Contemporary Mathematics Duality and Flat Base Change on Formal Schemes 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 subcat- egory 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 subcat- egory 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 2 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, x10]. 3See also [De, x0, x1] 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 abelianGgr2oupsA) F 2 • × 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 x in [H1, Appendix] (see also [De, 1.1.12]) to the category ~c(X), which presents itself as an appropriate generalizationx 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 JEREM´IAS, 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 ap- plied to injective resolutions of complexes.A .