version du 2016-11-14 à 13h36TU (19c1b56)
In this note, we sketch a short proof of Gabber’s hyper base change theorem exp.XIIA,2.2.5without using oriented products. We will first treat the Abelian case and then briefly comment on the non-Abelian case. We also include an analogue, due to Gabber, for topological spaces.
1. A descent formalism
We letS denote the category of schemes. We fix a diagramX→S←YinS and a complexK∈D+(Y´et,Z) whose cohomology sheaves are torsion (see [SGA 4IX1.1] for the definition of torsion sheaves).
We regardX → S ← Y as an object of the category of functors Fun(V,S), whereV denotes the diagram category•→• ←•. We letBdenote the full subcategory of Fun(V,S)spanned by diagramsX0 →S0 ←Y0 such thatY0 → S0 is coherent (namely, quasi-compact quasi-separated). We letC denote the overcategory B/(X→S←Y). We will drop the letterRin derived direct image functors.
Definition 1.1. — i. We say that an augmented simplicial objectf•inC corresponding to the diagram
(1.1.1) X• a• //
fX•
S•
fS•
Y•
b•
oo
fY•
X−1 a−1 //
x
S−1
Y−1
b−1
oo
y
X //Soo Y
isof descent(relative toK) if the composite map
(1.1.2) x?a?−1b−1?y?K−−A→Y x?a?−1b−1?fY•?f?Y•y?K'x?a?−1fS•?b•?f?Y•y?K−−C→X x?fX•?a?•b•?f?Y•y?K is an isomorphism inD+(X´et,Z), whereAYis the adjunction map induced byfY•andCXis the base change map induced by the top left square of (1.1.1).
ii. We say that a morphismf:T0→T−1inC isof descentif its ˇCech nerve(coskT0−1T0)•→T−1is of descent.
We say thatfisof universal descentif it is of descent after arbitrary base change inC.
iii. We say that an augmented simplicial objectT•→T−1inC is ahypercovering for universal descentif for everyn≥0, the morphismTn→(coskTn−1−1T•)nis of universal descent.
Remark 1.2. — The map (1.1.2) is equal to the composite map
x?a?−1b−1?y?K−−A→X x?fX•?f?X•a?−1b−1?y?K'x?fX•?a?•f?S•b−1?y?K−−C→Y x?fX•?a?•b•?f?Y•y?K,
whereAXis the adjunction map induced byfX•andCY is the base change map induced by the transpose of the top right square of (1.1.1). In fact, the diagram
x?a?−1b−1?fY•?f?Y•y?K ∼ //x?a?−1fS•?b•?f?Y•y?K
CX
**x?a?−1b−1?y?K AS //
AY
55
AX ))
x?a?−1fS•?f?S•b−1?y?K
CY
44
CX
**
x?fX•?a?•b•?f?Y•y?K
x?fX•?f?X•a?−1b−1?y?K ∼ //x?fX•?a?•f?S•b−1?y?K
CY
44
is commutative, whereASis the adjunction map induced byfS•.
We say that a morphismg:Z0 → ZinS isof cohomological descent relative to L ∈ D+(Z´et,Z) if the adjunction mapL →g•?g?•Lis an isomorphism, whereg•is the ˇCech nerve ofg. By classical cohomological descent [SGA 4Vbis4.3.2, 4.3.3],gis of cohomological descent relative toLif eithergis proper surjective and the cohomology sheaves ofLare torsion orgis flat surjective locally of finite presentation.
(i)Partially supported by China’s Recruitment Program of Global Experts; National Natural Science Foundation of China Grant 11321101.
1
Proposition 1.3. — Letfbe a morphism inC corresponding to the diagram
(1.3.1) X0
a0 //
fX
S0 fS
Y0 b0
oo
fY
X−1 a−1 //
x
S−1
Y−1 b−1
oo
y
X //Soo Y.
Thenfis of descent if it satisfies either of the following conditions:
i. fY is of cohomological descent relative toy?K,fSis proper, and the top left square of (1.3.1)is Cartesian;
ii. fXis of cohomological descent relative toa?−1b−1?y?K,fSis étale, and the top right square of(1.3.1)is Cartesian.
Note that the assumption thatbnis coherent implies thatbn?f?Y,ny?Khas torsion cohomology sheaves.
Proof. — In case (i), AY is an isomorphism by assumption and CX is an isomorphism by the proper base change theorem [SGA 4XII5.1]. In case (ii),AXis an isomorphism by assumption andCYis an isomorphism by étale base change, and we conclude by Remark1.2.
The following lemma on comparison of augmentations is similar to [SGA 4Vbis3.1.1].
Lemma 1.4. — Letf•: T• → T−1,f•0: T•0 → T−1 be augmented simplicial objects of C and let α•: T• → T•0 be a morphism overT−1. Then the diagram
x?a?−1b−1?y?K
--
x?fX•?0 a•0?b•?0 fY•0?y?K
AY
//x?f0X•?a0?•b0•?αY•?f?Y•y?K ∼ //x?f0X•?a0?• αS•?b•?f?Y•y?K
CX
//x?fX•?a?•b•?f?Y•y?K
where the vertical and oblique arrows are given by(1.1.2), is commutative. Moreover, the compositeηα•of the horizontal arrows depends only on the simplicial homotopy class ofα•.
Proof. — The first assertion is trivial: the diagram x?a?−1b−1?y?K
Af0 Y•
AfY•
**x?a?−1b−1?f0Y•?f0?Y•y?K AαY• //
Cf0 X•
x?a?−1b−1?fY•?f?Y•y?K
Cf0
X•
CfX•
**
x?fX•?0 a•0?b•?0 fY•0?y?K AαY•//x?fX•?0 a•0?b•?0 αY•?f?Y•y?KCαX• //x?fX•?a?•b•?f?Y•y?K
is commutative. To show the second assertion, letβ•:T•→T•0be a morphism overT−1and leth: α• →β•be a simplicial homotopy overT−1. As in the proof of [SGA 4Vbis3.1.1.9], consider the categoryDwhose objects are objects of[1]×∆and whose morphisms are pairs(ξ, δ) : (m,[n])→(m0,[n0]), whereξ: [n]→[m, m0]and δ: [n]→[n0]. Here[m, m0]denotes the subset of[1]spanned byiwithm≤i≤m0. The homotopyhdefines a functorThT0:Dopp→C, augmented overT−1. LetJbe a resolution of(aha0)?(bhb0)?(fhf0)?Yy?Ksuch that the sheavesJkm,[n]are flabby. We letrm:∆ → Ddenote the functor carrying[n]to(m,[n]). ThenJinduces a cosimplicial homotopy between the mapsfX•?0 r?0J ⇒fX•?r?1Jof cosimplicial complexes onX−1, which give rise toηα•andηβ•. Therefore,ηα•=ηβ•.
The proof of the following proposition is similar to [SGA 4Vbis3.3.1 a), b)] (see also Remark2.5below).
Proposition 1.5. — i. A morphismfinC is of universal descent if it admits a section.
ii. Let
T0 f
0 //
g0
U0
g
T f //U
be a Cartesian square inC such that the base change offby(U0/U)n →Uis of descent for alln≥1and the base change ofgby(T/U)n→Uis of descent for alln≥0. Thenfis of descent.
In (ii), the assumptions on fand gare satisfied ifgis of descent and f0,g0 are of universal descent. In particular, if in (ii)f0andgare of universal descent, thenfis ofuniversaldescent.
The following corollary is similar to [SGA 4Vbis3.3.1 c), d)].
Corollary 1.6. — LetT −→f U−→g Wbe a sequence of morphisms inC. i. Ifgfis of universal descent, thengis of universal descent.
ii. Iffis of universal descent andgis of descent, thengfis of descent.
Proof. — Consider the commutative diagram T
f
))
id
##
T ×WU
h //
g0
U
g
T gf //W.
Sinceg0admits a section,g0is of universal descent by Proposition1.5(i). Case (i) then follows from Proposition 1.5(ii). In case (ii),his of universal descent by (i), and the assertion follows again from Proposition1.5(ii).
The following analogue of [SGA 4Vbis3.3.1 f)] is obvious.
Proposition 1.7. — Let(fα:Tα→Uα)α∈Abe a family of morphisms ofC of descent. Then`
α∈Afα: `
α∈ATα→
`
α∈AUαis of descent.
The following is a consequence of Corollary1.6and Proposition1.7, applied to constant diagrams of the formZ→Z←Z.
Corollary 1.8. — Let (Zα → Z)α∈A be a covering family for the h-topology (exp.XIIA, 2.1.3) on S. Then the morphism`
α∈AZα →Zis of cohomological descent relative to everyL ∈D+(Z´et,Z)whose cohomology sheaves are torsion.
We consider the orientedh-topology onBgenerated by families of types (i) through (v) (exp.XIIA,2.1.4).
Corollary 1.9. — Let(Tα→T)α∈Abe a family of morphisms inC that is covering for the orientedh-topology onB.
Then the morphism`
α∈ATα → T is of universal descent. In particular, iffis a morphism inC corresponding to the diagram(1.3.1)such that both top squares are Cartesian andfSis a covering morphism for theh-topology, thenfis of universal descent.
Proof. — The second assertion follows from the first one, asfis a covering family of type (iii) for the oriented h-topology. To show the first assertion, we may assume, by Corollary1.6and Proposition1.7, that the family (Tα → T)α∈Ais of one of the five types in the definition of the orientedh-topology. Moreover, for type (iii), we may assume that the family is induced by either a proper surjective morphism or an étale covering. The case of a proper surjective morphism being a composition of types (ii) and (iv), we may further restrict to the case of an étale covering. We writeTα = (Xα →Sα ←Yα),T = (X0 →S0←Y0). For type (ii), localizing onS0, we may assume thatAis finite, because everyh-covering of a quasi-compact scheme admits a finite subfamily that is anh-covering. The finiteness ofAimplies that`
α∈ASα =`
α∈AS0 →S0is proper. We conclude by Proposition1.3(ii) for types (i), (iii), (v), and by Proposition1.3(i) for types (ii) and (iv).
The proof of the following proposition is similar to [SGA 4Vbis3.3.3].
Proposition 1.10. — A hypercovering for universal descent inC is an augmentation of descent.
2. Variants and counterexamples
Remark 2.1. — LetΛbe a ring annihilated by a positive integer. ForK∈D+(Y´et, Λ), Section1holds withB replaced by the larger category Fun(V,S), and the orientedh-topology in the first assertion of Corollary1.9 replaced by therestricted orientedh-topologyon Fun(V,S), generated by families of types (i) through (v), where for type (ii) we restrict to families of finite index sets. In view of the second assertion of Corollary1.9, Proposition1.10in this setting implies the hyper base change theorem exp.XIIA,2.2.5(without the assumption thatgis coherent).
Remark 2.2. — In Section1, we may replaceKby an ind-finite stack onY´et. For proper base change and proper cohomological descent in this setting, we refer to exp.XX,7.1and [Orgogozo, 2003, Proposition 2.5] (the latter extends easily to stacks not necessarily in groupoids). Thus we obtain another proof of the non-Abelian hyper base change theorem (see exp.XIIA,2.2.6.2).
Remark 2.3(Gabber). — Section1also holds withS replaced by the category of topological spaces,Bre- placed by the category Fun(V,S),Kreplaced by either a complex inD+(Y,Z)(not necessarily of torsion co- homology sheaves), or a stack onY (not necessarily ind-finite), and the orientedh-topology replaced by the restricted orientedh-topology. Proper morphisms are defined to be separated and universally closed, and étale morphisms are defined to be local homeomorphisms. The analogue of the proper base change theorem for topological spaces in the Abelian case is shown in [SGA 4Vbis4.1.1]. See Section3for the case of stacks.
Note that the first assertion of Corollary1.9in this setting does not hold for the orientedh-topology. Indeed, the family(Ti→T)i>0, whereTi= ({0}→[0, 1]←ξi),T = ({0}→[0, 1]←`
i>0ξi),ξi={1/i}, is covering for the orientedh-topology, but the morphism`
i>0Ti→T is not of descent for generalK.
Remark 2.4. — Classical cohomological descent and the descent formalism in this note can be dealt with uniformly using the language of∞-categories. In fact, Proposition1.5, Corollary 1.6, and Proposition 1.10 (resp. their non-Abelian analogues in Remark 2.2) incarnate abstract descent properties in ∞-categories ([Liu & Zheng, 2012, Lemma 3.1.2], [Liu & Zheng, 2014, Section 4.1]) applied to the functor from the nerve of C to the derived ∞-category D(X´et,Z) (resp. to the2-category of stacks onX´et) sending an object of C corresponding to the commutative diagram
X0 a //
x
S0
Y0
oo b
y
X //S Yoo
tox?a?b?y?K.
Remark 2.5. — We conclude this section by making corrections to [SGA 4Vbis3.2.1, 3.3.1 b)]. As the referee points out, [SGA 4Vbis 3.3.1 b)] would imply, by takingg = fand applying [SGA 4 Vbis 3.3.1 a)] tof0, that every morphism of cohomologicalG-descent is of universal cohomologicalG-descent. We have the following counterexample. We define a categoryBby taking for objects the pairs(M, w), whereMis a set andw:M→ {0, 1, 2}is a function, and for morphisms(M, w)→(M0, w0)the mapsµ: M→M0such thatw(m)≤w0(µ(m)) for allm∈M. The categoryBadmits small limits and small colimits. Small coproducts inBare disjoint and universal. Consider the functorF:B→Ensto the category of sets carrying(M, w)toMqw−1(1)and carrying µ: (M, w)→(M0, w0)to the mapMqw−1(1)→M0qw0−1(1)induced by the mapsM−→µ M0→M0qw0−1(1) andw−1(1)−→µ w0−1(2)qw0−1(1)−−−ιqId→M0qw0−1(1), whereι: w0−1(2)→M0is the inclusion. Fori∈{0, 1, 2}, we let ¯i denote the object ({?}, i)of B, where {?}denotes a singleton. By construction, F(0) =¯ F(2) =¯ {?} and F(1) =¯ {?}q{?}. We define a categoryE bifibered in duals of topoi overEnsby taking for objects the pairs(N,F), whereN is a set andF is a presheaf on the discrete categoryN, and for morphisms the pairs (ν, φ) : (N,F)→(N0,F0), whereν:N→N0is a map andφ:F0→ν?F is a morphism of presheaves. LetΛ be a nonzero ring. We consider the categoryE ×EnsBbifibered in duals of topoi overBand the corresponding categoryGbifibered in the categories ofΛ-modules over the opposite ofB. Then, in the Cartesian square
0¯ f
0 //
g0
0¯
g
1¯ f //2,¯
f0 is of universal effective cohomologicalG-descent,gis of effective cohomological G-descent, but neitherf norg0is of cohomologicalG-descent. The constant simplicial objects associated to ¯0and ¯1augmented over ¯2 also provide a counterexample to [SGA 4Vbis3.2.1].
In [SGA 4Vbis3.2.1], one should make the additional assumption thatvremains an augmentation of coho- mologicalG-descent after every base changeXn →S,n≥0. In [SGA 4Vbis3.3.1 b)],gshould be assumed to be ofuniversalcohomologicalG-i-descent.
3. Appendix: Proper base change for stacks on topological spaces
The results and proofs in this section are completely due to Gabber. Recall that a continuous mapf:X→Y between topological spaces is said to be separatedif the diagonal embedding X → X×Y Xis closed. We say that a continuous map between topological spaces isproperif it is separated and universally closed, or, equivalently, separated and closed with compact fibers. We donotassume stacks to be in groupoids.
Theorem 3.1. — Consider a Cartesian square
X0 g
0 //
f0
X
f
Y0 g //Y
in the category of topological spaces withfproper. Then, for every stackC onX, the base change morphism g?f?C →f?0g0?C
is an equivalence.
For a generalization to higher stacks in the case of locally compact Hausdorff spaces, we refer to [Lurie, 2009, Corollary 7.3.1.18].
The first reduction steps for Theorem3.1are similar to the Abelian case [SGA 4 Vbis 4.1.1]. Theorem3.1 is equivalent to Corollary3.2below. We obtain the corollary by takingY0 to be a point in the theorem. By [Giraud, 1971, III.2.1.5.8], to show the theorem, it suffices to show that the base change morphism is an equiv- alence after taking stalks, which follows from the corollary applied tof andf0 by considering the sequence (f?C)y→(f?0g0?C)y0 →(j?C)(f−1(y)), wherey0is a point ofY0andy=g(y0).
Corollary 3.2. — Letf:X→Y be a proper map between topological spaces and letybe a point ofY. Then, for every stackC onX, the functor(f?C)y→(j?C)(f−1(y))is an equivalence. Herej:f−1(y)→Xdenotes the inclusion.
We have (cf. [Giraud, 1971, VII.2.1.5.2])(f?C)y ' colimVC(f−1(V)), where colim means2-colimit, and V runs through open neighborhoods of yinY. Sincefis closed, the f−1(V)form a fundamental system of neighborhoods off−1(y). Therefore, Corollary3.2follows from case (i) of the following theorem.
Theorem 3.3. — LetXbe a topological space and letKbe a topological subspace satisfying any of the following condi- tions:
i. Kis compact and every pair of distinct points ofKhave disjoint neighborhoods inX; ii. Xis paracompact Hausdorff andKis closed inX;
iii. Xis metrizable.
We letj:K→Xdenote the inclusion. Then, for every stackC onX, the functor α: colimUC(U)→(j?C)(K),
induced by the restriction functorsC(U) → (j?C)(K), where Uruns through open neighborhoods ofK in X, is an equivalence.
The Abelian analogue of the theorem was proved in [SGA 4Vbis4.1.3] for case (i) and in [Godement, 1973, Théorème II.4.11.1] for cases (ii) and (iii).
Proof. — The theorem holds for sheaves of sets: for cases (ii) and (iii), this is [Godement, 1973, Théorème II.3.3.1, Corollaire 1]; for case (i), the proof forH0 of Abelian sheaves given in [SGA 4 Vbis 4.1.3] works in general. Since taking sheaves of morphisms commutes with direct and inverse images [Giraud, 1971, II.3.1.5.3, II.3.2.8] and filtered colimits, it follows that αis fully faithful. Thus it suffices to show that αis essentially surjective.
Next we perform two reduction steps similar to exp.XX,6.2. A given sectionsofj?C generates a maximal subgerbe, corresponding to a section of the sheaf of maximal subgerbes Ger(j?C). By [Giraud, 1971, III.2.1.5.5], Ger(j?C)'j?Ger(C). By the known case of sheaves of sets of the theorem, the section ofj?Ger(C)extends to a section of Ger(C)on an open neighborhoodU, corresponding to a maximal subgerbeG of the restriction of C toUsuch thatsis a section of the restriction ofG toK. If the theorem holds forG, then we obtain the desired extension ofs. Therefore, we may assume thatC is a gerbe.
We claim that ifβ:C →C0 is a faithful morphism of stacks onXand the theorem holds forC0, then the theorem holds forC. Lets∈(j?C)(K)be a section. By the theorem forC0, the image ofsin(j?C0)(K)extends to a section ofC0on an open neighborhoodU. The stack of liftingsU×C0 C is a sheaf by the faithfulness of β. Thus, by the known case of sheaves of the theorem, the section ofj?(U×C0C)'K×j?C0j?C defined bys extends to a section ofU×C0C on an open neighborhood ofKinU, which gives an extension ofsas claimed.
LetXδbeXwith the discrete topology and letε:Xδ→Xbe the identity map. For any sheaf of setsF onX, the adjunction mapF →ε?ε?F is a monomorphism. Since taking sheaves of morphisms commutes withε? andε?, the adjunction morphismC →ε?ε?C is faithful. By the above claim, it is enough to prove the theorem forε?ε?C. The gerbeε?C onXδis necessarily equivalent to the gerbe ofG-torsors for a sheaf of groupsGon Xδ. Thenε?ε?C is equivalent to the gerbe ofε?G-torsors, and it suffices to show that every j?ε?G-torsor on Kis trivial. Sinceε?G is flabby,j?ε?Gis soft in cases (i) and (ii) by [SGA 4Vbis 4.1.5] and [Godement, 1973, Théorème II.3.4.2] and flabby in case (iii) by [Godement, 1973, Théorème II.3.3.1, Corollaire 2]. We conclude by the following lemma.
Lemma 3.4. — LetGbe a flabby sheaf of groups on a topological spaceX, or a soft sheaf of groups on a paracompact Hausdorff spaceX. Then anyG-torsorPonXhas a section.
Proof. — Consider an open covering(Ui)i∈Isuch thatP(Ui)is nonempty for alli∈I. In the flabby case, consider the set of pairs(J, σ), whereJ⊂Iandσ∈P(S
i∈JUi), ordered by extension. By Zorn’s lemma, there exists a maximal element(J0, σ0). LetU0=S
i∈J0Ui. Leti ∈I. Chooseσi∈P(Ui). The element ofG(U0∩Ui)carrying the restriction ofσito the restriction ofσ0extends to an elementgi∈G(Ui). Patchingσ0andσigiproduces an extension ofσ0toU0∪Ui. By the maximality ofJ0, we geti∈J0. Therefore, J0=Iandσ0∈P(X).
The soft case is similar. SinceXis paracompact, we may assume(Ui)i∈I locally finite. By a usual lemma on normal spaces, there exists an open covering(Vi)i∈I withVi ⊂Uifor alli ∈ I. Consider the set of pairs (J, σ), whereJ⊂ Iandσis a section ofPrestricted toS
i∈JVi, ordered by extension. By local finiteness, the hypothesis of Zorn’s lemma is verified. Thus there exists a maximal element(J0, σ0). LetV=S
i∈J0Vi, which is closed inX. Leti ∈ I. Chooseσi ∈ P(Vi). The element ofG(V∩Vi)carrying the restriction ofσito the restriction ofσ0extends to an elementgi∈G(Vi). Patchingσ0andσigiproduces an extension ofσ0toV∪Vi. By the maximality ofJ0, we geti∈J0. Therefore,J0=Iandσ0∈P(X).
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