Sheaves on subanalytic sites

Sheaves on Subanalytic Sites

LUCAPRELLI(*)

ABSTRACT- In [7]the authors introduced the notion of ind-sheaves and defined the six Grothendieck operations in this framework. As a byproduct, they obtained subanalytic sheaves and the six Grothendieck operations on them.

The aim of this paper is to give a direct construction of the six Grothendieck operations in the framework of subanalytic sites avoiding the heavy machinery of ind-sheaves. As an application, we show how to recover the subanalytic sheavesO^tandO^wof temperate and Whitney holomorphic functions respec- tively.

Introduction.

LetXbe a real analytic manifold andka field. The spaces of functions which are not defined by local properties, such as tempered distributions, tempered and WhitneyC¹ functions, etc., are very useful in the study of systems of linear partial differential equations (Laplace transform, tem- pered holomorphic solutions ofD-modules etc.). Although these spaces do not define sheaves onX, they define sheaves on a site associated toX, the subanalytic site Xsa, where one just considers open subanalytic sets and locally finite coverings.

In [7], Kashiwara and Schapira, motivated by the construction of the microlocalization functor, treated a more general theory, namely that of ind-sheaves. They defined the category I(kX) of ind-sheaves onX as the category of ind-objects of the category Mod^c(k_X) of sheaves with compact support and they developped the six Grothendieck operations in this fra- mework. When restricting to R-constructible sheaves, they showed the equivalence between the category IR-c(kX)ˆInd(Mod^c_R-c(kX)) of ind-R- constructible sheaves onX and the category Mod(kXsa) of sheaves on the subanalytic site associated toX.

(*) Indirizzo dell'A.: Dipartimento di Matematica Pura ed Applicata, Via Trieste, 63, 35121 Padova.

In this way, tempered distributions, tempered and Whitney C¹ func- tions, etc., are obtained as a byproduct 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 paper is to give a direct, self-contained and elementary construction of the six Grothendieck operations on Mod(kXsa), without using the more sophisticated and much more difficult theory of ind- sheaves. Indeed, contrary to the category I(k_X), the category Mod(k_Xsa) is a Grothendieck category.

We will start by recalling some results of [7], the definition of a sub- analytic site, the natural functor of sitesr:X!Xsa, and the functorsr, r ¹ andr_! relating the categories of ``classical'' and subanalytic sheaves.

We also recall a very useful description of subanalytic sheaves as inductive limits ofR-constructible sheaves.

Then we go into the study of subanalytic sheaves, without using the notion of ind-sheaf. Letf :X!Y be a morphism of real analytic mani- folds. The functorsHom,,fand f ¹are well defined sinceXsais a site.

We introduce the proper direct image functorf_!!and we study the relations between the above operations and the functorsr, r ¹ andr_!. In the de- rived category we obtain an exceptional inverse image, denoted byf^!, right adjoint to Rf!!. This is obtained via Brown representability 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 re- spect to the above functors and we prove that the quasi-injective dimension of Mod(k_Xsa) is finite. We end this work giving some examples of sub- analytic sheaves.

In more details, the contents of this paper are as follows.

In Section 1 we construct the operations in Mod(k_Xsa). Inx1.1, we start by recalling the definitions of the functors r, r ¹ andr_! of [7]and their properties (a more detailed study of the functorr_!is done inx1.6). Inx1.2, we recall the internal operations and the functors of direct and inverse image (which are well defined on any site) and their relations withr, r ¹ andr_!. We also define (inx1.4) the proper direct image functorf_!!, where the notation f_!! follows from the fact that f_!!r6'rf_! in general. We study its properties and the relations with the other operations. While the functorsf ¹andare exact, the functorsHom,fandf!!are left exact, and we introduce the subcategory of quasi-injective objects which is injective with respect to these functors. This is done inx1.5.

In Section 2 we consider the derived category of Mod(kXsa). Inx2.1, we consider the subcategoryD^b_R-c(kXsa) consisting of bounded complexes with R-constructible cohomology and we prove the equivalence of derived ca- tegoriesD^b_R-c(k_X)'D^b_R-c(k_Xsa). Then, inx2.2, we study the derived func- tors ofHom,fandf_!!and we obtain the usual formulas (projection formula, base change formula, KuÈnneth formula, etc.) in the framework of sub- analytic sites. We also prove (inx2.3) some vanishing theorems for sub- analytic sheaves, in particular we prove that the quasi-injective dimension of Mod(k_Xsa) is finite. Using the Brown representability theorem we prove the existence of a right adjoint to the functor Rf_!!, denoted byf^!. This is done in x2.4. We calculate the functor f^! by decomposing f as the com- posite of a closed embedding and a submersion.

In Section 3 we give some examples of subanalytic sheaves. We start by recalling the definition of sheaves ofr_!D_X-modules, whereD_X denotes the sheaf of finite order differential operators on a complex analytic manifold X. Then inx3.3 we show how to recover the sheaves ofr_!D_X-modulesO_X^t and O^w_X of temperate and Whitney holomorphic functions of [7]respec- tively. We prove 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 Schapira who encouraged us to develop a theory of subanalytic sheaves independent 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. Re- ferences are made to [8]and [14]for an introduction to sheaves on Gro- thendieck topologies, to [5]for a complete exposition on classical sheaves and R-constructible sheaves and to [1]and [10]for the theory of sub- analytic sets (see also the Appendix for a short overview on some prop- erties of subanalytic subsets).

1.1 ±The subanalytic site. Notations and review.

We introduce the subanalytic site. The results of x1.1 have already been proved in [7], for sake of completeness we reproduce here the proofs.

Denote by Op(Xsa) the category of subanalytic subsets ofX. One en- dows Op(X_sa) with the following topology: SOp(X_sa) is a covering of

U2Op(Xsa) if for any compact subsetKofXthere exists a finite subsetS0

ofSsuch thatK\S

V2S0V ˆK\U. We will callXsathe subanalytic site, and forU2Op(Xsa) we denote byU_Xsa the category Op(Xsa)\Uwith the topology induced byX_sa. There is a natural morphism of sitesU_sa!U_Xsa: REMARK1.1.1. We use the notation UXsa to stress the difference from Usa, the subanalytic site associated to U. For example, let XˆR² and UˆR²n f0g. Let Vn ˆ fx2R²; jxj>_n¹g. ThenfVng_n2N2Cov(Usa)but fV_ng_n2N2=Cov(U_Xsa).

Let Mod(k_Xsa) denote the category of sheaves onX_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 particular as a Grothendieck category, Mod(k_Xsa) has enough injective objects.

REMARK1.1.2. Denote byOp^c(X_sa)the category of relatively compact subanalytic open subsets of X. One denotes by X^c_sa the categoryOp^c(Xsa) with the topology induced by Xsa. The forgetful functor gives an equiva- lence of categoriesMod(k_Xsa)! Mod(k_Xsa^c).

PROPOSITION 1.1.3. Let fF_ig_i2I be a filtrant inductive system in Mod(kXsa)and letU 2Op^c(Xsa). Then

lim!_i G(U;Fi)! G(U;lim!_i Fi):

PROOF. By Remark 1.1.2 it is enough to prove the assertion in the category Mod(kX_sa^c). Denote by ``lim''!<sub>i</sub> F<sub>i</sub>the presheafV7!lim!<sub>i</sub> G(V;F<sub>i</sub>) on X<sub>sa</sub><sup>c</sup> . LetU2Op<sup>c</sup>(X<sub>sa</sub>) and letSbe a finite covering ofU. Since lim!<sub>i</sub> com- mutes with finite projective limits we obtain the isomorphism ( ``lim''!

i

F_i)(S)! lim!

i

F_i(S) andF_i(U)! F_i(S) sinceF_i2Mod(k_X^c_sa) for each i. Moreover the family of finite coverings ofUis cofinal in Cov(U). Hence

``lim''!<sub>i</sub> F<sub>i</sub>! ( ``lim''!_i F_i)^‡. Applying once again the functor ()^‡we get

``lim''!

i

F_i'( ``lim''!

i

F_i)^‡'( ``lim''!

i

F_i)^‡‡'lim!

i

F_i:

Hence applying the functor G(U;) we obtain the isomorphism lim!_i G(U;F_i)! G(U;lim!_i F_i) for eachU2Op^c(Xsa): p

There is an easy way to construct sheaves on a subanalytic site PROPOSITION1.1.4. LetF be a presheaf onX_sa^c and assume that

(i) F(6)ˆ0;

(ii) For any U;V 2Op^c(Xsa) the sequence 0!F(U[V)!

!F(U)F(V)!F(U\V)is exact.

Then F2Mod(kX^c_sa)'Mod(kXsa).

PROOF. LetU2Op^c(Xsa) and letfU_jgⁿ_jˆ1be a finite covering ofU. Set for shortU_ijˆU_i\U_j. We have to show the exactness of the sequence

0!F(U)! _1knF(U_k)! _1i<jnF(U_ij);

where the second morphism sends (s_k)_1kn to (t_ij)_1i<jn by t_ijˆ

ˆs_ij_Uij s_jj_Uij. We shall argue by induction on n. Fornˆ1 the result is trivial, andnˆ2 is the hypothesis. Suppose that the assertion is true for jn 1 and setU⁰ˆS

1k<nUk. By the induction hypothesis the follow- ing commutative diagram is exact

Then the result follows. p

Let ModR-c(kX) be the abelian category ofR-constructible sheaves on X, and consider its subcategory Mod^c_R-c(kX) consisting of sheaves whose support is compact.

We denote by

r:X!X_sa

the natural morphism of sites. We have functors Mod^c_R-c(kX)ModR-c(kX)Mod(kX) ^r!

r ¹ Mod(kXsa):

(1:1)

We will still denote by r the restriction of r to Mod_R-c(k_X) and Mod^c_R-c(k_X).

REMARK 1.1.5. By Proposition 1.1.3 for each F2Mod(kX) and V 2Op^c(Xsa)one has

G(V;rF)'lim!_U G(V;rF_U)'G(V;lim!_U rF_U);

(1:2)

where U ranges through the family of relatively compact open subanalytic subsets of X. The first isomorphism follows since V is relatively compact and G(V;rFU)'G(V;rF) if VU. The isomorphism (1.2) implies lim!_U rFU ! rF.

REMARK 1.1.6. The functor r does not commute with filtrant in- ductive limits. For example consider the familyfVng_n2Nof Remark1.1.1.

We have rlim!_n k_Vn 'rk_R²_nf0g, while for each U2Op^c(R²_sa)with02@U we haveG(U;lim!_n rk_Vn)'lim!_n G(U;rk_Vn)ˆ0.

PROPOSITION 1.1.7. Let U be an open subanalytic subset of X and consider the constant sheafkU_Xsa 2Mod(kXsa). We havekU_Xsa 'rkU.

PROOF. LetFbe the presheaf onX_sa defined byF(V)ˆkifV U, F(V)ˆ0 otherwise. This is a separated presheaf andk_UXsa ˆF^‡‡. More- over there is an injective arrow F(V),!rkU(V) for each V 2Op(Xsa).

HenceF^‡‡,!rkU since the functor ()^‡‡is exact. LetT be the family of W2Op(Xsa) connected and such thatW does not contain any connected component of U. ThenT forms a basis for the topology ofX_sa since the connected components of U are locally finite. For each W2 T we have F(W)'rkU(W)'k if WU and F(W)ˆ0 otherwise. Then

F^‡‡'rkU. p

PROPOSITION1.1.8. The restriction ofr toMod_R-c(kX)is exact.

PROOF. (i) Let us consider an epimorphism G!!F in Mod^c_R-c(k_X), we have to prove that c:rG!rF is an epimorphism. Let

U2Op^c(Xsa) and let 06ˆs2G(U;rF)'HomkX(kU;F). Set G⁰ˆGFkU. Then G⁰2Mod^c_R-c(kX) and moreover G⁰!!kU. There exists a finitefU_ig_i2IOp^c(Xsa) withU_iconnected for eachisuch that _ik_Ui !!G⁰. The composition k_Ui!G⁰!k_U is given by the multi- plication by a_i2k. Set I₀ˆ fk_Ui; a_i6ˆ0g, we may assume a_iˆ1. We get a diagram

The composition kUi!G⁰!G defines ti2HomkX(kUi;G)ˆG(Ui;rG).

Hence for eachs2G(U;rF) there exists a finite coveringfU_igofUand t_i2G(U_i;rG) such thatc(t_i)ˆsj_Ui. This means thatcis surjective.

(ii) Let F2ModR-c(kX). By Remark 1.1.5 rF'lim!_U rFU, where U ranges through the family Op^c(Xsa). The result follows sinceris exact on Mod^c_R-c(k_X) and filtrant lim!are exact. p PROPOSITION1.1.9. The restriction ofrtoMod_R-c(k_X)is fully faithful.

PROOF. It is enough to prove that r ¹rF'F for each F2Mod_R-c(k_X). Since both functors are exact on Mod_R-c(k_X) we may reduce to the case Fˆk_U with U2Op(X_sa) and the result follows from

Proposition 1.1.7. p

NOTATIONS1.1.10. Since the functorr is fully faithful and exact on the category ModR-c(kX), we can identify ModR-c(kX) with its image in Mod(kXsa). When there is no risk of confusion we will write F instead of rF, forF 2Mod_R-c(kX).

The following theorem gives a fundamental characterization of sub- analytic sheaves and it will be used systematically in the following Sections.

THEOREM 1.1.11. (i) Let G2Mod^c_R-c(k_X) and let fF_ig be a filtrant inductive system inMod(k_Xsa). Then we have an isomorphism

lim!

i

Hom_kXsa(rG;F_i)! Hom_kXsa(rG;lim!

i

F_i):

(ii) Let F2Mod(kXsa). There exists a small filtrant inductive system fFig_i2IinMod^c_R-c(kX)such that F 'lim!_i rFi.

PROOF. (i) There exists an exact sequence G₁!G₀!G!0 with G₁;G₀finite direct sums of constant sheavesk_UwithU2Op^c(X_sa). Sincer is exact on ModR-c(kX) and commutes with finite sums, by Proposition 1.1.7 we are reduced to prove the isomorphism lim!_i G(U;Fi)! G(U;lim!_i Fi).

Then the result follows from Proposition 1.1.3.

(ii) LetF2Mod(k_Xsa), and define

I0:ˆf(U;s); U2Op^c(Xsa); s2G(U;F)g G0:ˆ _(U;s)2I0rk_U

The morphism rkU !F, where the section 12G(U;kU) is sent to s2G(U;F) defines un epimorphismW:G0!F. ReplacingFbykerWwe construct a sheaf G₁ˆ _(V;t)2I1rk_V and an epimorphism G₁!!kerW.

Hence we get an exact sequenceG₁!G₀!F!0. ForJ₀I₀ set for shortGJ0 ˆ (U;s)2J0rkU and define similarlyGJ1. Set

Jˆ f(J1;J0); J_k I_k; J_k is finite and imWj_GJ

1 GJ0g:

The categoryJis filtrant andF'_(Jlim!

1;J0)2J

coker(G_J1!G_J0). p PROPOSITION1.1.12. LetF 2Mod(kXsa), and letU2Op(X). Then

G(U;r ¹F)' lim

VU;V2Opc(Xsa)

G(V;F)

PROOF. By Theorem 1.1.11 we may assume Fˆlim!_i rF_i, with F_i2Mod^c_R-c(k_X). Then r ¹F'lim!_i r ¹rF_i'lim!_i F_i. We have the chain of isomorphisms

G(U;r ¹F) ' lim

VU

G(V;r ¹F) ' lim

VU

G(V;lim!_i F_i) ' lim

VU

lim!_i G(V;F_i) ' lim

VU

lim!_i G(V;F_i) ' lim

VU

lim!_i G(V;rF_i) ' lim

VU

G(V;F);

whereV 2Op^c(X_sa). The third isomorphism follows sinceVis compact and the last isomorphism follows from Proposition 1.1.3. p

PROPOSITION1.1.13. One has r ¹r! id, in particular the functor r is fully faithful.

PROOF. LetF2Mod(k_X). Everyx2Xhas a fundamental neighbor- hood system consisting of open subanalytic subsets. Hence we have the chain of isomorphisms

(r ¹rF)_x'lim_x2U!r ¹rF(U)'lim_x2U!rF(U)'lim_x2U!F(U)'F_x; whereUranges through the family of open subanalytic neighborhoods ofx.

The second isomorphism follows since, by Proposition 1.1.12 lim_x2U!r ¹rF(U)'lim_x2U!lim

VU

rF(V)'lim_x2U!rF(U):

p Now we will describe a left adjoint to the functor r ¹.

PROPOSITION1.1.14. The functor r ¹admits a left adjoint, denoted by r_!. It satisfies

(i) for F2Mod(k_X)and U2Op(X_sa),r_!F is the sheaf associated to the presheaf U7!G(U;F),

(ii) for U2Op(X)one hasr_!k_U ' lim!

VU;V2Opc(Xsa)

k_V.

PROOF. Let Fe2Psh(kXsa) be the presheaf U7!G(U;F), and let G2Mod(kXsa). We will construct morphisms

HomPsh(k_Xsa)(F;e G) !^j

w HomkX(F;r ¹G):

To define j, let W:Fe !G and U2Op(X). Then the morphism j(W)(U):F(U)! r ¹G(U) is defined as follows

F(U)' lim

VU;V2Opc(Xsa)

F(V) !^W lim

VU;V2Opc(Xsa)

G(V)' r ¹G(U):

On the other hand, let c:F! r ¹G and U2Op^c(X_sa). Then the morphismw(c)(U):F(U)e !G(U) is defined as follows

F(U)e ' lim!

UV2Opc(Xsa)

F(V) !^c lim!

UV2Opc(Xsa)

r ¹G(V)!G(U):

By construction one can check that the morphismsjandware inverse to each others. Then (i) follows from the chain of isomorphisms

Hom_kXsa(Fe^‡‡;G)'Hom_Psh(kXsa)(F;e G)'Hom_kXsa(F;r ¹G):

To show (ii), consider the following sequence of isomorphisms Hom_kXsa(r_!k_U;F) 'Hom_kX(k_U;r ¹F)

' lim

VU;V2Opc(Xsa)

Homk_Xsa(kV;F) 'Homk_Xsa( lim!

VU;V2Opc(Xsa)

kV;F);

where the second isomorphism follows from Proposition 1.1.12. p PROPOSITION1.1.15. The functor r_! is exact and commutes with lim! and.

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 Proposition 1.1.14, letF2Mod(k_X), and letFe 2Psh(k_Xsa) be the presheaf U7!G(U;F). Thenr_!F'Fe^‡‡, and the functors F7!Fe andG7!G^‡‡ are left exact.

Let us show that r_! commutes with . Let F;G2Mod(kX), the morphism

F(U)G(U)!(FG)(U) defines a morphism in Mod(k_Xsa)

r_!Fr_!G!r_!(FG)

by Proposition 1.1.14 (i). Sincer_!commutes with lim!we may suppose that FˆkU andGˆkV and the result follows from Proposition 1.1.14 (ii). p PROPOSITION1.1.16. The functorr_!is fully faithful. In particular one has r ¹r_!'id. Moreover, forF 2Mod(k_X)andG2Mod(k_Xsa)one has

r ¹Hom(r_!F; G)' Hom(F;r ¹G):

PROOF. ForF;G2Mod(k_X) we have by adjunction

HomkX(r ¹r_!F;G)'HomkX(F;r ¹rG)'HomkX(F;G):

This also implies thatr_!is fully faithful, in fact

Hom_kXsa(r_!F;r_!G)'Hom_kX(F; r ¹r_!G)'Hom_kX(F;G):

Now letK;F2Mod(k_X) andG2Mod(k_Xsa), we have

HomkX(K;r ¹Hom(r_!F;G))'Homk_Xsa(r_!K;Hom(r_!F;G)) 'Hom_kXsa(r_!Kr_!F;G) 'Hom_kXsa(r_!(KF);G) 'Hom_kX(KF;r ¹G) 'Hom_kX(K;Hom(F;r ¹G))

and the result follows. p

1.2 ±Operations on the subanalytic site.

Let X;Y be two real analytic manifolds, and letf :X!Y be a real analytic map. This defines a morphism of sites f :Xsa!Ysa. We have a diagram

The following functors are always well defined on a site Hom :Mod(kXsa)^opMod(kXsa)!Mod(kXsa);

:Mod(k^op_Xsa)Mod(k_Xsa)!Mod(k_Xsa);

f:Mod(k_Xsa)!Mod(k_Ysa);

f ¹:Mod(kYsa)!Mod(kXsa):

Let us summarize their properties:

the functorHomis left exact and commutes withr, the functoris exact and commutes with lim!, r ¹andr_!, the functorfis left exact and commutes withrand lim, the functor f ¹is exact and commutes with lim!,and r ¹, (f ¹;f) is a pair of adjoint functors.

LetZbe a subanalytic locally closed subset ofX. As in classical sheaf theory we define

G_Z:Mod(k_Xsa) !Mod(k_Xsa) F7! Hom(rk_Z;F) ()_Z:Mod(k_Xsa) !Mod(k_Xsa)

F7!Frk_Z: We have

the functorG_Z is left exact and commutes withrand lim, the functor ()_Z is exact and commutes with lim!,and r ¹, (()_Z;G_Z) is a pair of adjoint functors.

1.3 ±R-constructible sheaves on subanalytic sites.

Let us consider the category Mod_R-c(k_X). We prove that the sub- category of Mod(k_Xsa) consisting ofR-constructible sheaves is stable under inverse image and tensor product.

PROPOSITION 1.3.1. Let F; G2Mod_R-c(k_X). Then r(F G)'rF rG.

PROOF. We may reduce to the case FˆkU, GˆkV with U;V 2 Op(X_sa). In this caserk_U\V 'rk_Urk_V by Proposition 1.1.7. p COROLLARY 1.3.2. Let F 2Mod_R-c(k_X), and let Z be a subanalytic locally closed subset ofX. ThenrF_Z '(rF)_Z.

Let X;Y be two real analytic manifolds, and let f :X!Y be a real analytic map.

PROPOSITION 1.3.3. Let f:X!Y be a real analytic map. Let G2 Mod_R-c(k_Y). Thenrf ¹G' f ¹rG.

PROOF. Since the functor f ¹ is exact, we may reduce to the case GˆkV, with V 2Op(Ysa). In this case we have r f ¹kV 'rk_f ¹_(V)' ' f ¹rk_V, where the last isomorphism follows from Proposition 1.1.7. p We apply the above results to calculate the functorHomin the category Mod(k_Xsa).

PROPOSITION 1.3.4. Let F ˆlim!_i Fi, with Fi2Mod(kXsa) and let Gˆlim!_j rG_jwithG_j2Mod_R-c(k_X). One has

Hom(G; F)'lim

j

lim!_i Hom(rG_j; F_i):

PROOF. For eachU2Op^c(X_sa) one has the isomorphisms G(U;Hom(G;F)) 'Hom_kXsa(G_U;F)

'lim

j

lim!

i

Homk_Xsa(rGjU;Fi) 'lim

j

lim!_i G(U;Hom(rG_j;F_i)) 'G(U;lim

j

lim!_i Hom(rG_j;F_i)):

In the second isomorphism we used Corollary 1.3.2, and the last iso- morphism follows from Proposition 1.1.3 and because G(U;) commutes

with lim. p

COROLLARY 1.3.5. Let F ˆlim!_i rFi; Gˆlim!_j rGj with Fi; Gj2 2Mod_R-c(k_X). One has

Hom(G; F)'lim

j

lim!_i rHom(G_j; F_i):

PROOF. It follows from the fact thatHomcommutes withrand from

Proposition 1.3.4. p

COROLLARY 1.3.6. LetF ˆlim!_i rFi, with Fi2Mod^c_R-c(X) be a sheaf on Xsa. Let Z be a subanalytic locally closed subset of X. Then GZF 'lim!_i rGZFi:

1.4 ±Proper direct image onMod(k_Xsa).

In [7]the authors defined the functorf!!of proper direct image using ind-sheaves. Here we give a direct construction:

f_!!:Mod(k_Xsa)!Mod(k_Ysa)

F7! lim!_U fF_U 'lim!_K fG_KF

whereUranges through the family of relatively compact open subanalytic

subsets of X and K ranges through the family of subanalytic compact subsets of X. One shall be aware that lim! is taken in the category Mod(k_Ysa). Let V 2Op^c(Ysa). Then G(V;f_!!F)ˆlim!_U G(f ¹(V);F_U)' lim!_K G(f ¹(V);GKF); where U ranges through the family of relatively compact open subanalytic subsets ofXandKranges through the family of subanalytic compact subsets ofX. Iffis proper on supp(F) thenf'f!!and in this casef!!r'rf!.

REMARK 1.4.1. Remark that f_!!r6'rf_! in general. Indeed let V 2Op^c(Ysa), then

G(V;f!!rF)ˆlim!_K G(f ¹(V);G_KF);

G(V;rf!F)ˆlim!_Z G(f ¹(V);GZF);

where Z ranges through the family of closed subsets of f ¹(V)such that fj_Z:Z!V is proper. Then

G(V;f_!!rF)ˆfs2G(f ¹(V);F); supp (s)is compact in Xg;

G(V;rf_!F)ˆfs2G(f ¹(V);F); f :supp (s)!V is properg:

For example, let f :R²!Rbe the projection on the first coordinate, and let V ˆ(a;b)2Op^c(R_sa). Suppose that supp (s)ˆ f(x;y)2(a;b)R;

yˆ_{(x a)(b x)}¹ g. Then f :supp (s)!V is proper butsupp (s)is not compact.

PROPOSITION1.4.2. The functorf!!commutes with filtrant lim!. More- over r ¹f_!!'f_! r ¹.

PROOF. Let us show that f!! commutes with filtrant lim!. Let V 2Op^c(Ysa) and letfF_ig_ibe a filtrant inductive system in Mod(kXsa). Then

lim!

K

Homk_Xsa(k_f ¹_(V);GKlim!

i

Fi)'lim!

K

Homk_Xsa(k_f ¹_(V)\K;lim!

i

Fi) 'lim_i;K!Hom_kXsa(k_f ¹_(V)\K;F_i) 'lim_i;K!Hom_kXsa(k_f ¹_(V);G_KF_i) 'lim!

i

Hom_kYsa(k_V;f_!!F_i) 'Homk_Ysa(kV;lim!_i f!!Fi);

where the second isomorphism follows from the fact that k_f ¹_(V)\K2 2Mod^c_R-c(kX).

Let us show r ¹f_!!'f_! r ¹. Let Fˆlim!

i

rF_i. Sincef_!! commutes with filtrant lim! and F_i has compact support for each i we have f!!Fˆlim!_i rf!Fi. We have the chain of isomorphisms

f_!r ¹lim!_i rF_i'f_!lim!_i r ¹rF_i'f_!lim!_i F_i'lim!_i f_!F_i 'lim!

i

r ¹rf_!F_i' r ¹lim!

i

rf_!F_i:

p PROPOSITION1.4.3. The functorfcommutes with r ¹.

PROOF. Let F2Mod(k_Xsa). Then fF'lim

K

fF_K, where K ranges through the family of subanalytic compact subsets ofX. We have the chain of isomorphisms

fr ¹F'lim

K

f(r ¹F)_K'lim

K

f!!(r ¹F)_K 'lim

K

f!!r ¹F_K 'lim

K

r ¹f_!F_K ' r ¹lim

K

f_!F_K' r ¹lim

K

fF_K ' r ¹fF;

where the second and the sixth isomorphism follow from the fact thatf is

proper on a compact subset ofX. p

COROLLARY1.4.4. The functor f ¹ commutes withr_!.

PROOF. It follows immediately by adjunction. p PROPOSITION1.4.5. LetF 2Mod(k_Xsa)andG2Mod(k_Ysa). Then

f_!!FG'f_!!(F f ¹G):

PROOF. Let Fˆlim!_i rF_i,Gˆlim!_j rG_j. The functors,f!! and f ¹ commute with lim!_i . Moreover supp(F_i f ¹G_j) is compact for each i;j,

hencef is proper on it. Then f_!!lim!

i

rF_ilim!

j

rG_j' lim!

i;j

r(f_!F_iG_j) 'lim_i;j!r(f_!(F_if ¹G_j)) 'f!!( lim!_i rF_if ¹lim!_i rG_j):

In the first isomorphism we used Proposition 1.3.1 and in the last one we

used Propositions 1.3.1 and 1.3.3. p

Now let us consider a cartesian square

PROPOSITION1.4.6. LetF 2Mod(k_Xsa). Then g ¹f_!!F 'f_!!⁰g^{0 1}F. PROOF. LetFˆlim!_i rFi. All the functors in the above formula com- mute with lim!_i . Moreover since supp(Fi) is compact, f⁰ is proper on supp(g^{0 1}Fi) for eachi. Then

g ¹f_!!lim!_i rF_i'lim!_i rg ¹f_!F_i'lim!_i rf_!⁰g^{0 1}F_i'f_!!⁰g^{0 1}lim!_i rF_i; where the first and the last isomorphisms follow from Proposition 1.3.3. p

The following isomorphism is the analogue for subanalytic sheaves of Corollary 4.3.15 of [7].

PROPOSITION1.4.7. LetG2Mod_R-c(k_Y)and letF 2Mod(k_Xsa). Then the natural morphism

f!!Hom(f ¹G; F)! Hom(G; f!!F) is an isomorphism.

PROOF. Let us construct the morphism. By adjunction we have f ¹G Hom( f ¹G;F)!F;

hence, using the projection formula we get

Gf!!Hom( f ¹G;F)'f!!( f ¹G Hom( f ¹G;F))!f!!F;

then by adjunction we obtain the desired morphism. Let us show that it is an isomorphism. We have the chain of isomorphisms

f_!!Hom( f ¹G;F) 'lim!_K fG_KHom(f ¹G;F) 'lim!

K

fHom(f ¹G;G_KF)

'lim!_K Hom(G;fG_KF) ' Hom(G;lim!

K

fGKF) ' Hom(G;f_!!F);

where the fourth isomorphism follows from Proposition 1.3.4. p

1.5 ±Quasi-injective objects.

Let us introduce a category which is useful in order to find acyclic objects with respect to the functors defined in the previous Sections. Al- though the definition is inspired 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 proofs of Theorems 1.5.4 and 1.5.16 are independent of the theory of ind-objects.

DEFINITION 1.5.1. An object F2Mod(k_Xsa) is quasi-injective if the functor Hom_kXsa(;F)is exact in Mod^c_R-c(k_X) or, equivalently (see Theo- rem 8.7.2 of [8]) if for each U;V 2Op^c(X_sa)with V U the restriction morphismG(U;F)!G(V;F)is surjective.

It follows from the definition that injective sheaves belong toJ_Xsa. This implies thatJ_Xsa is cogenerating. Moreover the categoryJ_Xsa is stable by filtrant lim!andQ

.

PROPOSITION1.5.2. Let 0!F⁰!F !F⁰⁰!0be an exact sequence in Mod(kXsa) and assume that F⁰ is quasi-injective. Let U2Op^c(Xsa).

Then the sequence

0!G(U;F⁰)!G(U;F)!G(U;F⁰⁰)!0 is exact.

PROOF. Let s⁰⁰2G(U;F⁰⁰), and let fVigⁿ_iˆ1 be a finite covering of U such that there exists s_i2G(V_i;F) whose image is s⁰⁰j_Vi. For n2 on V₁\V₂ s₁ s₂ defines a section of G(V₁\V₂;F⁰) which extends to s⁰2G(U;F⁰). Replace s₁ with s₁ s⁰. We may suppose that s₁ˆs₂ on V₁\V₂. Then there existst2G(V₁[V₂) such thattj_Viˆs_i,iˆ1;2. Thus

the induction proceeds. p

PROPOSITION1.5.3. LetF⁰; F; F⁰⁰2Mod(k_Xsa), and consider the exact sequence

0!F⁰!F !F⁰⁰!0:

Suppose thatF⁰; F 2 J_Xsa. ThenF⁰⁰2 J_Xsa.

PROOF. LetU;V 2Op^c(Xsa) withVU and let us consider the dia- gram below

The morphismais surjective sinceFis quasi-injective andbis surjective by

Proposition 1.5.2. Thengis surjective. p

THEOREM1.5.4. The family of quasi-injective sheaves is injective with respect to the functorHom_kXsa(G;)for each G2Mod_R-c(k_X).

PROOF. (i) Let 0!F⁰!F!F⁰⁰!0 be an exact sequence in Mod(k_Xsa) and assume that F⁰2 J_Xsa. Let G2Mod^c_R-c(k_X). We have to show that the sequence

0!Hom_kXsa(G;F⁰)!Hom_kXsa(G;F)!Hom_kXsa(G;F⁰⁰)!0 is exact. There is an epimorphism W:_i2Ik_Ui!Gwhere I is finite and U_i2Op^c(X_sa) for eachi2I.

The sequence 0!kerW! _i2Ik_Ui!G!0 is exact. We set for short G1ˆkerWandG2ˆ i2IkUi. We get the following diagram where the first

column is exact

The second row is exact by Proposition 1.5.2, hence the top row is exact by the snake lemma.

(ii) Let G2Mod_R-c(k_X), let 0!F⁰!F!F⁰⁰!0 be an exact se- quence in Mod(kXsa) with F⁰2 JXsa. Let fVng_n2N 2Cov(Xsa) such that VnVn‡1. By (i), all the sequences

0!Hom_kXsa(GVn;F⁰)!Hom_kXsa(GVn;F)!Hom_kXsa(GVn;F⁰⁰)!0 are exact. Moreover since F⁰2 JXsa the morphism Hom_kXsa(GVn‡1;F⁰)!

!Hom_kXsa(G_Vn;F⁰) is surjective for all n. Then by the Mittag-Leffler property (see Proposition 1.12.3 of [5]) the sequence

0!lim

n

Hom_kXsa(G_Vn;F⁰)!lim

n

Hom_kXsa(G_Vn;F)!lim

n

Hom_kXsa(G_Vn;F⁰⁰)!0 is exact. Since lim

n

Homk_Xsa(GVn;)'Homk_Xsa(G;) the result follows. p PROPOSITION1.5.5. LetG2Mod_R-c(kX). Then quasi-injective sheaves are injective with respect to the functorHom(G;).

PROOF. Let G2ModR-c(kX). It is enough to check that for each U2Op(Xsa) and each exact sequence 0!F⁰!F!F⁰⁰!0 with F⁰2 J_Xsa, the sequence

0!G(U;Hom(G;F⁰))!G(U;Hom(G;F))!G(U;Hom(G;F⁰⁰))!0 is exact. We have G(U;Hom(G;))'Hom_kXsa(G_U;), and quasi-injective objects are injective with respect to the functor Hom_kXsa(G_U;) for each G2ModR-c(kX), and for eachU2Op(Xsa). p

COROLLARY1.5.6. Quasi-injective sheaves are injective with respect to the functorGZfor each locally closed subanalytic subsetZ ofX.

COROLLARY 1.5.7. Let F 2Mod(k_Xsa) be quasi-injective. Then the functorHom(; F)is exact onMod_R-c(k_X).

PROOF. LetF2Mod(kXsa) be quasi-injective. There is an isomorphism of functorsG(U;Hom(;F))'Hom_kXsa(()_U;F) for eachU2Op(Xsa). The functor Hom_kXsa(()_U;F) is exact on Mod_R-c(k_X) and the result follows. p PROPOSITION 1.5.8. Let F 2Mod(k_Xsa). Then F is quasi-injective if and only ifHom(G; F)is quasi-injective for eachG2ModR-c(kX).

PROOF. (i) LetFbe quasi-injective, and letG2Mod_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) Suppose thatHom(G;F) is quasi-injective for eachG2ModR-c(kX).

The result follows by settingGˆkX. p

COROLLARY 1.5.9. The functor G_Z sends quasi-injective objects to quasi-injective objects for each locally closed subanalytic subsetZ ofX.

Letf :X!Y be a morphism of real analytic manifolds.

PROPOSITION1.5.10. Quasi-injective sheaves are injective with respect to the functor f. The functor f sends quasi-injective objects to quasi- injective objects.

PROOF. (i) Let us consider V 2Op(Ysa). There is an isomorphism of functors G(V;f())'G(f ¹(V);). It follows from Proposition 1.5.4 that J_Xsa is injective with respect to the functor G(f ¹(V);)' 'Hom_kXsa(k_f ¹_(V);) for anyV 2Op(Y_sa).

(ii) LetF2 J_Xsa. For eachG2Mod^c_R-c(k_Y) we have Hom_kYsa(G;fF)' Homk_Xsa(f ¹G;F). Since f ¹is exact and sends Mod^c_R-c(kY) to ModR-c(kX), Proposition 1.5.4 implies that the functor Hom_kXsa(f ¹();F) is exact on

Mod^c_R-c(k_Y). p

PROPOSITION 1.5.11. The family of quasi-injective sheaves is f_!!-in- jective. The functor f!! sends quasi-injective objects to quasi-injective objects.

PROOF. (i) Let 0!F⁰!F!F⁰⁰!0 be an exact sequence in Mod(kXsa) and assume thatF⁰2 JXsa. We have to check that the sequence 0!f_!!F⁰!f_!!F!f_!!F⁰⁰!0 is exact. Since F⁰2 J_Xsa, we have G_KF⁰2 J_Xsa. MoreoverJ_Xsa is injective with respect to G_K and f. This implies that the sequence

0!fG_KF⁰!fG_KF!fG_KF⁰⁰!0 is exact. Applying the exact functor lim!

K

we find that the sequence 0! lim!_K fG_KF⁰! lim!_K fG_KF! lim!_K fG_KF⁰⁰!0 is exact.

(ii) LetKbe a compact subanalytic subset ofX. The functorsG_Kandf

send quasi-injective objects to quasi-injective objects, thenfG_KF2 J_Ysa. SinceJ_Ysa is stable by filtrant lim!, the result follows. p Let S be a closed subanalytic subset of X and let iS:S,!X be the closed embedding. Let Fˆlim!_i rF_i2Mod(kXsa) with F_i2Mod^c_R-c(kX).

We haveF_S 'lim!_i rF_iS'lim!_i ri_Si_S ¹F_i'i_Si_S ¹F.

LEMMA 1.5.12. Let S be a closed subanalytic subset of X and let U2Op^c(Xsa). Let F2Mod(kXsa). Then G(U;FS)' lim!

VS\U

G(V;F), with V 2Op^c(Xsa).

PROOF. LetF2Mod(k_Xsa). ThenF'lim!_i rF_iwithF_i2Mod^c_R-c(k_X).

We have the chain of isomorphisms

G(U;F_S)'lim!_i G(U;F_iS)

_i;VS\Ulim! G(V;F_i) ' lim!

VS\U

G(V;F);

whereVranges through the family of relatively compact open subanalytic subsets ofXcontainingS\U. The second isomorphism follows sinceFiis R-constructible for eachi. p REMARK1.5.13. The fact that F_i2Mod_R-c(k_X)is locally constant on a subanalytic stratification of X plays an essential role. In fact the iso-

morphism is not true in general inMod(kX), since the family of relatively compact open subanalytic subsets of X containing S\U is not cofinal to the family of open subsets of X containing S\U.

PROPOSITION1.5.14. LetSbe a closed subanalytic subset ofXand let F 2Mod(k_Xsa)be quasi-injective. ThenF_S is quasi-injective.

PROOF. Let U;V2Op^c(X_sa) with V U. Since F is quasi-injective and inductive limits are right exact, the morphism lim_U0S\U! G(U⁰;F)!

lim!

V0S\V

G(V⁰;F) withV⁰;U⁰2Op^c(X_sa), is surjective. Hence by Lemma 1.5.12 the morphismG(U;FS)!G(V;FS) is surjective and the result follows. p Recall thatF2Mod(kX) is c-soft if the natural morphismG(X;F)! G(K;F) is surjective for each compact KX. IfF is c-soft andZis a lo- cally closed subset of X, then F_Z is c-soft. Moreover c-soft sheaves are G(U;)-injective for eachU2Op(X).

PROPOSITION1.5.15. LetF 2Mod(k_Xsa)be quasi-injective. Then r ¹F is c-soft.

PROOF. LetKbe a compact subset ofX. Recall that ifU2Op(X) then G(U;r ¹F)'lim

VU

G(V;F), where V 2Op(Xsa). We have the chain of iso- morphisms

G(K; r ¹F)'lim!_U G(U;r ¹F) 'lim!

U

lim

VU

G(V;F) 'lim!_U G(U;F)

whereUranges through the family of subanalytic relatively compact open subsets ofXcontainingKandV 2Op(X_sa).

Since F is quasi-injective and filtrant inductive limits are exact, the morphism G(X;r ¹F)'G(X;F)!lim!_U G(U;F)'G(K;r ¹F), where U ranges through the family of subanalytic open subsets ofXcontainingK, is

surjective. p