Bousfield Localization on Formal Schemes

BOUSFIELD LOCALIZATION ON FORMAL SCHEMES

LEOVIGILDO ALONSO TARRÍO, ANA JEREMÍAS LOPEZ,´ AND MARÍA JOSE´ SOUTO SALORIO

Abstract. Let (X, OX) be a noetherian separated formal scheme and consider D(Aqct(X)), its associated derived category of quasi-coherent torsion sheaves. We show that there is a bijection between the set of rigid localizationsin D(Aqct(X)) and subsets in X. For a stable for specialization subset Z ⊂ X, the associated acyclization is RΓZ .If Z ⊂ X is generically stable, we provide a description of the associated localization.

Introduction

The techniques of localization have a long tradition in several areas of mathematics. They have the virtue of concentrate our attention on some part of the structure in sight allowing us to handle more manageable pieces of information. One of the clear examples of this technique is the localization in algebra where one studies a module centering the attention around a point of the spectrum of the base ring, i.e. a prime ideal. The idea was transported to topology by Adams and later, Bousfield proved that there are plenty of localizations in stable homotopy. In the past decade it became clear that one could successfully transpose homotopy techniques to the study of derived categories (over rings and schemes). In particular, in our previous work we have shown that for the derived category of a Grothendieck category we also have plenty of localizations. In that paper, [AJS], we applied the result to the existence of unbounded resolutions and we hinted that, in the case of the derived category of quasi-coherent sheaves over a nice scheme, there should be a connection between localizations in the derived category and the geometric structure of the underlying space. In this connection, there was already available the work of Neeman [N1] who classified all Bousfield localizations in the derived category of modules over a noetherian ring D(R) and related it to the chromatic tower in stable homotopy. Here we extend Neeman’s work to the case of noetherian separated schemes. It has turned out that one could also obtain similar results for a noetherian separated formal scheme. The interest of this generaliza- tion is the recent advances in the cohomological theory of this spaces. We are able to include the case quasi-coherent torsion sheaves that are a basic ingredient in Grothendieck duality [AJL2]. We do not have information about more general quasi-coherent sheaves, due in part to the bad behavior of quasi-coherent sheaves over affine formal schemes.

Date: July 9, 2003. 1991 Mathematics Subject Classiﬁcation. 18E30 (primary); 14F05, 16D90 (secondary). L.A.T. and A.J.L. partially supported by Spain’s MCyT and E.U.’s FEDER research project BFM2001-3241, supplemented by Xunta de Galicia grant PGDIT 01PX120701PR. 1 2 L. ALONSO, A. JEREM´IAS, AND M. J. SOUTO

Our results go beyond this. It become apparent that the monoidal structure of the derived category is essential. In fact the classification forced us to impose the condition of rigidity on the localizing categories. This is, roughly speaking, that the localizing subcategory is an ideal in the monoidal sense (see §2). This condition is needed in order to have compatibility with open sets. It holds always in the affine case, that is why it was not considered by Neeman. Moreover some localizations can be expressed in terms of the tensor product and others in terms of the internal Hom functor. We characterize both classes and establish an adjunction between them, giving another context to the results in [AJL1]. We also prove that localizations already present in the abelian category of sheaves are those that are tensor compatible. Moreover, we show that we can describe the Hom compatible localizations via a certain formal duality relation. While our work does not exhaust all the possible questions about these topics we believe that can be useful for the current program of extracting information on a space looking at its derived category. Now, let us describe briefly the contents of the paper. The first section recalls the concepts and notations used throughout and we give a detailed overview of the symmetric closed structure in the derived categories we are going to consider. Next, we specify the relationship between cohomology with supports and the algebraic version defined in terms of ext sheaves. In the third section we discuss the basic properties of rigid localizing subcategories. In section four we state and proof the classification theorem, the rigid localizing subcategories in the derived category of quasi-coherent torsion sheaves on a formal scheme X are in one to one correspondence with the subsets in the underlying space of X. The arguments are close in spirit to [N1], with the modifications needed to make them work in the present context. In the last section we give a description of the acyclization functor associated to a stable for specialization subset and the derived functor of the sections with support. By adjointness, we obtain also a description of the localization functor associated to generically stable subsets, i.e. those subsets which are complementary of the stable for specialization subsets. The question of describing localizations for subsets that are neither stable for specialization nor generically stable remains open for the moment.

1. Basic facts and set-up 1.1. Preliminaries. For formal schemes, we will follow the terminology of [EGA, Section 10] and of [AJL2]. Let us consider a noetherian separated formal scheme (X, OX). Let I be an ideal of definition. In what follows, we will identify an usual (noetherian separated) scheme with a formal scheme whose ideal of definition is 0. Denote by A(X) the category of all OX- modules. The powers of I define a torsion class (see [St, pp. 139-141]) whose associated torsion functor is n I F H O O I F Γ :=−→ lim om X( X/ , ) n>0 for F∈A(X). This functor does not depend on I but on the topology it determines in the rings of sections of OX, therefore we will denote it by A A F ΓX. Let t(X) be the full subcategory of (X) consisting of sheaves BOUSFIELD LOCALIZATION ON FORMAL SCHEMES 3 F F A such that ΓX = .Itisaplump subcategory of (X). This means it is closed for kernels, cokernels and extensions (cfr. [AJL2, beginning of §1]). Most important for us is the subcategory Aqct(X):=At(X) ∩Aqc(X). It is again a plump subcategory of A(X) by [AJL2, (5.1.3)] and it defines a triangulated subcategory of D(X):=D(A(X)), the derived category of A(X), it is Dqct(X), the full subcategory of D(X) formed by complexes whose homology lies in Aqct(X). If X = X is an usual scheme then At(X)=A(X) and Aqct(X)=Aqc(X). A →A t The inclusion functor qct(X) (X) has a right adjoint denoted QX (see [AJL2, Proposition 3.2.3]). By the existence of K-injective resolutions ([Sp, Theorem 4.5] or [AJS, Theorem 5.4]) it is possible to get right-derived functors from functors with source a category of sheaves, as a consequence t → A we have a functor RQX : D(X) D( qct(X)). This functor induces an equivalence between Dqct(X) and D(Aqct(X)) by [AJL2, Proposition 5.3.1]. This will allow us to identify freely D(Aqct(X)) and Dqct(X) something we will do from now on. The category Aqct(X) is a Grothendieck category so we can apply the machinery developed in [AJS]. In particular if L is the smallest localizing subcategory of D(Aqct(X)) that contains a given set, then there is a localization functor such that L is the full subcategory of D(Aqct(X)) whose objects are sent to 0 by (see [AJS, Theorem 5.7]). For the formalism of localization in triangulated categories the reader may consult [AJS, §1].

1.2. Monoidal structures. The categories A(X) and Aqct(X) are symmetric closed, in the sense of Eilenberg and Kelly, see [EK]. For every F∈K(A(X)) there is a K-flat resolution. This follows from [Sp, Proposition − 5.6] but let us give a simple argument. It is well-known that if F∈K (A(X)) there is a K-flat resolution constructed inductively (following the procedure dual to injective resolutions in [H, Lemma I.4.6] or [Iv, Corollary I.7.7]). ≤n Let Pn denote a K-flat resolution of τ F for every n ∈ N. Now, using the concept of (countable) homotopy direct limit from [BN], also explained in [AJS, 3.3 – 3.4], we have that F←− ≤nF←− P ˜ holim−→ τ ˜ holim−→ n, n∈N n∈N P P in D(X). Taking F := holim−→ n, we obtain the desired K-flat resolution n∈N of F. As a consequence, there exists a derived functor: F⊗L − → OX : D(X) D(X) F⊗L G P ⊗ G H · F − defined by O = F OX . Also the functor omO ( , ) has a X · · X right derived functor defined by R Hom (F, G)=Hom (F, JG) where OX OX G→JG denotes a K-injective resolution of G. The usual relations hold providing D(X) with the structure of symmetric closed category. Observe that the unit object is OX. The category D(Aqct(X)) = Dqct(X) inherits the structure of symmetric closed category from that of D(X). Let us see how. Note first that by [AJL3, 1.3.2] there is a K-flat resolution made up by ind-coherent sheaves1. And

1For the basic properties of these sheaves, see [AJL2, §3]. 4 L. ALONSO, A. JEREMÍAS, AND M. J. SOUTO using the fact that the functor F⊗L − : D (X) → D(X) takes values in OX qct Dqct(X) for every F∈Dqct(X). It provides an internal tensor product. For the internal hom we define: H · F G t H · F G omX( , ):=RQXR omOX( , ). F G∈ O for , Dqct(X). Now the unit object is RΓX X where by RΓX we O denote the right-derived functor of ΓX. We will denote this object by X for convenience. If the reader is only interested in usual schemes, then it is enough to get the quasi-coherence of the derived tensor product and consider quasi- coherent flat resolutions from [AJL1, Proposition 1.1]. For the internal hom-sheaf one uses the derived “coherator” functor RQ defined in [Il, §3] taking: H · F G H · F G omX ( , ):=RQR omOX ( , ). In this case the topology in the sections of the structural sheaf is discrete, O ΓX is the identity functor and so the unit object is X .

2. Cohomology with supports on formal schemes

2.1. Algebraic supports. For every F∈Dqct(X) and Z ⊂ X a closed subset, for the right derived functor of sheaf of sections with support along Z we have that RΓZ F∈Dqct(X) because in the distinguished triangle ∗ + RΓZ F→F→Rj∗j F →, ∗ where j : X \ Z→ X denotes the canonical open embedding, Rj∗j F∈ Dqct(X) [AJL2, Propositions 5.2.6 and 5.2.8]. On the other hand, the closed subset Z is the support of a coherent sheaf OX/Q where Q is an open coherent ideal in OX. The functor n Q H O O Q − ΓZ := Γ =−→ lim om X( X/ , ) n>0 of “sections with algebraic support along Z” does not depend on Q but → only on Z. The natural map ΓZ ΓZ is an isomorphism when applied to sheaves in Aqct(X). Furthermore the natural morphism in D(X) obtained F→ F F∈ by deriving θZ,F : RΓZ RΓZ is an isomorphism for all Dqct(X). Indeed, this is a local question, so we can assume that X is aﬃne with X = Spf(A) where A is a noetherian adic ring. Let κ : Spf(A) → Spec(A) be the canonical map. Let X := Spec(A). The set Z can be considered as a closed subset of either X or X. We will use ΓZ and ΓZ for the corresponding pair of endofunctors in A(X) and A(X). This will not cause any confusion, because the context will make it clear in which category we are working. By [AJL2, Proposition 5.2.4] the functor κ∗ converts quasi-coherent torsion sheaves in quasi-coherent sheaves, therefore it is enough to show that κ∗θZ,F is an isomorphism. But this is true because the diagram

κ∗θZ,F κ∗RΓ F −−−−→ κ∗RΓ F  Z  Z   F−−−→ F RΓZ κ∗ RΓZ κ∗ BOUSFIELD LOCALIZATION ON FORMAL SCHEMES 5 commutes and all the unlabeled maps are isomorphisms (for the map in the bottom use loc. cit. and [AJL1, Corollary 3.2.4]). Given E, F∈Dqct(X) there is a bifunctorial map E F E⊗L F→ E⊗L F ψZ ( , ): OX RΓZ RΓZ ( OX ) defined as follows. Assume E is K-flat and F is K-injective and choose a E⊗ F→J J quasi-isomorphism OX with K-injective. The composed map E⊗ F→E⊗ F→J J (of complexes) OX ΓZ OX has image into ΓZ and we define ψZ (E, F) to be the resulting factorization E F L ψZ ( , ) L E⊗ F→E ⊗O F −−−−−→ J → E⊗ F OX RΓZ ˜ X ΓZ ΓZ ˜ RΓZ ( OX ). This map is a quasi-isomorphism if Z is closed. The question is local so using again [AJL2, Proposition 5.2.4 and Proposition 5.2.8] we restrict to the analogous question for an ordinary scheme X and a closed subset Z ⊂ X. We conclude by [AJL1, Corollary 3.2.5]. As a consequence, if j : X \ Z → X denotes the canonical open embedding, the commutative diagram L L RΓZ O ⊗O F−−−→O ⊗O F X X X  X  

RΓZ F−−−→F can be completed to an isomorphism of distinguished triangles

L L ∗ L + RΓZ O ⊗O F−−−→O ⊗O F−−−→Rj∗j O ⊗O F −−−→ X X X  X X X   

∗ + RΓZ F −−−→ F −−−→ Rj∗j F −−−→ for every F∈Dqct(X).

2.2. Systems of supports on formal schemes. In general, a subset ⊂ Z X stable for specialization is a union Z = α∈I Zα of a directed system { ∈ } of closed subsets Zα /α I of X and ΓZ =−→ lim ΓZα , this corresponds α∈I to the classical case of a “system of supports”. Writing Γ = lim Γ the Z −→ Zα α∈I → F→ F canonical map ΓZ ΓZ induces natural maps θZ,F : RΓZ RΓZ for all F∈Dqct(X). If F→J is a K-injective resolution, we have that

lim→ θZα,F α∈I F F J J −−−−−−−→ J F θZ, : RΓZ = ΓZ =−→ lim ΓZα −→lim ΓZα = RΓZ α∈I α∈I therefore θZ,F is a quasi-isomorphism. Mimicking the case of a closed subset, for E, F∈Dqct(X) there is a bifunctorial map E F E⊗L F→ E⊗L F ψZ ( , ): OX RΓZ RΓZ ( OX ) that is a quasi-isomorphism. To check this fact we may assume E is K-ﬂat F E⊗ F→J J and is K-injective and choose a quasi-isomorphism OX with 6 L. ALONSO, A. JEREM´IAS, AND M. J. SOUTO a K-injective resolution and consider the commutativity of the diagram of complexes E F L ψZ ( , ) E⊗O ΓZ F −−−−−→ ΓZ J X    

E F lim→ ψZα ( , ) α∈I lim (E⊗L Γ F) −−−−−−−−−−→ lim Γ J . −→ OX Zα −→ Zα α∈I α∈I

3. Rigid localizing subcategories Let T be a triangulated category with all coproducts. This is the case for D(X) and D(Aqct(X)) for a formal scheme X [AJL2, Proposition 3.5.2], in particular for D(X) and D(Aqc(X)) for an usual scheme X. A triangulated subcategory L of T is called localizing if it is stable for coproducts in T. This does not ensure that L is well-behaved with respect to the tensorial structure. It turns out that we need such compatibility in order to localize on open subsets. So let us establish the following definition. A localizing subcategory L is called rigid if for every F∈Land G∈D(Aqct(X)), we have that F⊗L G∈L. This condition has been independently considered OX by Thomason for thick subcategories by the same reason (see [T, Definition 3.9], where they are called ⊗-subcategories). Our route to find this condition came from a paper by one of the authors where localizations are considered in the abelian context, see [JLV, 2.3].

Proposition 3.1. Let L be a localizing subcategory of D(Aqct(X)).IfL is ⊥ rigid, then, for every F, G∈D(Aqct(X)) such that G is L-local (i.e. G∈L ), H · F G L ⊥ L⊥ L then omX( , ) is -local. If moreover ( )= , the converse is true. Proof. Let H∈L, then, H H · F G H⊗L F G HomD(X)( , omX( , )) = HomD(X)( OX , )=0, (1) because G∈L⊥ and H⊗L F∈L. Conversely, if (1) holds for every G∈L⊥, OX then H⊗L F∈⊥(L⊥)=L. OX Remark. The condition ⊥(L⊥)=L holds if L is the localizing subcategory of objects whose image is 0 by a Bousﬁeld localization (see [AJS, Propo- sition 1.6]). We will see later (Corollary 4.14) that every rigid localizing subcategory of D(Aqct(X)) arises in this way.

Proposition 3.2. If X is aﬃne, every localizing subcategory of D(Aqct(X)) is rigid. Proof. Take X = Spf A where A is a noetherian adic ring. Every quasi- coherent torsion sheaf comes from an A-module and therefore it has a free resolution. Let κ : Spf A → Spec A the canonical morphism and X := Spec A. Let L be a localizing subcategory of D(Aqct(X)). The full subcategory T of D(Aqc(X)) deﬁned by {N ∈ A ∗N⊗L M∈L ∀M ∈ L} T = D( qc(X)) /κ OX , BOUSFIELD LOCALIZATION ON FORMAL SCHEMES 7 is triangulated and stable for coproducts. It is clear that OX ∈ T, therefore ∗ T = D(Aqc(X)). Now, given G∈D(Aqct(X)), G = κ κ∗G and κ∗G∈ D(A (X)) = T ([AJL2, Proposition 5.1.2]), therefore G⊗L M∈L, for qc OX every M∈L.

4. Localizing subcategories and subsets We keep denoting by X a noetherian, separated formal scheme and I its ideal of deﬁnition. Let x ∈ X. Consider the aﬃne formal scheme Xx := Spf(OX,x) where the adic topology in the ring OX,x is given by Ix and let κx : Spf(OX,x) → Spec OX,x be the canonical map. If X = Spf B and p O is the ideal corresponding to the point x, then X,x = B{p}. Denote by ix : Xx → X the canonical inclusion map. The functors

ix ∗ Aqct(Xx) Aqct(X) ∗ ix are exact because X is separated. Consider the induced functors

Rix ∗ D(Aqct(Xx)) D(Aqct(X)) ∗ ix ∗ A → A Moreover the functor ix : K( qct(X)) K( qct(Xx)) preserves acyclic complexes and, consequently, ix ∗ : K(Aqct(Xx)) → K(Aqct(X)) preserves K- injective complexes. Furthermore, the sequence → J→J→ ∗J→ 0 ΓX\Xx ix ∗ix 0 is exact for every injective object J∈Aqct(X), therefore for every F∈ D(Aqct(X)) one has a natural triangle F→F→ ∗F →+ RΓX\Xx Rix ∗ix . (2) Note that the canonical triangle O →O → ∗O →+ RΓX\Xx X X Rix ∗ix X tensored by F provides a triangle

L ∗ L + \ O ⊗ F→F→ ∗ O ⊗ F → RΓX Xx X OX Rix ix X OX that is isomorphic to (2) by 2.2. We will denote by κ(x) the residue field of the local ring OX,x, or, equivalently, of OX,x,byKx the quasi-coherent torsion sheaf over Spf(OX,x) associated to the OX,x-module κ(x) and K(x):=ix ∗(Kx)=Rix ∗(Kx). If X = X is a usual scheme and x is a closed point, K(x) has been denoted Ox in recent literature, but we will not use this notation to avoid potential confusions. Let Z be any subset of the underlying space of X. We define the subcategory LZ as the smallest localizing subcategory of D(Aqct(X)) that contains the set of quasi-coherent torsion sheaves {K(x)/x ∈ Z}.IfZ = {x} we will denote LZ simply by Lx. Note that if x ∈ Z, then Lx ⊂LZ . ∗ F∈ A ∈ ∗ F Lemma 4.1. If D( qct(X)) and x X, then RΓ{x}(Rix ix ) belongs to the localizing subcategory Lx. 8 L. ALONSO, A. JEREMÍAS, AND M. J. SOUTO

Proof. Let Q0 be a sheaf of coherent ideals in OX such that Supp(OX/Q0)= { } Q ∗Q x and denote := ix 0. Recall ∗ n ∗ F H O O Q ∗J RΓ{x}(ix ix ) =−→ lim om X( X/ 0 ,ix ) n>0 ∼ n ∗ H O O Q J = −→lim ix om Xx ( Xx / , ) n>0 ∗F→J where ix is a K-injective resolution. ∗ G H O O Qn J Let :=−→ lim om Xx ( Xx / ,ix ) and let us consider the ﬁltration n>0

0=G0 ⊂G1 ⊂G2 ⊂···⊂G

G H O O Qn G G where n := om Xx ( Xx / , ) i.e. the subcomplex of annihilated n by Q . The successive quotients Gn/Gn−1 are complexes of quasi-coherent Kx-modules and, therefore, isomorphic in D(Aqct(Xx)) to a direct sum of shiftings of Kx. The functor ix ∗ preserves coproducts, therefore every

ix ∗(Gn/Gn−1)=ix ∗Gn/ix ∗Gn−1 is an object of Lx. We deduce by induction, using the distinguished triangles + ix ∗Gn−1 → ix ∗Gn → ix ∗Gn/ix ∗Gn−1 → that every ix ∗Gn is in Lx for every n ∈ N. But we have ∗ ∼ ∗ F ∗G RΓ{x}(Rix ix ) = −→lim ix n, n>0 and the result follows from the fact that a localizing subcategory is stable for direct limits [AJS, Theorem 2.2 and Theorem 3.1].

Let Ex be an injective hull of the OX,x-module κ(x), then Ex is a Ix- torsion OX,x-module. Let then Ex be the sheaf in Aqct(Xx) determined by Γ(Xx, Ex)=Ex

Corollary 4.2. The object E(x):=ix ∗Ex ∈Aqct(X) belongs to Lx. Proof. It follows from the previous lemma, having in mind that E(x)is ∗ ∗ E quasi-coherent, injective and is equal to Γ{x}(ix ix (x)).

Lemma 4.3. Let M∈D(Aqct(X)) and L the smallest localizing subcategory of D(A (X)) that contains M.IfG∈D(A (X)) is such that M⊗L G =0 qct qct OX then F⊗L G =0, for every F∈L OX Proof. The ∆-functor −⊗L G preserves coproducts and therefore the full OX subcategory whose objects are those F∈Lsuch that F⊗L G =0is OX localizing, but it contains M, therefore it is L.

Proposition 4.4. The smallest localizing subcategory L of D(Aqct(X)) that contains K(x) for every x ∈ X is the whole D(Aqct(X)).

Proof. Let F∈D(Aqct(X)) and C denote the family of subsets Y ⊂ X stable for specialization such that RΓY F∈L.If{Wα}α∈I is a chain in C then

∪ F J RΓ Wα =−→ lim ΓWα , α∈I BOUSFIELD LOCALIZATION ON FORMAL SCHEMES 9 for a K-injective resolution F→J. By [AJS, Theorem 2.2 and Theorem F∈L F J∈L ∪ ∈C 3.1] RΓ∪Wα , because each RΓWα = ΓWα ,so Wα . The set C is stable for ﬁltered unions, therefore, there is a maximal element in C which we will denote by W . We will see that W = X from which it ∼ follows that F = RΓXF∈L. Indeed, otherwise suppose X \ W = ∅.AsX is noetherian the family of closed subsets C = {{z}/z ∈ X and {z}∩(X \ W ) = ∅} has a minimal subset {y}.Ifx ∈ {y}∩(X \ W ), then {x}∈C, but {y} is minimal, so x = y and W ∪ {y} = W ∪{y}. Consider now the inclusion iy : Xy → X and the distinguished triangle in D(Aqct(X)) ∗ + F−→ F−→ ∗ F −→ RΓW RΓW ∪{y} RΓ{y}(Riy iy ) obtained applying RΓW ∪{y} to the canonical triangle F−→F−→ ∗F −→+ RΓX\Xy Riy ∗iy . ∗ F∈L ∈C ∗ F ∈ We deduce that RΓW ∪{y} , because W and RΓ{y}(Riy iy ) Ly ⊂Lby Lemma 4.1, contradicting the maximality of W .

Corollary 4.5. Let G∈D(Aqct(X)). We have that G =0if, and only if, K G ∈ ∈ Z HomD(X)( (x)[n], )=0for all x X and n . Proof. Immediate from Proposition 4.4. Corollary 4.6. Let G∈D(A (X)) be such that K(x) ⊗L G =0for every qct OX x ∈ X, then G =0 Proof. It is a consequence of Proposition 4.4 and Lemma 4.3. Lemma 4.7. If x = y, then K(x) ⊗L K(y)=0. OX Proof. There exist an aﬃne open subset U ⊂ X such that it only contains one of the points, for instance assume that x ∈ U and y/∈ U. Denote by j : U → X the canonical inclusion map. Now we have L ∼ ∗ L K ⊗ K ∗ K ⊗ K (x) OX (y) = Rj j (x) OX (y) ∼ ∗ L L ∗ O ⊗ K ⊗ K = Rj j X OX (x) OX (y) ∼ L ∗ K ⊗ ∗ K = (x) OX Rj j (y) =0 because j∗K(y)=0.

Corollary 4.8. For every subset Z ⊂ X, the localizing subcategory LZ is rigid.

Proof. The full subcategory S⊂D(Aqct(X)) deﬁned by S {N ∈ A N⊗L M∈L ∀M ∈ L } = D( qct(X)) / OX Z , Z is a localizing subcategory of D(Aqct(X)). For every x ∈ X, we have that ∼ ∗ K(x)⊗L M RΓ (Ri ∗i K(x)⊗L M) therefore if x ∈ Z then K(x)⊗L OX = {x} x x OX OX M∈Lx ⊂LZ by Lemma 4.1, and for x/∈ Z, by Lemma 4.7 and Lemma 4.3, K(x) ⊗L M = 0 then it is also in L . Necessarily S = D(A (X)) by OX Z qct Proposition 4.4. 10 L. ALONSO, A. JEREM´IAS, AND M. J. SOUTO

Corollary 4.9. If Z and Y are subsets of X such that Z ∩ Y = ∅, then F⊗L G =0for every F∈L and G∈L . OX Z Y Proof. This follows from the previous lemma and Lemma 4.3.

Corollary 4.10. Given x ∈ X and F∈Lx we have that F ⇔F⊗L K =0 OX (x)=0.

Proof. By Lemma 4.7, given F∈Lx, for all y ∈ X, with y = x we have that F⊗L K(y) = 0, therefore if also F⊗L K(x) = 0, it follows that OX OX F⊗L G = 0 for all G∈D(A (X)), by Proposition 4.4. Thus, in particular OX qct F = F⊗L O =0. OX X

Corollary 4.11. Let L be a localizing subcategory of D(Aqct(X)) and F∈ D(A (X)).IfK(x) ⊗L F∈Lfor every x ∈ X, then F∈L. qct OX L Proof. Let L = {G ∈ D(Aqct(X)) / G⊗O F∈L}. The subcategory L is a X localizing subcategory of D(Aqct(X)) such that K(x) ∈L for all x ∈ X.By Proposition 4.4, we deduce that L = D(A (X)), in particular O ⊗L F = qct X OX F∈L. Remark. If the localizing subcategory L is rigid then K(x) ⊗L F∈Lfor OX all x ∈ X if, and only if, F∈L. Theorem 4.12. For a noetherian separated formal scheme X there is a bijection between the class of rigid localizing subcategories of D(Aqct(X)) and the set of all subsets of X.

Proof. Denote by Loc (D(Aqct(X))) the class of rigid localizing subcategories of D(Aqct(X)) and by P(X) the set of all subsets of X. Let us define a couple of maps: ψ Loc (D(Aqct(X))) P(X). φ and check that they are mutual inverses. Define for Z ⊂ X, φ(Z):=LZ which is rigid by Corollary 4.8, and for a rigid localizing subcategory L of D(A (X)), ψ(L):={x ∈ X/∃G ∈ L with K(x) ⊗L G =0}. qct OX Let us check first that ψ ◦ φ = id. Let Z ⊂ X and x ∈ Z, by definition K(x) ∈L and clearly K(x) ⊗L K(x) = 0 by Corollary 4.6 and Lemma Z OX 4.7, therefore x ∈ ψ(φ(Z)), so Z ⊂ ψ(φ(Z)). Conversely let x ∈ ψ(φ(Z)), by definition there is G∈L such that K(x) ⊗L G = 0, by Corollary 4.9, Z OX x ∈ Z. Now we have to prove that φ ◦ ψ = id. Let L be a rigid localizing A L ⊂L subcategory of D( qct(X)), we see first that ψ(L) and for this it is enough to check that K(x) ∈Lfor every x ∈ ψ(L). So let x ∈ ψ(L), there is a G∈Lsuch that K(x) ⊗L G = 0. On the other hand K(x) ⊗L G belongs OX OX to L because L is rigid and, moreover, we have that K ⊗L G ∼ F (x) OX = α α∈J where J is a set of indices and Fα = K(x)[si] with si ∈ Z. Indeed, it is M→ ∗G enough to take a free resolution ˜ ix of the complex of quasi-coherent BOUSFIELD LOCALIZATION ON FORMAL SCHEMES 11 O ∗G torsion Xx -modules ix and to consider the chain of natural isomorphisms L ∼ ∗ L K ⊗ G ∗ K ⊗ G (x) OX = Rix ix( (x) OX ) ∼ L ∗ = Ri ∗(K ⊗O i G) ([L, (3.2.4)]) x x Xx x ∼ L = Ri ∗(K ⊗O M) x x Xx L and use the fact that both functors −⊗O − and Rix ∗ commute with Xx coproducts. But L is localizing, so stable for coproducts and, as a consequence, for direct summands (see [BN] or [AJS, footnote, p. 227]). From F ∈L K ∈L this, α∈J α implies (x) , as required. Finally, let us see L⊂L F∈L F∈L that ψ(L). Let , by Corollary 4.11 to see that ψ(L) it L is enough to prove that K(x) ⊗ F∈L L for every x ∈ X. Suppose OX ψ( ) that K(x) ⊗L F = 0. In this case, x ∈ ψ(L), therefore we conclude that OX L L K(x) ⊗ F∈L L using Corollary 4.8 that tells us that K(x) ⊗ F OX ψ( ) OX belongs to the localizing subcategory generated by K(x). Corollary 4.13. For a noetherian separated scheme X there is a bijection between the class of rigid localizing subcategories of D(Aqc(X)) and the set of all subsets of X.

Corollary 4.14. Every rigid localizing subcategory of D(Aqct(X)) has associated a localization functor. Proof. The previous theorem says that a rigid localizing subcategory of D(Aqct(X)), L is the smallest localizing subcategory that contains the set {K(x)/x ∈ ψ(L)}. It follows from [AJS, Theorem 5.7] that there is an associated localization functor for L.

Corollary 4.15. Let L be a rigid localizing subcategory of D(Aqct(X)) and ∈ ∈ L K H · G F L z X.Ifz/ψ( ), then (z) and omX( , ) are -local objects for every F∈D(Aqct(X)) and G∈Lz. Proof. By Corollary 4.9 we have that N H · G F ∼ N⊗L G F HomD(X)( , omX( , )) = HomD(X)( OX , )=0 N∈L H · G F L for every , from which it follows that omX( , )is -local. Let N∈D(Aqct(X)) consider the natural map N K −→α N⊗L K K ⊗L K HomD(X)( , (z)) HomD(X)( OX (z), (z) OX (z)), and the map N⊗L K K ⊗L K −→β N K HomD(X)( OX (z), (z) OX (z)) HomD(X)( , (z)) induced by the canonical maps O →K K ⊗L K →K X (z) and (z) OX (z) (z). It is clear that β ◦ α = id. By Corollary 4.9 we have that N⊗L G = 0 for OX all N∈Land G∈Lz, and necessarily,

HomX(N , K(z))=0, therefore, K(z)isL-local. 12 L. ALONSO, A. JEREM´IAS, AND M. J. SOUTO

5. Compatibility of localization with the monoidal structure

Let L be a localizing subcategory of D(Aqct(X)) with associated Bous- ﬁeld localization functor . For every F∈D(Aqct(X)) there is a canonical distinguished triangle:

γ F−→F−→F −→+ (3) such that γ F∈Land F∈L⊥ (in other words, F is L-local). The functor γ is called the acyclization or colocalization associated to L and was denoted a in [AJS]. Here we have changed the notation for clarity. The endofunctors γ and are idempotent in a functorial sense as explained in §1ofloc. cit. For all F, G∈D(Aqct(X)) we have the following canonical isomorphisms F G −→ F G HomD(X)(γ ,γ )˜ HomD(X)(γ , )

F G −→ F G HomD(X)( , )˜ HomD(X)( , ) induced by γ G→Gand F→F, respectively. Lemma 5.1. With the previous notation, the following are equivalent (i) The localizing subcategory L is rigid. (ii) The natural transformation γ G→Ginduces isomorphisms H · F G ∼ H · F G omX(γ ,γ ) = omX(γ , )

for every F, G∈D(Aqct(X)). (iii) The natural transformation F→F induces isomorphisms H · F G ∼ H · F G omX( , ) = omX( , )

for every F, G∈D(Aqct(X)).

Proof. Let us show (i) ⇒ (ii). Let N∈D(Aqct(X)), we have the following chain of isomorphisms N H · F G ∼ N⊗L F G HomD(X)( , omX(γ ,γ )) = HomD(X)( OX γ ,γ ) a ∼ N⊗L F G = HomD(X)( OX γ , ) ∼ N H · F G = HomD(X)( , omX(γ , )); where a is an isomorphism because L is rigid and therefore N⊗L γ F = OX γ (N⊗L γ F). Having an isomorphism for every N∈D(A (X)) forces OX qct the target complexes to be isomorphic. We will see now (ii) ⇒ (iii). From (3), we have a distinguished triangle H · F G −→ H · F G −→ H · F G −→+ omX( , ) omX( , ) omX(γ , ) but its third point is null, considering

(ii) H · F G ∼ H · F G omX(γ , ) = omX(γ ,γ )=0, because γG =0. Finally, let us see that (iii) ⇒ (i). Take F∈Land N∈D(Aqct(X)). To see that N⊗L F∈Lit is enough to check that Hom (N⊗L F, G)=0 OX D(X) OX BOUSFIELD LOCALIZATION ON FORMAL SCHEMES 13 for every G∈L⊥ because ⊥(L⊥)=L. But this is true: N⊗L F G ∼ N H · F G HomD(X)( OX , ) = HomD(X)( , omX( , )) b ∼ N H · F G = HomD(X)( , omX( , )) =0, where b is an isomorphism, as follows from (iii) and the fact that G = G, and the last equality holds because F∈Land so F =0. Example. Let Z be a closed subset of X, or more generally, a set stable for 2 specialization . Recall the functor sections with support ΓZ : Aqct(X) → Aqct(X). From paragraph 2.2, we see that RΓZ : D(Aqct(X)) → D(Aqct(X)), its derived functor, together with the natural transformation RΓZ → id posses the formal properties of an acyclization such that the associated localizing subcategory

L = {M ∈ D(Aqct(X)) / RΓZ (M)=M} is rigid. Also, the functor RΓ has the following easily-checked property: Z 0ifx/∈ Z RΓZ (K(x)) = K(x)ifx ∈ Z. so L has to agree with LZ by Theorem 4.12 and, consequently, RΓZ is γZ , the acyclization functor associated to the localizing subcategory LZ . This acyclization functor satisﬁes a special property, namely, γ (F⊗L G) and Z OX F⊗L γ G are canonically isomorphic. OX Z

5.2. Let L be a rigid localizing subcategory of D(Aqct(X)) and F, G∈ D(A (X)). The morphism F⊗L γ G→F⊗L G induced by γ G→G qct OX OX factors naturally through γ (F⊗L G) giving a natural morphism OX F⊗L G−→ F⊗L G t : OX γ γ ( OX ). Let us denote by F⊗L G−→ F⊗L G p : OX ( OX ) a morphism such that the diagram

L L L + F⊗O γ G−−−→F⊗O G−−−→F⊗O G −−−→ X X X   t p

γ (F⊗L G) −−−→ F ⊗ L G−−−→(F⊗L G) −−−→+ OX OX OX is a morphism of distinguished triangles. In fact, the triangle is functorial in the sense that the map p is uniquely determined by t due to the fact that Hom (F⊗L γ G,(F⊗L G)[−1]) = 0. D(X) OX OX We say that the localization is ⊗-compatible (or that L is ⊗-compatible) if the canonical morphism t, or equivalently p, is an isomorphism. O O We remind the reader our convention that X denotes RΓX X.

2See 2.2. 14 L. ALONSO, A. JEREM´IAS, AND M. J. SOUTO

Proposition 5.3. In the previous hypothesis we have the following equivalent statements: (i) The localization associated to L is ⊗-compatible. (ii) For every E∈L⊥ and F∈D(A (X)) we have that F⊗L E∈L⊥. qct OX (iii) The functor preserves coproducts. (iv) A coproduct of L-local objects is L-local. (v) The set Z := ψ(L) is stable for specialization and its associated acyclization functor is γ = RΓZ . Proof. Let us begin proving the non-trivial part of (i) ⇔ (ii). Indeed, suppose that (ii) holds and consider the triangle F⊗L G−→F⊗L G−→F⊗L G −→+ OX γ OX OX we have that F⊗L γ G∈Lbecause L is rigid, on the other hand F⊗L G∈ OX OX L⊥ because G∈L⊥. The fact that the natural maps F⊗L G −→t F⊗L G F⊗L G −→p F⊗L G OX γ γ ( OX ) and OX ( OX ) are isomorphisms follow from [AJS, Proposition 1.6, (vi) ⇒ (i)]. Let us see now that (i) ⇒ (iii). If the localization associated to L is ⊗-compatible we have that F ∼ F⊗L O = OX X from which is clear that preserves coproducts. ∼ The implication (iii) ⇒ (iv) is obvious because F = F if, and only if, F∈L⊥. To see that (iv) ⇒ (v), we will use an argument similar to the one in the aﬃne case ([N1, Lemma 3.7]). Assume that preserves coproducts. Let x ∈ Z and z ∈ {x}.Ifz/∈ Z, then K(z) ∈L⊥ (Corollary 4.15) and ⊥ ⊥ Lz ⊂L because by (iv) L is localizing, and it follows by Corollary 4.2 that also E(z) ∈L⊥. But E(x) ∈Lwhich contradicts the existence on a non-zero map E(x) →E(z) because z ∈ {x}. Therefore Z is stable for specialization ∼ and γ = RΓZ by the example on page 13. The same example shows that (v) ⇒ (i). Remark. In the category of stable homotopy, HoSp, the localizations for which condition (iii) holds are called smashing. This can be characterized by a condition analogous to (i) in terms of its monoidal structure via the smash product, ∧. So, the previous result classiﬁes smashing localizations in D(Aqct(X)). Corollary 5.4. For a noetherian separated scheme X there is a bijection between the class of ⊗-compatible localizations of D(Aqc(X)) and the set of subsets stable for specialization of X.

5.5. Let L be a rigid localizing subcategory of D(Aqct(X)) and F, G∈ A H · F G → H · F G G→ D( qct(X)). The morphism omX( , ) omX( , ) induced by G H · F G factors through omX( , ) by Proposition 3.1. So, it gives a natural morphism H · F G −→ H · F G q : omX( , ) omX( , ). Let us denote by H · F G −→ H · F G h : γ omX( , ) omX( ,γ ) BOUSFIELD LOCALIZATION ON FORMAL SCHEMES 15 the morphism such that the diagram

Hom· (F,γG) −−−−→ Hom· (F, G) −−−−→ Hom· (F,G) −−−−→+ X X X   h q H · F G −−−−→ H · F G −−−−→ H · F G −−−−→+ γ omX( , ) omX( , ) omX( , ) is a morphism of distinguished triangles. Again, h and q determine each other. With the notation of the previous remark, we say that the localization is Hom-compatible (or that L is Hom-compatible) if the canonical morphism q, or equivalently h, is an isomorphism.

5.6. Let L be a ⊗-compatible localizing subcategory of D(Aqct(X)) whose associated (stable for specialization) subset is Z ⊂ X. Let us apply the H · − F F∈ A functor omX( , ), where D( qct(X)), to the canonical triangle O −→ O −→ O −→+ γZ X X Z X associated to L. We have added the associated subsets as subindices for clarity. We obtain: H · O F −→ F −→ H · O F −→+ omX(Z X, ) omX(γZ X, ) . Proposition 5.7. the canonical natural transformations: → H · O − H · O − → id omX(γZ X, ) and omX(Z X, ) id, correspond to a Hom-compatible localization and its corresponding acyclization in D(Aqct(X)), respectively. Its associated subset of X is X \ Z. Proof. We will consider two possibilities depending on the point being or ∈ \ K ∈L⊥ not in Z. First, if z X Z, it follows that (z) Z by Corollary 4.15 and therefore we have that H · O K ← H · O K omX(γ Z X, (z)) ˜ omX(γ Z X,γZ (z)) = 0, and that H · O K →K omX(Z X, (z)) ˜ (z). ∈ H · O K For z Z we will show that omX(Z X, (z)) = 0. By Proposition 4.4 and Theorem 4.12 it is enough to prove that K H · O K ∀ ∈ HomD(X)( (y), omX(Z X, (z))) = 0, y X, equivalently that K ⊗L O K ∀ ∈ HomD(X)( (y) OX Z X, (z)) = 0, y X. ∼ The localization functor is ⊗-compatible so K(y) ⊗L O K(y) Z OX Z X = Z will be zero if y ∈ Z. On the other hand, if y ∈ X \ Z we conclude because K(y) ⊗L O ∈L and K(z) ∈L⊥ (Corollary 4.15). OX Z X y y 5.8. Note that the following adjunction is completely formal H · F G −→ H · F G omX(γZ , )˜ omX( ,X\Z ). 16 L. ALONSO, A. JEREM´IAS, AND M. J. SOUTO

Indeed, H · F G ∼ H · O ⊗L F G omX(γZ , ) = omX(γZ X OX , ) ∼ H · F H · O G = omX( , omX(γZ X, )) ∼ H · F G = omX( ,X\Z ). Example. Let now Z be a closed subset of X which we assume it is an ordinary (noetherian separated) scheme. and LΛZ : D(Aqc(X)) → D(A(X)) the left-derived functor of the completion along the closed subset Z (which exist because it can be computed using quasi-coherent flat resolutions, as proved in [AJL1]). In loc. cit. it is also shown there is a natural isomorphism H · O G −→ G omX (RΓZ X , )˜RQLΛZ ( ). This result together with the previous adjunction is often referred to as Greenlees-May duality because generalizes a result from [GM] in the affine case. 5.9. In general, if Z ∈ X is a stable for specialization subset of X we will define for every G∈D(Aqct(X)): G H · O G ΛZ ( ):= omX(RΓZ X, )).

Proposition 5.10. For a rigid localizing subcategory L⊂D(Aqct(X)), the following are equivalent: (i) The localization associated to L is Hom-compatible. N∈L F∈ A H · F N ∈L (ii) For every and D( qct(X)) we have that omX( , ) . (iii) The set Y := ψ(L) is generically stable3 and its associated localization functor is ΛZ with Z = X \ Y . Proof. Let us see ﬁrst that (ii) ⇒ (iii). Let z ∈ Y and x ∈ X such that z ∈ {x}. With the notation of Corollary 4.2, if x/∈ Y then by Corollary H · E E ∈L⊥ 4.15, omX( (x), (z)) . H · E E L E ∈L By (ii), omX( (x), (z)) belongs to because (z) . Therefore H · E E omX( (x), (z)) = 0 and we have E E ∼ O H · E E HomD(X)( (x), (z)) = HomD(X)( X, omX( (x), (z))) = 0 a contradiction. Necessarily, the set Z = X \ Y is stable for specialization and Y = ΛZ . The implication (iii) ⇒ (i) follows from the previous remarks and the bijective correspondence established in Theorem 4.12. To ﬁnish, (i) ⇒ (ii) is straightforward because for every N∈L, we have that (i) H · F N H · F N ∼ H · F N ∈L omX( , )= omX( ,γ ) = γ omX( , )

Corollary 5.11. The functor γ associated to a Hom-compatible localization in D(Aqct(X)) commutes with products, in particular, the corresponding localizing class L is closed for products.

3i.e. an arbitrary intersection of open subsets. BOUSFIELD LOCALIZATION ON FORMAL SCHEMES 17

Proof. It is an immediate consequence of Proposition 5.10 (iii) and that every complex in K(Aqct(X)) admits a K-ﬂat resolution and a K-injective resolution.

Corollary 5.12. For a noetherian separated scheme X there is a bijection between the class of Hom-compatible localizations of D(Aqc(X)) and the set of generically stable subsets of X.

References

[AJL1] Alonso Tarr´ıo, L.; Jerem´ıasLópez, A.; Lipman, J.: Local homology and cohomology on schemes, Ann. Scient. Ec.´ Norm. Sup. 30 (1997), 1–39. [AJL2] Alonso Tarr´ıo, L.; Jerem´ıasLópez, A.; Lipman, J.: Duality and flat base change on formal schemes, in Studies in duality on noetherian formal schemes and non- noetherian ordinary schemes. Providence, RI: American Mathematical Society. Contemp. Math. 244, 3–90 (1999). [AJL3] Alonso Tarr´ıo, L.; Jerem´ıasLópez, A.; Lipman, J.: Greenlees-May duality of formal schemes, in Studies in duality on noetherian formal schemes and non- noetherian ordinary schemes. Providence, RI: American Mathematical Society. Contemp. Math. 244, 93–112 (1999). [AJS] Alonso Tarr´ıo, L.; Jerem´ıasLópez, A.; Souto Salorio, Ma. José: Localization in categoriesof complexesand unbounded resolutions Canad. J. Math. 52 (2000), no. 2, pp. 225–247. [Bo] Bourbaki, N.: Algèbre Commutative, Actualités Sci. et Industrielles, no. 1293, Hermann, Paris, 1961. [BN] Bökstedt, M.; Neeman, A.: Homotopy limits in triangulated categories, Compo- sitio Math. 86 (1993), 209–234. [GP] Popesco, N.; Gabriel, P.: Caractérisation des catégoriesabéliennesavec généra- teurset limitesinductivesexactes. C. R. Acad. Sci. Paris 258 (1964), 4188–4190. [GM] Greenlees, J.P.C.; and May, J.P.: Derived functors of I-adic completion and local homology J. Algebra 149 (1992), 438–453. [Gr] Grothendieck, A.: Sur quelquespointsd’alg` ebre homologique. Tôhoku Math. J. (2), 9 (1957), 119–221. [EGA] Grothendieck, A.; Dieudonné, J. A.: Eléments de Géométrie Algébrique I, Grundlehren der math. Wiss. 166, Springer–Verlag, Heidelberg, 1971. [EK] Eilenberg, Samuel; Kelly, G. Max: Closed categories. Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) pp. 421–562 Springer, New York, 1966 [H] Hartshorne, R.: Residues and Duality, Lecture Notesin Math., no. 20, Springer- Verlag, New York, 1966. [Il] Illusie, L.: Existence de résolutions globales, in Théorie des Intersections et Théorème de Riemann-Roch (SGA 6), Lecture Notesin Math., no. 225, Springer- Verlag, New York, 1971, pp. 160–222. [Iv] Iversen, Birger: Cohomology of sheaves. Universitext. Springer-Verlag, Berlin- New York, 1986. [JLV] Jerem´ıasLópez, A.; López López, M. P.; Villanueva Nóvoa, E.: Localization in symmetric closed Grothendieck categories. Third Week on Algebra and Algebraic Geometry (SAGA III) (Puerto de la Cruz, 1992). Bull. Soc. Math. Belg. Sér. A 45 (1993), no. 1-2, 197–221. [K] Keller, B.: Deriving DG categories. Ann. Sci. Ecole´ Norm. Sup. (4) 27 (1994), no. 1, 63–102. [L] Lipman, J.: Notes on Derived Categories. Preprint, Purdue University, available at http://www.math.purdue.edu/~lipman/ [N1] Neeman, A.: The chromatic tower for D(R). Topology 31 (1992), no. 3, 519–532. [N2] Neeman, A.: The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Ann. Sci. Ecole´ Norm. Sup. (4) 25 (1992), no. 5, 547–566. 18 L. ALONSO, A. JEREMÍAS, AND M. J. SOUTO

[Sp] Spaltenstein, N.: Resolutions of unbounded complexes. Compositio Math. 65 (1988), no. 2, 121–154. [St] Stenström, B.: Rings of quotients. An introduction to methods of ring theory. Grundlehren der math. Wiss. 217. Springer-Verlag, Berlin-Heidelberg-New York, 1975. [T] Thomason, R. W.: The classification of triangulated subcategories. Compositio Math. 105 (1997), no. 1, 1–27. [V] Verdier, J.-L.: Catégoriesdérivées. Quelques resultats (Etat 0). Semin. Geom. 1 algebr. Bois-Marie, SGA 4 2 , Lect. NotesMath. 569, 262-311 (1977).

(L. A. T.) Departamento de Alxebra,´ Facultade de Matematicas,´ Universi- dade de Santiago de Compostela, E-15782 Santiago de Compostela, SPAIN E-mail address: [email protected]

(A. J. L.) Departamento de Alxebra,´ Facultade de Matematicas,´ Universi- dade de Santiago de Compostela, E-15782 Santiago de Compostela, SPAIN E-mail address: [email protected]

(M. J. S. S.) Facultade de Informatica,´ Campus de Elvina,˜ Universidade da Coruna,˜ E-15071 A Coruna,˜ SPAIN E-mail address: [email protected]