On the homological dimension of o-minimal and subanalytic sheaves

ON THE HOMOLOGICAL DIMENSION arXiv:0911.1748v1 [math.AG] 9 Nov 2009 OF O-MINIMAL AND SUBANALYTIC SHEAVES Abstra t Here we prove that the homologi a...

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ON THE HOMOLOGICAL DIMENSION

arXiv:0911.1748v1 [math.AG] 9 Nov 2009

OF O-MINIMAL AND SUBANALYTIC SHEAVES

Abstra t Here we prove that the homologi al dimension of the ategory of sheaves on a topologi al spa e satisfying some suitable onditions is nite. In parti ular, we nd onditions to bound the homologi al dimension of o-minimal and subanalyti sheaves.

1 Introdu tion In [9℄ we studied the ategory Mod(kXsa ) of sheaves on the subanalyti site Xsa asso iated to a real analyti manifold X . We dened the sub ategory of quasi-inje tive sheaves (i.e. F is quasi-inje tive if the restri tion Γ(U ; F ) → Γ(V ; F ) is surje tive for ea h U ⊇ V open subanalyti relatively

ompa t) and we saw that quasi-inje tive are inje tive with respe t to the fun tors of dire t image, proper dire t image and Hom(G, ·) when G is a R- onstru tible sheaf on X . Moreover we proved that the quasi-inje tive dimension of Mod(kXsa ) is nite, whi h implies that the ohomologi al dimension of the above fun tors is nite. However we had no answer on erning the homologi al dimension of Mod(kXsa ) (see Remark 2.3.5 of [9℄). The aim of this paper is to show that under some onditions ( on erning the ardinality 2ℵ0 of the reals) it is possible to bound this dimension. The key point is the fa t that lo ally the ategory of subanalyti sheaves is equivalent to the e . We are redu ed to work with

ategory of sheaves on a topologi al spa e X k-sheaves on a topologi al spa e, and in this ase the homologi al dimension is equal to the abby dimension. Hen e we are redu ed to bound the ohoe . In order to do mologi al dimension of Γ(U ; ·), for any open subset U of X that we need to assume that the ardinality of subanalyti subsets (whi h is equal to 2ℵ0 ) is smaller than ℵk , k < ∞. 1

In more details the ontents of this paper are as follows. In § 2 we re all some notions as the denitions of inje tive and abby sheaf and the homologi al and abby dimension of the ategory of sheaves. We refer to [6℄ for more details. In § 3 we study the general ase of sheaves on a topologi al spa e X with a basis T whose elements are Lindelöf and stable under nite unions and interse tions. We dene the sub ategory of T -abby sheaves on X by saying that F is T -abby if the restri tion Γ(U ; F ) → Γ(V ; F ) is surje tive for ea h U, V ∈ T with U ⊇ V . Sin e T forms a basis of the topology of X , we have that ea h open U of X has a overing {Ui }i∈I with Ui ∈ T . Hen e RΓ(U ; F ) ≃ Rlim ←−RΓ(Ui ; F ), i∈I

and to bound the ohomologi al dimension of the fun tor Γ(U ; ·) is su ient to bound the ohomologi al dimension of the fun tors Γ(V ; ·), V ∈ T and the ohomologi al dimension of the proje tive limit. The ohomologi al dimension of the proje tive limit is bounded if the ardinality of the index set I is smaller than ℵk for some k < ∞, (see [8℄ or [10℄ in the more general setting of quasi-abelian ategories). Hen e it is bounded if the ardinality of T is smaller than ℵk , k < ∞. e be the In § 4 we onsider an o-minimal stru ture M = (M, <, . . .). Let X o-minimal spe trum of a denable spa e X . In this ase T is the family of e . In the ase of an o-minimal expansion of open onstru tible subsets of X e is nite. an ordered group, by a result of [4℄, the T -abby dimension of X Moreover the ardinality of T is bounded by the produ t of the ardinality of M and the ardinality of the language of the stru ture M. The ase of subanalyti sheaves on a real analyti manifold is studied in § 5. We rst redu e to Mod(kUXsa ) where U is a relatively ompa t subanalyti subset of X isomorphi to RN endowed with the Grothendie k topology indu ed by X . This ategory is equivalent to Mod(kUe ), where Ue is the o-minimal spe trum of U . In this ase T is the family of open globally subanalyti subsets of U , and its ardinality is 2ℵ0 . Hen e if we assume that 2ℵ0 is smaller or equal than ℵk , k < ∞ (in the ase k = 1 this is the

ontinuum hypothesis), then the homologi al dimension of Mod(kUe ) (and hen e of Mod(kUXsa )) is bounded. e there are open We end this work with an example showing that in X subsets whi h do not admit ountable overs.

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2 Some preliminaries We introdu e some fundamental results about sheaves we will use in the rest of the paper. We refer to [6℄ for more details. Let X be a topologi al spa e and let k be a eld. As usual, we denote by Mod(kX ) the ategory of sheaves of k-ve tor spa es.

Denition 2.1 Let F

∈ Mod(kX ).

- F is inje tive if the fun tor Hom(·, F ) is exa t on Mod(kX ). - F is abby if for any open subset U of X the restri tion morphism Γ(X; F ) → Γ(U ; F ) is surje tive. In general inje tive ⇒ abby. When k is a eld we have inje tive ⇔ abby ([6℄, Exer ise II.10).

Denition 2.2 The homologi al (resp. abby) dimension of the ategory

is the smallest N ∈ N ∪ {∞} su h that for any F ∈ Mod(kX ) there exists an exa t sequen e Mod(kX )

0 → F → I0 → · · · → I N → 0

with I j inje tive (resp. abby) for 0 ≤ j ≤ N . We shall need the following results. - The homologi al dimension of Mod(kX ) is nite if and only if there exists N ∈ N su h that Rj Hom(G, F ) = 0 for any F, G ∈ Mod(kX ) and any j > N . - The abby dimension of Mod(kX ) is nite if and only if there exists N ∈ N su h that for any open subsets U of X , any F ∈ Mod(kX ) and any j > N we have Rj Γ(U ; F ) = 0 (i.e. if the fun tor Γ(U ; ·) has nite

ohomologi al dimension). In parti ular, when k is a eld, sin e inje tive ⇔ abby, the homologi al dimension is nite if and only if the fun tor Γ(U ; ·) has nite ohomologi al dimension for any open subsets U of X .

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3 T -abby sheaves and homologi al dimension Denition 3.1 A Lindelöf spa e is a topologi al spa e in whi h every open

over has a ountable sub over.

Denition 3.2 The Lindelöf degree

l(X) of a topologi al spa e X is the smallest ardinal κ su h that every open over of the spa e X has a sub over of size at most κ. In this notation, X is Lindelöf i l(X) = ℵ0 .

Let k be a eld. Let X be a topologi al spa e and suppose that X admits a family T of open subsets su h that T1: ea h element of T is Lindelöf, T2: T is stable under nite unions and interse tions, T3: T forms a basis for the topology of X .

Denition 3.3 A sheaf F ∈ Mod(kX ) is said to be T -abby if the restri tion morphism Γ(U ; F ) → Γ(V ; F ) is surje tive for ea h U, V ∈ T with V ⊆ U . Remark that abby ⇒ T -abby.

Denition 3.4 The T -abby dimension of N∪∞

Mod(kX ) is the smallest N ∈ su h that for any F ∈ Mod(kX ) there exists an exa t sequen e 0 → F → I0 → · · · → I N → 0

with I j T -abby for 0 ≤ j ≤ N .

Proposition 3.5 Let with the morphism

Mod(kX )

F′

be an exa t sequen e in Then for any open subset U whi h is Lindelöf

0 → F ′ → F → F ′′ → 0

T -abby.

Γ(U ; F ) → Γ(U ; F ′′ )

is surje tive.

Proof. We rst onsider a se tion

s′′ S ∈ Γ(U ; F ′′ ). Sin e F → F ′′ is surje tive, we may nd a overing U = i∈I Ui , Ui ∈ T and si ∈ Γ(Ui ; F ) whoseSimage is s′′ |Ui . Sin e U is Lindelöf, we may nd a ountable sub over ′′ U = n∈N US n , Un ∈ T and sn ∈ Γ(Un ; F ) whose image is s |Un . n Set Vn = i=1 Ui . We prove by indu tion on n that there exists a se tion tn+1 ∈ Γ(Vn+1 ; F ) whose image is s′′ |Vn+1 and tn+1 |Vn = tn .

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This is lear for n = 0, 1 and we assume it is proved for n. By the indu tion hypothesis there exists a se tion tn ∈ Γ(Vn ; F ) whose image is s′′ |Vn and tn |Vm = tm if m < n. We set for short Vn = W1 and Un+1 = W2 . We have seen that there exist tj ∈ Γ(Wj ; F ) whose image is s′′ |Wj for j = 1, 2. On W1 ∩ W2 t1 − t2 denes a se tion of Γ(W1 ∩ W2 ; F ′ ) whi h extends to t′ ∈ Γ(Vn+1 ; F ′ ) be ause F ′ is T -abby. Repla e t2 with t2 + t′ . We may suppose that t1 = t2 on W1 ∩ W2 . Then there exists tn+1 ∈ Γ(Vn+1 ; F ) su h that tn+1 |Wj = tj , j = 1, 2. The tn 's glue together into a se tion s ∈ lim ←− Γ(Vn ; F ) ≃ Γ(U ; F ) n∈N

whi h is sent to s′′ , whi h proves the surje tivity of the morphism.

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Proposition 3.6 Let F ′ , F, F ′′ ∈ Mod(kX ), and onsider the exa t sequen e 0 → F ′ → F → F ′′ → 0.

Suppose that F ′ is T -abby. Then F is T -abby if and only if F ′′ is T -abby.

Proof. Let U, V 0

∈ T with V ⊆ U and let us onsider the diagram below / Γ(U ; F ′ )

/ Γ(U ; F )

α



0

/ Γ(V ; F ′ )

/ Γ(U ; F ′′ ) γ

β

 / Γ(V ; F )

/0



/ Γ(V ; F ′′ )

/0

where the row are exa t by Proposition 3.5 and the morphism α is surje tive sin e F ′ is T -abby. It follows from the ve lemma that β is surje tive if 2 and only if γ is surje tive.

Proposition 3.7 Let

be T -abby. Then F is Γ(U ; ·)-inje tive for ea h open U ⊆ X whi h is Lindelöf. F

Proof. The family of T -abby sheaves ontains inje tive sheaves, hen e it is ogenerating. Then the result follows from Propositions 3.5 and 3.6.

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Now let us onsider the ase of open subsets of Lindelöf degree ℵk , k < ∞. In order to do that we need the following result. 5

Proposition 3.8 Let

be a topologi al spa e admitting a family T satisfying T1-T3. Suppose that X has nite T -abby dimension. Let U be an open subset of X of Lindelöf degree ℵk , k < ∞. Then the ohomologi al dimension of Γ(U ; ·) is nite. X

Proof. We have RΓ(U ; F ) ≃ Rlim ←−RΓ(Ui ; F ) i∈I

S

where U = i∈I Ui with Ui ∈ T and ♯I smaller than ℵk . Sin e X has nite T -abby dimension and using Proposition 3.7 for ea h i we may repla e RΓ(Ui ; F ) with Γ(Ui ; I • ), where I • is a T -abby resolution of F of length N < ∞. Sin e ohomologi al dimension of lim ←− is nite if ♯I is smaller than i∈I

ℵk , k < ∞ (see [8, 10℄), then the j -th ohomology of Rlim ←−RΓ(Ui ; F ) vanishes i∈I

for j > N + M . Sin e M, N < ∞ are independent of F and i, the result follows. 2

Theorem 3.9 Let X be a topologi al spa e admitting a family T satisfying

T1-T3. Suppose that X has nite T -abby dimension. Suppose that there exists k < ∞ su h that every open subset of X has Lindelöf degree ℵk . Then Mod(kX ) has nite homologi al dimension.

Proof. By Proposition 3.8

X has nite abby dimension. In the ase of sheaves of k-ve tor spa es we have abby ⇔ inje tive and the result follows. 2

4 Homologi al dimension of o-minimal sheaves Let M = (M, <, . . .) be an o-minimal stru ture. In [4℄ the authors studied e , the o-minimal spe trum of a denable sheaf ohomology of sheaves on X e is equivalent to the ategory of spa e X . The ategory of sheaves on X sheaves on the o-minimal site, onsisting of open denable subsets of X and

overings admitting a nite renement. e . Then T satises Let T be the family of open onstru tible subsets of X T1-T3, indeed e, - T forms a basis for the topology of X

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- every element of T is quasi- ompa t, - T is stable under nite unions and interse tions. e and ea h F ∈ Mod(k e ) we have For ea h open subset U of X X RΓ(U ; F ) ≃ Rlim ←−RΓ(Ui ; F ) i∈I

S

where U = i∈I Ui and Ui ∈ T for ea h i ∈ I . In order to apply Theorem 3.9 we need that: e is nite, - the T -abby dimension of X

- ♯I ≤ ℵk , k < ∞.

If every open onstru tible subset U of X is normal (this is true in the ase of an o-minimal expansion of an ordered group), then by Proposition 4.2 of [4℄ we have Rj Γ(U ; F ) = 0 for all j > dimU = dimX . Con erning ♯I , we have that ♯I ≤ ♯T , and ♯T ≤ ♯L ∪ M = ♯L · ♯M , where L denotes the language of the stru ture M.

5 Homologi al dimension of subanalyti sheaves Let X be a real analyti manifold, let Xsa the asso iated subanalyti site and denote by Mod(kXsa ) the ategory of sheaves of k-ve tor spa es on Xsa . We refer to [7, 9℄ for the theory of subanalyti sheaves. We will see that under suitable hypothesis the homologi al dimension of Mod(kXsa ) is nite, i.e. there exists N ∈ N su h that Rj Hom(G, F ) = 0 for any F, G ∈ Mod(kXsa ) and any j > N . Let X be a real analyti manifold. Then X has an atlas (Un , φn )n∈N , φn : ∼ Un → RM , Un relatively ompa t open subanalyti subset. Let us onsider the subanalyti site Xsa asso iated to X . In order to bound RHom(F, G) for any F, G ∈ Mod(kXsa ) it su es to bound RHom(F, G). Indeed we have RHom(F, G) ≃ RΓ(X; RHom(F, G)) and Γ(X; ·) has nite ohomologi al dimension (see [9℄). We have Rj Hom(F, G) = 0 if Rj Hom(F, G)|Un = 0 for ea h n ∈ N. Then we may redu e to the ase Un with the Grothendie k topology indu ed by the subanalyti site, i.e. the ategory of globally subanalyti open subsets of RM with overings admitting a nite sub over. Globally subanalyti geometry is dened by the o-minimal stru ture given by the ordered eld of real numbers expanded by globally analyti fun tions Ran = (R, <, 0, 1, +, ·, (f )f ∈an ), 7

where f is a restri tion to [−1, 1]m of a onvergent power series on some neighborhood of [−1, 1]m . e be the o-minimal spe trum of X . The family T of Set X = Un and let X open sets Ue where U is open globally subanalyti satises T1-T3. Moreover the T -abby dimension of Mod(kXe ) is nite. Indeed by Proposition 4.2 of [4℄ the ohomologi al dimension of Γ(Ue ; ·) is nite for ea h Ue ∈ T . Let us see that ♯T ≤ 2ℵ0 . In order to see that we have to show that ♯L = ℵ is true if ♯{(f )f ∈an } = 2ℵ0 . 2 0 , where L denotes the language P of Ran . This m This is seen by identifying f = I∈Nm aI xI with (aI )I∈Nm ∈ RN . Then m ♯{(f )f ∈an } ≤ ♯RN = (2ℵ0 )ℵ0 = 2ℵ0 ·ℵ0 = 2ℵ0 (see [5℄). e . Then U = S Ui with Ui ∈ T and ♯I is Let U be an open subset of X i∈I smaller than ♯T = 2ℵ0 . Hen e every open subset of X has Lindelöf degree at most 2ℵ0 . If we assume that 2ℵ0 is smaller than ℵk for some k < ∞ (if we suppose ≤ ℵ1 it is the ontinuum hypothesis), then Theorem 3.9 implies that the homologi al dimension of Mod(kXe ) is nite.

Remark 5.1 In

there are open subsets whi h are not Lindelöf. For exf2 with respe t to Ran ) ample let us onsider the open set (in Xe = R e X

U=

[

r∈(0,1)\Q

where

Ver ,

Vr = {(x, y) ∈ R2 ; 0 < x < 1, r < y < r + x}.

We prove that {Vr }r∈(0,1)\Q has no ountable sub over. We argue by ontradi tion. Suppose that there exists a ountable sub over Let 0+ be the ultralter dened by 0+ = {S ⊃ (0, ε)}, globally subanalyti . Let π : R2 → R be the proje tion onto the rst oordinate. We have e. {Vern }n∈N of U in X where ε > 0 and S is

U ∩π e−1 (0+ ) =

G

r∈(0,1)\Q



 Ver ∩ π e−1 (0+ ) .

Indeed, let x ∈ Ver ∩ πe−1 (0+ ), and let s 6= r. Let ε < |r − s|, then Vs ∩ ] ] e−1 (0+ ) ⊂ Ves ∩ π e−1 ((0, ε)) sin e 0+ ⊂ (0, π −1 ((0, ε)) ∩ Vr = ∅ and Ves ∩ π ε) −1 + e e for any ε > 0. Hen e Vr ∩ πe (0 ) ∩ Vs = ∅ if r 6= s. Moreover G

r∈(0,1)\Q

  [ Vern = U. Ver ∩ π e−1 (0+ ) ⊂ n∈N

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Sin e ♯((0, 1) \ Q) = 2ℵ0 > ℵ0 , there exists n ∈ N su h that o n ♯ t ∈ (0, 1) \ Q ; Vet ∩ π e−1 (0+ ) ⊆ Vern ≥ 2.

But Vet ∩ πe−1 (0+ ) ⊆ Vern implies that Vt = Vrn , so t = rn whi h is a ontradi tion.

Remark 5.2 More generally, as in Remark 5.1, given an o-minimal ex-

pansion of an ordered group (R, <, +, . . .), for any denable manifold X of dimension ≥ 2, the spa e Xe has open subsets whi h have Lindelöf degree 2ℵ0 .

Remark 5.3 It seems that the right setting is that of lo ally denable spa es

in an o-minimal stru ture. In that setting the o-minimal (denable) ase and the subanalyti ase would be treated uniformly. However, to work on that setting one has to take into a

ount the (spe ial ase of the) theory lo ally semi-algebrai spa es developed in [2, 3℄.

Referen es [1℄ M. Coste An introdu tion to o-minimal geometry Dip. Mat. Univ. Pisa, Dottorato di Ri er a in Matemati a, Istituti Editoriali e Poligra i Internazionali, Pisa (2000). [2℄ H. Delfs Homology of lo ally semialgebrai spa es Le ture Notes in Math. 1484 Springer-Verlag (1991). [3℄ H. Delfs and M. Knebus h Lo ally semi-algebrai spa es Le ture Notes in Math. 1173 Springer-Verlag (1985). [4℄ M. Edmundo, G. Jones and N. Peateld Sheaf ohomology in o-minimal stru tures J. Math. Logi 6 pp. 163-179 (2006). [5℄ T. Je h Introdu tion to set theory Dekker (1984). [6℄ M. Kashiwara and P. Shapira Sheaves on manifolds Grundlehren der Math. 292 Springer Verlag (1990). [7℄ M. Kashiwara and P. Shapira Ind-sheaves Astérisque 271 (2001). [8℄ C. U. Jensen Les fon teurs dérivés de lim ←− et leurs appli ations en théorie des modules Le ture Notes in Math. 254 Springer-Verlag (1972). 9

[9℄ L. Prelli Sheaves on subanalyti sites Rend. Sem. Mat. Univ. Padova 120 pp. 167-216 (2008). [10℄ F. Prosmans Derived limits in quasi-abelian ategories Bull. So . Roy. S i. Liège 68 pp. 335-401 (1999).

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