < n(X), giving rise to the natural commutativ→e diagram κ V =: V V V /Z∩V −−−−→ vˆ v   Xy Xy −−κ−X−→ then there are natural isomorphisms, for D (X) and vqc : (V ) (X) F ∈ ~c ∗ Aqc → Aqc the restriction of v∗ : RQ κ Rvˆ vˆ∗ = RQ Rv κ vˆ∗ = RvqcRQ κ vˆ∗ , X X∗ ∗ F ∼ X ∗ V∗ F ∼ ∗ V V∗ F ∗ and the functor RQV κV∗vˆ is bounded-above, by the inductive hypothesis on qc n(V ) < n(X), as is Rv∗ , by the proof of [AJL, p. 12, Proposition (1.3)].

3.5. Here are some examples of how boundedness is used. Proposition 3.5.1. Let f : X Y be a proper map of noetherian formal schemes. Then → Rf D (X) D (Y) and Rf D (X) D (Y). ∗ c ⊂ c ∗ ~c ⊂ ~c Proof. For a coherent X-module , Rf∗ Dc(Y) [EGA, p. 119, (3.4.2)]. O M Mi ∈ Since X is noetherian, the homology functors H Rf commute with lim on X- ∗ −−→ O modules, whence Rf D (Y) for all (X). Rf being bounded on D (X) ∗N ∈ ~c N ∈ A~c ∗ ~c (Proposition 3.4.3(b)), way-out reasoning [H1, p. 74, (iii)] completes the proof. Proposition 3.5.2. Let f : X Y be a map of quasi-compact formal schemes, X → with noetherian. Then the functor Rf∗ D~c(X) commutes with small direct sums, i.e., for any small family ( ) in D (X) the| natural map Eα ~c (Rf ) Rf ( ) ⊕α ∗Eα → ∗ ⊕α Eα is a D(Y)-isomorphism. Proof. It suffices to look at the induced homology maps in each degree, i.e., setting Rif := HiRf (i Z), we need to show that the natural map ∗ ∗ ∈ (Rif ) ∼ Rif ( ). ⊕α ∗Eα −→ ∗ ⊕α Eα is an isomorphism. For any D~c(X) and any integer e 0, the vertex of a triangle based on F ∈ ≥ ≥i−e G j the natural map ti−e from to the truncation (see (13)) satisfies H ( ) = 0 F F i−1G for all j i e 1; so if e is the integer in Proposition 3.4.3(b), then R f∗ = Rif = ≥0, and− the− map induced by t is an isomorphism G ∗G i−e Rif ∼ Rif ≥i−e. ∗F −→ ∗F ≥i−e We can therefore replace each α by α , i.e., we may assume that the α are uniformly bounded below. E E E 42 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

We may assume further that each complex α is injective, hence f∗-acyclic (i.e., ∼E the canonical map is an isomorphism f∗ α Rf∗ α). Since X is noetherian, Rif commutes with direct sums; and so eacE h−→componenE t of is an f -acyclic ∗ ⊕α Eα ∗ X-module. This implies that the bounded-below complex α α is itself f∗-acyclic. ThO us in the natural commutative diagram ⊕ E (f ) f ( ) ⊕α ∗Eα −−−−→ ∗ ⊕α Eα f ' '   (Ryf ) Rf (y ) ⊕α ∗Eα −−−−→ ∗ ⊕α Eα the top and both sides are isomorphisms, whence so is the bottom.

The following Proposition generalizes [EGA, p. 92, Theorem (4.1.5)].

Proposition 3.5.3. Let f0 : X Y be a proper map of locally noetherian schemes, let W Y be a closed subset,→ let Z := f −1W, let κ : Y = Y Y ⊂ 0 Y /W → and κ : X = X X be the respective (flat) completion maps, and let f : X Y X /Z → → be the map induced by f0 . Then for Dqc(X) the map θE adjoint to the natural composition E ∈ Rf Rf κ κ∗ ∼ κ Rf κ∗ 0∗E −→ 0∗ X∗ XE −→ Y∗ ∗ XE is an isomorphism θ : κ∗ Rf ∼ Rf κ∗ . E Y 0∗E −→ ∗ XE Proof. We may assume Y affine, say Y = Spec(A), and then W = Spec(A/I) for some A-ideal I. Let Aˆ be the I-adic completion of A, so that there is a natural cartesian diagram k X Aˆ =:X X X ⊗A 1 −−−−→ f1 f0   y y Spec(Aˆ) =: Y1 Y −−k−Y−→ Here k is flat, and the natural map is an isomorphism k∗ Rf ∼ Rf k∗ : Y Y 0∗E −→ 1∗ X E since Rf0∗ (resp. Rf1∗) is bounded-above on Dqc(X) (resp. Dqc(X1)), see Proposi- tion 3.4.3(b), way-out reasoning reduces this assertion to the well-known case where is a single quasi-coherent X -module. Simple considerations show then that we E O ∗ ˆ can replace f0 by f1 and by kX ; in other words, we can assume A = A. From Proposition 3.5.1E it folloEws that Rf D (Y ) and Rf κ∗ D (Y). 0∗E ∈ qc ∗ XE ∈ ~c Recalling the equivalences in Proposition 3.3.1, we see that any D~c(Y) is ∗ ∗ F ∈ isomorphic to κY 0 for some 0 Dqc(Y ) (so that Lf0 0 Dqc(X)), and that there is a sequenceF of natural isomorphismsF ∈ F ∈

∗ ∼ HomY( , κ Rf ) Hom ( , Rf ) F Y 0∗E −→ Y F0 0∗E ∼ Hom (Lf ∗ , ) −→ X 0 F0 E ∼ ∗ ∗ ∗ HomX(κ Lf , κ ) −→ X 0 F0 XE ∼ ∗ ∗ ∗ ∼ ∗ HomX(Lf κ , κ ) HomY( , Rf κ ). −→ YF0 XE −→ F ∗ XE The conclusion follows. DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 43

4. Global Grothendieck Duality. Theorem 4.1. Let f : X Y be a map of quasi-compact formal schemes, with X noetherian, and let j : D→( (X)) D(X) be the natural functor. Then the A~c → ∆-functor Rf∗ ◦ j has a right ∆-adjoint. In fact there is a bounded-below ∆-functor f × : D(Y) D ( (X)) and a map of ∆-functors τ : Rf jf × 1 such that for → A~c ∗ → all D( ~c(X)) and D(Y), the composed map (in the derived category of abelianG ∈groupsA ) F ∈

• × natural • × RHom ( , f ) RHom (Rf∗ j , Rf∗ jf ) A~c(X) G F −−−−−→ A(Y) G F via τ RHom• (Rf j , ) −−−−→ A(Y) ∗ G F is an isomorphism. With Corollary 3.3.4 this gives:

Corollary 4.1.1. If X is properly algebraic, the restriction of Rf∗ to D~c(X) has a right ∆-adjoint (also to be denoted f × when no confusion results). Remarks. 1. Recall that over any abelian category in which each complex has a K-injective resolution ρ( ), we can set A F F RHom• ( , ):= Hom• ( , ρ( )) ( , D( )); A G F A G F G F ∈ A and there are natural isomorphisms HiRHom• ( , ) = Hom ( , [i]) (i Z). A G F ∼ D(A) G F ∈ 2. Application of homology to the second assertion in the Theorem reveals that it is equivalent to the first one. 3. We do not know in general (when X is not properly algebraic) that the × functor j is fully faithful—j has a right adjoint (identity) ∼= RQX (see Proposi- tion 3.2.3), but it may be that for some ~c(X) the natural map RQXj is not an isomorphism. E ∈ A E → E 4. For a proper map f0 : X Y of ordinary schemes it is customary to ! × → write f0 instead of f0 . (Our extension of this notation to maps of formal schemes— introduced immediately after Definition 7.3—is not what would be expected here.) 5. Theorem 4.1 includes the case when X and Y are ordinary noetherian schemes. (In fact the proof below applies with minor changes to arbitrary maps of quasi- compact, quasi-separated schemes, cf. [L4, Chapter 4].) The next Corollary relates the formal situation to the ordinary one. Corollary 4.1.2. Let A be a noetherian adic ring with ideal of definition I, set Y := Spec(A) and W := Spec(A/I) Y . Let f0 : X Y be a proper map and −1 ⊂ → set Z := f0 W, so that there is a commutative diagram κ X:= X X X /Z −−−−→ f f0   Y:= Spfy(A) Yy −−−κY−→ with κX and κY the respective (flat) completion maps, and f the (proper) map induced by f0 . 44 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

Then the map adjoint to the natural composition Rf κ∗ f !κ 3.5.3 κ∗ Rf f !κ κ∗ κ 1 ∗ X 0 Y∗ −−−→ Y 0∗ 0 Y∗ −→ Y Y∗ −→ is an isomorphism of functors—from D(Y) to D~c(X), see Corollary 4.1.1— κ∗ f !κ ∼ f ×. X 0 Y∗ −→ Proof. For any D~c(X) set 0 := jX RQX κX∗ Dqc(X) (see Section 3.3). Using Proposition 3.3.1E ∈we have thenE for any D(EY)∈the natural isomorphisms F ∈ ∗ ! ∼ ! Hom X ( , κ f κ ) Hom ( , f κ ) D( ) E X 0 Y∗F −→ D(X) E0 0 Y∗F ∼ Hom (Rf , κ ) −→ D(Y ) 0∗E0 Y∗F ∼ ∗ Hom Y (κ Rf , ) −→ D( ) Y 0∗E0 F ∼ ∗ ∼ HomD(Y)(Rf∗κX 0 , ) HomD(Y)(Rf∗ , ). −3.→5.3 E F −→ E F ∗ ! Thus κ f κ is right-adjoint to Rf X , whence the conclusion. X 0 Y∗ ∗|D~c( )

Proof of Theorem 4.1. 1. Following Deligne [H1, p. 417, top], we begin by considering for (X) the functorial flasque Godement resolution M ∈ A 0 G0( ) G1( ) . → M → M → M → · · · Here, with G−2( ) := 0, G−1( ) := , and for i 0, Ki( ) the cokernel of Gi−2( ) Gi−M1( ), the sheafMGi( M) is specified inductiv≥ elyMby M → M M Gi( )(U):= Ki( ) (U open in X). M Y M x x∈U One shows by induction on i that all the functors Gi and Ki (from (X) to itself) are exact. Moreover, for i 0, Gi( ), being flasque, is f -acyclic, i.e.,A ≥ M ∗ Rjf Gi( ) = 0 for all j > 0. ∗ M The category ~c(X) has small colimits (Proposition 3.2.2), and is generated by its coherent memAbers, of which there exists a small set containing representa- tives of every isomorphism class. The Special Adjoint Functor Theorem ([F, p. 90] or [M1, p. 126, Corollary]) guarantees then that a right-exact functor F from ~c into an abelian category 0 has a right adjoint iff F is continuous in the senseA that it commutes with filteredA direct limits, i.e., for any small directed system ( , ϕ : ) in , with lim = ( , ϕ : ) it holds that α αβ β α ~c −−→ α α α M M → M A α M M M → M (F ( ), F (ϕ )) = lim (F ( ), F (ϕ )). α −−→ α αβ M α M Accordingly, for constructing right adjoints we need to replace the restrictions of Gi and Ki to (X) by continuous functors. A~c Lemma 4.1.3. Let X be a locally noetherian formal scheme and let G be a 0 functor from c(X) to a category in which direct limits exist for all small directed systems. Let Aj : (X) , (X) Abe the inclusion functor. Then: Ac → A~c (a) There exists a continuous functor G : (X) 0 and an isomorphism ~c A~c → A of functors ε: G ∼ G ◦ j such that for any map of functors ψ : G F ◦ j with −→ ~c → F continuous, there is a unique map of functors ψ~c : G~c F such that ψ factors as → ε via ψ~c G G ◦ j F ◦ j. −→ ~c −−−−→ DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 45

(b) Assume that 0 is abelian, and has exact filtered direct limits (i.e., satisfies A Grothendieck’s axiom AB5). Then if G is exact, so is G~c . Proof. (a) For (X), let ( ) be the directed system of coherent M ∈ A~c Mα X-submodules of , and set O M G ( ):= lim G( ). ~c −−→ α M α M For any (X)-map ν : and any α, there exists a coherent submodule A~c M → N Nβ ⊂ such that ν Mα factors as α β , (Corollary 3.1.7 and Lemma 3.2.1, Nwith q = 0); and| the resultingMcomp→ositionN → N ν0 : G( ) G( ) G ( ) α Mα → Nβ → ~c N does not depend on the choice of . We define the map Nβ G (ν): G ( ) = lim G( ) G ( ) ~c ~c −−→ α ~c M α M → N 0 to be the unique one whose composition with G( α) G~c( ) is να for all α. Verification of the rest of assertion (a) is straightforwM ard.→ M π (b) Let 0 0 be an exact sequence in ~c(X). Let ( β) be the filtered→systemM →ofNcoheren−→ Qt →submodules of , so that =A lim (Corol-N N N −−→ Nβ lary 3.1.7). Then ( ) is a filtered system of coherent X-modules whose lim M ∩ Nβ O −−→ is , and (π ) is a filtered system of coherent X-modules whose lim is (see M Nβ O −−→ Q Corollary 3.1.6). The exactness of G is then made apparent by application of lim ~c −−→β to the system of exact sequences 0 G( ) G( ) G(π ) 0. → M ∩ Nβ → Nβ → Nβ →

Now for (X), the lim of the system of Godement resolutions of all the M ∈ A~c −−→ coherent submodules is a functorial resolution Mα ⊂ M 0 G0( ) G1( ) ; → M → ~c M → ~c M → · · · and the cokernel of Gi−2( ) Gi−1( ) is Ki( ):= lim Ki( ). By (b) above ~c ~c ~c −−→ α M → i M i M M i i (applied to the exact functors G and K ), the continuous functors G~c and K~c are exact; and Gi ( ) = lim Gi( ) is f -acyclic since Gi( ) is, and—X being ~c M −−→ Mα ∗ Mα noetherian—the functors Rjf commute with lim . Proposition 3.4.3(b) implies then ∗ −−→ that there is an integer e 0 such that for all (X), Ke( ) is f -acyclic. ≥ M ∈ A~c ~c M ∗ So if we define the exact functors i : (X) (X) by D A~c → A Gi ( ) (0 i < e)  ~c M ≤ i( )= Ke( ) (i = e) D M  ~c M 0 (i > e)  then for (X), each i( ) is f -acyclic and the natural sequence M ∈ A~c D M ∗ δ(M) δ0(M) δ1(M) 0 0( ) 1( ) 2( ) e( ) 0 −→ M −−−→ D M −−−−→ D M −−−−→ D M −→ · · · −→ D M −→ is exact. In short, the sequence 0 1 2 e 0 is an exact, continuous, f -acyclic, finite resolutionD →ofDthe→inclusionD → · functor· · → D →(X) , (X). ∗ A~c → A 46 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

• 2. We have then a ∆-functor ( , Id): K( ~c(X)) K(X) which assigns an D A p → f -acyclic resolution to each (X)-complex = ( ) Z : ∗ A~c G G p∈ ( • )m := q( p) (m Z, 0 q e), D G M D G ∈ ≤ ≤ p+q=m the differential ( • )m ( • )m+1 being defined on q( p) (p + q = m) to be d0 + ( 1)pd00 whereD Gd0 : →q( Dp)G q( p+1) comes fromDtheG differential in and d00 = δ−q( p): q( p) D q+1G ( →p). D G G It isGelemenD taryG to→ Dcheck Gthat the natural map δ( ): • is a quasi- isomorphism. The canonical maps are D(Y)-isomorphismsG G → D G • ∼ • ∼ (12) f∗ ( ) Rf∗ ( ) Rf∗ , D G −→ D G R←f∗δ−(G) G i i • i • i.e., the natural map α : H (f∗ ( )) H (Rf∗ ( )) is an isomorphism for D G → D• G all i Z: this holds for bounded-below because ( ) is a complex of f∗-acyclic objects;∈ and for arbitrary since for anyG n Z, withD G≥n denoting the truncation G ∈ G (13) 0 0 coker( n−1 n) n+1 n+2 · · · → → → G → G → G → G → · · · there is a natural commutative diagram

i Hi(f •( )) α Hi(Rf •( )) ∗D G −−−−→ ∗D G i i βn γn   i y• ≥n i y• ≥n H (f∗ ( )) H (Rf∗ ( )) i D G −−α−n−→ D G

i ≥n in which, when n i, βn is an isomorphism (since and are identical in all i G G degrees > n), γn is an isomorphism (by Proposition 3.4.3(b) applied to the mapping • ∼ ≥n ∼ • ≥n i cone of the natural composition ( ) ( )), and αn is an isomorphism (since ≥n is boundedD G b−elo→w).G −→ G −→ D G G • • Thus we have realized Rf∗ ◦ j at the homotopy level, via the functor := f∗ ; and our task is now to ﬁnd a right adjoint at this level. C D 3. Each functor p = f p : (X) (Y) is exact, since R1f ( p( )) = 0 C ∗D A~c → A ∗ D M for all (X). p is continuous, since p is and, X being noetherian, f M ∈ A~c C D ∗ commutes with lim . As before, the Special Adjoint Functor Theorem yields that −−→ p has a right adjoint p : (Y) ~c(X). C C A → Ap For each (Y)-complex = ( ) Z let be the (X)-complex with A F F p∈ C• F A~c ( )m := p (m Z, 0 q e), C• F Y CqF ∈ ≤ ≤ p−q = m m m+1 and with diﬀerential ( • ) ( • ) the unique map making the following diagram commute for allC Fr, s with→ Cr Fs = m +1: − p p q q p−qQ= mC F −−−−−−−−→ p−q Q= m+1C F

    r−1 y r y r s s+1 s r C F ⊕ C F −d−0−+−(−−1)−−d→00 C F where: (i) the vertical arrows come from projections, DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 47

r−1 r (ii) d0 : s s corresponds to the differential in , and (iii) withC Fδ : → C F corresponding by adjunction to fF(δs): s s+1, s Cs+1 → Cs ∗ C → C d := ( 1)sδ ( r): r r. 00 − s F Cs+1F → CsF This construction leads naturally to a ∆-functor ( • , Id): K(Y) K( ~c(X)). The adjunction isomorphism C → A ∼ p Hom X ( , ) Hom Y ( , ) ( (X), (Y)) A~c( ) M Cp N −→ A( ) C M N M ∈ A~c N ∈ A applied componentwise produces an isomorphism of complexes of abelian groups • ∼ • • (14) Hom ( , • ) Hom ( , ) A~c(X) G C F −→ A(Y) C G F for all (X)-complexes and (Y)-complexes . A~c G A F 4. The isomorphism (14) suggests that we use to construct f ×, as fol- C• lows. Recall that a complex J K( ~c(X)) is K-injective iff for each exact complex K( (X)), the complex∈ HomA • ( , J) is exact too. By (12), • is G ∈ A~c A~c(X) G C G exact if is; so it follows from (14) that if is K-injective in K(Y) then is G F C•F K-injective in K( (X)). Thus if KI( ) K( ) is the full subcategory of all A~c − ⊂ − K-injective complexes, then we have a ∆-functor ( • , Id): KI(Y) KI( ~c(X)). Associating a K-injective resolution to each complex Cin (Y) leads to→ a ∆-functorA 16 A (ρ, Θ): D(Y) KI(Y). This ρ is bounded below: an (Y)-complex such that Hi( ) = 0 for→all i < n is quasi-isomorphic to its truncationA ≥n (see (13)E ), which is quasi-isomorphicE to an injective complex which vanishesEin all degrees below n. (Such an is K-injective.) F FinallyF, one can define f × to be the composition of the functors

ρ C• natural D(Y) KI(Y) KI( (X)) D( (X)), −→ −→ A~c −−−−−→ A~c and check, via (12) and (14) that Theorem 4.1 is satisﬁed. (This involves some tedium with respect to ∆-details.)

5. Torsion sheaves. Refer to 1.2 for notation and ﬁrst sorites regarding torsion sheaves. Paragraphs§ 5.1 and 5.2 develop properties of quasi-coherent torsion sheaves and their derived categories on locally noetherian formal schemes—see e.g., Propo- sitions 5.2.1, 5.2.4, 5.2.6, and Corollary 5.2.11. (There is some overlap here with 4 in [Y].) Such properties will be needed throughout the rest of the paper. For instance,§ Paragraph 5.3 establishes for a noetherian formal scheme X, either sepa- ≈ rated or ﬁnite-dimensional, an equivalence of categories D( qct(X)) Dqct(X), thereby enabling the use of D (X)—rather than D( A(X))—in −Theorem→ 6.1 qct Aqct ( ∼= Theorem 2 of Section 1). Also, Lemma 5.4.1, identifying the derived functor RΓJ ( ) (for any X -ideal , where X is a ringed space) with the homotopy colimit of−the functorsO R om•J( / n, ), plays a key role in the proof of the H OX J − Base Change Theorem 7.4 ( ∼= Theorem 3).

16In fact (ρ, Θ) is an equivalence of ∆-categories, see [L4, §1.7]. But note that Θ need not be the identity morphism, i.e., one may not be able to ﬁnd a complete family of K-injective resolutions commuting with translation. For example, we do not know that every periodic complex has a periodic K-injective resolution. 48 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

5.1. This paragraph deals with categories of quasi-coherent torsion sheaves on locally noetherian formal schemes. Proposition 5.1.1. Let f : X Y be a map of noetherian formal schemes, → and let qct(X). Then f∗ qct(Y). Moreover, if f is pseudo-proper (see 1.2.2)M ∈andA is coherent thenM ∈f A is coherent. § M ∗ M Proof. Let J X and I Y be ideals of deﬁnition such that I X J, and let ⊂ O ⊂ O O ⊂ n fn n X := (X, X/J ) (Y, Y/I ) =: Y (n > 0) n O −→ O n be the scheme-maps induced by f, so that if jn and in are the canonical closed n immersions then fj = i f . Let := om( X/J , ), so that n n n Mn H O M = Γ 0 = lim = lim j j∗ . X −−→ n −−→ n∗ n n M M n M n M n Since J is a coherent X-ideal [GD, p. 427], therefore n is quasi-coherent (Corol- O M ∗ lary 3.1.6(d)), and it is straightforward to check that in∗fn∗jn n qct(Y). Thus, X being noetherian, and by Corollary 5.1.3 below, M ∈ A f = f lim lim f j j∗ = lim i f j∗ (Y). ∗ ∗ −−→ n = −−→ ∗ n∗ n n −−→ n∗ n∗ n n qct M n M ∼ n M n M ∈ A

When f is pseudo-proper every fn is proper; and if qct(X) is coherent then so is f , because for some n, f = f j j∗ =Mi ∈fA j∗ . ∗ M ∗ M ∗ n∗ n Mn n∗ n∗ n Mn Proposition 5.1.2. Let Z be a closed subset of a locally noetherian scheme X, ∗ and let κ: X X be the completion of X along Z. Then the functors κ and κ∗ restrict to inverse→ isomorphisms between the categories (X) and (X), and AZ At between the categories qcZ (X) and qct(X); and if qct(X) is coherent, then so is κ . A A M ∈ A ∗ M Proof. Let be a quasi-coherent X -ideal such that the support of X / is Z. Applying limJ to the natural isomorphismsO O J −−→ n ∗ n ∼ n ∗ κ om ( / , ) omX( X/ X , κ ) ( (X), n > 0) H X OX J N −→ H O J O N N ∈ A ∗ 0 ∼ 0 ∗ ∗ we get a functorial isomorphism κ ΓZ ΓXκ , and hence κ ( Z (X)) t(X). Applying lim to the natural isomorphisms−→ A ⊂ A −−→ n n ∼ n om ( / , κ ) κ omX( X/ X , ) ( (X), n > 0) H X OX J ∗M −→ ∗H O J O M M ∈ A 0 ∼ 0 we get a functorial isomorphism ΓZ κ∗ κ∗ΓX , and hence κ∗( t(X)) Z (X). As κ is a pseudo-proper map of lo−→cally noetherian formalAschemes⊂ A((0) being an ideal of deﬁnition of X), we see as in the proof of Proposition 5.1.1 that for (X), κ is a lim of quasi-coherent -modules, so is itself quasi- qct ∗ −−→ X M ∈ A M 17 O coherent, and κ∗ is coherent whenever is. Finally, examiningM stalks (see 1.2) wMe ﬁnd that the natural transformations 1 κ κ∗ and κ∗κ 1 induce isomorphisms§ → ∗ ∗ → Γ 0 ∼ κ κ∗Γ 0 ( (X)), Z N −→ ∗ Z N N ∈ A κ∗κ Γ 0 ∼ Γ 0 ( (X)). ∗ XM −→ XM M ∈ A

17 The noetherian assumption in Lemma 5.1.1 is needed only for commutativity of f∗ with lim−→ , a condition clearly satisﬁed by f = κ in the present situation. DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 49

Corollary 5.1.3. If X is a locally noetherian formal scheme then qct(X) is plump in (X) and closed under small (X)-colimits. A A A

Proof. The assertions are local, and so, since t(X) is plump ( 1.2.1), Propo- sition 5.1.2 (where κ∗ commutes with lim ) enablesAreduction to well-kno§ wn facts −−→ about (X) (X) with X an aﬃne noetherian (ordinary) scheme. AqcZ ⊂ A Lemma 5.1.4. Let X be a locally noetherian formal scheme. If is a quasi- 0 M coherent X-module then Γ (X) is the lim of its coherent submodules. X qct −−→ In particular,O (X) (XM). ∈ A Aqct ⊂ A~c

Proof. Let J be an ideal of deﬁnition of X. For any positive integer n, let Xn n be the scheme (X, X/J ), let jn : Xn X be the canonical closed immersion, and O n 0 → let n := om( X/J , ) ΓX( ), so that n qct(X) (Corollary 3.1.6(d)). ThenM the Hquasi-coherenO tM ⊂-moduleM j∗ isMthe∈ limA of its coherent submod- OXn nMn −−→ ules [GD, p. 319, (6.9.9)], hence so is = j j∗ (since j∗ and j preserve Mn n∗ nMn n n∗ both lim and coherence [GD, p. 115, (5.3.13) and (5.3.15)]), and therefore so is −−→ Γ 0 = lim . That lim (X) results from Corollary 5.1.3. X −−→ n −−→ n qct M n M n M ∈ A Corollary 5.1.5. For a locally noetherian formal scheme X, the inclusion t t functor jX : qct(X) , (X) has a right adjoint QX. If moreover X is noetherian then Qt commutesA with→ Alim. X −−→ t Proof. To show that jX has a right adjoint one can, in view of Corollary 5.1.3 and Lemma 5.1.4, simply apply the Special Adjoint Functor theorem. More speciﬁcally, since Γ 0 is right-adjoint to the inclusion (X) , (X), and X At → A ~c(X) qc(X) (Corollary 3.1.5), it follows from Lemma 5.1.4 that the restriction A 0 ⊂ A of ΓX to ~c(X) is right-adjoint to qct(X) , ~c(X); and by Proposition 3.2.3, A A →t A 0 t ~c(X) , (X) has a right adjoint QX ; so QX := ΓX ◦ QX is right-adjoint to jX . A → A 0 t (Similarly, QX ◦ ΓX is right-adjoint to jX .) Commutativity with lim means that for any small directed system ( ) in (X) −−→ Gα A and any (X), the natural map M ∈ Aqct φ: Hom( , lim Qt ) Hom( , Qt lim ) −−→ X α X−−→ α M α G → M α G is an isomorphism. This follows from Lemma 5.1.4, which allows us to assume that is coherent, in which case φ is isomorphic to the natural composed isomorphism M lim Hom( , Qt ) ∼ lim Hom( , ) ∼ Hom( , lim ). −−→ X α −−→ α −−→ α α M G −→ α M G −→ M α G

t Remark. For an ordinary noetherian scheme X we have QX = QX (see 3.1). t ∗ § More generally, if κ: X X is as in Proposition 5.1.2, then QX = κ ΓZ QX κ∗. Hence Proposition 5.1.1 →(applied to open immersions X , Y with X aﬃne) lets t → us construct the functor QY for any noetherian formal scheme Y by mimicking the construction for ordinary schemes (cf. [I, p. 187, Lemme 3.2].)

5.2. The preceding results carry over to derived categories. From Corollary 5.1.3 it follows that on a locally noetherian formal scheme X, Dqct(X) is a triangulated subcategory of D(X), closed under direct sums. 50 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

Proposition 5.2.1. For a locally noetherian formal scheme X, set := (X), At At the category of torsion X-modules, and let i: D( t) D(X) be the natural functor. Then: O A → 0 (a) An X-complex is in Dt(X) iff the natural map iRΓX is a D(X)- isomorphism.O E E → E 0 (b) If Dqc(X) then RΓX Dqc( t). E ∈ E ∈ A 0 (c) The functor i and its right adjoint RΓX induce quasi-inverse equivalences between D( ) and D (X) and between D ( ) and D (X).18 At t qc At qct Proof. (a) For D( ) (e.g., := RΓ 0 ), any complex isomorphic to i F ∈ At F XE F is clearly in Dt(X). Suppose conversely that D (X). The assertion that iRΓ 0 = is local, E ∈ t XE ∼ E so we may assume that X = Spf(A) where A = Γ(X, X) is a noetherian adic ring, so that any defining ideal J of X is generated by a finiteO sequence in A. Then iRΓ 0 = • , where • is a bounded flat complex—a lim of Koszul complexes XE ∼ K∞ ⊗ E K∞ −−→ on powers of the generators of J—see [AJL, p. 18, Lemma 3.1.1]. 0 So iRΓX is a bounded functor, and the usual way-out argument reduces the question to where is a single torsion sheaf. But then it is immediate from the • E • construction of ∞ that ∞ = . K K ⊗ E E 0 (b) Again, we can assume that X = Spf(A) and RΓX is bounded, and since qc(X) is plump in (X) (Proposition 3.2.2) we can reduce to where is a single A A E + quasi-coherent X-module, though it is better to assume only that Dqc(X), for then we may alsoO assume injective, so that E ∈ E RΓ 0 = Γ 0 = lim om( /Jn, ). X ∼ X −−→ E E n>0 H O E n From Corollary 3.1.6(d) it follows that om( /J , ) Dqct(X)—for this assertion another way-out argument reduces usHagainOto whereE ∈ is a single quasi-coherent E X-module—and since homology commutes with lim and is closed under lim −−→ qct −−→ O(Corollary 5.1.3), therefore RΓ 0 has quasi-coherent homologyA . XE Assertion (c) results now from the following simple lemma.

Lemma 5.2.2. Let be an abelian category, let j : be the inclusion of A A[ → A a plump subcategory such that j has a right adjoint Γ, and let j : D( [) D( ) be the derived-category extension of j. Suppose that every -complexA has→ a AK- A injective resolution, so that the derived functor RΓ : D( ) D( [) exists. Then RΓ is right-adjoint to j. Furthermore, the following conditionsA → arAe equivalent.

(1) j induces an equivalence of categories from D( [) to D[( ), with quasi- inverse R Γ := RΓ . A A [ |D[(A) (2) For every D ( ) the natural map jRΓ is an isomorphism. E ∈ [ A E → E (3) The functor R[Γ is bounded, and for 0 [ the natural map jRΓ 0 is a D( )-isomorphism. E ∈ A E → E0 A When these conditions hold, every -complex has a K-injective resolution. A[ Proof. Since Γ has an exact left adjoint, it takes K-injective -complexes to K-injective -complexes, whence there is a bifunctorial isomorphismAin the derived A[ 18 R 0 iR 0 We may therefore sometimes abuse notation and write ΓX instead of ΓX; but the meaning should be clear from the context. DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 51 category of abelian groups

RHom• (j , ) ∼ RHom• ( , RΓ ) ( D( ), D( )). A G E −→ A[ G E G ∈ A[ E ∈ A (To see this, one can assume to be K-injective, and then drop the R’s. . .) Apply homology H0 to this isomorphismE to get adjointness of j and RΓ. The implications (1) (3) (2) are straightforward. For (2) (1), one needs ⇒ ⇒ ⇒ that for D( [) the natural map RΓ j is an isomorphism, or equivalently (look at Ghomology),∈ A that the correspGonding→ mapG j jRΓ j is an isomorphism. But the composition of this last map with the isomorphismG → jRGΓ j ∼ j (given by (2)) is the identity, whence the conclusion. G −→ G Finally, if is an [-complex and j is a K-injective -resolution, then as before Γ is aGK-injectivA e -complex; Gand→(1)J implies that theAnatural composition J A[ Γj Γ ( = RΓ j ) G → G → J ∼ G is a D( )-isomorphism, hence an -K-injective resolution. A[ A[

Corollary 5.2.3. For any complexes Dt(X) and D(X) the natural map RΓ 0 induces an isomorphism E ∈ F ∈ XF → F R om•( , RΓ 0 ) ∼ R om•( , ). H E XF −→ H E F Proof. Consideration of homology presheaves shows it suﬃcient that for each aﬃne open U X, the natural map ⊂ 0 Hom U ( U , (RΓ ) U) Hom U ( U , U) D( ) E| XF | → D( ) E| F| 0 be an isomorphism. But since RΓX commutes with restriction to U, that is a direct consequence of Proposition 5.2.1(c) (with X replaced by U).

Parts (b) and (c) of the following Proposition will be generalized in parts (d) and (b), respectively, of Proposition 5.2.8.

Proposition 5.2.4. Let Z be a closed subset of a locally noetherian scheme X, and let κ: X X be the completion of X along Z. Then: → ∗ (a) The exact functors κ and κ∗ restrict to inverse isomorphisms between the categories DZ (X) and Dt(X), and between the categories DqcZ (X) and Dqct(X); and if D (X) has coherent homology, then so does κ . M ∈ qct ∗ M (b) There is a unique derived-category isomorphism

RΓ 0κ ∼ κ RΓ 0 ( D(X)) Z ∗E −→ ∗ XE E ∈ 0 whose composition with the natural map κ∗RΓX κ∗ is just the natural map RΓ 0κ κ . E → E Z ∗E → ∗E (c) There is a unique derived-category isomorphism

κ∗RΓ 0 ∼ RΓ 0 κ∗ ( D(X)) Z F −→ X F F ∈ 0 ∗ ∗ whose composition with the natural map RΓXκ κ is just the natural map κ∗RΓ 0 κ∗ . F → F Z F → F 52 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

Proof. The assertions in (a) follow at once from Proposition 5.1.2. ∗ (b) Since κ∗ has an exact left adjoint (namely κ ), therefore κ∗ transforms K-injective (X)-complexes into K-injective (X)-complexes, and consequently A A 0 ∼ 0 the isomorphism in (b) results from the isomorphism ΓZ κ∗ κ∗ΓX in the proof of Proposition 5.1.2. That the composition in (b) is as asserted−→ comes down then to the elementary fact that the natural composition n ∼ n om ( / , κ ) κ omX( X/ X , ) κ H X OX J ∗M −→ ∗H O J O M −→ ∗M 0 (see proof of Proposition 5.1.2) is just the obvious map. Since κ∗RΓX DZ (X) (by (a) and Proposition 5.2.1(a)), the uniqueness assertion (for the inverseE ∈ isomor- 0 phism) results from adjointness of RΓZ and the inclusion DZ (X) , D(X). (The proof is similar to that of Proposition 5.2.1(c)). → (c) Using (b), we have the natural composed map κ∗RΓ 0 κ∗RΓ 0κ κ∗ ∼ κ∗κ RΓ 0 κ∗ RΓ 0 κ∗ . Z F → Z ∗ F −→ ∗ X F → X F Showing this to be an isomorphism is a local problem, so assume X = Spec(A) with A a noetherian adic ring. Let K• be the usual lim of Koszul complexes on ∞ −−→ powers of a ﬁnite system of generators of an ideal of deﬁnition of A ([AJL, 3.1]); • § and let K∞ be the corresponding quasi-coherent complex on Spec(A), so that the complexe • in the proof of Proposition 5.2.1(a) is just κ∗K• . Then one checks K∞ ∞ via [AJL, p. 18, Lemma (3.1.1)] that the map in questioneis isomorphic to the natural isomorphism of complexes κ∗(K• ) ∼ κ∗K• κ∗ . ∞ ⊗OX F −→ ∞ ⊗OX F That the compositionein (c) is as assertede results from the following natural commutative diagram, whose bottom row composes to the identity: κ∗RΓ 0 κ∗RΓ 0κ κ∗ κ∗κ RΓ 0 κ∗ RΓ 0 κ∗ Z F −−−→ Z ∗ F −−−→ ∗ X F −−−→ X F f   (b)       κy∗ κ∗κyκ∗ κ∗κyκ∗ κy∗ F −−−→ ∗ F ∗ F −−−→ F Uniqueness is shown as in (b).

Corollary 5.2.5. The natural maps are isomorphisms ∗ ∗ ∗ Hom ( , ) = Hom ( , κ κ ) = HomX(κ , κ ) ( D (X), D(X)), X E F ∼ X E ∗ F ∼ E F E ∈ Z F ∈ ∗ ∗ ∗ Hom ( , ) = Hom ( , κ κ ) = HomX(κ , κ ) ( D(X), D (X)), X E F ∼ X E ∗ F ∼ E F E ∈ F ∈ Z ∗ HomX( , ) = HomX(κ κ , ) = Hom (κ , κ ) ( D (X), D(X)). G H ∼ ∗G H ∼ X ∗G ∗H G ∈ t H ∈ Proof. For the ﬁrst line, use Proposition 5.2.1 and its analogue for DZ (X), Lemma 5.2.2, and Proposition 5.2.4 to get the equivalent sequence of natural isomorphisms Hom ( , ) = Hom ( , RΓ 0 ) X E F ∼ X E Z F ∗ ∗ 0 = HomX(κ , κ RΓ ) ∼ E Z F ∗ 0 ∗ = HomX(κ , RΓ κ ) ∼ E X F ∗ ∗ = HomX(κ , κ ) ∼ E F = Hom ( , κ κ∗ ). ∼ X E ∗ F The rest is immediate from Proposition 5.2.4(a). DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 53

The next series of results concerns the behavior of Dqct with respect to maps of formal schemes. Proposition 5.2.6. Let f : X Y be a map of noetherian formal schemes. → Then Rf X is bounded, and ∗|Dqct( ) Rf (D (X)) D (Y). ∗ qct ⊂ qct Moreover, if f is pseudo-proper and Dt(X) has coherent homology, then so does Rf D (Y). F ∈ ∗ F ∈ t Proof. Since D (X) D (X) (Lemma 5.1.4), the boundedness assertion is qct ⊂ ~c given by Proposition 3.4.3(b). (Clearly, Rf∗ is bounded-below.) It suffices then for the next assertion (by the usual way-out arguments [H1, p. 73, Proposition 7.3]) to show for any qct(X) that Rf∗ Dqct(Y). Let be Man injectiv∈ A e resolution Mof ∈ , let J be an ideal of definition of X, and let beEthe flasque complex := omM( /Jn, ). Then by Proposition 5.2.1(a), En En H O E = RΓ 0 = lim . Since X is noetherian, lim’s of flasque sheaves are f -acyclic ∼ X ∼ −−→n n −−→ ∗ Mand lim commM utes withE f ; so with notation as in the proof of Proposition 5.1.1, −−→ ∗

Rf Rf RΓ 0 f lim lim f j j∗ lim i f j∗ . ∗ = ∗ X = ∗−−→ n = −−→ ∗ n∗ n n = −−→ n∗ n∗ n n M ∼ M ∼ n E ∼ n E ∼ n E Since D+ (X), therefore E ∈ qc j j∗ = om( /Jn, ) D (X), n∗ nEn H O E ∈ qc as we see by way-out reduction to where is a single quasi-coherent sheaf and ∗ E then by Corollary 3.1.6(d); and hence jn n Dqc(Xn) (see [GD, p. 115, (5.3.15)]). ∗ E ∈ Now jn n is a flasque bounded-below Xn -complex, so by way-out reduction to (for example)E [Ke, p. 643, corollary 11], O f j∗ = Rf j∗ D (Y ); n∗ nEn ∼ n∗ nEn ∈ qc n and finally, in view of Corollary 5.1.3, Rf i lim f j∗ D (Y). ∗ = n∗−−→ n∗ n n qct M ∼ n E ∈ For the last assertion, we reduce as before to showing for each coherent torsion p p X-module and each p 0 that R f := H Rf is a coherent Y-module. O M ≥ ∗M ∗M O With notation remaining as in Proposition 5.1.1, the maps in and jn are exact, and for some n, = j j∗ . So M n∗ nMn Rpf = Rpf j j∗ = i Rpf j∗ , ∗ M ∗ n∗ nMn n∗ n∗ nMn ∗ which is coherent since jn n is a coherent Xn -module and fn : Xn Yn is a proper scheme-map. M O → Corollary 5.2.7 (cf. Corollary 3.5.3). Let f : X Y be a map of locally 0 → noetherian schemes, let W Y and Z f −1W be closed subsets, with associated ⊂ ⊂ 0 (flat) completion maps κY : Y = Y/W Y , κX : X = X/Z X, and let f : X Y be the map induced by f . For D(→X) let → → 0 E ∈ θ : κ∗ Rf Rf κ∗ E Y 0∗E → ∗ XE be the map adjoint to the natural composition Rf Rf κ κ∗ ∼ κ Rf κ∗ . 0∗E −→ 0∗ X∗ XE −→ Y∗ ∗ XE Then θ is an isomorphism for all D (X) . E E ∈ qcZ 54 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

Proof. θE is the composition of the natural maps κ∗ Rf κ∗ Rf κ κ∗ ∼ κ∗ κ Rf κ∗ Rf κ∗ . Y 0∗E → Y 0∗ X∗ XE −→ Y Y∗ ∗ XE → ∗ XE By Proposition 5.2.4, the first map and (in view of Proposition 5.2.6) the third map are both isomorphisms. Proposition 5.2.8. Let f : X Y be a map of locally noetherian formal → schemes. Let I be a coherent Y-ideal, and let DI(Y) be the triangulated subcategory of D(Y) whose objects arOe the complexes with I-torsion homology (i.e., i i Z F ΓI H = H for all i —see 1 and 1.2.1). Then: F ∗ F ∈ §§ (a) Lf (DI(Y)) DI (X). ⊂ OX (b) There is a unique functorial isomorphism ξ( ): Lf ∗RΓ ∼ RΓ Lf ∗ ( D(Y)) E I E −→ IOX E E ∈ whose composition with the natural map RΓ Lf ∗ Lf ∗ is the natural map IOX E → E Lf ∗RΓ Lf ∗ . I E → E (c) The natural map is an isomorphism RΓ 0 Lf ∗RΓ 0 ∼ RΓ 0 Lf ∗ ( D(Y)). X YE −→ X E E ∈ (d) If X is noetherian, there is a unique functorial isomorphism RΓ Rf ∼ Rf RΓ ( D+(X)) I ∗ G −→ ∗ IOX G G ∈ whose composition with the natural map Rf RΓ Rf is the natural map ∗ IOX G → ∗G RΓ Rf Rf . I ∗G → ∗G ∗ Proof. (a) Let DI(Y). To show that Lf DIOX(X) we may assume that is K-injective.F Let∈ x X, set y := f(x), andF ∈let P • be a flat resolution F ∈ x of the Y,y-module X,x . Then, as in the proof of Proposition 5.2.1(a), there is a canonicalO D(Y)-isomorphismO • n ∼ lim om ( Y/I , ) = Γ = RΓ , −−→ I I n H O F F F −→ F and it follows that for any i the stalk at x of the homology H iLf ∗ is F Hi(P • ) = lim Hi(P • Hom• ( /In , )). x OY,y y −−→ x OY,y OY,y Y,y y y ⊗ F n ⊗ O F Hence each element of the stalk is annihilated by a power of I X , and (a) results. O ,x (b) The existence and uniqueness of a functorial map ξ( ) satisfying everything E except the isomorphism property result from (a) and the fact that RΓIOX is right- adjoint to the inclusion DIOX(X) , D(X). To show that ξ( ) is an isomorphism→ we may assume that Y is affine and that is K-flat, and then proE ceed as in the proof of (the special case) Proposition 5.2.4(c),E • via the bounded flat complex K∞ . (c) Let I, J be defining ideals of Y and X respectively, so that := I X J. The natural map RΓ 0 RΓ := RΓ RΓ RΓ =: RΓ 0 is an isomorphism,K O as⊂one X K J K → J X checks locally via [AJL, p. 20, Corollary (3.1.3)]. So for any D(Y), (b) gives E ∈ RΓ 0 Lf ∗ = RΓ 0 RΓ Lf ∗ = RΓ 0 Lf ∗ RΓ 0 . X E ∼ X K E ∼ X Y E (d) may be assumed bounded-below and injective, so that G • n := om ( X/I X , ) Gn H O O G is flasque. DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 55

Then, since X is noetherian, Γ = lim is flasque too, and IOX G −−→n Gn Rf RΓ Rf Γ f lim lim f DI(Y). ∗ IOX = ∗ IOX = ∗ −−→ n = −−→ ∗ n G ∼ G ∼ n G ∼ n G ∈ By Lemma 5.2.2, RΓ (resp. RΓ ) is right-adjoint to the inclusion DI(Y) , D(Y) I IOX → (resp. DI (X) , D(X)), whence, in particular, the uniqueness in (d). Moreover, OX → in view of (a), for any DI(Y) the natural maps are isomorphisms E ∈ ∼ ∼ ∗ HomY( , RΓ Rf ) HomY( , Rf ) HomX(Lf , ) E I ∗ G −→ E ∗ G −→ E G ∼ ∗ ∼ HomX(Lf , RΓ ) HomY( , Rf RΓ ). −→ E IOX G −→ E ∗ IOX G It follows formally that the image under this composed isomorphism of the identity map of RΓI Rf∗ is an isomorphism as asserted. (In fact this isomorphism is adjoint G ∗ ∗ to the composition Lf RΓI Rf∗ RΓIOX Lf Rf∗ RΓIOX .) G −ξ−(R−−f∗−G→) G −nat’l−−→ G Definition 5.2.9. For a locally noetherian formal scheme X, 0 −1 Dqc(X):= RΓX (Dqc(X)) is the ∆-subcategory of D(eX) whose objects are those complexes such that RΓ 0 D (X)—or equivalently, RΓ 0 D (X). F XF ∈ qc XF ∈ qct Remarks 5.2.10. (1) By Proposition 5.2.1(b), D (X) D (X). Hence qc ⊂ qc e RΓ 0 (D (X)) D (X). X qc ⊂ qc 0 e e (2) Since RΓX is idempotent (see Proposition 5.2.1), the vertex of any triangle based on the canonical map RΓ 0 ( D(X)) is annihilated by RΓ 0 . It XE → E E ∈ X follows that Dqc(X) is the smallest ∆-subcategory of D(X) containing Dqct(X) and 0 all complexese such that RΓX = 0. F F (3) The functor RΓ 0 : D(X) D(X) has a right adjoint X → • 0 ΛX( ):= R om (RΓ , ). − H XOX − Indeed, there are natural functorial isomorphisms for , D(X), E F ∈ 0 ∼ 0 Hom X (RΓ , ) Hom X ( RΓ , ) D( ) X D( ) = X X (15) E F −→ E ⊗ O F ∼ • 0 Hom X ( , R om (RΓ , )). −→ D( ) E H XOX F 0 ∼ 0 (Whether the natural map RΓX X RΓX is an isomorphism is a local E ⊗= O −→ E question, dealt with e.g., in [AJL, p. 20, Corollary (3.1.2)]. The second isomorphism is given, e.g., by [Sp, p. 147, Proposition 6.6 (1)].) 0 ∼ 0 There is a natural isomorphism RΓX RΓXΛX (see (d) in Remark 6.3.1 below), and consequently −→ Λ (D (X)) D (X). X qc ⊂ qc e e (4) If D−(X) and D (X) then R om•( , ) D (X), and hence E ∈ c F ∈ qc H E F ∈ qc R om•(RΓ 0 , ) D (X). Indeed,e the natural map e H XE F ∈ qc RΓe0 R om•( , RΓ 0 ) RΓ 0 R om•( , ) X H E XF → X H E F is an isomorphism, since for any in D (X), D (X) (an assertion which t = t can be checked locally, using PropositionG 5.2.1(a)G ⊗andE ∈the complex • in its proof), K∞ 56 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN so that there is a sequence of natural isomorphisms (see Proposition 5.2.1(c)): Hom( , RΓ 0 R om•( , RΓ 0 )) ∼ Hom( , R om•( , RΓ 0 )) G X H E XF −→ G H E XF ∼ 0 Hom( , RΓX ) −→ G ⊗= E F ∼ Hom( , ) −→ G ⊗= E F ∼ Hom( , R om•( , )) −→ G H E F ∼ Hom( , RΓ 0 R om•( , )). −→ G X H E F Since (X) is plump in (X) (Corollary 5.1.3), Proposition 3.2.4 shows that Aqct A RΓ 0 R om•( , RΓ 0 ) D (X), whence R om•( , ) D (X). X H E XF ∈ qct H E F ∈ qc From (3) and the natural isomorphisms e • 0 • 0 • R om (RΓX , ) = R om (RΓX X , ) = ΛXR om ( , ) H E F ∼ H O ⊗= E F ∼ H E F we see then that R om•(RΓ 0 , ) D (X). H XE F ∈ qc e (5) For D(X) it holds that F ∈ • D (X) R om ( X/J, ) D (X) for all defining ideals J of X. F ∈ qc ⇐⇒ H O F ∈ qct e The implication = is given, in view of Corollary 5.2.3, by Proposition 3.2.4; and ⇒ the converse is given by Lemma 5.4.1, since Corollary 5.1.3 implies that Dqct(X) is a ∆-subcategory of D(X) closed under direct sums. (6) Let f : X Y be a map of locally noetherian formal schemes. For any → D (Y), Lemma 5.1.4 and Proposition 3.3.5 give F ∈ qc e Lf ∗ RΓ 0 Lf ∗(D (Y)) Lf ∗(D (Y)) D (X) D (X), Y F ∈ qct ⊂ ~c ⊂ qc ⊂ qc 0 ∗ 0 ∗ 0 e and so RΓX Lf = RΓX Lf RΓY Dqct(X). Thus F 5.2.8(c)∼ F ∈ Lf ∗(D (Y)) D (X). qc ⊂ qc e e Corollary 5.2.11. Let f : X Y be an adic map of locally noetherian formal schemes. Then: → (a) Lf ∗(D (Y)) D (X). t ⊂ t (b) Lf ∗(D (Y)) D (X). qct ⊂ qct (c) There is a unique functorial isomorphism Lf ∗RΓ 0 ∼ RΓ 0 Lf ∗ ( D(Y)) YE −→ X E E ∈ 0 ∗ ∗ whose composition with the natural map RΓX Lf Lf is the natural map Lf ∗RΓ 0 Lf ∗ . There results a conjugate isomorphismE → of Eright-adjoint functors YE → E ∼ Rf ΛX ΛYRf ( D(X)). ∗ G −→ ∗G G ∈ whose composition with the natural map Rf Rf ΛX is the natural map ∗G → ∗ G Rf ΛYRf . ∗G → ∗G DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 57

(d) If X is noetherian then there is a unique functorial isomorphism RΓ 0 Rf ∼ Rf RΓ 0 ( D+(X) or D (X)) Y ∗ G −→ ∗ X G G ∈ G ∈ qc 0 e whose composition with the natural map Rf∗RΓX Rf∗ is the natural map RΓ 0 Rf Rf . G → G Y ∗G → ∗ G (e) If X is noetherian then Rf (D (X)) D (Y). ∗ qc ⊂ qc e e Proof. To get (a) and (c) take I in Proposition 5.2.8 to be an ideal of deﬁnition of Y. (The second assertion in (c) is left to the reader.) As Dqct(Y) = D~c(Y) Dt(Y) (Corollary 3.1.5 and Lemma 5.1.4), (b) follows from (a) and Proposition∩ 3.3.5. The same choice of I gives (d) for D+(X)—and the argument also works for D (X) once one notes that G ∈ G ∈ qc e 0 Rf∗RΓX(Dqc(X)) Rf∗(Dqct(X)) Dqct(Y) Dt(Y). ⊂ 5.2.6⊂ ⊂ e The isomorphism in (d) gives (e) via Proposition 5.2.6.

−1 Corollary 5.2.12. In Corollary 5.2.7, if X is noetherian and Z = f0 W then for all D (X) the map θ0 := RΓ 0(θ ) is an isomorphism F ∈ qc F Y F θ0 : RΓ 0κ∗ Rf ∼ RΓ 0 Rf κ∗ . F Y Y 0∗ F −→ Y ∗ X F

Proof. Arguing as in Proposition 5.2.1, we ﬁnd that RΓZ DqcZ (X), so that we have the isomorphism θ of Corollary 5.2.7. F ∈ RΓZ F Imitating the proof of Corollary 5.2.11, we get an isomorphism α : Rf RΓ ∼ RΓ Rf F 0∗ Z F −→ W 0∗ F whose composition with the natural map RΓ Rf Rf is the natural map W 0∗ F → 0∗ F Rf0∗RΓZ Rf0∗ . ConsiderF →then theF diagram

nat’l κ∗ Rf RΓ κ∗ RΓ Rf RΓ 0κ∗ Rf κ∗ Rf Y 0∗ Z ∗ Y W 0∗ Y Y 0∗ Y 0∗ F −−κY−(−α−F→) F −−5.2.4(c)−−−→ F −−−−→ F f f 0 θRΓ F ' (1) θ θ Z   F  F    ∗y 0 ∗ 0 y ∗ y∗ Rf∗κXRΓZ Rf∗RΓXκX RΓY Rf∗κX Rf∗κX F −−5.2.4(c)−−−→ F −−5.2.11(d)−−−−→ F −−nat’l−−→ F f f 0 It suﬃces to show that subdiagram (1) commutes; and since RΓY is right-adjoint to the inclusion Dt(Y) , D(Y) it follows that it’s enough to show that the outer border of the diagram comm→ utes. But it is straightforward to check that the top and bottom rows compose to the maps induced by the natural map RΓZ 1, whence the conclusion. →

5.3. From the following key Proposition 5.3.1—generalizing the noetherian case of [AJL, p. 12, Proposition (1.3)]—there will result, for complexes with quasi- coherent torsion homology, a stronger version of the Duality Theorem 4.1, see Section 6. Recall what it means for a noetherian formal scheme X to be separated ( 3.4.1). t § Recall also from Corollary 5.1.5 that the inclusion functor jX : qct(X) , (X) has t A → A a right adjoint QX. 58 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

Proposition 5.3.1. Let X be a noetherian formal scheme. t (a) The extension of jX induces an equivalence of categories jt : D+( (X)) ≈ D+ (X), X Aqct −→ qct t with bounded quasi-inverse RQX D+ X . | qct( ) t (b) If X is separated, or of finite Krull dimension, then the extension of jX induces an equivalence of categories jt : D( (X)) ≈ D (X), X Aqct −→ qct t with bounded quasi-inverse RQ X . X|Dqct( ) Proof. (a) The asserted equivalence is given by [Y, Theorem 4.8]. The idea is that qct(X) contains enough t(X)-injectives [Y, Proposition 4.2], so by [H1, p. 47, A + A + Proposition 4.8], D ( qct(X)) is equivalent to Dqc( t(X)), which is equivalent + A A to Dqct(X) (Proposition 5.2.1(c)). t t + Since RQX is right-adjoint to jX (Lemma 5.2.2), its restriction to Dqct(X) is t quasi-inverse to j + X . From the resulting isomorphism X|D (Aqct( )) ι : jt RQt ∼ ( D+ (X)) E X XE −→ E E ∈ qct i i t t we see that if H = 0 then H RQX = 0, so that RQX D+ X is bounded. E E | qct( ) (b) By Lemma 5.2.2, and having the isomorphism ιE , we need only show that t RQX is bounded on Dqct(X). Suppose that X is the completion of a separated ordinary noetherian scheme X along some closed subscheme, and let κ: X X be the completion map, so that t ∗ → QX = κ ΓZ QX κ∗ (see remark following Corollary 5.1.5). The exact functor κ∗ preserves K-injectivity, since it has an exact left adjoint, namely κ∗. Similarly QX transforms K-injective (X)-complexes into K-injective qc(X)-complexes. qc A qc A Hence RQt = κ∗RΓ RQ κ , where Γ : (X) (X) is the restric- X ∼ Z X ∗ Z Aqc → AqcZ tion of ΓZ . Now by the proof of [AJL, p. 12, Proposition (1.3)], RQX is bounded on Dqc(X) κ∗Dqct(X) (Proposition 5.2.4). Also, by [AJL, p. 24, Lemma (3.2.3)], ⊃ qc RΓZ is bounded; and hence by [AJL, p. 26, Proposition (3.2.6)], so is ΓZ . Thus t RQX is bounded on Dqct(X). In the general separated case, one proceeds by induction on the least number of affine open subsets covering X, as in the proof of [AJL, p. 12, Proposition (1.3)] (which is Proposition 5.3.1 for X an ordinary scheme), mutatis mutandis—namely, substitute “X” for “X,” “qct” for “qc,” “Qt” for “Q,” and recall for a map v : V X of noetherian formal schemes that v ( (V)) (X) (Proposition 5.1.1),→and ∗ Aqct ⊂ Aqct furthermore that if v is affine then v∗ Aqct(V) is exact (Lemmas 5.1.4 and 3.4.2). A similar procedure works when |the Krull dimension dim X is finite, but now the induction is on n(X) := least n such that X has an open covering X = n U ∪i=1 i where for each i there is a separated ordinary noetherian scheme Ui such that Ui is isomorphic to the completion of Ui along one of its closed subschemes. (This property of Ui is inherited by any of its open subsets). The case n(X) = 1 has just been done. Consider, for any open immersion qct v : V , X, the functor v := v V . To complete the induction as in the → ∗ ∗|Aqct( ) proof of [AJL, p. 12, Proposition (1.3)], one needs to show that the derived functor qct Rv∗ : D( (V)) D(X) is bounded above. Aqct → DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 59

For (V), let • be an -injective—hence flasque—resolution N ∈ Aqct N → J Aqct [Y, Proposition 4.2]. Now H iRvqct( ) is the sheafification of the presheaf send- ∗ N ing an open W X to HiΓ(W V, •) = Hi(W V, ), which vanishes when i > dim X, whence⊂ the conclusion∩([L4J, Proposition∩(2.7.5)]).N

5.4. Let (X, ) be a ringed space, and let be an -ideal. The next OX J OX Lemma, expressing RΓJ as a “homotopy colimit,” lifts back to D(X) the well- known relation HiRΓ = lim xti ( / n, ) ( D(X)). J −−→ OX X G n E O J G G ∈ Deﬁne h : D(X) D(X) by n → h ( ):= R om•( / n, ) (n 1, D(X)). n G H OX J G ≥ G ∈ There are natural functorial maps s : h h and ε : h RΓ , satisfying n n → n+1 n n → J εn+1sn = εn. The family (1, s ): h h h h (m 1) − m m → m ⊕ m+1 ⊂ ⊕n≥1 n ≥ deﬁnes a natural map s: n≥1 hn n≥1 hn. There results, for each D(X), a map of triangles ⊕ → ⊕ G ∈

h s h ?? + ⊕n≥1 n G −−−−→ ⊕n≥1 n G −−−−→ −−−−→  Pεn ε    y0 RΓy RΓy + −−−−→ J G J G −−−−→ Lemma 5.4.1. The map ε is a D(X)-isomorphism, and so we have a triangle

Pε h s h n RΓ + ⊕n≥1 n G −−−−→ ⊕n≥1 n G −−−−→ J G −−−−→ Proof. In the exact homology sequence

i Hi( h ) σ Hi( h ) Hi(??) Hi+1( h ) · · · → ⊕n≥1 n G −→ ⊕n≥1 n G −→ −→ ⊕n≥1 n G → · · · the map σi is injective, as can be veriﬁed stalkwise at each x X. Assuming, as one may, that is K-injective, one deduces that ∈ G Hi(??) = lim Hi(h ) = Hi lim (h ) = Hi lim om•( / n, ) = Hi(RΓ ), −−→ n −−→ n −−→ X J n G n G n H O J G G whence the assertion.

6. Duality for torsion sheaves. Paragraph 6.1 contains the proof of Theorem 2 (section 1), that is, of two essentially equivalent forms of Torsion Duality on formal schemes—Theorem 6.1 and Corollary 6.1.4. The rest of the paragraph deals with numerous relations among the functors which have been introduced, and with compatibilities among dualizing functors occurring before and after completion of maps of ordinary schemes. More can be said for complexes with coherent homology, thanks to Greenlees- May duality. This is done in paragraph 6.2. 60 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

0 Paragraph 6.3 discusses additional relations involving RΓX : D(X) D(X) and its right adjoint R om•(RΓ 0 , ) on a locally noetherian formal sc→heme X. H XOX − Theorem 6.1. (a) Let f : X Y be a map of noetherian formal schemes. Assume that f is separated, or X→has finite Krull dimension, or else restrict to 5.2.6 bounded-below complexes. Then the ∆-functor Rf∗ : Dqct(X) Dqct(Y) , D(Y) has a right ∆-adjoint. −−−→ → × In fact there is a bounded-below ∆-functor ft : D(Y) Dqct(X) and a map × → of ∆-functors τt : Rf∗ft 1 such that for all Dqct(X) and D(Y), the composed map (in the derive→d category of abelianGgr∈oups) F ∈ RHom• ( , f × ) natural RHom• (Rf , Rf f × ) X G t F −−−−−→ Y ∗G ∗ t F via τ t RHom• (Rf , ) −−−−−→ Y ∗G F is an isomorphism. (b) If g : Y Z is another such map then there is a natural isomorphism → (gf)× ∼ f ×g×. t −→ t t × × × Proof. Assertion (b) follows from (a), which easily implies that (gf)t and ft gt are both right-adjoint to the restriction of R(gf)∗ = Rg∗Rf∗ to Dqct(X). As for (a), assuming first that X is separated or finite-dimensional, or that only bounded-below complexes are considered, we can replace Dqct(X) by the equivalent category D( qct(X)) (Proposition 5.3.1). The inclusion k : qct(X) , ~c(X) has A 0 0 A → A the right adjoint ΓX. (ΓX( ~c(X)) qct(X), by Lemma 5.1.4 and Corollary 3.1.5.) A 0 ⊂ A 0 So for all qct(X)-complexes and ~c(X)-complexes there is a natural isomorphism of abA elian-group complexesG A F Hom• ( 0, Γ 0 0 ) ∼ Hom• (k 0, 0 ). Aqct G XF −→ A~c G F 0 0 0 Note that if is K-injective over ~c(X) then ΓX is K-injective over qct(X), 0 F A F A because ΓX has an exact left adjoint. Combining this isomorphism with the isomorphism (14) in the proof of Theorem 4.1, we can conclude just as in part 4 at the × end of that proof, with the functor ft defined to be the composition 0 ρ C• ΓX natural D(Y) KI(Y) KI( (X)) KI( (X)) D( (X)). −→ −→ A~c −−→ Aqct −−−−−→ Aqct (We have in mind here simply that the natural functor D( qct(X)) D( ~c(X)) has a right adjoint. That is easily seen to be true once oneAknows the→ existenceA of K-injective resolutions in D( ~c(X)); but we don’t know how to prove the latter other than by quoting the generalizationA to arbitrary Grothendieck categories [Fe, Theorem 2], [AJS, Theorem 5.4]. The preceding argument avoids this issue. One could also apply Brown Representability directly, as in the proof of Theorem 1 described in the Introduction.) Now suppose only that the map f is separated. If Y is separated then so is X, and the preceding argument holds. For arbitrary noetherian Y the existence of a bounded-below right adjoint for Rf∗ : Dqct(X) D(Y) results then from the following Mayer-Vietoris pasting argument, by induction→ on the least number of separated open subsets needed to cover Y. Finally, to dispose of the assertion • × about the RHom ’s apply homology to reduce it to ft being a right adjoint. To reduce clutter, we will abuse notation—but only in the rest of the proof of × × Theorem 6.1—by writing “f ” in place of “ft .” DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 61

Lemma 6.1.1. Let f : X Y = Y1 Y2 (Yi open in Y) be a map of formal schemes, with X noetherian. Consider→ the∪ commutative diagrams

qi x X := X X X i X 12 1 ∩ 2 −−−−→ i −−−−→ f12 fi f (i = 1, 2)    y y y Y12 := Y1 Y2 Yi Y ∩ −−−pi−→ −−−y−i →

−1 where Xi := f Yi and all the horizontal arrows represent inclusions. Suppose that for i = 1, 2, 12, the functor Rf : D (X ) D(Y ) has a right adjoint f ×. Then i∗ qct i → i i Rf : D (X) D(Y) has a right adjoint f ×; and with the inclusions y := y ◦ p , ∗ qct → 12 i i x := x ◦ q , there is for each D(Y) a natural D(X)-triangle 12 i i F ∈ λ f × Rx f ×y∗ Rx f ×y∗ F Rx f × y∗ (f × )[1] . F → 1∗ 1 1 F ⊕ 2∗ 2 2F −−→ 12∗ 12 12F → F × ∗ × × ∗ Remark. If we expect f to exist, and the natural maps xi f fi yi to be isomorphisms, then there should be such a triangle—the Mayer-Vietoris→ triangle of f × . This suggests we ﬁrst deﬁne λ , then let f × be the vertex of a triangle F F F based on λF , and verify . . .

Proof. There are natural maps

τ : Rf f × 1, τ : Rf f × 1, τ : Rf f × 1. 1 1∗ 1 → 2 2∗ 2 → 12 12∗ 12 → ∗ × × ∗ For i = 1, 2, deﬁne the “base-change” map βi : qi fi f12 pi to be adjoint under Theorem 6.1 to the map of functors →

∗ × ∗ × τi ∗ Rf12∗qi fi pi Rfi∗fi pi . natural−−−→ −−→ f 0 × × ∗ This βi corresponds to a functorial map βi : fi Rqi∗f12 pi , from which we obtain a functorial map →

Rx f ×y∗ Rx Rq f × p∗y∗ ∼ Rx f × y∗ , i∗ i i −→ i∗ i∗ 12 i i −→ 12∗ 12 12 and hence a natural map, for any D(Y): F ∈ λ Dˇ 0( ):= Rx f ×y∗ Rx f ×y∗ F Rx f × y∗ =: Dˇ 1( ). F 1∗ 1 1F ⊕ 2∗ 2 2 F −−→ 12∗ 12 12F F Embed this map in a triangle Dˇ( ), and denote the third vertex by f ×( ): F F λ Dˇ( ): f × Dˇ 0( ) F Dˇ 1( ) (f × )[1]. F F → F −−→ F → F 0 1 × Since Dˇ ( ) and Dˇ ( ) are in Dqct(X) (see Proposition 5.2.6), therefore so is f (CorollaryF5.1.3). F F This is the triangle in Lemma 6.1.1. Of course we must still show that this f × is functorial, and right-adjoint to Rf∗. (Then by uniqueness of adjoints such a triangle will exist no matter which right adjoint f × is used.) 62 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

Let us next construct a map τ : Rf f × ( D(Y)). Set F ∗ F → F F ∈ Cˇ0( ):= Ry y∗ Ry y∗ , Cˇ1( ):= Ry y∗ . F 1∗ 1 F ⊕ 2∗ 2 F F 12∗ 12F We have then the Mayer-Vietoris D(Y)-triangle µ Cˇ( ): Cˇ0( ) F Cˇ1( ) [1], F F → F −−→ F → F arising from the usual exact sequence (Cecˇ h resolution) 0 y y∗ y y∗ y y∗ 0, → F → 1∗ 1 F ⊕ 2∗ 2 F → 12∗ 12F → where may be taken to be K-injective. Checking commutativity of the following naturalFdiagram is a purely category-theoretic exercise (cf. [L4, Lemma (4.8.1.2)] : Rf λ Rf Dˇ 0( ) ∗ F Rf Dˇ 1( ) ∗ F −−−−−−−−−−−−−−−−→ ∗ F

Rf (Rx f ×y∗ Rx f ×y∗ ) Rf Rx f × y∗ ∗ 1∗ 1 1 F ⊕ 2∗ 2 2 F ∗ 12∗ 12 12F ' '   Ry Rf f ×y∗ y Ry Rf f ×y∗ Ry Rfy f × y∗ 1∗ 1∗ 1 1 F ⊕ 2∗ 2∗ 2 2 F 12∗ 12∗ 12 12F τ τ1⊕τ2  12   Ry y∗ y Ry y∗ Ry yy∗ 1∗ 1 F ⊕ 2∗ 2 F 12∗ 12F

Cˇ0 ( ) Cˇ1 ( ) F −−−−−−−−µ−F−−−−−−−→ F

This commutative diagram extends to a map τˇF of triangles: Rf f × Rf Dˇ 0( ) Rf Dˇ 1( ) Rf f × [1] ∗ F −−−−→ ∗ F −−−−→ ∗ F −−−−→ ∗ F τ F    τF [1]     y Cˇ0y( ) Cˇ1y( ) y[1] F −−−−→ F −−−−→ F −−−−→ F The map τF is not necessarily unique. But the next Lemma will show, for ﬁxed , that the pair (f × , τ ) represents the functor F F F Hom Y (Rf , ) ( D (X)). D( ) ∗E F E ∈ qct × × It follows formally that one can make f into a functor and τ : Rf∗f 1 into a morphism of functors in such a way that the pair (f ×, τ) is a right adjoin→ t for Rf∗ : Dqct(X) D(Y) (cf. [M1, p. 83, Corollary 2]); and that there is a unique → × ∼ × isomorphism of functors Θ: f T2 T1f (where T1 and T2 are the respective −→ × translations on Dqct(X) and D(Y)) such that (f , Θ) is a ∆-functor ∆-adjoint to Rf∗ (cf. [L4, Proposition (3.3.8)]). That will complete the proof of Lemma 6.1.1.

Lemma 6.1.2. For D (X), and with f × , τ as above, the composition E ∈ qct F F × natural × via τF Hom X ( , f ) Hom Y (Rf , Rf f ) Hom Y (Rf , ) Dqct( ) E F −−−−→ D( ) ∗E ∗ F −−−−→ D( ) ∗E F is an isomorphism. DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 63

Proof. In the following diagram, to save space we write HX for HomDqct(X), HY for HomD(Y), and f∗ for Rf∗ :

0 0 0 HX( , (Dˇ )[ 1]) HY(f , f ((Dˇ )[ 1])) HY(f , (Cˇ )[ 1]) E F − −−−−→ ∗E ∗ F − −−−−→ ∗E F −       y1 y 1 y 1 HX( , (Dˇ )[ 1]) HY(f , f ((Dˇ )[ 1])) HY(f , (Cˇ )[ 1]) E F − −−−−→ ∗E ∗ F − −−−−→ ∗E F −       y × y × y HX( , f ) HY(f , f f ) HY(f , ) E F −−−−→ ∗E ∗ F −−−−→ ∗E F       y 0 y 0 y 0 HX( , Dˇ ) HY(f , f Dˇ ) HY(f , Cˇ ) E F −−−−→ ∗E ∗ F −−−−→ ∗E F       y 1 y 1 y 1 HX( , Dˇ ) HY(f , f Dˇ ) HY(f , Cˇ ) E F −−−−→ ∗E ∗ F −−−−→ ∗E F

The ﬁrst column maps to the second via functoriality of f∗ , and the second to the third via the above triangle map τˇF ; so the diagram commutes. The columns are exact [H1, p. 23, Prop. 1.1 b)], and thus if each of the ﬁrst two and last two rows is shown to compose to an isomorphism, then the same holds for the middle row, proving Lemma 6.1.2. Let’s look at the fourth row. With notation as in Lemma 6.1.1 (and again, with all the appropriate R’s omitted), we want the left column in the following natural diagram to compose to an isomorphism:

× ∗ ∗ × ∗ HX( , x f y ) HX (x , f y ) E i∗ i i F −−−−→ i i E i i F f     y × ∗ ∗ y × ∗ HY(f , f x f y ) HY (f x , f f y ) ∗E ∗ i∗ i i F i i∗ i E i∗ i i F ' '   y × ∗ ∗ y × ∗ HY(f , y f f y ) HY (y f , f f y ) ∗E i∗ i∗ i i F −−−−→ i i ∗E i∗ i i F f via τi via τi   y ∗ ∗ y ∗ HY(f , y y ) HY (y f , y ) ∗E i∗ i F −−−−→ i i ∗E i F f Here the horizontal arrows represent adjunction isomorphisms. Checking that the diagram commutes is a straightforward category-theoretic exercise. By hypothesis, the right column composes to an isomorphism. Hence so does the left one. The argument for the fifth row is similar. Using the (easily checked) fact 0 0 that the morphism f∗Dˇ Cˇ is ∆-functorial, we find that the first row is, up to isomorphism, the same→as the fourth row with [ 1] in place of , so it too composes to an isomorphism. Similarly, isomorphismF −for the secondFrow follows from that for the fifth. 64 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

Examples 6.1.3. (1) Let f : X Y be a map of quasi-compact formal schemes with X properly algebraic, and let f→× be the right adjoint given by Corollary 4.1.1. × 0 × Using Proposition 5.2.1 we find then that ft := RΓX ◦ f is a right adjoint for the restriction of Rf∗ to Dqct(X). ! × (2) For a noetherian formal scheme X, Theorem 6.1 gives a right adjoint 1 := 1t 0 to the inclusion Dqct(X) , D(X). If Dqc(X) (i.e., RΓX Dqct(X), see → G ∈ 0 ! G ∈ Definition 5.2.9), then the natural Dqct(X)-mape RΓX 1 (corresponding to the natural D(X)-map RΓ 0 ) is an isomorphism,Gsee→PropG osition 5.2.1. X G → G (3) If X is separated or if X is finite-dimensional, then we have the equivalence t ≈ ! t t jX : D( qct(X)) Dqct(X) of Proposition 5.3.1, and we can take 1 := jX ◦ RQX, see CorollaryA 5.1.5−→and Lemma 5.2.2. (4) Let f : X Y be a closed immersion of noetherian formal schemes (see [GD, p. 442]). The→functor f : (X) (Y) is exact, so Rf = f . Let I be the ∗ A → A ∗ ∗ kernel of the surjective map Y f∗ X and let Y be the ringed space (Y, Y/I), O f¯ O O so that f factors naturally as X Y i Y, the map f¯ being flat. The inverse ¯ ¯ f∗ → → f∗ isomorphisms (X) (Y) extend to inverse isomorphisms D(X) D(Y) A −←−f¯→∗− A −←−f¯→∗− The functor I : (Y) (Y) defined by I(F ) := om( Y/I, F ) has an H A → A H H O exact left adjoint, namely i : (Y) (Y), so I preserves K-injectivity and ∗ A → A H R I is right-adjoint to i : D(Y) D(Y) (see proof of Lemma 5.2.2). Hence the H ∗ → functor f \ : D(Y) D(Y) defined by → \ ∗ ∗ • (16) f ( ):= f¯ R I( ) = f¯ R om ( Y/I, ) ( D(Y)) F H F H O F F ∈ is right-adjoint to f = i f¯, and f : D (X) D(Y) has the right adjoint ∗ ∗ ∗ ∗ qct → × ! ! \ ft := f := 1 ◦ f .

We recall that (X) is quasi-coherent iff f¯∗ qc(Y) iff f∗ qc(Y), see [GD, p. 115, (5.3.15),G ∈ A(5.3.13)]. Also, by lookingGat∈stalksA (see 1.2.1)G ∈ Awe find that f (Y) (X). Hence Remark 5.2.10(4) together with§ the isomor- ∗G ∈ At ⇒ G ∈ At phism Rf RΓ 0 = RΓ 0Rf of Corollary 5.2.11(d) yields that f \D+ (Y) D+ (X); ∗ X ∼ Y ∗ qc ⊂ qc and given Corollary 5.1.3, Proposition 3.2.4 yields f \D+ (Y) eD+ (X). eThus if qct ⊂ qct D+ (Y) then by (2) above, f ! = RΓ 0 f \ ; and if D+ (Y) then f ! = f \ . F ∈ qc F ∼ X F F ∈ qct F ∼ F (5)e Let f : X Y be any map satisfying the hypotheses of Theorem 6.1. Let → J X and I Y be ideals of definition such that I X J, and let ⊂ O ⊂ O O ⊂ n fn n X := (X, X/J ) (Y, Y/I ) =: Y (n > 0) n O −−→ O n be the scheme-maps induced by f, so that each fn also satisfies the hypotheses × of Theorem 6.1. As the target of the functor (fn)t is Dqct(Xn) = Dqc(Xn), we × × write fn for (fn)t (see (1) above). If jn : Xn , X and in : Yn , Y are the → ! × × ! → canonical closed immersions then fjn = infn, and so jnft = fn in. The functor j\ : D(X) D(X ) being as in (16), we have, using (4), n → n h := R om•( /Jn, ) = j j\ = j j! ( D+ (X)). nG H OX G n∗ nG ∼ n∗ nG G ∈ qc e Hence for := f × ( D+(Y)), Lemma 5.4.1 gives a “homotopy colimit” triangle G t F F ∈ j f ×i! j f ×i! f × + ⊕n≥1 n∗ n n F −→ ⊕n≥1 n∗ n n F −→ t F −→ DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 65

0 −1 Once again, Dqc(X):= (RΓX) Dqct(X) (Deﬁnition 5.2.9). e Corollary 6.1.4. (a) Let f : X Y be a map of noetherian formal schemes. Suppose that f is separated or that X→has ﬁnite Krull dimension, or else restrict to bounded-below complexes. Let ΛX : D(X) D(X) be the bounded-below ∆-functor → • 0 ΛX( ):= R om (RΓ , ), − H XOX − # # × and let f : D(Y) D (X) be the ∆-functor f := ΛXf (see Example 5.2.10(3)). → qc t The functor f # ise bounded-below, and is right-adjoint to

5.2.6 Rf RΓ 0 : D (X) D (Y) , D(Y). ∗ X qc −−−→ qct → e (In particular with j : D( (X)) D(X) the natural functor, the functor A~c → Rf RΓ 0 j : D( (X)) D(Y) ∗ X A~c → # has the bounded-below right adjoint RQXf —see Proposition 3.2.3.) In fact there is a map of ∆-functors

τ # : Rf RΓ 0 f # 1 ∗ X → such that for all D (X) and D(Y), the composed map G ∈ qc F ∈ e RHom• ( , f # ) natural RHom• (Rf RΓ 0 , Rf RΓ 0 f # ) X G F −−−−−→ Y ∗ XG ∗ X F # via τ RHom• (Rf RΓ 0 , ) −−−−→ Y ∗ XG F is an isomorphism. (b) If g : Y Z is another such map then there is a natural isomorphism → (gf)# ∼ f #g#. −→ 0 Proof. (a) The functor ΛX is bounded below because RΓX X is locally iso- • O morphic to the bounded complex ∞ in the proof of Proposition 5.2.1(a), hence K 0 homologically bounded-above. Since ΛX is right-adjoint to RΓX (see (15)), (a) follows directly from Theorem 6.1.

(b) Propositions 5.2.6 and 5.2.1(a) show that for any Dqct(X) we have 0 × × G ∈ RΓ Rf = Rf , and hence the functors f ΛY and f are both right-adjoint Y ∗G ∼ ∗G t t to Rf∗ Dqct(X), so they are isomorphic. Then Theorem 6.1(b) yields functorial isomorphisms|

# × ∼ × × ∼ × × # # (gf) = ΛX(gf) ΛXf g ΛXf ΛYg = f g . t −→ t t −→ t t

Here are some “identities” involving the dualizing functors f × (Theorem 4.1), × # × ft (Theorem 6.1), and f := ΛXft (Corollary 6.1.4). 0 Note that ΛX is right-adjoint to RΓX, see (15). Simple arguments show that ∼ 0 ∼ 0 the natural maps are isomorphisms ΛX ΛXΛX , RΓX RΓXΛX , see (b) and (d) in Remark 6.3.1(1). −→ −→ 66 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

Corollary 6.1.5. With the notation of Corollary 6.1.4, (a) There are natural isomorphisms 0 # ∼ × # ∼ × RΓ f f , f ΛXf , X −→ t −→ t 0 × ∼ × # ∼ # RΓ f f , f ΛXf . X t −→ t −→ 0 (b) The natural functorial maps RΓ 1 ΛY induce isomorphisms Y → → × 0 ∼ × ∼ × f RΓ f f ΛY, t Y −→ t −→ t # 0 ∼ # ∼ # f RΓ f f ΛY. Y −→ −→ (c) There are natural pairs of maps α α f × 1 RΓ 0 jf × 2 f ×, t −−→ X −−→ t # β1 × β2 # f ΛX jf f , −→ −→ each of which composes to an identity map. If X is properly algebraic then all of these maps are isomorphisms. (d) If f is adic then the isomorphism Rf RΓ 0 j ∼ RΓ 0 Rf j in 5.2.11(d) ∗ X ←− Y ∗ induces an isomorphism of the right adjoints (see Theorem 4.1, Proposition 3.2.3) × ∼ # f ΛY RQ f . −→ X Proof. (a) The second isomorphism (first row) is the identity map. Proposi- tion 5.2.1 yields the third. The first is the composition 0 # 0 × ∼ 0 × ∼ × RΓ f = RΓ ΛXf RΓ f f . X X t −→ X t −→ t The fourth is the composition # × ∼ × ∼ # f = ΛXf ΛXΛXf ΛXf . t −→ t −→ 0 (b) The first isomorphism results from RΓY being right adjoint to the inclu- × sion Dt(Y) , D(Y) (see Proposition 5.2.1(c)). For the second, check that ft × → and ft ΛY are both right-adjoint to Rf∗ Dqct(X) . . . (Or, consider the composition × ∼ × 0 ∼ × 0 ∼ ×| ft ft RΓY ft RΓYΛY ft ΛY.) Then apply ΛX to the first row to get the−→second ro−w.→ −→ (c) With k : D( (X)) D( (X)) the natural functor, let Aqct → A~c α: kRQt f × f × X t → t × 5.3.1 × τt × × be adjoint to Rf∗jkRQXft = Rf∗ft 1. By Corollary 5.2.3, j(α): ft jf factors naturally as −→ → α f × 1 RΓ 0 jf × jf ×. t −→ X → Let α be the map adjoint to the natural composition Rf RΓ 0 jf × Rf jf × 1. 2 ∗ X → ∗ → One checks that τt ◦ Rf∗(α2α1) = τt (τt as in Theorem 6.1), whence α2α1 = identity. The pair β1 , β2 is obtained from α1 , α2 by application of the functor ΛX— see Corollary 5.2.3. (Symmetrically, the pair α1 , α2 is obtained from β1 , β2 by 0 application of the functor RΓX.) When X is properly algebraic, the functor j is fully faithful (Corollary 3.3.4); 0 × × and it follows that RΓ jf and f are both right-adjoint to Rf X . X t ∗|Dqct( ) (d) Straightforward. DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 67

The next three corollaries deal with compatibilities between formal (local) and ordinary (global) Grothendieck duality. Corollary 6.1.6. Let f : X Y be a map of noetherian ordinary schemes. 0 → Suppose either that f0 is separated or that X is ﬁnite-dimensional, or else restrict −1 to bounded-below complexes. Let W Y and Z f0 W be closed subsets, κY : Y = Y Y and κ : X = X X the⊂ respective⊂ completion maps, and f : X Y /W → X /Z → → the map induced by f0. κ X:= X X X /Z −−−−→ f f0   y y Y:= Y/W Y −−−κY−→ With f× := (f )× right-adjoint to Rf : D (X) D(Y ), let τ 0 be the composition 0 0 t ∗ qc → t ∗ × ∼ ∗ × ∗ × ∗ Rf∗κXRΓZ f κY∗ κYRf0∗RΓZ f κY∗ κYRf0∗f κY∗ κYκY∗ 1. 0 5−.2→.7 0 −→ 0 −→ −→ Then for all D (X) and D(Y), the composed map E ∈ qct F ∈ ∗ × ∗ × α( , ): Hom X ( , κ RΓ f κ ) Hom Y (Rf , Rf κ RΓ f κ ) E F D( ) E X Z 0 Y∗F −→ D( ) ∗E ∗ X Z 0 Y∗F Hom Y (Rf , ) 0 D( ) ∗ via−→τt E F 0 is an isomorphism. Hence the map adjoint to τt is an isomorphism of functors κ∗ RΓ f×κ ∼ f ×. X Z 0 Y∗ −→ t Proof. For any Dqct(X), set 0 := κX∗ DqcZ (X) (Proposition 5.2.4). Proposition 5.2.4 andE[AJL∈ , p. 7, LemmaE (0.4.2)]Egiv∈ e natural isomorphisms ∗ ∼ ∼ Hom X ( , κ RΓ ) Hom ( , RΓ ) Hom ( , ) D( ) E X Z G −→ D(X) E0 Z G −→ D(X) E0 G ( Dqc(X)). ∗ × G ∈ (In other words, κXRΓZ = (κX)t .) One checks then that the map α( , ) factors as the sequence ofGnatural isomorphismsG E F ∗ × ∼ × Hom X ( , κ RΓ f κ ) Hom ( , f κ ) D( ) E X Z 0 Y∗F −→ D(X) E0 0 Y∗F ∼ Hom (Rf , κ ) −→ D(Y ) 0∗E0 Y∗F ∼ ∗ Hom Y (κ Rf , ) −→ D( ) Y 0∗E0 F ∼ ∗ Hom Y (Rf κ , ) (Corollary 5.2.7) −→ D( ) ∗ XE0 F ∼ Hom Y (Rf , ). −→ D( ) ∗E F Corollary 6.1.7. With hypotheses as in Corollary 6.1.6: (a) There are natural isomorphisms RΓ 0 κ∗ f×κ = (κ )×f×κ ∼ f ×, X X 0 Y∗ X t 0 Y∗ −→ t ∗ × # × ∼ # ΛXκ f κ = κ f κ f ; X 0 Y∗ X 0 Y∗ −→ −1 × and if f0 is proper, Y = Spec(A) (A adic), Z = f0 W, then with f as in Corol- lary 4.1.1: κ∗ f×κ ∼ f ×. X 0 Y∗ −→ (b) The functor f × := RΓ f× : D(Y ) D (X) is right-adjoint to the func- 0,Z Z 0 → qcZ tor Rf ; and there is an isomorphism ∗|DqcZ (X) κ∗ f × κ ∼ f ×. X 0,Z Y∗ −→ t 68 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

(c) If X is separated then, with notation as in Section 3.3, the functor f # := j RQ R om• (RΓ , f× ): D(Y ) D (X) 0,Z X X H X Z OX 0 − → qc is right-adjoint to Rf RΓ ; and if X is properly algebraic, so that we have 0∗ Z |Dqc(X) the equivalence j : D( (X)) D (X) (Corollary 3.3.4), there is an isomorphism X A~c → ~c κ∗ f # κ ∼ j RQ f #. X 0,Z Y∗ −→ X X

Proof. (a) The first isomorphism combines Corollary 6.1.6 (in proving which ∗ × we noted that κXRΓZ = (κX)t for Dqc(X)) and Proposition 5.2.4. The #G × G G ∈ second follows from f = ΛXft . The third is Corollary 4.1.2. (b) The first assertion is easily checked; and the isomorphism is given by Corol- lary 6.1.6. (c) When X is separated, jX is an equivalence [AJL, p. 12, Proposition (1.3)], and then the first assertion is easily checked. From Corollary 6.1.6 and Proposition 5.2.4 we get an isomorphism RΓ f×κ ∼ κ f ×. Z 0 Y∗ −→ X∗ t As in Corollary 5.2.3, the natural map is an isomorphism R om• (RΓ , ) ∼ R om• (RΓ , RΓ ) ( D (X)). H X Z OX G −→ H X Z OX Z G G ∈ qc X j ∗ j When is properly algebraic, XRQX ∼= κX X RQX κX∗ (Proposition 3.2.3). So then we have a sequence of natural isomorphisms κ∗ f # κ κ∗ j RQ R om• (RΓ , f×κ ) X 0,Z Y∗ X X X H X Z OX 0 Y∗− ∼ κ∗ j RQ R om• (RΓ , RΓ f×κ ) −→ X X X H X Z OX Z 0 Y∗− ∼ κ∗ j RQ R om• (RΓ , κ f × ) −→ X X X H X Z OX X∗ t − ∼ κ∗ j RQ κ R om• (κ∗ RΓ , f × ) −→ X X X X∗ H X X Z OX t − ∼ • 0 × j RQ R om (RΓ X , f ) −→ X X H X XO t − # jXRQXf .

The following instance of “ﬂat base change” will be needed in the proof of the general base-change Theorem 3. −1 Corollary 6.1.8. In Corollary 6.1.6, assume further that Z = f0 W. Then the natural map is an isomorphism × ∼ × ∗ RΓ f RΓ f κY κ ( D(Y )), Z 0 F −→ Z 0 ∗ YF F ∈ and so there is a composed isomorphism 0 ∗ × ∼ ∗ × ∼ ∗ × ∗ ∼ × ∗ ζ : RΓXκXf0 κXRΓZ f0 κXRΓZ f0 κY∗κY ft κY . F 5.2.4(c)−→ F −→ F 6.1.7(b)−→ F

Proof. First, Rf0∗(DqcZ (X)) DqcW (Y ). For, by [L4, Proposition (3.9.2)], Rf (D (X)) D (Y ); and then ⊂the assertion follows from the natural isomor- 0∗ qc ⊂ qc phism of functors (from D (X) to D (Y )) RΓ Rf Rf RΓ −1 , because qc qc W 0∗ ∼= 0∗ f W DqcZ (X) (resp. DqcW (Y )) iﬀ RΓZ ∼= (resp. RΓW ∼= ), cf. Propo- Gsition∈ 5.2.1(a) and itsH ∈proof. (The said functorialG G isomorphismH arisesH from the corresponding one without the R’s, since Rf0∗ preserves K-ﬂabbiness, see [Sp, 5.12, 5.15(b), 6.4, 6.7]. DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 69

Now Corollary 5.2.5 gives that the natural map is an isomorphism ∼ ∗ Hom (Rf , ) Hom (Rf , κY κ ) ( D (X)), D(Y ) 0∗E F −→ D(Y ) 0∗E ∗ YF E ∈ qcZ and the conclusion follows from the adjunction in Corollary 6.1.7(b).

6.2. The next Proposition is a special case of Greenlees-May Duality for formal schemes (see [AJL0, Proposition 0.3.1]). It is the key to many statements in this paper concerning complexes with coherent homology. Proposition 6.2.1. Let X be a locally noetherian formal scheme, D(X). Then for all D (X) the natural map RΓ 0 induces an isomorphismE ∈ F ∈ c XE → E R om•( , ) ∼ R om•(RΓ 0 , ). H E F −→ H XE F Proof. The canonical isomorphism (cf. (15)) R om•(RΓ 0 , ) ∼ R om•( , R om•(RΓ 0 , )) H XE F −→ H E H XOX F reduces the question to where = X . It suﬃces then—as in the proof of Corol- lary 5.2.3—that for aﬃne X = ESpf(AO), the natural map be an isomorphism

∼ 0 Hom X ( X , ) Hom X (RΓ X , ) ( D (X)). D( ) O F −→ D( ) XO F F ∈ c Let I be an ideal of definition of the adic ring A, set Z := Supp(A/I), and let κ: X X := Spec(A) be the completion map. The categorical equivalences in Proposition→ 3.3.1 and the isomorphism κ∗RΓ 0 ∼ RΓ 0 in Proposition 5.2.4 Z OX −→ XOX make the problem whether for all F Dc(X) (e.g., F = RQκ∗ := jX RQX κ∗ ) the natural map is an isomorphism ∈ F F Hom ( , F ) ∼ Hom (RΓ , F ). D(X) OX −→ D(X) Z OX Now, the canonical functor jX : D( qc(X)) D(X) induces an equivalence of categories D( (X)) ≈ D (X) (seeAbeginning→ of 3.3), and so we may assume Aqc −→ qc § that F is a K-flat quasi-coherent complex. Lemma 5.2.2 shows that jX RQX is right-adjoint to the inclusion D (X) , D(X). The natural map qc → R om•( , F ) R om•(RΓ , F ) H OX → H Z OX factors then as • ∼ ∗ R om ( X , F ) = F jX RQX κ∗κ F H O 3−.→3.1 (17) κ κ∗F −→ ∗ ∼ lim F/(I )nF ∼ R om•(RΓ , F ), ←−− X Z X −λ→ n O −Φ→ H O ∗ where the map λ, obtained by applying κ∗ to the natural map from κ F to the completion F/Z , is a D(X)-isomorphism by [AJL, p. 6, Proposition (0.4.1)]; and Φ is the isomorphism Φ(F, X ) of [AJL, 2]. (The fact that Φ is an isomorphism is essentially the main resultO in [AJL].) Also,§ by adjointness, the natural map is an isomorphism Hom ( , j RQ κ κ∗F ) ∼ Hom ( , κ κ∗F ). D(X) OX X X ∗ −→ D(X) OX ∗ Conclude now by applying the functor H0RΓ(X, ) to (17). − 70 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

Corollary 6.2.2. Let f : X Y be as in Corollary 6.1.4, and assume further → that f is adic. Then for all Dc(Y) the map corresponding to the natural composition Rf RΓ 0 jf × RFf j∈f × (see Theorem 4.1) is an isomorphism ∗ X F → ∗ F → F f × ∼ RQ f # . F −→ X F Proof. • 0 By Proposition 6.2.1, ∼= ΛY := R om (RΓY Y , ); so this Corollary is a special case of CorollaryF 6.1.5(d).F H O F Corollary 6.2.3. In Corollary 6.1.6, suppose Y = Spec(A) (A adic) and that ! × the the map f0 is proper. Then with the customary notation f0 for f0 we have, for any D+(Y), a natural isomorphism F ∈ c κ∗ f !κ ∼ f # D+(X). X 0 Y∗F −→ F ∈ c ! ! Proof. The natural map f0 jY RQY κY∗ f0κY∗ is an isomorphism of func- → ∗ tors from D(Y) to Dqc(X), both being right-adjoint to κYRf0∗. Proposition 3.3.1 + ! + 19 gives jY RQY κY∗ Dc (Y ); so by [V, p. 396, Lemma 1], f0κY∗ Dc (X). Hence PropositionF6.2.1∈ and Corollary 6.1.7(a) yield isomorphisms F ∈ ∗ ! ∼ • 0 ∗ ! ∗ ! ∼ # κ f κ R om (RΓ , κ f κ ) =: ΛXκ f κ f . X 0 Y∗F −→ H XOX X 0 Y∗F X 0 Y∗F −→ F

0 • 0 6.3. More relations, involving the functors RΓX and ΛX := R om (RΓX X, ) on a locally noetherian formal scheme X, will now be summarized.H O − Remarks 6.3.1. Let X be a locally noetherian formal scheme. 0 γ (1) The functor Γ := RΓX : D(X) → D(X) admits a natural map Γ −→ 1, which induces a functorial isomorphism (A) Hom(Γ E, Γ F ) −∼→ Hom(Γ E, F ) (E, F ∈ D(X)), see Proposition 5.2.1(c). Moreover Γ has a right adjoint, viz. Λ:= ΛX (see (15)). The rest of (1) consists of (well-known) formal consequences of these properties. Since γ is functorial, it holds that γ(F ) ◦ γ(Γ F ) = γ(F ) ◦ Γ (γ(F )): Γ Γ F → F, so injectivity of the map in (A) (with E = Γ F ) yields γ(Γ F ) = Γ (γ(F )): Γ Γ F → Γ F; and one ﬁnds after setting F = Γ G in (A) that this functorial map is an isomorphism (a) γ(Γ ) = Γ (γ): Γ Γ −∼→ Γ . Conversely, given (a) one can deduce that the map in (A) is an isomorphism, whose Γ inverse takes α: Γ E → F to the composition Γ E −∼→ Γ Γ E −−α→ Γ F .20 The composed Λ(γ) functorial map λ: 1 → ΛΓ −−−→ Λ induces an isomorphism (B) Hom(ΛE, ΛF ) −∼→ Hom(E, ΛF ) (E, F ∈ D(X)),

19 D+ ! D+ For G ∈ c (Y ) one has f0G ∈ c (X): The question being local on X one reduces to where Pn ! ∗ n D+ either X is a projective space Y and f0 is projection, so that f0G = f0 G ⊗ΩX/Y [n] ∈ c (X), or ! R • D+ H1 f0 is a closed immersion and f0∗f0G = HomY (f0∗OX , F ) ∈ c (Y ) [ , p. 92, Proposition 3.3] ! D+ GD whence, again, f0G ∈ c (X) [ , p. 115, (5.3.13)]. 20The idempotence of Γ, expressed by (a) or (A), can be interpreted as follows. q Γ Set D := D(X), S := { E ∈ D | Γ (E) = 0 }, so that Γ factors uniquely as D → D/S → D where q is the “Verdier quotient” functor. Then Γ is left-adjoint to q, so that S ⊂ D admits a “Bousfield colocalization.” It follows from (c) and (d) below that S = { E ∈ D | Λ(E) = 0 }, and (b) below means that the functor Λ¯ : D/S → D defined by Λ = Λ¯ ◦ q is right-adjoint to q; thus S ⊂ D also admits a “Bousfield localization.” And D/S is equivalent, via Γ and Λ¯ respectively, ⊥ ⊥ to the categories Dt ⊂ D and Dˆ ⊂ D introduced below—categories denoted by S and S in [N2, Chapter 8]. DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 71 or equivalently (as above), λ induces an isomorphism (b) λ(Λ) = Λ(λ): Λ −∼→ ΛΛ. Moreover, the isomorphism (A) transforms via adjointness to an isomorphism Hom(E, ΛΓ F ) −∼→ Hom(E, ΛF ) (E, F ∈ D(X)), whose meaning is that γ induces an isomorphism (c) ΛΓ −∼→ Λ. Similarly, (B) means that λ induces the conjugate isomorphism (d) Γ Λ ←∼− Γ . Similarly, that Λ(λ(F ))—or γ(Γ (E))—is an isomorphism (respectively that λ(Λ(F ))— or Γ (γ(E))—is an isomorphism) is equivalent to the first (respectively the second) of the following maps (induced by λ and γ respectively) being an isomorphism: (AB) Hom(Γ E, F ) −∼→ Hom(Γ E, ΛF ) ←∼− Hom(E, ΛF ). That (c) is an isomorphism also means that the functor Λ factors, via Γ, through the essential image Dt(X) of Γ (i.e., the full subcategory Dt(X) whose objects are isomorphic to Γ E for some E); and similarly (d) being an isomorphism means that Γ factors, via Λ, through the essential image Dˆ(X) of Λ; and the isomorphisms Γ ΛΓ =∼ Γ and ΛΓ Λ =∼ Λ deduced from (a)–(d) signify that Λ and Γ induce quasi-inverse equivalences between the categories Dt(X) and Dˆ(X).

(2) If X is properly algebraic, the natural functor j : D(A~c(X)) → D~c(X) is an equivalence, and the inclusion D~c(X) ,→ D(X) has a right adjoint Q := jRQX (Corollary 3.3.4.) Then (easily checked, given Corollary 3.1.5 and Proposition 5.2.1) all of (1) holds with ~c D, Dt, and Λ replaced by D~c , Dqct , and Λ := QΛ, respectively.

(3) As in (1), Λ induces an equivalence from Dqct(X) to Dˆqc(X), the essential image ∼ of Λ|Dqct(X)—or, since Λ = ΛΓ, of Λ|Dqc(X) (Proposition 5.2.1). So for any f : X → Y as in Corollary 6.1.5, the functor 0 ΛYRf∗ RΓX : Dˆqc(X) → Dˆqc(Y) × 0 × # has the right adjoint ΛXft RΓY = ΛXft = f . There result two “parallel” adjoint pseudofunctors [L4, (3.6.7)(d)] (where “3.6.6” should be “3.6.2”): × 0 # (Rf∗ , ft ) (on Dqct) and (ΛYRf∗ RΓX, f ) (on Dˆqc).

Both of these correspond to the same adjoint pseudofunctor on the quotient Dqc/(S∩Dqc), see footnote under (1). If f is adic then Rf∗ΛX =∼ ΛYRf∗ (Corollary 5.2.11(c)), and so Proposition 5.2.6 gives that Rf∗(Dˆqc(X)) ⊂ Dˆqc(Y). Moreover, there are functorial isomorphisms 0 0 ΛYRf∗ RΓXΛX =∼ Rf∗ ΛXRΓXΛX =∼ Rf∗ ΛX . 0 Thus for adic f, ΛYRf∗ RΓX can be replaced above by Rf∗. When f is proper more can be said, see Theorem 8.4.

7. Flat base change. A ﬁber square of adic formal schemes is a commutative diagram V v X −−−−→ g f   Uy yY −−−u−→ 72 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

∼ such that the natural map is an isomorphism V X Y U. If I, J, K are ideals −→ × of definition of Y, X, U respectively, then := J V + K V is an ideal of definition L O O of V, and the scheme V := (V, V/ ) is the fiber product of the (Y, Y/I)-schemes O L O (X, X/J) and (U, U/K), see [GD, p. 417, Proposition (10.7.3)]. By [GD, p. 414, O O 2 Corollaire (10.6.4)], if V is locally noetherian and the V -module / is of finite type then V is locally noetherian. That happens whenevO er X, Y andL LU are locally noetherian and either u or f is of pseudo-finite type. Our goal is to prove Theorem 7.4 (= Theorem 3 of the Introduction). That is, given a fiber square as above, with X, Y, U and V noetherian, f pseudo-proper, and u flat, we want to establish a functorial isomorphism β : RΓ 0 v∗f × ∼ g× RΓ 0 u∗ (= g×u∗ ) ( D+ (Y)). F V t F −→ t U F ∼ t F F ∈ qc Some consequences of this theorem will be given in Section 8.e In order to define βF (Definition 7.3) we first need to set up a canonical iso- 0 ∗ ∼ 0 ∗ morphism RΓUu Rf∗ RΓURg∗v . This is done in Proposition 7.2. (When u is −0→ adic as well as flat, RΓU can be omitted.) Our proof of Theorem 7.4 has the weakness that it assumes the case when f is a proper map of noetherian ordinary schemes. As far as we know, the published proofs of this latter result make use of finite-dimensionality hypotheses on the schemes involved (see [V, p. 392, Thm. 2], [H1, p. 383, Cor. 3.4]), or projectivity hypotheses on f [H1, p. 191, 5]). There is however an outline of a proof for the general case, even without noetherian hypotheses, in [L5]—see Corollary 4.3 there.21 To begin with, here are several properties of formal-scheme maps (see 1.2.2) which propagate across fiber squares. § Proposition 7.1. (a) Let f : X Y and u : U Y be maps of locally noe- → → therian formal schemes, such that the fiber product X Y U is locally noetherian (a condition which holds, e.g., if either f or u is of pseudo-finite× type, see [GD, p. 414, Corollaire (10.6.4)]). If f is separated (resp. affine, resp. pseudo-proper, resp. pseudo-finite, resp. of pseudo-finite type, resp. adic) then so is the projection X Y U U. × → (b) With f : X Y and u : U Y as in (a), assume either that u is adic or → → that f is of pseudo-finite type. If u is flat then so is the projection X Y U X. × → (c) Let f : X Y, u : U Y be maps of locally noetherian formal schemes, with u flat and locally→over Y the→completion of a finite-type map of ordinary schemes. Then X Y U is locally noetherian, and the projection X Y U X is flat. × × → Proof. (a) The adicity assertion is obvious, and the rest follows from corresponding assertions for the ordinary schemes obtained by factoring out defining ideals. (b) It’s enough to treat the case when Y, X, and U are the formal spectra, respectively, of noetherian adic rings (A, I), (B, J) and (C, K) such that B and C are A-algebras with J IB and K IC, and such that B C is noetherian ⊃ ⊃ ⊗A (since X Y U is locally noetherian, see [GD, p. 414, Corollaireb(10.6.5)]). By the following×Lemma 7.1.1, the problem is to show that if C is A-flat and either K = IC (u adic), or B/J is a finitely-generated A-algebra (f of pseudo-finite type), then B C is B-flat. ⊗A b 21Details may eventually appear in [L4]. It is quite possible that the argument can be adapted to give a direct proof for formal schemes too. DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 73

The local criterion of flatness [B, p. 98, 5.2, Thm. 1 and p. 101, 5.4, Prop. 2] § §n reduces the problem further to showing that for all n > 0, (B A C)/J (B A C) is n n n ⊗ ⊗ (B/J )-flat, i.e., that (B/J ) A C is (B/J )-flat. But, C beingb A-flat, if Kb = IC n n ⊗ n n then (B/J ) A C = (B/J ) bA/In (C/I C) is clearly B/J -flat; while if B/J is ⊗ ⊗ n n a finitely-generatedb A-algebra, then (B/J ) A C is noetherian and (B/J )-flat, ⊗ whence so is its K-adic completion (B/J n) C. ⊗A (c) Proceeding as in the proof of (b), bwe may assume C to be the K 0-adic completion of a finite-type A-algebra C0 (K0 a C0-ideal). If C is A-flat then by [B, 5.4, Proposition 4], the localization C00 := C0[(1+K0)−1] is A-flat, so the noetherian § B-algebra B C00 is B-flat, as is its (noetherian) completion B C. ⊗A ⊗A Lemma 7.1.1. Let ϕ : A C be a continuous homomorphismb of noetherian adic rings. Then C is A-flat iff→ the corresponding map Spf(ϕ): Spf(C) Spf(A) → is flat, i.e., iff for each open prime q C, C is A −1 -flat. ⊂ {q} {ϕ q} Proof. Recall that if K is an ideal of definition of C and q K is an open prime ideal in C, then with C q ordered by divisibility, ⊃ \ C := = lim C {q} Spf(C),q −−→ {f} O f∈C\q where C{f} is the K-adic completion of the localization Cf . Now for each f / q and n > 0 the canonical map C /KnC C /KnC is ∈ f f → {f} {f} bijective, so the lim of these maps is an isomorphism C /KnC ∼ C /KnC , −−→ q q −→ {q} {q} whence so is the K-adic completion C ∼ C of the canonical map C C . q −→ {q} q → {q} We can therefore apply [B, 5.4, Propcositiond4] twice to get that Cq is A −1 -flat § ϕ q iff C{q} is A{ϕ−1q}-flat. So if C is A-flat then Spf(ϕ) is flat; and the converse holds because C is A-flat iff Cm is Aϕ−1m-flat for every maximal ideal m in C, and every such m is open since C is complete. Proposition 7.2. (a) Consider a fiber square of noetherian formal schemes V v X −−−−→ g  f   Uy yY −−−u−→ with u and v flat. Let ψ : Rg RΓ 0 v∗ RΓ 0 Rg v∗ ( D (X)) G ∗ V G → U ∗ G G ∈ qc 0 ∗ ∗ be the unique map whose composition with the natural map RΓU Rg∗v Rg∗v 0 ∗ ∗ G → G is the natural map Rg∗RΓV v Rg∗v . (The existence of ψG is given by Propositions 5.2.1 and 5.2.6.) ThenG → for all G D (X), ψ is an isomorphism. E ∈ qct E In particular, if u (hence v) is adic then ψE can be identified with the identity map of Rg v∗ . ∗ E (b) Let X, Y, U be noetherian formal schemes, let f : X Y and u: U Y be maps, with u flat, and assume further that one of the following→ holds: → (i) u is adic, and V:= X Y U is noetherian, (ii) f is of pseudo-finite typ×e, (iii) u is locally the completion of a finite-type map of ordinary schemes; so that by Proposition 7.1 we have a fiber square as in (a). Let θ : u∗ Rf Rg v∗ ( D(X)) G ∗G → ∗ G G ∈ 74 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN be adjoint to the canonical map Rf Rf Rv v∗ = Ru Rg v∗ . ∗G → ∗ ∗ G ∗ ∗ G Then for all D (X), the map θ0 := RΓ 0 (θ ) is an isomorphism E ∈ qct E U E θ0 : RΓ 0 u∗ Rf ∼ RΓ 0 Rg v∗ . E U ∗E −→ U ∗ E In particular, if u (hence v) is adic then θE itself is an isomorphism. 0 (c) Under the hypotheses of (a) resp. (b), if f (hence g) is adic then ψE resp. θE is an isomorphism for all D (X) (see Definition 5.2.9). E ∈ qc e Proof. (a) Let J be an ideal of definition of X, and of U, so that J V + V V K O KO is an ideal of definition of . The obvious equality ΓJOV+KOV = ΓKOV ΓJOV , applied to K-injective V-complexes, leads to a natural functorial map O 0 def RΓV = RΓJO +KO RΓKO RΓJO 1.2.1 V V −→ V V which is an isomorphism, as one checks locally via [AJL, p. 20, Corollary (3.1.3)]. Also, there are natural isomorphisms ∗ ∼ ∗ 0 ∗ ∼ ∗ RΓJO v v RΓX = v RΓJ v ( Dqct(X)). V E5.2.8(b)−→ E E5.2.1(a)−→ E E ∈ Thus the natural map RΓ 0 RΓ induces an isomorphism—the composition V → KOV Rg RΓ 0v∗ ∼ Rg RΓ RΓ v∗ ∼ Rg RΓ v∗ . ∗ V E −→ ∗ KOV JOV E −→ ∗ KOV E Since ( ): Rg RΓ v∗ = Rg RΓ 0v∗ D (U) (Propositions 5.2.1 and 5.2.6) ∗ ∗ KOV E ∼ ∗ V E ∈ t therefore we can imitate the proof of Proposition 5.2.8(d)—without the boundedness imposed there on , since that would be needed only to get ( )—to see that the map Rg RΓ v∗G Rg v∗ induced by RΓ 1 factors∗uniquely as ∗ KOV E → ∗ E KOV → Rg RΓ v∗ ∼ RΓ 0 Rg v∗ Rg v∗ , ∗ KOV E −→ U ∗ E −→ ∗ E with the first map an isomorphism. It follows that ψE is the composed isomorphism Rg RΓ 0 v∗ ∼ Rg RΓ RΓ v∗ ∼ Rg RΓ v∗ ∼ RΓ 0 Rg v∗ . ∗ V E −→ ∗ KOV JOV E −→ ∗ KOV E −→ U ∗ E The last statement in (a) (for adic u) results then from Corollary 5.2.11(b) and Propositions 5.2.6 and 5.2.1(a). 0 (b) Once θE is shown to be an isomorphism, the last statement in (b) (for adic u) follows from Corollary 5.2.11(b), and Propositions 5.2.6 and 5.2.1(a). 0 To show that θE is an isomorphism, it suffices to show that the composition ψ−1θ0 : RΓ 0 u∗ Rf Rg RΓ 0 v∗ ( D (X)). E E U ∗E → ∗ V E E ∈ qct is an isomorphism. We use Lemma 5.4.1 to reduce the problem, as follows. ∗ ∗ 0 0 First, the functors u , v , RΓU and RΓV are bounded, and commute with ∗ ∗ 0 0 direct sums: for u and v that is clear, and for RΓU and RΓV it holds because they can be realized locally by tensoring with a bounded flat complex (see proof of Proposition 5.2.1). Furthermore, Lemma 5.1.4, Proposition 5.2.1, and Proposi- tion 3.3.5 show that RΓ 0 v∗ D (X) D (V); and the functor Rg (resp. Rf ) V qct ⊂ qct ∗ ∗ is bounded on, and commutes with direct sums in, Dqct(V) (resp. Dqct(X)), see Propositions 5.1.4, 3.4.3 and 3.5.2. Hence, standard way-out reasoning allows us to + assume that Dqct(X). E ∈ n Next, let J be an ideal of definition of X, Xn (n > 0) the scheme (X, X/J ), and j : X , X the associated closed immersion. The functor j : (X O) (X) is n n → n∗ A n → A DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 75

\ exact, so it extends to a functor D(Xn) D(X). The functor jn : D(X) D(Xn) being defined as in (16), we have → → • n \ h ( ):= R om ( X/J , ) = j j ( D(X)). n G H O G n∗ nG G ∈ If D+ (X) then = RΓ (Proposition 5.2.1(a)), and, as noted just af- E ∈ qct E J E ter (16), j\ D (X ). Hence, from the triangle in Lemma 5.4.1 (with replaced nE ∈ qc n G by an D+ (X)) we derive a diagram of triangles E ∈ qct RΓ 0 u∗ Rf ( h ) RΓ 0 u∗ Rf ( h ) RΓ 0 u∗ Rf (RΓ ) + U ∗ ⊕n≥1 n E −−→ U ∗ ⊕n≥1 n E −−→ U ∗ J E −−→ ' ' '    RΓ 0yu∗ Rf h RΓ 0yu∗ Rf h RΓ 0 uy∗ Rf + ⊕n≥1 U ∗ n E −−→ ⊕n≥1 U ∗ n E −−→ U ∗E −−→ −1 0 −1 0 −1 0 ⊕ ψ θh ⊕ ψ θh ψ θ  hnE nE  hnE nE  E E    + Rg yRΓ 0 v∗h Rg yRΓ 0 v∗h Rg RyΓ 0 v∗ ⊕n≥1 ∗ V n E −−→ ⊕n≥1 ∗ V n E −−→ ∗ V E −−→ ' ' '    Rg RΓ 0 v∗y( h ) Rg RΓ 0v∗y( h ) Rg RΓ 0yv∗(RΓ ) + ∗ V ⊕n≥1 n E −−→ ∗ V ⊕n≥1 n E −−→ ∗ V J E −−→ From this diagram we see that if each ψ−1 θ0 is an isomorphism, then so is ψ−1 θ0 . hnE hnE E E \ So we need only prove (b) when = jn∗ with := jn Dqc(Xn). Let us show E F F 0 E ∈ that in fact for any n > 0 and any Dqc(Xn), θ is an isomorphism. F ∈ jn∗F The assertion (b) is local both on Y and on U. Indeed, for (b) to hold it suffices, for every diagram of fiber squares

0 0 V j V0 v X0 j X ←−−−− −−−−→ −−−−→ g 0 0  g  f f     Uy Uy0 Yy0 yY ←−−i0−− −−−u−0 → −−−i−→ where Y0 ranges over a base of open subsets of Y, U0 ranges over a base of open −1 0 0 0 0∗ 0 0 subsets of u Y , u is induced by u, and i, i are the inclusions, that i θ (= θ U0 ) E E | be an isomorphism. Now when u is an open immersion, θG is an isomorphism for all D(X). (One may assume to be K-injective and note that v∗, having the G ∈ G exact left adjoint “extension by zero,” preserves K-injectivity, so that θG becomes ∗ ∼ ∗ the usual isomorphism u f∗ g∗v ). Thus there are functorial isomorphisms 0∗ ∼ 0 0∗ ∗ G −→∼ G0 ∗ i Rg∗ Rg∗ j and i Rf∗ Rf∗ j ; and similarly there is an isomorphism 0∗ 0 −→∼ 0 0∗ −→ i RΓ RΓ 0 i . So it suffices that the composition U −→ U 0∗ 0 0∗ 0 ∗ i θE 0∗ 0 ∗ ∼ 0 0∗ ∗ ∼ 0 0 0∗ ∗ i RΓ u Rf i RΓ Rg v RΓ 0 i Rg v RΓ 0 Rg j v U ∗E −−−→ U ∗ E −→ U ∗ E −→ U ∗ E be an isomorphism; and with a bit of patience one identifies this composition with 0 θ ∗ 0 0∗ ∗ ∼ 0 0∗ 0 ∗ j E 0 0 0∗ ∗ RΓ 0 u i Rf RΓ 0 u Rf j RΓ 0 Rg v j , U ∗E −→ U ∗ E −−→ U ∗ E 0 thereby reducing to showing that θj∗E is an isomorphism. Thus one may assume that both Y and U are affine, say Y = Spf(A) and U = Spf(C) with C a flat A-algebra (Lemma 7.1.1). Suppose next that X and Y are ordinary schemes, so that Y = Spec(A). In cases (i) and (ii) of (b), set C00 := C, and in case (iii) let C00 be as in the proof of 76 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN part (c) of Proposition 7.1. In any case, C00 is A-flat, C is the K-adic completion 00 00 00 of C for some C -ideal K, X Y Spf(C) is the K-adic completion of X Y Spec(C ), and we have a natural commutativ× e diagram ×

v2 00 v1 X Y Spf(C) X Y Spec(C ) X × −−−−→ × −−−−→ g g  1 f    Spfy(C) Spec(yC00) yY −−−u2−→ −−−u1−→ With Γ 0 denoting Γ 0 , θ0 =: θ0( , f, u) factors naturally as the composition Spf(C) E E RΓ 0u∗(θ(E,f,u )) θ0(v∗E, g , u ) RΓ 0u∗u∗ Rf 2 1 RΓ 0u∗ Rg v∗ 1 1 2 RΓ 0v∗ Rg v∗v∗ . 2 1 ∗E −−−−−−−−−−−→ 2 1∗ 1 E −−−−−−−−→ 2 ∗ 2 1 E Here θ( , f, u1) is an isomorphism because all the schemes involved are ordinary schemes.E (One argues as in [H1, p. 111, Prop. 5.12], using [AHK, p. 35, (6.7)]; for a fussier treatment see [L4, Prop. (3.9.5)].) Also, θ0(v∗ , g , u ) is an isomorphism, 1 E 1 2 in case (i) of (b) since then u2 and v2 are identity maps, and in cases (ii) and (iii) 00 by Corollary 5.2.12 since then X Y Spec(C ) is noetherian. Thus: × Lemma 7.2.1. Proposition 7.2 holds when X and Y are both ordinary schemes. We will also need the following special case of Proposition 7.2:

n Lemma 7.2.2. Let I be an ideal of definition of Y, Y the scheme (Y, Y/I ), n O and i : Y , Y the canonical closed immersion. Let u : Y Y U Y and n n → n n × → n pn : Yn Y U U be the projections (so that un is flat and pn is a closed immersion, see [GD×, p.→442, (10.14.5)(ii)]). Then the natural map is an isomorphism u∗i ∼ p u∗ ( D (Y )). n∗G −→ n∗ nG G ∈ qc n ∗ ∗ Proof. Since the functors u , in∗ , pn∗ , and un are all exact, we may assume that is a quasi-coherent -module; and since those functors commute with lim G OYn −−→ we may further assume coherent, and then refer to [GD, p. 443, (10.14.6)]. G Finally, for general noetherian formal schemes X and Y, and I and Yn as above, n let J I X be an ideal of definition of X, let X be the scheme (X, X/J ), and let ⊃ O n O fn : Xn Yn be the map induced by f. Then for any Dqc(Xn), it holds that Rf →D (Y ). (See Proposition 5.2.6—though theFsimpler∈ case D+ (X ) n∗F ∈ qc n F ∈ qc n would do for proving Proposition 7.2.) Associated to the natural diagram

vn X Y U X n× n qn jn v V X fn

gn (7.2.3) f g Yn Y U Yn × un pn in U Y u DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 77 there is a composed isomorphism

RΓ 0 u∗ Rf j ∼ RΓ 0 u∗i Rf ( D (X )) U ∗ n∗ F −→ U n∗ n∗ F F ∈ qc n ∼ RΓ 0 p u∗ Rf (Lemma 7.2.2) −→ U n∗ n n∗ F ∼ RΓ 0 p Rg v∗ (Lemma 7.2.1) −→ U n∗ n∗ nF ∼ RΓ 0 Rg q v∗ −→ U ∗ n∗ nF ∼ RΓ 0 Rg v∗j (Lemma 7.2.2), −→ U ∗ n∗ F 0 which—the conscientious reader will verify—is just θjn∗F . 0 Thus θjn∗F is indeed an isomorphism. 0 (c) By deﬁnition RΓX(Dqc(X)) Dqct(X), and so by (a) and (b) it’s enough ⊂ 0 to see, as follows, that the enatural map RΓX induces isomorphisms of the 0 E → E source and target of both ψE and θE . 0 ∗ 0 ∼ 0 ∗ Proposition 5.2.8(c) gives the isomorphism Rg∗RΓV v RΓX Rg∗RΓV v , as well as the second of the following isomorphisms, the ﬁrstEand−→third of whicEh follow from Corollary 5.2.11(d):

RΓ 0 Rg v∗RΓ 0 = Rg RΓ 0 v∗RΓ 0 = Rg RΓ 0 v∗ = RΓ 0 Rg v∗ . U ∗ X E ∼ ∗ V X E ∼ ∗ V E ∼ U ∗ E Likewise, there are natural isomorphisms

RΓ 0 u∗ Rf RΓ 0 = RΓ 0 u∗ RΓ 0 Rf = RΓ 0 u∗ Rf . U ∗ XE ∼ U Y ∗E ∼ U ∗E

Notation and assumptions stay as in Proposition 7.2(a). Assume that f and g satisfy the hypotheses of Theorem 6.1, so that the functor Rf∗ : Dqct(X) D(Y) × → has a right adjoint ft , and similarly for g. Recall from Corollary 6.1.5(b) that there is a natural isomorphism g×RΓ 0 ∼ g×. t U −→ t Definition 7.3. With conditions as in Proposition 7.2(b), the base-change map

β : RΓ 0 v∗f × g× RΓ 0 u∗ ( D(Y)) F V t F → t U F F ∈ is deﬁned to be the map adjoint to the natural composition

0 ∗ × ∼ 0 ∗ × ∼ 0 ∗ × 0 ∗ Rg∗RΓV v ft RΓU Rg∗v ft RΓUu Rf∗ft RΓUu F −ψ→ F −θ0→−1 F → F

0 0 where ψ := ψf ×F and θ := θ × . In particular, if u (hence v) is adic then t ft F

β : v∗f × g×u∗ F t F → t F is the map adjoint to the natural composition

∗ × ∼ ∗ × ∗ Rg∗v ft u Rf∗ft u F −θ−→1 F → F where θ := θ × . ft F Notation. For a pseudo-proper (hence separated) map f (see 1.2.2), we write f ! × § instead of ft . 78 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

Theorem 7.4. Let X, Y and U be noetherian formal schemes, let f : X Y be a pseudo-proper map, and let u: U Y be flat, so that in any fiber square → → V v X −−−−→ g f   Uy yY −−−u−→ the formal scheme V is noetherian, g is pseudo-proper, and v is flat (Proposi- tion 7.1). Then for all D+ (Y):= D (Y) D+(Y) the base-change map β is F ∈ qc qc ∩ F an isomorphism e e β : RΓ 0 v∗f ! ∼ g! RΓ 0 u∗ (= g!u∗ ). F V F −→ U F ∼ F Remark. In [N1, p. 233, Example 6.5] Neeman gives an example where f is a finite − map of ordinary schemes, u is an open immersion, F ∈ Dc (Y), and βF is not an isomorphism.

Proof. Recall diagram (7.2.3), in which, I and J I X being defining n ⊃ O ideals of Y and X respectively, Yn is the scheme (Y, Y/I ) and Xn is the scheme n O (X, X/J ). Let I U be a defining ideal of U, let := J V + V , a defining O K ⊃ O n L O KO ideal of V, let Vn (n > 0) be the scheme (V, V/ ), and let ln : Vn , V be the canonical closed immersion. Then by ExampleO6.1.3(4),L → ! \ n + l l = l l = R om( V/ , ) =: h ( ) ( D (V)). n∗ nG n∗ nG H O L G n G G ∈ qct 0 ! ∗ ∼ ! ∗ So in view of the natural isomorphism RΓVg u g u (Proposition 5.2.1(a)), Lemma 5.4.1 shows it sufficient to prove that theF −maps→ F h (β ): l l! RΓ 0v∗f ! l l! g!u∗ (n > 0) n F n∗ n V F → n∗ n F are all isomorphisms. Moreover, the closed immersion ln factors uniquely as

rn qn V X Y U V, n −−→ n × −−→ ! ! ! so we can replace ln by rnqn (Theorem 6.1(b)). Thus it will suffice to prove that the maps q! (β ): q! RΓ 0v∗f ! q! g!u∗ (n > 0) n F n V F → n F are all isomorphisms. In the cube (7.2.3), the front, top, rear, and bottom faces are fiber squares, denoted, respectively, by , t , r and b ; and we have the “composed” fiber square c : vn X Y U X n × −−−−→ n pngn= gqn infn= fjn   Uy yY −−−u−→ The proper map fn and the closed immersions in and jn are all of pseudo-finite type. Also, it follows from Proposition 7.1(b) that in addition to u, the maps u, un, v and vn are all flat. So corresponding to the fibre squares • we have base-change maps β• . DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 79

Consider the following diagram of functorial maps where, to save space, we set N:= X Y U and H:= Y Y U. n × n ×

! 0 ∗ ! βt 0 ∗ ! ! 0 ∗ ! 0 ∗ ! 0 ∗ ! ! q RΓ v f RΓNv j f RΓNv (fj ) RΓNv (i f ) RΓNv f i n V ←−− n n ←− n n n n n −→ n n n ! e β e β qn(β) c  r    ! y! ∗ ! ∗ y ! ∗ ! ! ∗ ! y0 ∗ ! q g u (gqn) u (pngn) u g p u g RΓHu i n n n ! n n n −→ ←− gn←(−βb) e e ! As above, we want to see that qn(β) is an isomorphism (in the category of functors + from D (Y) to D(X Y U)). For that the following assertions clearly suﬃce: qc n× (a)e The preceding diagram commutes. (b) The base-change maps βt and βb are isomorphisms. (c) The base-change map βr is an isomorphism.

Assertion (a) results from part (b) of the transitivity lemma 7.5.2 below. Since in and jn are closed immersions, assertion (b) results from Lemma 7.6.1, which is just Theorem 7.4 for the case when f is a closed immersion. Since f is pseudo-proper therefore fn is proper, and assertion (c) is essentially the case of Theorem 7.4— established in Lemma 7.7.1—when X and Y are ordinary schemes. Thus these three Lemmas will complete the proof of Theorem 7.4.

0 7.5. We will need some “transitivity” properties of the maps θE and βF relative to horizontal and vertical composition of ﬁber squares of noetherian formal schemes, i.e., diagrams of the form

V v2 V v1 X −−−−→ 1 −−−−→ g g (7.5.0a)  1 f    y y y U U1 Y −−−u2−→ −−−u1−→

V v X −−−−→ g 2 f2   y w y (7.5.0b) W Z −−−−→ g 1 f1   Uy yY −−−u−→ where all squares are fiber squares, and the maps u, ui , v, vi , and w are all flat. As we will be dealing with several fiber squares simultaneously we will indicate the square with which, for instance, the map θG in Proposition 7.2 is associated, by writing θf,u( ) instead. The transitivitG y properties begin with: 80 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

Lemma 7.5.1. Coming out of the ﬁber square diagrams (7.5.0a) and (7.5.0b), the following natural diagrams commute for all D(X): G ∈ θf,u u (G) (u u )∗ Rf 1 2 Rg (v v )∗ 1 2 ∗G −−−−−−−−−−−−−−−−−−−−−−−−−−−→ ∗ 1 2 G ' '   ∗ ∗y ∗ ∗ y∗ ∗ u2u1 Rf∗ u2 Rg1∗v1 Rg∗v2 v1 u∗(θ (G)) θ (v∗G) G −−2−−f−,u−1−−→ G −−g1−,−u2−−1−→ G

θf f ,u(G) u∗ R(f f ) 1 2 R(g g ) v∗ 1 2 ∗G −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ 1 2 ∗ G ' '   ∗ y ∗ y ∗ u Rf1∗ Rf2∗ Rg1∗ w Rf2∗ Rg1∗ Rg2∗v G −θ−f1−,u−(−R−f−2∗−G→) G −−R−g1−∗−(θ−f2−,−w−(G−→)) G

∗ Proof. This is a formal exercise, based on adjointness of u and Ru∗, etc. Details are left to the reader.

Lemma 7.5.2. (a) In the fiber square diagram (7.5.0a) (with u1 , v1 , u2 and v2 0 0 × 0 0 × ∗ flat), let D(Y) be such that the maps θ1 := θ (ft ), θ2 := θg ,u ((g1)t u1 ) F ∈ f,u1 F 1 2 F and θ00 := θ0 (RΓ 0 v∗f × ) of Proposition 7.2 are isomorphisms. Then the map 2 g1,u2 V1 1 t 0 0 × F θ := θ (ft ) is an isomorphism, so the base-change maps β1 := βf,u ( ), f,u1u2 F 1 F β := β (u∗ ) and β := β ( ) can all be defined as in Definition 7.3; and 2 g1,u2 1F f,u1u2 F the following natural diagram, all of whose uparrows are isomorphisms, commutes:

β RΓ 0(v v )∗f × g×RΓ 0 (u u )∗ V 1 2 t F −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ t U 1 2 F 'x x'    β2  RΓ 0v∗v∗f × RΓ 0 v∗(g )×u∗ g×RΓ 0 u∗u∗ V 2 1 t F V 2 1 t 1F −−−−→ t U 2 1F 'x5.2.8(c) 'x6.1.5(b) 5.2.8(c)x'    RΓ 0 v∗ RΓ 0 v∗f × RΓ 0 v∗(g )×RΓ 0 u∗ g×RΓ 0 u∗RΓ 0 u∗ V 2 V1 1 t 0 ∗ V 2 1 t U1 1 t U 2 U1 1 F −R−−ΓV−v−2−(β−1→) F −−−β2−→ F (b) In the fiber square diagram (7.5.0b)—where u, v and w are assumed flat— 0 0 × set f := f1f2 and g := g1g2. Let D(Y) be such that the maps θ := θ ((f ) ), F ∈ 1 f1,u 1 t F 0 0 × and 0 0 × of Proposition are isomorphisms, so that θ2 := θf2,w(ft ) θ := θf,u(ft ) 7.2 F F × the base-change maps β1 := βf1,u( ), β2 := βf2 ,w((f1)t ) and β := βf,u( ) are all defined. Then the following diagrF am, whose two uparrF ows are isomorphisms,F commutes: β RΓ 0v∗f × g×RΓ 0 u∗ V t F −−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ t U F 'x x'   0 ∗ × × × 0 ∗ × × × 0 ∗ RΓVv (f2)t (f1)t (g2)t RΓW w (f1)t (g2)t (g1)t RΓUu β2 × F −−−−→ F −(−g−2)−t −(β−1→) F DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 81

Proof. (a) The map 0 ∗ × 0 ∗ ∗ × 0 ∗ ∗ × γ := RΓ u (θf,u (f )): RΓ u u Rf∗f RΓ u Rg v f U 2 1 t F U 2 1 t F −→ U 2 1∗ 1 t F is isomorphic, by Proposition 5.2.8(c), to RΓ 0 u∗(θ0 ): RΓ 0 u∗RΓ 0 u∗Rf f × RΓ 0 u∗RΓ 0 Rg v∗f × , U 2 1 U 2 U1 1 ∗ t F −→ U 2 U1 1∗ 1 t F 0 and so is an isomorphism (since θ1 is). The map 0 ∗ × 0 ∗ ∗ × 0 ∗ ∗ × θ (v f ): RΓUu Rg v f RΓU Rg∗v v f g1,u2 1 t F 2 1∗ 1 t F → 2 1 t F is also an isomorphism, as it is isomorphic to θ00 : RΓ 0 u∗ Rg RΓ 0 v∗f × RΓ 0 Rg v∗RΓ 0 v∗f × , 2 U 2 1∗ V1 1 t F → U ∗ 2 V1 1 t F 0 ∗ 0 ∗ × 0 ∗ ∗ × because the natural map RΓUu2 Rg1∗RΓV1v1ft RΓUu2 Rg1∗v1ft is the composed isomorphism F → F RΓ 0u∗ Rg RΓ 0 v∗f × RΓ 0 u∗ RΓ 0 Rg v∗f × U 2 1∗ V1 1 t 0 ∗ U 2 U1 1∗ 1 t RΓ u ψ × F −−U−−2−−f →F F f t 0 ∗ ∗ × RΓUu2 Rg1∗v1ft −5.2.8(c)−−−→ F f 0 ∗ 0 ∗ × 0 ∗ ∗ × (see Proposition 7.2(a)); and because RΓ Rg∗v RΓ v f RΓ Rg∗v v f U 2 V1 1 t F → U 2 1 t F is one of the maps in the commutative diagram (B) below, all of whose other maps are isomorphisms. Thus in the next diagram, whose commutativity results easily from that of the ﬁrst diagram in Lemma 7.5.1, all the maps other than θ0 are isomorphisms, whence so is θ0. 0 RΓ 0 Rg (v v )∗f × θ RΓ 0 u∗u∗Rf f × U ∗ 1 2 t F ←−−−−−−−−− U 2 1 ∗ t F γ ' (A) '   0 y ∗ ∗ × 0 ∗ y ∗ × RΓU Rg∗v2 v1ft RΓUu2 Rg1∗v1ft 0 ∗ × F ←θ−g −,u−−(v−1 −ft−F−−) F 1 2f Now it suﬃces to show that the diagram which is adjoint to the diagram in (a) without its southeast (bottom right) corner, commutes. That adjoint diagram is the outer border of the following one, where, to reduce clutter, we omit all occurrences × × of the symbols R and , write f for ft , etc., and leave some obvious maps unlabeled: F ψ θ0−1 g Γ 0 (v v )∗f× Γ 0 g (v v )∗f× Γ 0 u∗u∗f f× Γ 0 u∗u∗ ∗ V 1 2 −−−→ U ∗ 1 2 −−−−−−−−→ U 2 1 ∗ −−−→ U 2 1 (A) γ−1 x x x    0 ∗ ∗ × 0 ∗ ∗ × 0 ∗  ∗ × g∗Γ v v f (B) Γ g∗v v f Γ u g v f V 2 1 U 2 1 0 ∗ × U 2 1∗ 1 ←θ−g −,u−−(v−1−f−−) 1 2 'x5.2.8(c) x x (C)    0 ∗ 0 ∗ × 0 ∗ 0 ∗ × 0 ∗  0 ∗ × g∗Γ v Γ v f Γ g∗v Γ v f Γ u g Γ v f V 2 V1 1 U 2 V1 1 00 −1 U 2 1∗ V1 1 −−ψ−→ −−−−θ2−−−−→

β1 β1 β1    0 y∗ × ∗ 0 y∗ × ∗ 0 ∗ y × ∗ 0 ∗ ∗ g∗Γ v g u Γ g∗v g u Γ u g g u Γ u u V 2 1 1 U 2 1 1 0 −1 U 2 1∗ 1 1 U 2 1 −−ψ−→ −−−−θ−2 −−−→ −−−→ It suﬃces then that each one of the subrectangles commute. 82 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

For the three unlabeled subrectangles commutativity is clear. As before, commutativity of subrectangle (A) follows from that of the ﬁrst diagram in Lemma 7.5.1. Commutativity of (B) is easily checked after composition with the natural map 0 ∗ × ∗ × ΓU g∗(v1v2) f g∗(v1v2) f . (See the characterization of ψ in Proposition 7.2(a).) Commutativit→ y of (C) results from that of the following diagram:

θf,u g v∗f× g v∗f× 1 u∗f f× u∗ 1∗ 1 1∗ 1 ←−−−− 1 ∗ −−−−→ 1 x (D) x x x     0 ∗ × 0  ∗ × 0 ∗ × 0 ∗ g Γ v f Γ g v f Γ u f∗f Γ u 1∗ V1 1 U1 1∗ 1 0 −1 U1 1 U1 1 −−−ψ−→ −−θ−1 −→ −−−−→

β1 (E)    g gy×u∗ uy∗ 1∗ 1 1 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ 1 Here subrectangle (D) commutes by the characterization of ψ in Proposition 7.2(a); and (E) commutes by the very definition of the base-change map β1. (b) As in (a), we consider the adjoint diagram, essentially the outer border of the following diagram (7.5.2.1). 0 ∗ × × 0 ∗ × × (Note: The map ψ : g1∗ΓW w f2∗f2 f1 ΓU g1∗w f2∗f2 f1 in the middle of × ×→ × × diagram 7.5.2.1 is defined because f2∗f2 f1 := Rf2∗(f2)t (f1)t Dqc(Z), by Proposition 5.2.6.) F ∈ For diagram 7.5.2.1, commutativity of subrectangle (B) (resp. (D)) is given by the definition of β2 (resp. β1.) Commutativity of (C) follows from that of the second diagram in Lemma 7.5.1. Commutativity of (A) is left as an exercise. (It is helpful 0 ∗ × × ∗ × × to compose with the natural map ΓU g1∗g2∗v f2 f1 g1∗g2∗v f2 f1 and to use the characterization of ψ in Proposition 7.2(a).) The rest→ is straightforward.

via β g Γ 0 v∗f×f× g g Γ 0v∗f×f× 2 g g g×Γ 0 w∗f× ∗ V 2 1 −−−−−−−→ 1∗ 2∗ V 2 1 −−−−−−−−−−−−−→ 1∗ 2∗ 2 W 1 ' f ψ       0y ∗ × 0 y ∗ × ×  g∗ΓV v f g1∗ΓWg2∗v f2 f1 (B)   ψ ' 0 −1  (A) g1∗(θ2 )       y Γ 0 g yv∗f× g Γ 0 wy∗f f×f× g Γ 0 w∗f× U ∗ 1∗ W 2∗ 2 1 −−−−−−−−−−−−−−−−→ 1∗ W 1 ' ψ ×  via θf ,w(f )  ψ Γ 0 g g yv∗f×f× 2 Γ 0 g w∗yf f×f× Γ 0 g w∗f× g Γ 0 w∗f× U 1∗ 2∗ 2 1 ←−−−−−−− U 1∗ 2∗ 2 1 −−→ U 1∗ 1 ←−− 1∗ W 1 1 θ0 (f f×)− θ0−1   f1,u 2∗  1      0 ∗ y × × 0 ∗y × '  (C) Γ u f f f f Γ u f f (D) g (β1)  U 1∗ 2∗ 2 1 U 1∗ 1  1∗  −−→     '       0 y ∗ × 0 y∗ × y0 ∗ ×y 0 ∗ ΓU g∗v f ΓU u f∗f ΓUu g1∗g1 ΓUu −−−θ−0−−1 −→ −−→ ←−− (7.5.2.1) DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 83

7.6. This subsection, proving Lemma 7.6.1, is independent of the preceding one. Lemma 7.6.1. Theorem 7.4 holds when f is a closed immersion. Proof. The natural isomorphisms RΓ 0 v∗f !RΓ 0 ∼ RΓ 0 v∗f ! and V YF −→ V F ! ∗ 0 ∼ ! 0 ∗ 0 ∼ ! 0 ∗ ∼ ! ∗ g u RΓY g RΓU u RΓY g RΓUu g u F −→ F 5.2.8(c)−→ F −→ F (see Corollary 6.1.5(b)) let us replace by RΓ 0 , i.e., we may assume D+ (Y). F YF F ∈ qct Recall from Example 6.1.3(4) that Rf∗ = f∗ : D(X) D(Y) has a right adjoint f \ such that f \(D+ (Y)) D+ (X); and that there is→a natural isomorphism qct ⊂ qct j X : RΓ 0 f \ ∼ 1!f \ = f ! ( D+ (Y)). G X G −→ G ∼ G G ∈ qc The canonical map f f ! 1 is the natural composition ∗ → ! ∼ 0 \ \ f∗f f∗RΓXf f∗f 1. (j−X→)−1 → → Similar remarks hold for g—also a closed immersion [GD, p. 442, (10.14.5)(ii)]. ∗ ∼ ∗ As in the proof of Lemma 7.2.2, the map θE : u f∗ g∗v of Proposi- tion 7.2 is an isomorphism for all D (X). (Recall LemmaE −→ 3.1.5.)E This being E ∈ qct so, the base-change map βF is easily seen to factor naturally as 0 ∗ ! ! 0 ∗ ! ! ∗ ! ∼ ! ∗ ! ! ∗ RΓVv f g g∗RΓVv f g g∗v f g u f∗f g u . F → F → F −θ−→1 F → F \ Also, we can define the functorial map βC to be the natural composition ∗ \ \ ∗ \ ∼ \ ∗ \ \ ∗ v f g g∗v f g u f∗f g u ( Dqc(Y)). C → C −θ−→1 C → C C ∈ \ The maps βF and βF are related by commutativity of the following diagram, in which J is an ideal of definition of Y (so that J X is an ideal of definition of X): O R 0 \ 0 ∗ 0 \ 0 ∗ \ 0 ∗ \ ΓV(β ) 0 \ ∗ RΓVv RΓXf RΓV RΓJO v f RΓVv f RΓV g u −5.2.8(b)−−−−→ V −−−−→ −−−−−→ 0 ∗ X V RΓVv (j )' f f 'j   y0 ∗ ! !y ∗ RΓVv f g u −−−−−−−−−−−−−−−−−−−−β−−−−−−−−−−−−−−−−−−−→ (For the unlabeled isomorphism, see the beginning of the proof of Proposition 7.2.) 0 Since RΓV is right-adjoint to the inclusion Dqct(V) , D(V) (Proposition 5.2.1), we → ! ∗ ∼ 0 \ ∗ \ ∗ can verify this commutativity after composing with g u RΓV g u g u , at which point the verification is straightforward. −→ → \ Thus to prove Lemma 7.6.1 we need only show that βF is an isomorphism, i.e. \ (since g is a closed immersion), that g∗(βF ) is an isomorphism. For that purpose, consider the unique functorial map σ = σ( , ): u∗R om• ( , ) R om• (u∗ , u∗ ) ( D−(Y), D+(Y)) E G H Y E G → H U E G E ∈ c G ∈ which for bounded-below injective complexes is the natural composition G u∗R om• ( , ) = u∗ om• ( , ) om• (u∗ , u∗ ) R om• (u∗ , u∗ ). H Y E G ∼ H Y E G → H U E G → H U E G This map is an isomorphism. Indeed, it commutes with localization, so we need only check for affine Y, and then, since every coherent Y-module is a homomor- phic image of a finite-rank free one ([GD, p. 427, (10.10.2)]),O a standard way-out argument reduces the problem to the trivial case = Y. E O 84 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

Take := f X = (say) Y/I. The source and target of σ( Y/I, ) are E ∗O O O F ∗ • ∗ \ ∗ \ u R om ( Y/I, ) = u f f = g v f , H O F ∗ F ∼ ∗ F • ∗ ∗ \ ∗ R om (u ( Y/I), u ) = g g u . H O F ∗ F ∗ Let be a K-injective U-complex quasi-isomorphic to u . Since the com- K ∗ • O • ∗ •F ∗ ∗ plexes u om ( Y/I, ) and om (u Y/I, ) = R om (u Y/I, u ) are H Y O F H U O K ∼ H U O F both annihilated by I U, we see that the isomorphism σ( Y/I, ) is isomorphic O ∗ \ \ ∗ O F to a map of the form g∗(ς) where ς : v f g u is a map in D(V). It suﬃces \ F → \ F then to show that ς = βF , i.e. (by deﬁnition of βF ), that the natural composition

\ g (ς) τ ∗ u∗f f \ ∼ g v∗f \ ∗ g g\u∗ u F u∗ ∗ F −→ ∗ F −−−→ ∗ F −−−→ F is induced by the natural map \ \ • • τ : f f = R om ( Y/I, ) R om ( Y, ) = . F ∗ F H O F → H O F F \ From Example 6.1.3(4) one sees, for injective , that τF takes any homomor- F \ phism ϕ: Y/I over an open subset of Y to ϕ(1); and similarly for τ ∗ . The O → F u F conclusion follows from the above definition of σ( Y/I, ) = g (ς). O F ∗ 7.7. In this subsection we prove Theorem 7.4 in case f : X Y is a proper map of ordinary noetherian schemes, by reduction to the case where→ X, Y, U and V are all ordinary schemes—a case which we take for granted (see the introductory remarks for section 7). Of course when u is adic then U is already an ordinary scheme, and no reduction is needed at all. Lemma 7.7.1. Let f : X Y be a proper map of ordinary noetherian schemes. For Theorem 7.4 to hold with→this f it suffices that it hold whenever U and V are ordinary schemes as well. Proof. Without yet assuming that X and Y are ordinary schemes, we can reduce Theorem 7.4 to the special case where the formal scheme U is affine and u(U) is contained in an affine open subset of Y. Indeed, for the base-change map βF = βf,u( ) of Theorem 7.4 to be an isomorphism, it clearly suffices that for any compositionF of fiber squares V v0 V v X 0 −−−−→ −−−−→ g g 0  f    y y y U0 U Y −−−u0−→ −−−u−→ with u0 the inclusion of an affine open U0 U such that u(U0) is contained in an affine open subset of Y, the map ⊂ v∗(β ): v∗RΓ 0v∗f ! v∗g!u∗ 0 F 0 V F → 0 F + ∗ + be an isomorphism. Remark 5.2.10(6) yields that Dqc(Y) u Dqc(U). F ∈ ⇒ F ∈ ∗ So if we assume the above-specified special case, then βf,uue ( ) and βg,u (ue ) are 0 F 0 F both isomorphisms. From Proposition 5.2.1(a) we have a natural isomorphism v∗(β ) = RΓ 0 v∗(β ( )), 0 F ∼ V0 0 f,u F ∗ so Lemma 7.5.2(a) shows that v0 (βF ) is in fact an isomorphism. DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 85

With reference to the remarks just preceding Section 7.5, (a) and (b) having already been proved, only (c) remains, i.e., we need only prove Theorem 7.4 for the rear face of diagram (7.2.3). In other words, with the notation of diagram (7.2.3), we may assume in proving Theorem 7.4 that f = fn (a proper map of ordinary schemes), and that u = un. Moreover Yn is a closed subscheme of Y, and so if U is affine and u(U) is contained in an affine open subset of Y, then Y Y U is affine and u (Y Y U) is contained n × n n × in an affine open subset of Y . It follows that Y Y U is the completion of an n n × ordinary affine Yn-scheme. (That can be seen via the one-one correspondence from maps between affine formal schemes to continuous homomorphisms between their associated rings [GD, p. 407, (10.4.6)]). Theorem 7.4 is thus reduced to the case depicted in the following diagram, where f : X Y is now a proper map of ordinary noetherian schemes, U is an ordinary affine Y→-scheme, κ: U U is a completion map, and u: U Y factors as shown. → →

X Y U X Y U X × −−−−→ × −−−−→ g (1)  (2) f    Uy Uy yY −−−κ−→ −−−−→ We will show that Theorem 7.4 holds for subdiagram (1) by identifying the base-change map associated to κ with the isomorphism ζ in Corollary 6.1.8. As subdiagram (2) is a ﬁber square of ordinary schemes, Lemma 7.7.1 will then result from the preceding reduction and the transitivity Lemma 7.5.2(a). It is convenient to re-represent subdiagram (1) in the notation of Corollary 6.1.8. Consider then a diagram κ X:= X X X /Z −−−−→ f f0   y y Y:= Y/W Y −−−κY−→ −1 as in Corollary 6.1.6, with Z = f0 W. That ζ is the base-change map means that ζ is adjoint to the natural composition

0 ∗ × ∼ 0 ∗ × ∼ 0 ∗ × 0 ∗ ∗ Rf∗RΓXκXf0 RΓYRf∗κXf0 RΓYκYRf0∗f0 RΓYκY κY. −ψ→ −θ0→−1 −→ −→ But by deﬁnition, ζ is adjoint to the natural composition

0 ∗ × ∼ ∗ × ∗ × ∗ ∗ Rf RΓ κ f Rf κ RΓ f Rf κ RΓ f κ ∗ κ κ ∗ X X 0 ∗ X Z 0 ∗ X Z 0 Y Y 0 ∗ Y 5.2.4(c)−→ −→ τ−t (→κY)

0 0 ∗ with τt as in Corollary 6.1.6—so that τt(κY) factors naturally as

∗ × ∗ ∼ ∗ × ∗ Rf∗κXRΓZ f κY∗ κY κYRf0∗RΓZ f κY∗ κY 0 −5.2.7→ 0 ∗ × ∗ κ Rf f κ ∗ κ −→ Y 0∗ 0 Y Y ∗ ∗ κ κ ∗ κ −→ Y Y Y π κ∗ . −−→ Y 86 LEOVIGILDO ALONSO, ANA JEREM´IAS, AND JOSEPH LIPMAN

It will suﬃce then to verify that the following natural diagram commutes (where, again, we omit all occurrences of R):

ψ 0−1 f Γ 0 κ∗ f× Γ 0f κ∗ f× θ Γ 0κ∗ f f× Γ 0κ∗ ∗ X X 0 −−−→ Y ∗ X 0 −−−→ Y Y 0∗ 0 −−−→ Y Y 5.2.4(c) (A)      ∗y × ∗ × ∗ y × y∗ f∗κXΓ f κYf0∗Γ f κYf0∗f κY Z 0 −−5.2.7−→ Z 0 −−−→ 0 −−−→ f ι    πx     ∗ y× ∗ ∗ y× ∗ ∗ y× ∗ ∗ y ∗ f∗κXΓ f0 κY∗ κY κYf0∗Γ f0 κY∗ κY κYf0∗f0 κY∗ κY κYκY∗ κY Z −−5.2.7−→ Z −−−→ −−−→ Given that πι = 1, thef veriﬁcation of commutativity is straightforward, except for subrectangle (A). Now there is a functorial isomorphism α: Rf RΓ ∼ RΓ Rf which arises 0∗ Z −→ W 0∗ in the obvious way, via “K-ﬂabby” resolutions, from the equality f0∗ΓZ = ΓW f0∗ (see the last paragraph in the Remark following (3.2.5) in [AJL, p. 25]), and whose composition with the natural map RΓW Rf0∗ Rf0∗ is the natural map → 0 ∗ ∼ ∗ Rf0∗RΓZ Rf0∗. And, again, we have the isomorphism RΓYκY κYRΓW of → ∗ −→ ∗ Proposition 5.2.4(c), whose composition with the natural map κYRΓW κY is the 0 ∗ ∗ → natural map RΓYκY κY. Hence commutativity of (A) follows from that of the outer border—consisting→ entirely of isomorphisms—of the following diagram:

ψ θ0 f Γ 0 κ∗ Γ 0f κ∗ Γ 0κ∗ f ∗ X X −−−−→ Y ∗ X ←−−−−−−−−−−−−−−−−− Y Y 0∗ −− x −−  −− x  −−  −−  → y∗ ∗ −− 5.2.4(c)  ' f∗κX κYf0∗ ← '  5.2.4(c)  ←−−θ −−   ←   −→ −−   − x −−   −−  −−  ∗ −− ∗  ∗  f∗κXΓZ κYf0∗ΓZ κYΓW f0∗ −−−−−−−−−5.2.7−−−−−−−−→ −−−α−→ 0 f f Since RΓY is right-adjoint to the inclusion Dt(Y) , D(Y) (Proposition 5.2.1), we can check commutativity after composing the outer→border with the natural map RΓ 0f κ∗ f κ∗ , so that it suﬃces to check commutativity of all the subdiagrams Y ∗ X → ∗ X of the preceding one. This is easy to do, as, with := f× , the maps denoted E 0 F by θE (= θf ,κ ( )) in Corollary 5.2.7 and in Proposition 7.2 are the same. 0 Y E This completes the proof of Lemma 7.7.1, and of Theorem 7.4.

8. Consequences of the flat base change isomorphism. # ! We begin with a flat-base-change theorem for the functor f = ΛXf associated to a pseudo-proper map f : X Y of noetherian formal schemes. (As before, → f ! := f ×, and f # is right-adjoint to the functor Rf RΓ 0 : D (X) D(Y), where t ∗ X qc → Dqc(X) is the (full) ∆-subcategory of D(X) such that e e D (X) RΓ 0 D (X), F ∈ qc ⇔ XF ∈ qct see Corollary 6.1.4.) e We deduce a sheafified version Theorem 8.2 of Theorem 2 of the Introduction (= Theorem 6.1 + Corollary 6.1.4). This is readily seen equivalent to the case of DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 87

flat base change where u: U Y is an open immersion; in other words, it expresses the local nature, over Y, of f→! and f #. Section 8.3 establishes the local nature of f ! and f # over X. From this we obtain # + + that f (Dc (Y)) Dc (X) (Proposition 8.3.2). This leads further to an improved base-change theorem⊂ for bounded-below complexes with coherent homology, and to Theorem 8.4, a duality theorem for such complexes under proper maps. We consider as in Theorem 7.4 a fiber square of noetherian formal schemes V v X −−−−→ g f   Uy yY −−−u−→ with f and g pseudo-proper, u and v flat. For any D+ (Y) we have the composed isomorphism F ∈ qc 0 e∗ # ∼ 0 ∗ 0 # ∼ 0 ∗ ! ∼ ! ∗ ϑ: RΓV v f RΓV v RΓXf RΓV v f g u . F 5.2.8(c)−→ F 6.1.5(a)−→ F −7.4→ F In particular, v∗f # D (V). F ∈ qc Theorem 8.1. Undere the preceding conditions, let β# : v∗f # g#u∗ ( D+ (Y)) F F → F F ∈ qc be the map adjoint to the natural composition e 0 ∗ # ∼ ! ∗ ∗ (8.1.1) Rg∗RΓV v f Rg∗g u u . F R−g→∗ϑ F → F # Then the map ΛV(βF ) is an isomorphism # ∗ # ∼ # ∗ # ∗ ΛV(βF ): ΛV v f ΛV g u = g u . F −→ F 6.1.5(a)∼ F # Moreover, if u is an open immersion then βF itself is an isomorphism. Proof. The map β# factors naturally as ∗ # ∗ # ∼ 0 ∗ # ∼ ! ∗ # ∗ (8.1.2) v f ΛV v f ΛVRΓV v f ΛVg u = g u . → 6.3.1(c)−→ −ΛV→ϑ To see this, one needs to check that (8.1.2) is adjoint to (8.1.1). The natural map 0 1 ΛV factors naturally as 1 ΛVRΓV ΛV (easy check), and hence the adjoin→ tness in question amounts to→the readily-v→erified commutativity of the outer border of the following diagram (with all occurrences of R left out): 0 0 ∗ # 0 ∗ # ! ∗ g∗ΓVΛVΓV v f g∗ΓV v f g∗g u ←−−−− −−via−−ϑ→ ' f  x x    0 y ∗ # 0  0 ∗ # 0  ! ∗ g∗ΓVΛV v f g∗ΓVΛVΓV v f g∗ΓVΛVg u −−−−→ −−via−−ϑ→ # f f That ΛV(βF ) is an isomorphism results then from the idempotence of ΛV (Remark 6.3.1(b)). When u—hence v—is an open immersion, we have isomorphisms (the first of which is obvious): ∗ # ∼ ∗ # ∼ ∗ # ΛV v f v ΛXf v f , −→ 6.1.5(a)−→ and the last assertion follows. 88 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

Next comes the sheaﬁﬁcation of Theorem 2. Let f : X Y be a map of locally noetherian formal schemes. For and D(X) we have natural→ compositions G E ∈ • • ∗ • Rf∗R omX( , ) Rf∗R omX(Lf Rf∗ , ) R omY(Rf∗ , Rf∗ ) H G E → H G E [Sp−−,−p.−147,−−−→6.7] H G E f and

• • 0 • 0 0 R omX( , ) R omX(RΓX , ) R omX(RΓX , RΓX ). H G E → H G E −5.2.3−−→ H G E f Theorem 8.2. Let X and Y be noetherian formal schemes and let f : X Y be a pseudo-proper map. Then the following natural compositions are isomorphisms:→

δ#: Rf R om• ( , f # ) R om• (Rf RΓ 0 , Rf RΓ 0 f # ) ∗ H X G F → H Y ∗ X G ∗ X F R om• (Rf RΓ 0 , ) ( D (X), D+ (Y)); → H Y ∗ X G F G ∈ qc F ∈ qc e e δ!: Rf R om• ( , f ! ) R om• (Rf , Rf f ! ) R om• (Rf , ) ∗ H X G F → H Y ∗G ∗ F → H Y ∗G F ( D (X), D+ (Y)). G ∈ qct F ∈ qc e Proof. The map δ# is an isomorphism iff the same is true of RΓ(U, δ#) for all open U Y. (For if —which may be assumed K-injective—is the vertex of ⊂ E a triangle based on δ#, then δ# is an isomorphism = 0 Hi( ) = 0 for ⇔ E ∼ ⇔ E all i Z the sheaf associated to the presheaf U HiΓ(U, ) = HiRΓ(U, ) vanishes∈ for⇔ all i.) Set V := f −1U, and let u: U , 7→ Y and vE: V , X be theE respective inclusions. We have then the fiber square → → V v X −−−−→ g f   Uy yY, −−−u−→ and need only verify that RΓ(U, δ#) is the composition of the following sequence of isomorphisms: RΓ(U, Rf R om• ( , f # )) ∼ RΓ(V, R om• ( , f # )) [Sp, 6.4, 6.7, 5.15] ∗ H X G F −→ H X G F ∼ RHom• (v∗ , v∗f # ) [Sp, 5.14, 5.12, 6.4] −→ V G F ∼ RHom• (v∗ , g#u∗ ) (Theorem 8.1) −→ V G F ∼ RHom• (Rg RΓ 0 v∗ , u∗ ) (6.1.4, 5.2.10(6)) −→ U ∗ V G F ∼ RHom• (Rg v∗RΓ 0 , u∗ ) (elementary) −→ U ∗ X G F ∼ RHom• (u∗Rf RΓ 0 , u∗ ) (elementary) −→ U ∗ X G F ∼ RΓ(U, R om• (Rf RΓ 0 , )) (as above). −→ H Y ∗ X G F This somewhat tedious verification is left to the reader (who may e.g., refer to the proof of (4.3)o (4.2) near the end of [L5]). That δ! is ⇒an isomorphism can be shown similarly—or be deduced via the natural map f ! = RΓ 0 f # f # (Corollary 6.1.5), which for D (X) induces ∼ X → G ∈ qct an isomorphism R om• ( , f ! ) ∼ R om• ( , f # ) (Corollary 5.2.3). H X G F −→ H X G F DUALITY AND FLAT BASE CHANGE ON FORMAL SCHEMES 89

! × # 8.3. For pseudo-proper f : X Y, the functors f := ft and f are local on X, in the following sense. → Proposition 8.3.1. Let there be given a commutative diagram U i1 X −−−−→ 1 i2 f1   y y X2 Y −−−f2−→ of noetherian formal schemes, with f1 and f2 pseudo-proper and i1 and i2 open immersions. Then there are functorial isomorphisms i∗f ! ∼ i∗f ! , i∗f # ∼ i∗f #. 1 1 −→ 2 2 1 1 −→ 2 2 Proof. The second isomorphism results from the ﬁrst, since for any D(Y) and for j = 1, 2, F ∈

∗ # 6.1.4 ∗ • 0 ! • ∗ 0 ∗ ! i f = i R om (RΓ X , f ) = R om (i RΓ X , i f ) j j F j H Xj Xj O j jF ∼ H U j Xj O j j jF • 0 ∗ ! = R om (RΓ U, i f ). ∼ H U UO j j F For the first isomorphism, Verdier’s proof of [V, p. 395, Corollary 1]—a special case of Proposition 8.3.1—applies verbatim, modulo the following extensions (a), (b) and (c) of some elementary properties of schemes to formal schemes. (a) Since pseudo-proper maps are separated, the graph of ij is a closed immersion γ : U , X YU (see [GD, p. 445, (10.15.4)], where the “finite-type” hypothesis → j× is used only to ensure that Xj Y U is locally noetherian, a condition which holds here by the first paragraph in Section× 7. And if U Y is an open immersion, then → so is γ (since then both π : X Y U X and i = π γ are open immersions). j j × → j j j (b) If s: U V is an open and closed immersion, then the exact functors s∗ and s∗ are adjoin→t, and by Example 6.1.3(4) there is a functorial isomorphism s! = s\ = s∗ ( D (V)). F ∼ F ∼ F F ∈ qct γ w (c) (Formal extension of [GD, p. 325, (6.10.6)].) Let U , W , Z be maps of locally noetherian formal schemes such that γ is a closed →immersion→ and w is an open immersion. (We are interested specifically in the case W := X Y U and 2 × Z := X Y X , see (a).) Set u := wγ. Then the closure U of u(U) is a formal 2 × 1 subscheme of Z, and the map U U induced by u is an open immersion. → Indeed, U is the support of Z/I where I is the kernel of the natural map O Z u∗ U ; and it follows from [GD, p. 441, (10.14.1)] that we need only show thatO →I is cOoherent. The question being local, we may assume that Z is affine, say Z = Spf(A). Cover U by a finite number of affine open subschemes Ui (1 i n), with inclusions u : U , U. Then there is a natural injection ≤ ≤ i i → n n u U , u ( u U ) = (uu ) U , ∗O → ∗ ⊕i=1 i∗O i ∼ ⊕i=1 i ∗O i so that I is the intersection of the kernels of the natural maps Z (uui)∗ Ui , giving us a reduction to the case where U itself is affine, say U =OSpf→(B). NowOif I is the kernel of the ring-homomorphism ρ: A B corresponding to u, then for any → f A the kernel of the induced map ρ{f} : A{f} B{f} is I{f}; and one deduces ∈ ∆ → that I is the coherent Z-module denoted by I in [GD, p. 427, (10.10.2)]. O 90 LEOVIGILDO ALONSO, ANA JEREMÍAS, AND JOSEPH LIPMAN

Proposition 8.3.2. If f : X Y is a pseudo-proper map of noetherian formal schemes then → f #(D+(Y)) D+(X). c ⊂ c Proof. Since f # commutes with open base change (Theorem 8.1) we may assume Y to be aﬃne, say Y = Spf(A). Since f is of pseudo-ﬁnite type, every point of X has an open neighborhood U such that f U factors as | i p U , Spf(B) Spf(A) = Y → −→ where B is the completion of a polynomial ring P := A[T0, T1, . . . , Tn] with respect to an ideal I whose intersection with A is open, i is a closed immersion, and p corresponds to the obvious continuous ring homomorphism A B (see footnote in Section 1.2.2). This Spf(B) is an open subscheme of the completion→ P of the n projective space PA along the closure of its subscheme Spec(P/I). Thus by Propo- sition 8.3.1 and item (c) in its proof, we can replace X by a closed formal subscheme of P having U as an open subscheme. In other words, we may assume that f factors

i1 p1 as X , P Spf(A) = Y with i1 a closed immersion and p1 the natural map. →# −#→# Then f = i1p1, and we need only consider the two cases (a) f = p1 and (b) f = i1. Case (a) is given by Corollary 6.2.3. In case (b) we see as in example 6.1.3(4) that for D+(Y) we have f \ D+(X) and F ∈ c F ∈ c # 0 \ 6.3.1(c) \ 6.2.1 \ + f = ΛXRΓXf === ΛXf == f Dc (X). F F or 5.2.3 F F ∈ + # Corollary 8.3.3. For all Dc (Y) the base-change map βF of Theorem 8.1 is an isomorphism F ∈ β# : v∗f # ∼ g#u∗ . F F −→ F ∗ # ∼ ∗ # Proof. Proposition 6.2.1 gives an isomorphism v f ΛV v f . F −→ F We have now the following duality theorem for proper maps and bounded-below complexes with coherent homology. Theorem 8.4. Let f : X Y be a proper map of noetherian formal schemes, + + → # + + so that Rf∗(Dc (X)) Dc (Y) and f (Dc (Y)) Dc (X) (see Propositions 3.5.1 and 8.3.2). Then for ⊂ D+(X) and D+(Y)⊂there are functorial isomorphisms G ∈ c F ∈ c • # ∼ • 0 Rf∗R om ( , f ) R om (Rf∗RΓX , ) H G F −8.2→ H G F ∼ • 0 ∼ • R om (RΓY Rf∗ , ) R om (Rf∗ , ). 5.2.11(d)−→ H G F −6.2.1→ H G F In particular, f # : D+(Y) D+(X) is right-adjoint to Rf : D+(X) D+(Y). c → c ∗ c → c If X is properly algebraic we can replace f # by the functor f× of Corollary 4.1.1. Proof Left to reader. (For the last assertion see Corollaries 6.2.2 and 3.3.4.)

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Universidade de Santiago de Compostela, Facultade de Matematicas,´ E-15771 San- tiago de Compostela, SPAIN E-mail address: [email protected] URL: http://web.usc.es/~lalonso/

Universidade de Santiago de Compostela, Facultade de Matematicas,´ E-15771 San- tiago de Compostela, SPAIN E-mail address: [email protected]

Dept. of Mathematics, Purdue University, W. Lafayette IN 47907, USA E-mail address: [email protected] URL: www.math.purdue.edu/~lipman/