Sheaves on Subanalytic Sites

Transcription

1 REND. SEM. MAT. UNIV. PADOVA, Vol ) Sheaves on Subanalytic Sites LUCA PRELLI *) ABSTRACT - In [7]the authors introduced the notion of ind-sheaves and defined the six Grothendieck oerations in this framework. As a byroduct, they obtained subanalytic sheaves and the six Grothendieck oerations on them. The aim of this aer is to give a direct construction of the six Grothendieck oerations in the framework of subanalytic sites avoiding the heavy machinery of ind-sheaves. As an alication, we show how to recover the subanalytic sheaves O t and O w of temerate and Whitney holomorhic functions resectively. Introduction. Let X be a real analytic manifold and k a field. The saces of functions which are not defined by local roerties, such as temered distributions, temered and Whitney C 1 functions, etc., are very useful in the study of systems of linear artial differential equations Lalace transform, temered holomorhic solutions of D-modules etc.). Although these saces do not define sheaves on X, they define sheaves on a site associated to X, the subanalytic site X sa, where one just considers oen subanalytic sets and locally finite coverings. In [7], Kashiwara and Schaira, motivated by the construction of the microlocalization functor, treated a more general theory, namely that of ind-sheaves. They defined the category I k X ) of ind-sheaves on X as the category of ind-objects of the category Mod c k X ) of sheaves with comact suort and they develoed the six Grothendieck oerations in this framework. When restricting to R-constructible sheaves, they showed the equivalence between the category I R-c k X ) ˆ Ind Mod c R-c k X)) of ind-rconstructible sheaves on X and the category Mod k Xsa ) of sheaves on the subanalytic site associated to X. *) Indirizzo dell'a.: Diartimento di Matematica Pura ed Alicata, Via Trieste, 63, Padova.

2 168 Luca Prelli In this way, temered distributions, temered and Whitney C 1 functions, etc., are obtained as a byroduct of the whole theory of ind-sheaves. It turns out to be useful to have a more straightforward introduction of these sheaves. Our aim in this aer is to give a direct, self-contained and elementary construction of the six Grothendieck oerations on Mod k Xsa ), without using the more sohisticated and much more difficult theory of indsheaves. Indeed, contrary to the category I k X ), the category Mod k Xsa )isa Grothendieck category. We will start by recalling some results of [7], the definition of a subanalytic site, the natural functor of sites r : X! X sa, and the functors r, r 1 and r! relating the categories of ``classical'' and subanalytic sheaves. We also recall a very useful descrition of subanalytic sheaves as inductive limits of R-constructible sheaves. Then we go into the study of subanalytic sheaves, without using the notion of ind-sheaf. Let f : X! Y be a morhism of real analytic manifolds. The functors Hom,, f and f 1 are well defined since X sa is a site. We introduce the roer direct image functor f!! and we study the relations between the above oerations and the functors r, r 1 and r!. In the derived category we obtain an excetional inverse image, denoted by f!, right adjoint to Rf!!. This is obtained via Brown reresentability thanks to the existence of quasi-injective objects. We study quasi-injective objects without references to ind-sheaves, we show that they are acyclic with resect to the above functors and we rove that the quasi-injective dimension of Mod k Xsa ) is finite. We end this work giving some examles of subanalytic sheaves. In more details, the contents of this aer are as follows. In Section 1 we construct the oerations in Mod k Xsa ). In x 1.1, we start by recalling the definitions of the functors r, r 1 and r! of [7]and their roerties a more detailed study of the functor r! is done in x 1.6). In x 1.2, we recall the internal oerations and the functors of direct and inverse image which are well defined on any site) and their relations with r, r 1 and r!. We also define in x 1.4) the roer direct image functor f!!, where the notation f!! follows from the fact that f!! r 6' r f! in general. We study its roerties and the relations with the other oerations. While the functors f 1 and are exact, the functors Hom, f and f!! are left exact, and we introduce the subcategory of quasi-injective objects which is injective with resect to these functors. This is done in x 1.5.

3 Sheaves on Subanalytic Sites 169 In Section 2 we consider the derived category of Mod k Xsa ). In x 2.1, we consider the subcategory D b R-c k X sa ) consisting of bounded comlexes with R-constructible cohomology and we rove the equivalence of derived categories D b R-c k X) ' D b R-c k X sa ). Then, in x 2.2, we study the derived functors of Hom, f and f!! and we obtain the usual formulas rojection formula, base change formula, KuÈnneth formula, etc.) in the framework of subanalytic sites. We also rove in x 2.3) some vanishing theorems for subanalytic sheaves, in articular we rove that the quasi-injective dimension of Mod k Xsa ) is finite. Using the Brown reresentability theorem we rove the existence of a right adjoint to the functor Rf!!, denoted by f!. This is done in x 2.4. We calculate the functor f! by decomosing f as the comosite of a closed embedding and a submersion. In Section 3 we give some examles of subanalytic sheaves. We start by recalling the definition of sheaves of r! D X -modules, where D X denotes the sheaf of finite order differential oerators on a comlex analytic manifold X. Then in x 3.3 we show how to recover the sheaves of r! D X -modules O t X and O w X of temerate and Whitney holomorhic functions of [7]resectively. We rove the relations between the above sheaves and the functors of moderate and formal cohomology of [4]and [6]. Acknowledgments. We would like to thank Prof. Pierre Schaira who encouraged us to develo a theory of subanalytic sheaves indeendent of that of ind-sheaves, and for his many useful remarks. 1. Sheaves on subanalytic sites. In the following X will be a real analytic manifold and k a field. References are made to [8]and [14]for an introduction to sheaves on Grothendieck toologies, to [5]for a comlete exosition on classical sheaves and R-constructible sheaves and to [1]and [10]for the theory of subanalytic sets see also the Aendix for a short overview on some roerties of subanalytic subsets). 1.1 ± The subanalytic site. Notations and review. We introduce the subanalytic site. The results of x 1.1 have already been roved in [7], for sake of comleteness we reroduce here the roofs. Denote by O X sa ) the category of subanalytic subsets of X. One endows O X sa ) with the following toology: S O X sa ) is a covering of

4 170 Luca Prelli U 2 O X sa ) if for any comact subset K of X there exists a finite subset S 0 of S such that K \ S V2S 0 V ˆ K \ U. We will call X sa the subanalytic site, and for U 2 O X sa ) we denote by U Xsa the category O X sa ) \ U with the toology induced by X sa. There is a natural morhism of sites U sa! U Xsa : REMARK We use the notation U Xsa to stress the difference from U sa, the subanalytic site associated to U. For examle, let X ˆ R 2 and U ˆ R 2 nf0g. Let V n ˆfx2 R 2 ; jxj > 1 n g. Then fv ng n2n 2 Cov U sa ) but fv n g n2n =2 Cov U Xsa ). Let Mod k Xsa ) denote the category of sheaves on X sa. Then Mod k Xsa )is a Grothendieck category, i.e. it admits a generator and small inductive limits, and small filtrant inductive limits are exact. In articular as a Grothendieck category, Mod k Xsa ) has enough injective objects. REMARK Denote by O c X sa ) the category of relatively comact subanalytic oen subsets of X. One denotes by Xsa c the category Oc X sa ) with the toology induced by X sa. The forgetful functor gives an equivalence of categories Mod k Xsa )! Mod k X c sa ). PROPOSITION Let ff i g i2i be a filtrant inductive system in Mod k Xsa ) and let U 2 O c X sa ). Then lim G U; F i )! G U; lim F i ): i i PROOF. By Remark it is enough to rove the assertion in the category Mod k X c sa ). Denote by ``lim'' F i the resheaf V7! lim G V; F i )on i i Xsa c. Let U 2 Oc X sa ) and let S be a finite covering of U. Since lim commutes with finite rojective limits we obtain the isomorhism i ``lim'' F i ) S)! lim F i S) and F i U)! F i S) since F i 2 Mod k X c sa ) for each i i i. Moreover the family of finite coverings of U is cofinal in Cov U). Hence ``lim'' i F i! ``lim'' i ``lim'' i F i ). Alying once again the functor ) we get F i ' ``lim'' i F i ) ' ``lim'' F i ) ' lim F i : i i Hence alying the functor G U; ) we obtain the isomorhism lim G U; F i )! G U; lim F i ) for each U 2 O c X sa ): i i

5 Sheaves on Subanalytic Sites 171 There is an easy way to construct sheaves on a subanalytic site PROPOSITION Let F be a resheaf on Xsa c and assume that i) F 6) ˆ 0; ii) For any U; V 2 O c X sa ) the sequence 0! F U [ V)!! F U) F V)! F U \ V) is exact. Then F 2 Mod k X c sa ) ' Mod k Xsa ). PROOF. Let U 2 O c X sa ) and let fu j g n jˆ1 be a finite covering of U. Set for short U ij ˆ U i \ U j. We have to show the exactness of the sequence 0! F U)! 1kn F U k )! 1i<jn F U ij ); where the second morhism sends s k ) 1kn to t ij ) 1i<jn by t ij ˆ ˆ s i j Uij s j j Uij. We shall argue by induction on n. For n ˆ 1 the result is trivial, and n ˆ 2 is the hyothesis. Suose that the assertion is true for j n 1 and set U 0 ˆ S1k<n U k. By the induction hyothesis the following commutative diagram is exact Then the result follows. Let Mod R-c k X ) be the abelian category of R-constructible sheaves on X, and consider its subcategory Mod c R-c k X) consisting of sheaves whose suort is comact. We denote by r : X! X sa

6 172 Luca Prelli the natural morhism of sites. We have functors 1:1) Mod c R-c k r X) Mod R-c k X ) Mod k X ) Mod k Xsa ): r 1 We will still denote by r the restriction of r to Mod R-c k X ) and Mod c R-c k X). REMARK By Proosition for each F 2 Mod k X ) and V 2 O c X sa ) one has 1:2) G V; r F) ' lim G V; r F U ) ' G V; lim r F U ); U U where U ranges through the family of relatively comact oen subanalytic subsets of X. The first isomorhism follows since V is relatively comact and G V; r F U ) ' G V; r F) if V U. The isomorhism 1.2) imlies lim U r F U! r F. REMARK The functor r does not commute with filtrant inductive limits. For examle consider the family fv n g n2n of Remark We have r lim k Vn ' r n k R 2 nf0g, while for each U 2 Oc R 2 sa ) with 0 we have G U; lim n r k Vn ) ' lim n G U; r k Vn ) ˆ 0. PROPOSITION Let U be an oen subanalytic subset of X and consider the constant sheaf k UXsa 2 Mod k Xsa ). We have k UXsa ' r k U. PROOF. Let F be the resheaf on X sa defined by F V) ˆ k if V U, F V) ˆ 0 otherwise. This is a searated resheaf and k UXsa ˆ F. Moreover there is an injective arrow F V),! r k U V) for each V 2 O X sa ). Hence F,!r k U since the functor ) is exact. Let T be the family of W 2 O X sa ) connected and such that W does not contain any connected comonent of U. Then T forms a basis for the toology of X sa since the connected comonents of U are locally finite. For each W 2T we have F W) ' r k U W) ' k if W U and F W) ˆ 0 otherwise. Then F ' r k U. PROPOSITION The restriction of r to Mod R-c k X ) is exact. PROOF. i) Let us consider an eimorhism G! F in Mod c R-c k X), we have to rove that c : r G! r F is an eimorhism. Let

7 Sheaves on Subanalytic Sites 173 U 2 O c X sa ) and let 0 6ˆ s 2 G U; r F) ' Hom kx k U ; F). Set G 0 ˆ G F k U. Then G 0 2 Mod c R-c k X) and moreover G 0! k U. There exists a finite fu i g i2i O c X sa )withu i connected for each i such that i k Ui! G 0. The comosition k Ui! G 0! k U is given by the multilication by a i 2 k. SetI 0 ˆfk Ui ; a i 6ˆ 0g, wemayassumea i ˆ 1. We get a diagram The comosition k Ui! G 0! G defines t i 2 Hom kx k Ui ; G) ˆ G U i ; r G). Hence for each s 2 G U; r F) there exists a finite covering fu i g of U and t i 2 G U i ; r G) such that c t i ) ˆ sj Ui. This means that c is surjective. ii) Let F 2 Mod R-c k X ). By Remark r F ' lim r F U, where U U ranges through the family O c X sa ). The result follows since r is exact on Mod c R-c k X) and filtrant lim are exact. PROPOSITION The restriction of r to Mod R-c k X ) is fully faithful. PROOF. It is enough to rove that r 1 r F ' F for each F 2 Mod R-c k X ). Since both functors are exact on Mod R-c k X ) we may reduce to the case F ˆ k U with U 2 O X sa ) and the result follows from Proosition NOTATIONS Since the functor r is fully faithful and exact on the category Mod R-c k X ), we can identify Mod R-c k X ) with its image in Mod k Xsa ). When there is no risk of confusion we will write F instead of r F, for F 2 Mod R-c k X ). The following theorem gives a fundamental characterization of subanalytic sheaves and it will be used systematically in the following Sections. THEOREM i) Let G 2 Mod c R-c k X) and let ff i g be a filtrant inductive system in Mod k Xsa ). Then we have an isomorhism lim Hom kxsa r G; F i )! Hom kxsa r G; lim F i ): i i

8 174 Luca Prelli ii) Let F 2 Mod k Xsa ). There exists a small filtrant inductive system ff i g i2i in Mod c R-c k X) such that F ' lim i r F i. PROOF. i) There exists an exact sequence G 1! G 0! G! 0 with G 1 ; G 0 finite direct sums of constant sheaves k U with U 2 O c X sa ). Since r is exact on Mod R-c k X ) and commutes with finite sums, by Proosition we are reduced to rove the isomorhism lim G U; F i )! G U; lim F i ). i i Then the result follows from Proosition ii) Let F 2 Mod k Xsa ), and define I 0 :ˆf U; s); U 2 O c X sa ); s 2 G U; F)g G 0 :ˆ U;s)2I0 r k U The morhism r k U! F, where the section 1 2 G U; k U ) is sent to s 2 G U; F) defines un eimorhism W : G 0! F. Relacing F by ker W we construct a sheaf G 1 ˆ V;t)2I1 r k V and an eimorhism G 1! ker W. Hence we get an exact sequence G 1! G 0! F! 0. For J 0 I 0 set for short G J0 ˆ U;s)2J0 r k U and define similarly G J1. Set J ˆf J 1 ; J 0 ); J k I k ; J k is finite and im Wj GJ1 G J0 g: The category J is filtrant and F ' lim coker G J1! G J0 ). J 1 ; J 0 ) 2 J PROPOSITION Let F 2 Mod k Xsa ), and let U 2 O X). Then G U; r 1 F ) ' lim G V ; F ) V U;V 2O c Xsa) PROOF. By Theorem we may assume F ˆ lim r F i, with i F i 2 Mod c R-c k X). Then r 1 F ' lim i isomorhisms r 1 r F i ' lim F i. We have the chain of i G U; r 1 F) ' lim G V; r 1 F) ' lim G V; lim F VU VU i ) i ' lim lim G V; F i ) ' lim lim G V; F i i ) i ' VU VU lim lim G V; r VU F i ) ' lim G V; F); i VU where V 2 O c X sa ). The third isomorhism follows since V is comact and the last isomorhism follows from Proosition

9 Sheaves on Subanalytic Sites 175 PROPOSITION r is fully faithful. One has r 1 r! id, in articular the functor PROOF. Let F 2 Mod k X ). Every x 2 X has a fundamental neighborhood system consisting of oen subanalytic subsets. Hence we have the chain of isomorhisms r 1 r F) x ' lim x2u r 1 r F U) ' lim x2u r F U) ' lim x2u F U) ' F x ; where U ranges through the family of oen subanalytic neighborhoods of x. The second isomorhism follows since, by Proosition lim x2u r 1 r F U) ' lim lim r F V) ' lim r F U): x2u VU x2u Now we will describe a left adjoint to the functor r 1. PROPOSITION The functor r 1 admits a left adjoint, denoted by r!. It satisfies i) for F 2 Mod k X ) and U 2 O X sa ), r! F is the sheaf associated to the resheaf U7!G U; F), ii) for U 2 O X) one has r! k U ' lim VU;V2O c Xsa) k V. PROOF. Let F e 2 Psh kxsa ) be the resheaf U7!G U; F), and let G 2 Mod k Xsa ). We will construct morhisms Hom Psh kxsa ) F; e j G) Hom kx F; r 1 G): w To define j, let W : F e! G and U 2 O X). Then the morhism j W) U) : F U)! r 1 G U) is defined as follows F U) ' lim F V) W VU;V2O c Xsa) lim G V) ' r 1 G U): VU;V2O c Xsa) On the other hand, let c : F! r 1 G and U 2 O c X sa ). Then the morhism w c) U) : e F U)! G U) is defined as follows ef U) ' lim F V) c UV2O c Xsa) lim r 1 G V)! G U): UV2O c Xsa)

10 176 Luca Prelli By construction one can check that the morhisms j and w are inverse to each others. Then i) follows from the chain of isomorhisms Hom kxsa e F ; G) ' Hom Psh kxsa ) e F; G) ' Hom kxsa F; r 1 G): To show ii), consider the following sequence of isomorhisms Hom kxsa r! k U ; F) ' Hom kx k U ; r 1 F) ' lim VU;V2O c Xsa) Hom kxsa k V ; F) ' Hom kxsa lim VU;V2O c Xsa) k V ; F); where the second isomorhism follows from Proosition PROPOSITION and. The functor r! is exact and commutes with lim PROOF. It follows by adjunction that r! is right exact and commutes with lim, so let us show that it is also left exact. With the notations of Proosition , let F 2 Mod k X ), and let F e 2 Psh k Xsa ) be the resheaf U 7! G U; F). Then r! F ' F e, and the functors F 7! F e and G 7! G are left exact. Let us show that r! commutes with. Let F; G 2 Mod k X ), the morhism defines a morhism in Mod k Xsa ) F U) G U)! F G) U) r! F r! G! r! F G) by Proosition i). Since r! commutes with lim we may suose that F ˆ k U and G ˆ k V and the result follows from Proosition ii). PROPOSITION The functor r! is fully faithful. In articular one has r 1 r! ' id. Moreover, for F 2 Mod k X ) and G 2 Mod k Xsa ) one has r 1 Hom r! F; G) 'Hom F; r 1 G): PROOF. For F; G 2 Mod k X ) we have by adjunction Hom kx r 1 r! F; G) ' Hom kx F; r 1 r G) ' Hom kx F; G):

11 Sheaves on Subanalytic Sites 177 This also imlies that r! is fully faithful, in fact Hom kxsa r! F; r! G) ' Hom kx F; r 1 r! G) ' Hom kx F; G): Now let K; F 2 Mod k X ) and G 2 Mod k Xsa ), we have Hom kx K; r 1 Hom r! F; G)) ' Hom kxsa r! K; Hom r! F; G)) ' Hom kxsa r! K r! F; G) ' Hom kxsa r! K F); G) ' Hom kx K F; r 1 G) ' Hom kx K; Hom F; r 1 G)) and the result follows. 1.2 ± Oerations on the subanalytic site. Let X; Y be two real analytic manifolds, and let f : X! Y be a real analytic ma. This defines a morhism of sites f : X sa! Y sa. We have a diagram The following functors are always well defined on a site Hom : Mod k Xsa ) o Mod k Xsa )! Mod k Xsa ); : Mod k o X sa ) Mod k Xsa )! Mod k Xsa ); f : Mod k Xsa )! Mod k Ysa ); f 1 : Mod k Ysa )! Mod k Xsa ): Let us summarize their roerties: the functor Hom is left exact and commutes with r, the functor is exact and commutes with lim, r 1 and r!, the functor f is left exact and commutes with r and lim, the functor f 1 is exact and commutes with lim, and r 1, f 1 ; f ) is a air of adjoint functors.

12 178 Luca Prelli Let Z be a subanalytic locally closed subset of X. As in classical sheaf theory we define G Z : Mod k Xsa )! Mod k Xsa ) F 7! Hom r k Z ; F) ) Z : Mod k Xsa )! Mod k Xsa ) F 7! F r k Z : We have the functor G Z is left exact and commutes with r and lim, the functor ) Z is exact and commutes with lim, and r 1, ) Z ; G Z ) is a air of adjoint functors. 1.3 ± R-constructible sheaves on subanalytic sites. Let us consider the category Mod R-c k X ). We rove that the subcategory of Mod k Xsa ) consisting of R-constructible sheaves is stable under inverse image and tensor roduct. r G. PROPOSITION Let F; G 2 Mod R-c k X ). Then r F G) ' r F PROOF. We may reduce to the case F ˆ k U, G ˆ k V with U; V 2 O X sa ). In this case r k U\V ' r k U r k V by Proosition COROLLARY Let F 2 Mod R-c k X ), and let Z be a subanalytic locally closed subset of X. Then r F Z ' r F ) Z. Let X; Y be two real analytic manifolds, and let f : X! Y be a real analytic ma. PROPOSITION Let f : X! Y be a real analytic ma. Let G 2 Mod R-c k Y ). Then r f 1 G ' f 1 r G. PROOF. Since the functor f 1 is exact, we may reduce to the case G ˆ k V, with V 2 O Y sa ). In this case we have r f 1 k V ' r k f 1 V) ' ' f 1 r k V,wherethelastisomorhismfollows from Proosition We aly the above results to calculate the functor Hom in the category Mod k Xsa ).

13 Sheaves on Subanalytic Sites 179 PROPOSITION Let F ˆ lim F i, with F i 2 Mod k Xsa ) and let i G ˆ lim r G j with G j 2 Mod j R-c k X ). One has Hom G; F ) ' lim lim Hom r G j ;F i ): j i PROOF. For each U 2 O c X sa ) one has the isomorhisms G U; Hom G; F)) ' Hom kxsa G U ; F) ' lim j lim Hom kxsa r i G ju ; F i ) ' lim j lim G U; Hom r G j ; F i )) i ' G U; lim j lim Hom r G j ; F i )): i In the second isomorhism we used Corollary 1.3.2, and the last isomorhism follows from Proosition and because G U; ) commutes with lim. COROLLARY Let F ˆ lim i 2 Mod R-c k X ). One has Hom G; F ) ' lim j r F i ;Gˆ lim j r G j with F i ;G j 2 lim r Hom G j ;F i ): i PROOF. It follows from the fact that Hom commutes with r and from Proosition COROLLARY Let F ˆ lim i r F i, with F i 2 Mod c R-c X) be a sheaf on X sa. Let Z be a subanalytic locally closed subset of X. Then G Z F ' lim i r G Z F i : 1.4 ± Proer direct image on Mod k Xsa ). In [7]the authors defined the functor f!! of roer direct image using ind-sheaves. Here we give a direct construction: f!! : Mod k Xsa )! Mod k Ysa ) F 7! lim f F U ' lim f G K F U K where U ranges through the family of relatively comact oen subanalytic

14 180 Luca Prelli subsets of X and K ranges through the family of subanalytic comact subsets of X. One shall be aware that lim is taken in the category Mod k Ysa ). Let V 2 O c Y sa ). Then G V; f!! F) ˆ lim G f 1 V); F U ) ' U lim G f 1 V); G K F); where U ranges through the family of relatively K comact oen subanalytic subsets of X and K ranges through the family of subanalytic comact subsets of X.Iff is roer on su F) then f ' f!! and in this case f!! r ' r f!. REMARK Remark that f!! r 6' r f! in general. Indeed let V 2 O c Y sa ), then G V; f!! r F) ˆ lim G f 1 V); G K F); K G V; r f! F) ˆ lim G f 1 V); G Z F); Z where Z ranges through the family of closed subsets of f 1 V) such that f j Z : Z! V is roer. Then G V; f!! r F) ˆfs 2 G f 1 V); F); su s) is comact in Xg; G V; r f! F) ˆfs 2 G f 1 V); F); f : su s)! V is roerg: For examle, let f : R 2! R be the rojection on the first coordinate, and let V ˆ a; b) 2 O c R sa ). Suose that su s) ˆf x; y) 2 a; b) R; 1 y ˆ x a) b x) g. Then f : su s)! V is roer but su s) is not comact. PROPOSITION The functor f!! commutes with filtrant lim. Moreover r 1 f!! ' f! r 1. PROOF. Let us show that f!! commutes with filtrant lim. Let V 2 O c Y sa ) and let ff i g i be a filtrant inductive system in Mod k Xsa ). Then lim K Hom kxsa k f 1 V) ; G K lim i F i ) ' lim Hom kxsa k K f 1 V)\K ; lim F i ) i ' lim Hom kxsa k f 1 V)\K ; F i) i;k ' lim Hom kxsa k f 1 V) ; G KF i ) i;k ' lim Hom kysa k V ; f!! F i ) i ' Hom kysa k V ; lim i f!! F i );

15 Sheaves on Subanalytic Sites 181 where the second isomorhism follows from the fact that k f 1 V)\K 2 2 Mod c R-c k X). Let us show r 1 f!! ' f! r 1. Let F ˆ lim r F i. Since f!! commutes i with filtrant lim and F i has comact suort for each i we have f!! F ˆ lim r f! F i. We have the chain of isomorhisms i f! r 1 lim i r F i ' f! lim i r 1 r F i ' f! lim i F i ' lim f! F i i ' lim i r 1 r f! F i ' r 1 lim i r f! F i : PROPOSITION The functor f commutes with r 1. PROOF. Let F 2 Mod k Xsa ). Then f F ' lim f K F K, where K ranges through the family of subanalytic comact subsets of X. We have the chain of isomorhisms f r 1 F ' lim K ' lim K f r 1 F) K ' lim K f!! r 1 F) K ' lim f K!! r 1 F K r 1 f! F K ' r 1 lim f K! F K ' r 1 lim f K F K ' r 1 f F; where the second and the sixth isomorhism follow from the fact that f is roer on a comact subset of X. COROLLARY The functor f 1 commutes with r!. PROOF. It follows immediately by adjunction. PROPOSITION Let F 2 Mod k Xsa ) and G 2 Mod k Ysa ). Then f!! F G ' f!! F f 1 G): PROOF. Let F ˆ lim i r F i, G ˆ lim r G j. The functors, f!! and f 1 j commute with lim i. Moreover su F i f 1 G j ) is comact for each i; j,

16 182 Luca Prelli hence f is roer on it. Then f!! lim i r F i lim r G j ' lim r f! F i G j ) j i; j ' lim i; j r f! F i f 1 G j )) ' f!! lim i r F i f 1 lim i r G j ): In the first isomorhism we used Proosition and in the last one we used Proositions and Now let us consider a cartesian square PROPOSITION Let F 2 Mod k Xsa ). Then g 1 f!! F ' f 0!! g0 1 F. PROOF. Let F ˆ lim r i F i. All the functors in the above formula commute with lim. Moreover since su F i ) is comact, f 0 is roer on i su g 0 1 F i ) for each i. Then g 1 f!! lim i r F i ' lim r i g 1 f! F i ' lim r i f! 0 g0 1 F i ' f!! 0 g0 1 lim r i F i ; where the first and the last isomorhisms follow from Proosition The following isomorhism is the analogue for subanalytic sheaves of Corollary of [7]. PROPOSITION the natural morhism Let G 2 Mod R-c k Y ) and let F 2 Mod k Xsa ). Then f!! Hom f 1 G; F )!Hom G; f!! F ) is an isomorhism. PROOF. Let us construct the morhism. By adjunction we have f 1 G Hom f 1 G; F)! F;

17 Sheaves on Subanalytic Sites 183 hence, using the rojection formula we get G f!! Hom f 1 G; F) ' f!! f 1 G Hom f 1 G; F))! f!! F; then by adjunction we obtain the desired morhism. Let us show that it is an isomorhism. We have the chain of isomorhisms f!! Hom f 1 G; F) ' lim K f G K Hom f 1 G; F) ' lim K f Hom f 1 G; G K F) ' lim K Hom G; f G K F) 'Hom G; lim K f G K F) 'Hom G; f!! F); where the fourth isomorhism follows from Proosition ± Quasi-injective objects. Let us introduce a category which is useful in order to find acyclic objects with resect to the functors defined in the revious Sections. Although the definition is insired to the definition of quasi-injective objects of [7] or, more generally, to the definition of quasi-injective objects of a ind-category, see [8]for more details), the roofs of Theorems and are indeendent of the theory of ind-objects. DEFINITION An object F 2 Mod k Xsa ) is quasi-injective if the functor Hom kxsa ; F) is exact in Mod c R-c k X) or, equivalently see Theorem of [8]) if for each U; V 2 O c X sa ) with V U the restriction morhism G U; F)! G V; F) is surjective. It follows from the definition that injective sheaves belong to J Xsa. This imlies that J Xsa is cogenerating. Moreover the category J Xsa is stable by filtrant lim and Q. PROPOSITION Let 0! F 0! F! F 00! 0 be an exact sequence in Mod k Xsa ) and assume that F 0 is quasi-injective. Let U 2 O c X sa ).

18 184 Luca Prelli Then the sequence 0! G U; F 0 )! G U; F)! G U; F 00 )! 0 is exact. PROOF. Let s 00 2 G U; F 00 ), and let fv i g n iˆ1 be a finite covering of U such that there exists s i 2 G V i ; F) whose image is s 00 j Vi. For n 2on V 1 \ V 2 s 1 s 2 defines a section of G V 1 \ V 2 ; F 0 ) which extends to s 0 2 G U; F 0 ). Relace s 1 with s 1 s 0. We may suose that s 1 ˆ s 2 on V 1 \ V 2. Then there exists t 2 G V 1 [ V 2 ) such that tj Vi ˆ s i, i ˆ 1; 2. Thus the induction roceeds. PROPOSITION sequence Let F 0 ;F;F 00 2 Mod k Xsa ), and consider the exact 0! F 0! F! F 00! 0: Suose that F 0 ;F 2J Xsa. Then F 00 2J Xsa. PROOF. Let U; V 2 O c X sa ) with V U and let us consider the diagram below The morhism a is surjective since F is quasi-injective and b is surjective by Proosition Then g is surjective. THEOREM The family of quasi-injective sheaves is injective with resect to the functor Hom kxsa G; ) for each G 2 Mod R-c k X ). PROOF. i) Let 0! F 0! F! F 00! 0 be an exact sequence in Mod k Xsa ) and assume that F 0 2J Xsa. Let G 2 Mod c R-c k X). We have to show that the sequence 0! Hom kxsa G; F 0 )! Hom kxsa G; F)! Hom kxsa G; F 00 )! 0 is exact. There is an eimorhism W : i2i k Ui! G where I is finite and U i 2 O c X sa ) for each i 2 I. The sequence 0! ker W! i2i k Ui! G! 0 is exact. We set for short G 1 ˆ ker W and G 2 ˆ i2i k Ui. We get the following diagram where the first

19 Sheaves on Subanalytic Sites 185 column is exact The second row is exact by Proosition 1.5.2, hence the to row is exact by the snake lemma. ii) Let G 2 Mod R-c k X ), let 0! F 0! F! F 00! 0 be an exact sequence in Mod k Xsa ) with F 0 2J Xsa. Let fv n g n2n 2 Cov X sa ) such that V n V n 1. By i), all the sequences 0! Hom kxsa G Vn ; F 0 )! Hom kxsa G Vn ; F)! Hom kxsa G Vn ; F 00 )! 0 are exact. Moreover since F 0 2J Xsa the morhism Hom kxsa G Vn 1 ; F 0 )!! Hom kxsa G Vn ; F 0 ) is surjective for all n. Then by the Mittag-Leffler roerty see Proosition of [5]) the sequence 0! lim n Hom kxsa G Vn ; F 0 )! lim n Hom kxsa G Vn ; F)! lim n Hom kxsa G Vn ; F 00 )!0 is exact. Since lim n Hom kxsa G Vn ; ) ' Hom kxsa G; ) the result follows. PROPOSITION Let G 2 Mod R-c k X ). Then quasi-injective sheaves are injective with resect to the functor Hom G; ). PROOF. Let G 2 Mod R-c k X ). It is enough to check that for each U 2 O X sa ) and each exact sequence 0! F 0! F! F 00! 0 with F 0 2J Xsa, the sequence 0! G U; Hom G; F 0 ))! G U; Hom G; F))! G U; Hom G; F 00 ))! 0 is exact. We have G U; Hom G; )) ' Hom kxsa G U ; ), and quasi-injective objects are injective with resect to the functor Hom kxsa G U ; ) for each G 2 Mod R-c k X ), and for each U 2 O X sa ).

20 186 Luca Prelli COROLLARY Quasi-injective sheaves are injective with resect to the functor G Z for each locally closed subanalytic subset Z of X. COROLLARY Let F 2 Mod k Xsa ) be quasi-injective. Then the functor Hom ;F) is exact on Mod R-c k X ). PROOF. Let F 2 Mod k Xsa ) be quasi-injective. There is an isomorhism of functors G U; Hom ; F)) ' Hom kxsa ) U ; F) for each U 2 O X sa ). The functor Hom kxsa ) U ; F) is exact on Mod R-c k X ) and the result follows. PROPOSITION Let F 2 Mod k Xsa ). Then F is quasi-injective if and only if Hom G; F ) is quasi-injective for each G 2 Mod R-c k X ). PROOF. i) Let F be quasi-injective, and let G 2 Mod R-c k X ). We have Hom kxsa ; Hom G; F)) ' Hom kxsa G; F), and Hom kxsa G; F) is exact on Mod c R-c k X). ii) Suose that Hom G; F) is quasi-injective for each G 2 Mod R-c k X ). The result follows by setting G ˆ k X. COROLLARY The functor G Z sends quasi-injective objects to quasi-injective objects for each locally closed subanalytic subset Z of X. Let f : X! Y be a morhism of real analytic manifolds. PROPOSITION Quasi-injective sheaves are injective with resect to the functor f. The functor f sends quasi-injective objects to quasiinjective objects. PROOF. i) Let us consider V 2 O Y sa ). There is an isomorhism of functors G V; f )) ' G f 1 V); ). It follows from Proosition that J Xsa is injective with resect to the functor G f 1 V); ) ' ' Hom kxsa k f 1 V) ; ) for any V 2 O Y sa). ii) Let F 2J Xsa. For each G 2 Mod c R-c k Y) we have Hom kysa G; f F) ' Hom kxsa f 1 G; F). Since f 1 is exact and sends Mod c R-c k Y) to Mod R-c k X ), Proosition imlies that the functor Hom kxsa f 1 ); F) is exact on Mod c R-c k Y). PROPOSITION The family of quasi-injective sheaves is f!! -in- sends quasi-injective objects to quasi-injective jective. The functor f!! objects.

21 Sheaves on Subanalytic Sites 187 PROOF. i) Let 0! F 0! F! F 00! 0 be an exact sequence in Mod k Xsa ) and assume that F 0 2J Xsa. We have to check that the sequence 0! f!! F 0! f!! F! f!! F 00! 0 is exact. Since F 0 2J Xsa, we have G K F 0 2J Xsa. Moreover J Xsa is injective with resect to G K and f. This imlies that the sequence 0! f G K F 0! f G K F! f G K F 00! 0 is exact. Alying the exact functor lim K 0! lim K we find that the sequence f G K F 0! lim f G K F! lim f G K F 00! 0 K K is exact. ii) Let K be a comact subanalytic subset of X. The functors G K and f send quasi-injective objects to quasi-injective objects, then f G K F 2J Ysa. Since J Ysa is stable by filtrant lim, the result follows. Let S be a closed subanalytic subset of X and let i S : S,! X be the closed embedding. Let F ˆ lim r i F i 2 Mod k Xsa ) with F i 2 Mod c R-c k X). We have F S ' lim r F is ' lim r i S i 1 S F i ' i S i 1 S F. i i LEMMA Let S be a closed subanalytic subset of X and let U 2 O c X sa ). Let F 2 Mod k Xsa ). Then G U; F S ) ' lim G V; F), with VS\U V 2 O c X sa ). PROOF. Let F 2 Mod k Xsa ). Then F ' lim r i F i with F i 2 Mod c R-c k X). We have the chain of isomorhisms G U; F S ) ' lim i G U; F is ) lim G V; F i;vs\u i ) ' lim G V; F); VS\U where V ranges through the family of relatively comact oen subanalytic subsets of X containing S \ U. The second isomorhism follows since F i is R-constructible for each i. REMARK The fact that F i 2 Mod R-c k X ) is locally constant on a subanalytic stratification of X lays an essential role. In fact the iso-

22 188 Luca Prelli morhism is not true in general in Mod k X ), since the family of relatively comact oen subanalytic subsets of X containing S \ U is not cofinal to the family of oen subsets of X containing S \ U. PROPOSITION Let S be a closed subanalytic subset of X and let F 2 Mod k Xsa ) be quasi-injective. Then F S is quasi-injective. PROOF. Let U; V 2 O c X sa ) with V U. Since F is quasi-injective and inductive limits are right exact, the morhism lim G U 0 ; F)! U 0 S\U lim G V 0 ; F) with V 0 ; U 0 2 O c X sa ), is surjective. Hence by Lemma V 0 S\V the morhism G U; F S )! G V; F S ) is surjective and the result follows. Recall that F 2 Mod k X ) is c-soft if the natural morhism G X; F)! G K; F) is surjective for each comact K X. IfF is c-soft and Z is a locally closed subset of X, then F Z is c-soft. Moreover c-soft sheaves are G U; )-injective for each U 2 O X). PROPOSITION is c-soft. Let F 2 Mod k Xsa ) be quasi-injective. Then r 1 F PROOF. Let K be a comact subset of X. Recall that if U 2 O X) then G U; r 1 F) ' lim G V; F), where V 2 O X VU sa ). We have the chain of isomorhisms G K; r 1 F) ' lim U G U; r 1 F) ' lim U lim VU G V; F) ' lim U G U; F) where U ranges through the family of subanalytic relatively comact oen subsets of X containing K and V 2 O X sa ). Since F is quasi-injective and filtrant inductive limits are exact, the morhism G X; r 1 F) ' G X; F)! lim G U; F) ' G K; r 1 F), where U U ranges through the family of subanalytic oen subsets of X containing K,is surjective.

23 Sheaves on Subanalytic Sites 189 Let us consider the following subcategory of Mod k Xsa ): P Xsa :ˆfG 2 Mod k Xsa ); G is Hom kxsa ; F)-acyclic for each F 2J Xsa g: This category is generating, in fact if fg j g j are R-constructible sheaves j r G j 2P Xsa by Theorem Moreover P Xsa is stable by K, where K 2 Mod R-c k X ). In fact if G 2P Xsa and F 2J Xsa we have Hom kxsa G K; F) ' Hom kxsa G; Hom K; F)) and Hom K; F) 2J Xsa by Proosition THEOREM The category P o X sa ; J Xsa ) is injective with resect to the functor Hom kxsa ; ). PROOF. i) Let G 2P Xsa and consider an exact sequence 0! F 0!! F! F 00! 0 with F 0 2J Xsa. We have to rove that the sequence 0! Hom kxsa G; F 0 )! Hom kxsa G; F)! Hom kxsa G; F 00 )! 0 is exact. Since the functor Hom kxsa G; ) is acyclic on quasi-injective sheaves we obtain the result. ii) Let F 2J Xsa, and let 0! G 0! G! G 00! 0 be an exact sequence on P Xsa. Since the objects of P Xsa are Hom kxsa ; F)-acyclic the sequence 0! Hom kxsa G 00 ; F)! Hom kxsa G; F)! Hom kxsa G 0 ; F)! 0 is exact. COROLLARY to the functor Hom ; ). The category P o X sa ; J Xsa ) is injective with resect PROOF. Let G 2P Xsa, and let 0! F 0! F! F 00! 0 be an exact sequence with F 0 2J Xsa. We shall show that for each U 2 O X sa ) the sequence 0! G U; Hom G; F 0 ))! G U; Hom G; F))! G U; Hom G; F 00 ))! 0 is exact. This is equivalent to show that for each U 2 O X sa ) the sequence 0! Hom kxsa G U ; F 0 )! Hom kxsa G U ; F)! Hom kxsa G U ; F 00 )! 0 is exact. This follows since G U 2P Xsa. The roof of the exactness in P o X sa is similar.

24 190 Luca Prelli 1.6 ± The functor r!. We have seen in x 1.1 that the functor r 1 : Mod k Xsa )! Mod k X ) has a left adjoint r! : Mod k X )! Mod k Xsa ). The functor r! is fully faithful and exact. In articular, for U 2 O X) one has r! k U ' lim r k V, where V 2 O X sa ). VU PROPOSITION Let S be a closed subset of X. Then r! k S ' lim r k W, where W 2 O X sa ). WS PROOF. i) Let U ˆ X n S. Since r! is exact we have an exact sequence 0! r! k U! r! k X! r! k S! 0: On the other hand, let V 2 O c X sa ) and V U. We have an exact sequence 0! k V! k X! k XnV! 0: Since r is exact on Mod R-c k X ) the sequence 0! r k V! r k X! r k XnV! 0 is exact. Alying the exact lim we obtain an exact sequence 0! lim r k V! r k X! lim r k XnV! 0: VU VU We have lim r k V ' r! k U and r k X ' r! k X. Hence r! k S ' lim r k XnV : 1:3) VU VU ii) We shall show that for each U 0 2 O c X sa ) the natural morhism lim VU G U 0 ; k XnV )! lim WS G U 0 ; k W ) VU is an isomorhism. We shall see that for each W 2 O X sa ) with W S there exists W 0 2 O X sa ) such that X n W 0 U and W \ U 0 ˆ W 0 \ U 0. Set W 0 ˆ W [ X n U 0 ). Since U 0 is relatively comact, X n W 0 U, and W \ U 0 ˆ W 0 \ U 0 by construction. Then lim G U 0 ; k XnV ) ' VU lim XnW)U G U 0 ; k W ) ' lim WS G U 0 ; k W ): NOTATIONS Let Z ˆ U \ S, where U 2 O X) and let S be a closed subset of X. LetF 2 Mod k Xsa ). We set for short Z F ˆ F r! k Z LEMMA subset of X. Let F 2 Mod k Xsa ).LetU2 O X) and let S be a closed

25 Sheaves on Subanalytic Sites 191 i) One has U F ' lim F V ' lim G V F, V 2 O c X sa ). VU VU ii) One has S F ' lim F W ' lim G WS W F, W 2 O X sa ). WS PROOF. i) The first isomorhism is obvious. Let us show the second isomorhism. We have the chain of isomorhisms lim F V VU lim G V 0F) V! V;V 0 U lim G V 0F; V 0 U where V; V 0 range through the family of subanalytic oen subsets of X. The roof of ii) is similar. PROPOSITION Let Z be a locally closed subset of X. Let G 2 Mod R-c k X ) and F 2 Mod k Xsa ). Then Z Hom G; F ) 'Hom G; Z F). PROOF. i) Let U 2 O X). We have the chain of isomorhisms UHom G; F) ' lim G V Hom G; F) VU ' lim Hom G; G V F) VU 'Hom G; lim G V F) VU 'Hom G; U F); where V 2 O X sa ). The third isomorhism follows from Proosition ii) If S is a closed subset of X the roof is similar. PROPOSITION Let F 2 Mod k Xsa ) be quasi-injective. Then r! K F is quasi-injective for each K 2 Mod k X ). PROOF. i) Let us show the result when K ˆ k Z, for a locally closed subset Z of X. Let G 2 Mod c R-c k X). We have Hom kxsa G; Z F) ' G X; Hom G; Z F)) ' G X; Z Hom G; F)) ' G X; r 1 ZHom G; F)) ' G X; r 1 Hom G; F)) Z ): Since F is quasi-injective, Hom G; F) is quasi-injective. Then by Pro-

26 192 Luca Prelli osition the sheaf r 1 Hom G; F)) Z is c-soft and it is injective with resect to the functor G X; ). Hence the functor G X; r 1 Hom ; F) Z )is exact on Mod c R-c k X). ii) Let K 2 Mod k X ). There exists an eimorhism i2i k Ui! K with U i 2 O X sa ) for each i. Let K J be the image of i2j k Ui, with J I finite. We have K ' lim J K J, hence r! K ' lim J r! K J since r! commutes with lim.it is enough to rove the result for K J. We argue by induction on the cardinal of J. Set K ˆ K J.IfjJj ˆ1 then K ' k Z with Z locally closed subset of X and the result follows from i). Let us show n 1 ) n. There is an eimorhism n iˆ1 k U i! K. Let K 1 be the image of k U1! K and let K 2 ˆ K=K 1. We have a commutative diagram where the vertical arrows are surjective, and the rows are exact. By the exactness of r! and we obtain the exact sequence 0! r! K 1 F!! r! K F! r! K 2 F! 0. By the inductive hyothesis r! K 1 F and r! K 2 F are quasi-injective, then r! K F is quasi-injective. PROPOSITION Let F 2 Mod k Xsa ), G 2 Mod R-c k X ) and let K 2 Mod k X ). One has the isomorhism Hom G; F )r! K 'Hom G; F r! K). PROOF. Both sides are left exact with resect to F. Hence we may assume that F is quasi-injective. Since quasi-injective sheaves are Hom G; )-injective, both sides are exact with resect to K. Moreover as a consequence of Proosition both sides commute with filtrant lim with resect to K. We may reduce to the case K ˆ k U, with U 2 O X sa ). Then the result follows from Proosition Derived category. As usual, we denote D k Xsa ) the derived category of Mod k Xsa ) and its full subcategory consisting of bounded res. bounded below, res. bounded above) comlexes is denoted by D b k Xsa ) res. D k Xsa ), res. D k Xsa )).

27 Sheaves on Subanalytic Sites ± The category D b R-c k X sa ). As usual we denote by D b R-c k X) res. D b R-c k X sa )) the full subcategory of D b k X ) res. D b k Xsa )) consisting of objects with R-constructible cohomology. Recall that r : X! X sa is the natural morhism of sites. It induces the functor r : Mod k X )! Mod k Xsa ). LEMMA Let F 2 Mod R-c k X ). Then R j r F ˆ 0 for each j 6ˆ 0. PROOF. The sheaf R j r F is the sheaf associated to the resheaf V! R j G V; F). We have to show that R j G V; F) ˆ 0 for j 6ˆ 0 on a family of generators of the toology of X sa. This means that for each V 2 O c X sa ) and for each j 6ˆ 0, there exists I finite and fv i g i2i 2 Cov V sa ) such that R j G V i ; Rr F) ' R j G V i ; F) ˆ 0. We use the notation of [5]. There exists a locally finite stratification fx i g i2i of X consisting of subanalytic subsets such that for all j 2 Z and all i 2 I the sheaf Fj Xi is locally constant. By the triangulation theorem there exist a simlicial comlex S; D) and a subanalytic homeomorhism c : jsj! X comatible with the stratification and such that V is a finite union of the images by c of oen subsets V s) ofjsj, where V s) ˆ St2D;ts jtj. By Proosition of [5]we have R j G c V s)); F) ˆ 0 for each s and for each j 6ˆ 0. The result follows because V ˆ Sc jsj)v c V s)). Since R-constructible sheaves are injective with resect to the functor r, the following diagram of derived categories is quasi-commutative. 2:1) THEOREM One has the equivalence of categories D b R-c k X) ' D b Mod R-c k X )) ' D b R-c k X sa ):

28 194 Luca Prelli PROOF. BydeÂvissage, to rove the equivalence between D b Mod R-c k X )) and D b R-c k X sa ) it is enough to check that the functor r in 2.1) is fully faithful. We have r 1 r ' id and the result follows. The equivalence between D b Mod R-c k X )) and D b R-c k X) was shown by Kashiwara in [4]. 2.2 ± Oerations in the derived category. Let us study the oerations in the derived category of Mod k Xsa ). Let f : X! Y be an analytic ma. We will use quasi-injective objects and the results of x 1.5 to derive the formulas of Section 1. Since Mod k Xsa ) has enough injectives, then the derived functors are well defined. PROPOSITION RHom : D k Xsa ) o D k Xsa )! D k Xsa ); Rf : D k Xsa )! D k Ysa ); Rf!! : D k Xsa )! D k Ysa ); Let f : X! Y be an analytic ma. Then i) The functors Rf and RHom commute with Rr. ii) The functors Rf and Rf!! commute with r 1. iii) We have R g f ) ' Rg Rf and R g f )!! ' Rg!! Rf!! : iv) The functor R k f!! : Mod k Xsa )! Mod k Ysa ) commutes with filtrant inductive limits for each k 2 Z. v) If F 2 D k Xsa ) and f is roer on su F), then Rf!! ' Rf. PROOF. i) The functor r sends injective sheaves to injective sheaves, then Rf and RHom commute with Rr. ii) Since r 1 has an exact left adjoint it sends injective sheaves to injective sheaves. Then Rf and Rf!! commute with r 1. iii) The functor f res. f!! ) sends injective sheaves to injective res. quasi-injective) sheaves. Then R g f ) ' Rg Rf and R g f )!! ' ' Rg!! Rf!! : iv) Quasi-injective objects of Mod k Xsa ) are stable by filtrant lim, and the functor f!! commutes with such limits. Then R k f!! commutes with filtrant lim for each k 2 Z.

29 Sheaves on Subanalytic Sites 195 v) We can find a reresentative F 0 of F in K J Xsa ) with f roer on su F 0 ). Then the result follows from the non derived case. PROPOSITION Let F ˆ lim F i with F i 2 Mod k Xsa ) and let i G 2 D b R-c k X). One has R k Hom G; F) ' lim R k Hom G; F i ) for each k 2 Z. i PROOF. There exists see [8], Corollary 9.6.7) an inductive system of injective resolutions Ii of F i. Then lim I i i is a comlex of quasi-injective objects quasi-isomorhic to F. Each object of Mod R-c k X ) o ; J Xsa ) is Hom ; )-acyclic. Proosition imlies the isomorhism Hom G; lim I i i ) ' lim Hom G; Ii ) i and the result follows. PROPOSITION Let F 2D k Xsa ), G2D b R-c k X) and let K 2D k X ). One has the isomorhism RHom G; F ) r! K ' RHom G; F r! K). PROOF. Let I be a quasi-injective resolution of F. By Proosition we have that I r! K is a comlex of quasi-injective objects. Each object of Mod R-c k X ) o ; J Xsa )ishom ; )-acyclic. Hence we are reduced to rove the isomorhism Hom G; I ) r! K 'Hom G; I r! K). The result follows from Proosition PROPOSITION Let U 2 O c X sa ). Let F 2 Mod k Xsa ) be quasiinjective. Then F U is G V; )-acyclic for each V 2 O X sa ). PROOF. Since F U has comact suort, we may suose that V is relatively comact. Let S ˆ X n U. Since F is quasi-injective and filtrant lim are exact, the morhism G V; F)! lim G W; F)! G V; F WS\V S ) is surjective. Consider the exact sequence 0! F U! F! F S! 0. We get the exact sequence 0! G V; F U )! G V; F)! G V; F S )! 0: By Proosition F and F S are quasi-injective, hence G V; )-acyclic. This imlies that F U is G V; )-acyclic. COROLLARY Let f : X! Y be a real analytic ma and

30 196 Luca Prelli let U 2 O c X sa ) Let F 2 Mod k Xsa ) be quasi-injective. Then F U is f!! -acyclic. PROOF. Since F U has comact suort, Rf!! F U ' Rf F U. The result follows because F U is G f 1 V); )-acyclic for each V 2 O Y sa ). LEMMA Let F be quasi-injective object of Mod k Xsa ) and let G 2 Mod c R-c k X). Then F r Gisf!! -acyclic. PROOF. Let G 2 Mod c R-c k X). Then G has a resolution 0! i1 2I 1 k Ui1!...! in 2I n k Uin! W G! 0; where I j is finite and U ij 2 O c X sa ) for each i j 2 I j, j 2f1;...; ng. Let us argue by induction on the length n of the resolution. If n ˆ 1, then G is isomorhic to a finite sum i k Ui, with U i 2 O c X sa ), and the result follows from Corollary Let us show n 1 ) n. The sequence 0! ker W! in 2I n k Uin! G! 0 is exact. The sheaf ker W belongs to Mod c R-c k X) and it has a resolution of length n 1. Alying F r ) we get the exact sequence 0! F r ker W! in 2I n F Uin! F r G! 0: By the induction hyothesis F r ker W is f!! -acyclic. Moreover in 2I n F Uin is f!! -acyclic, then F r G is f!! -acyclic. PROPOSITION Let F be quasi-injective object of Mod k Xsa ) and let G 2 Mod k Xsa ). Then F G is f!! -acyclic. PROOF. Let G ' lim r i G i with G i 2 Mod c R-c k X) for each i. Since the functors and R k f!! commute with filtrant lim we have R k f!! F lim r G i ) ' lim R k f!! F r G i ) ˆ 0 i i if k 6ˆ 0 by Lemma PROPOSITION Let F 2 D k Xsa ) and G 2 D k Ysa ). Then Rf!! F G ' Rf!! F f 1 G):

31 Sheaves on Subanalytic Sites 197 PROOF. First assume that F 2 Mod k Xsa ) is injective. By Proosition F f 1 G is f!! -acyclic. Now let F 2 D k Xsa ) and G 2 D k Ysa ). Let F 0 be a comlex of injective sheaves quasi-isomorhic to F. Then Rf!! F G ' f!! F 0 G ' f!! F 0 f 1 G) ' Rf!! F f 1 G); where the second isomorhism follows from Proosition Now let us consider a cartesian square PROPOSITION Let F 2 D k Xsa ). Then g 1 Rf!! F ' Rf 0!! g0 1 F. PROOF. We have an isomorhism f!! 0g0 1 ' g 1 f!!, and R g 1 f!! ) ' ' g 1 Rf!! since g 1 is exact. Then we obtain a morhism g 1 Rf!!! Rf!! 0g0 1.Itisenoughtorovethatforanyk2Zand for any F 2 Mod k Xsa )wehave g 1 R k f!! F! R k f!! 0g0 1 F. Since both sides commute with filtrant lim,wemayassumefˆr G with G 2 Mod c R-c k X). Moreover since su G) iscomact,f 0 is roer on su g 0 1 G). Then both sides commute with r and the result follows from the corresonding one for classical sheaves. As in classical sheaf theory, the KuÈnneth formula follows from the rojection formula and the base change formula. PROPOSITION Consider a cartesian square where d ˆ fg 0 ˆ gf 0. There is a natural isomorhism for F 2 D k Xsa ) and G 2 D k Y 0 sa ). Rd!! g 0 1 F f 0 1 G) ' Rf!! F Rg!! G

32 198 Luca Prelli PROOF. Using the rojection formula and the base change formula we deduce Rf!! 0 g0 1 F f 0 1 G) ' Rf!! 0 g0 1 F G ' g 1 Rf!! F G: Using the rojection formula once again we find Rg!! Rf!! 0 g0 1 F f 0 1 G) ' Rg!! g 1 Rf!! F G) ' Rf!! F Rg!! G and the result follows since Rd!! ' Rg!! Rf!! 0. The following isomorhism is the analogue for subanalytic sheaves of Lemma of [7] PROPOSITION natural morhism Let G 2 D b R-c k Y ) and let F 2 D k Xsa ). Then the is an isomorhism. Rf!! RHom f 1 G; F )! RHom G; Rf!! F) PROOF. The morhism is obtained as in the non derived case. Let us show that it is an isomorhism. Let F 0 be a comlex of injective sheaves quasi-isomorhic to F. Then Rf!! Hom f 1 G; F) ' f!! Hom f 1 G; F 0 ) 'Hom G; f!! F 0 ) ' RHom G; Rf!! F); where the second isomorhism follows from Proosition ± Vanishing theorems on Mod k Xsa ). In x 2.3 we rove some results on the vanishing of the cohomology of sheaves on a subanalytic site. DEFINITION The quasi-injective dimension of the category Mod k Xsa ) is the smallest n 2 N [f1g such that for any F 2 Mod k Xsa ) there exists an exact sequence with I j quasi-injective for 0 j n. 0! F! I 0!!I n! 0

33 Sheaves on Subanalytic Sites 199 PROPOSITION dimension. The category Mod k Xsa ) has finite quasi-injective PROOF. Let F 2 Mod k Xsa ). Then F ˆ lim r i F i, with F i 2 Mod c R-c k X). There exists see [8], Corollary 9.6.7) an inductive system of injective resolutions Ii of F i. By Proosition of [5], the category Mod k X ) has finite homological dimension. Then we may assume that Ii has length N 0 < 1 for each i. SinceF i is r -injective for each i, r Ii is an injective resolution of r F i of length N 0. Taking the inductive limit we find that lim r I i i is a resolution of F of length N 0,andlimr I j i i 2J X sa for each j. COROLLARY Let f : X! Y be a real analytic ma, and let F 2 Mod R-c k X ). The functors f, f!! and Hom F; ) have finite cohomological dimension. PROPOSITION Let F 2 Mod k X ) and let G 2 Mod k Xsa ). There exists a finite j 0 2 N such that R j Hom r! F; G) ˆ 0 for j > j 0 : PROOF. Let U 2 O X sa ). We have the chain of isomorhisms RG U; RHom r! F; G)) ' RHom kxsa r! F; RG U G) ' RHom kx F; r 1 RG U G): The functor G U has finite cohomological dimension, and the homological dimension of the category Mod k X ) is finite. Hence we can find a finite j 0 2 N such that R j G U; RHom r! F; G)) vanishes for j > j 0 and for each U 2 O X sa ). This shows the result. REMARK We have seen that the functor Hom F; ) has finite cohomological dimension when F is R-constructible and when F ˆ r! G with G 2 Mod k X ). We do not know if the cohomological dimension is finite for any F 2 Mod k Xsa ). Indeed we do not know if the homological dimension of Mod k Xsa ) is finite or not.