Passage to the limit in non-abelian Čech cohomology

Benedictus Margaux

Abstract: We give a detailed proof of the good behaviour of non-abelian cohomology under passage to the limit on the base scheme. Mathematics Subject Classification: Primary 14F20, 14L15. Secondary 17B65. Key words and phrases: Limit of schemes. Non-abelian cohomology.

1 Introduction

One of the central technical results in étale cohomology ([SGA4] théorème VII 5.7 and its corollaries) is the good behaviour of H^{q}(S, G) for S = \underleftarrow{lim}S_{λ} and G = \underleftarrow{lim}G_{λ} under reasonable assumptions on the schemes S_{λ} and the abelian group schemes G_{λ}. It is also remarked without proof therein (ibid. remarque 5.14(a)) that similar results hold for nonabelian H^{1} defined à la Čech, i.e., for sheaf torsors. This passage to the limit appears as a crucial ingredient in the study of Galois cohomology of local and henselian rings, as well as in the study of infinite dimensional Lie theory by cohomological methods (see [CTO], [GP1], [GP2] and [GP3] for example). We have considered it useful to write down a detailed proof of this important fact. For clarity of exposition we have chosen not to restrict our attention to the case when the S_{λ} are affine (which would be sufficient for the work of Gille and Pianzola under consideration).

2 Passage to to the limit for non-abelian H^{1}

If X is a scheme and G is a group scheme over X, then H_{fppf}^{1}(X, G) will denote the pointed set of Čech cohomology for the fppf topology of X (see [SGA3] Exp IV.6 for details. See also [Gi] and [M]). H_{ét}^{1} and H_{Zar}^{1} are defined analogously.

Let S_{0} be a scheme. Throughout we assume that (S_{λ})_{λ∈Λ} is a projective system of S_{0}-schemes based on some non-empty directed set Λ such that for all λ ≥ μ the transition morphisms u_{λμ} : S_{μ} → S_{λ} are affine. We can then form the projective limit S = \underleftarrow{lim}S_{λ} in the category of S_{0}-schemes ([EGAIV] §8.2). By construction, we have for each λ ∈ Λ a canonical morphism u_{λ} : S → S_{λ}.

Let G_{0} be a group scheme over S_{0}. For λ ∈ Λ let G_{λ} = G_{0} ×_{S_{0}} S_{λ}, and G = G_{0} ×_{S_{0}} S. If U^{α} = (U_{i}^{α} → S_{α}) is a covering of S_{α} in the fppf topology, then the base change

(U_{i}^{α} ×_{S_{α}} S) ×_{S} (U_{j}^{α} ×_{S_{α}} S) ≃ (U_{i}^{α} ×_{S_{α}} U_{j}^{α}) ×_{S_{α}} S → U_{j}^{α} ×_{S_{α}} U_{j}^{α}

maps cocycles in Z_{fppf}^{1}(U^{α}, G_{α}) into cocycles in Z_{fppf}^{1}(U^{α} ×_{S_{α}} S, G). This leads to a map H_{fppf}^{1}(U^{α}, G_{α}) → H_{fppf}^{1}(U^{α} ×_{S_{α}} S, G) which, by passing to the limit over all coverings of S_{α}, yields a map ψ_{α} : H_{fppf}^{1}(S_{α}, G_{α}) → H_{fppf}^{1}(S, G). By considering \underrightarrow{lim}ψ_{α} we obtain a canonical map

ψ : \underrightarrow{lim}_{λ∈Λ} H_{fppf}^{1}(S_{λ}, G_{λ}) → H_{fppf}^{1}(S, G).

Completely analogous considerations hold for the Zariski and étale topology.

The main result is as follows.

Theorem 2.1. Assume that S_{0} and the (S_{λ})_{λ∈Λ} are all quasicompact and quasiseparated, and that the group G_{0} → S_{0} is locally of finite presentation. Then the canonical map

\underrightarrow{lim}_{λ∈Λ} H_{fppf}^{1}(S_{λ}, G_{λ}) → H_{fppf}^{1}(S, G)

is bijective. Similarly for H_{ét}^{1} and H_{Zar}^{1} .

We begin by establishing two preliminary results that will be used in the proof of the Theorem. The notation is chosen to closely match that of [EGAIV] (to which all references henceforth belong).

Lemma 2.2. Let S be a quasicompact scheme. Then any covering U = (U_{i} →^{ϕ_{i}} S)_{i∈I} of S (for either of our three topologies) admits a refinement V = (V_{ℓ} →^{ψ_{ℓ}} S)_{ℓ∈L} where L is finite and the V_{ℓ} are affine. If in addition S is quasiseparated, then the morphisms ψ_{ℓ} of the refinement V may be assumed to be of finite presentation.

Proof. The ϕ_{i} are open maps (being flat and locally of finite presentation. See théorème 2.4.6). Given that S is quasicompact, there exists a finite subcovering (in particular a refinement) of U.

Assume henceforth that I is finite. Let S = ⋃_{j∈J}Y_{j} be a finite open covering of S by

affine Y_{j} . Let W_{ij} = ϕ_{i}^{−1}(Y_{j}), and let W_{ij} = ∪_{k∈K}V_{ijk} be an open affine cover of W_{ij} (where K is some index set). Consider the morphisms ψ_{ijk} : V_{ijk} → S defined by

ψ_{ijk} : V_{ijk} ↪ W_{ij} →^{ϕ_{i}|_{W_{ij}}} Y_{j} ↪ S.

The V_{ijk} form a covering of S. Since any covering admits a finite subcovering, we may assume that K is finite. Set L = I × J × K, and define τ : L → I by τ : (i, j, k) ↦ i. Then V = (V_{ijk} →^{ψ_{ijk}} S)_{(i,j,k)∈L} together with τ yield a finite refinement by affine schemes of our original U.

We claim that if S is quasiseparated, then the morphism ψ_{ijk} above are of finite presentation, i.e., quasicompact, quasiseparated, and locally of finite presentation. The ψ_{ijk} are locally of finite presentation since V is a covering in one of our three topologies. That ψ_{ijk} is quasiseparated is automatic since the V_{ijk}, being affine, are quasiseparated (cor. 1.2.3(i)). Finally, since any morphism from a quasicompact scheme into a quasiseparated scheme is quasicompact (prop. 1.2.4), the ψ_{ijk} are quasicompact.

Proposition 2.3. Assume S_{0} is quasicompact and quasiseparated. Let f : X → S be a morphism of S_{0}-schemes which is of finite presentation. Then.

(i) There exist α ∈ Λ, and a scheme morphism f_{α} : X_{α} → S_{α} of finite presentation, such that X_{α} ×_{S_{α}} S ≃ X as S-schemes. (ii) If α is as in (i), then for f to be surjective (resp. an open immersion, flat, faithfully flat, étale), it is necessary and sufficient that there exist λ ≥ α for which f_{λ} = f_{α} × I_{S_{λ}} : X_{α} ×_{S_{α}} S_{λ} → S_{α} ×_{S_{α}} S_{λ} ≃ S_{λ} is surjective (resp. an open immersion, flat, faithfully flat, étale).

Proof. The existence of f_{α} : X_{α} → S_{α} as in (i) is given by théorème 8.8.2. In view of (i), to establish (ii) we may assume with no loss of generality that X = X_{α} ×_{S_{α}} S and f = f_{α} ×I_{S}. For all λ ≥ α set

X_{λ} = X_{α} ×_{S_{α}} S_{λ} and f_{λ} = f_{α} × I_{S_{λ}}.

The existence of λ as prescribed in (ii) follows from théorème 8.10.5 (for f surjective or an open immersion), théorème 11.2.6 (for f flat), and proposition 17.7.8 (for f étale). Combining surjectivity with flatness yields a λ for which f_{λ} is faithfully flat.

We are now ready to establish our main result.

Proof (of Theorem 2.1). For future use we begin with an observation. For α ∈ Λ the morphism u_{α} : S → S_{α} is affine, hence quasicompact and quasiseparated. Thus S itself is quasicompact and quasiseparated.

In what follows we fix one of our three topologies on S. All coverings, cocycles, and H^{1} will refer to this chosen topology.

ψ is surjective. Let c ∈ H^{1}(S, G), and choose a covering U = (U_{i} → S)_{i∈I} so that c corresponds to a cocycle z ∈ Z^{1}(U, G). Taking into account that Čech cohomology is defined by passing to the limit of all refinements of covers of S, we may by Lemma 2.2 assume with no loss of generality that I is finite and that the U_{i} → S are of finite presentation. Given that Λ is directed, Proposition 2.3 now yields the existence of an α ∈ Λ and a covering U^{α} = (U_{i}^{α} → S_{α}) which induces U under the base change u_{α} : S → S_{α}. Thus,

z_{ij} ∈ G((U_{i}^{α} ×_{S_{α}} S) ×_{S} (U_{j}^{α} ×_{S_{α}} S))

≃ G(U_{i}^{α} ×_{S_{α}} U_{j}^{α} ×_{S_{α}} S) = G(U_{i}^{α} ×_{S_{α}} U_{j}^{α} ×_{S_{α}} \underleftarrow{lim}_{λ≥α} S_{λ})

= \underrightarrow{lim}_{λ≥α} G_{λ}(U_{i}^{α} ×_{S_{α}} U_{j}^{α} ×_{S_{α}} S_{λ})

= \underrightarrow{lim}_{λ≥α} G_{λ}((U_{i}^{α} ×_{S_{α}} S_{λ}) ×_{S_{λ}} (U_{j}^{α} ×_{S_{α}} S_{λ}))

(this penultimate equality of limits because G is locally of finite presentation. See théorème 8.8.2(i)). This yields the existence of a β ≥ α for which there exists elements z_{ij}^{β} ∈ G_{β}((U_{i}^{α} ×_{S_{α}} S_{β}) ×_{S_{β}} (U_{j}^{α} ×_{S_{α}} S_{β})) such that z_{ij}^{β} ↦ z_{ij} under the base change u_{β} : S → S_{β}. We do not know whether the z_{ij}^{β} satisfy the cocycle condition, but since the z_{ij} do, we can again use the fact that G is locally of finite presentation to conclude that there exists γ ≥ β such that the image z_{ij}^{γ} of the z_{ij}^{β} under the base change u_{βγ} : S_{γ} → S_{β} form a cocyle. Since z_{ij}^{γ} ↦ z_{ij}, our map ψ is surjective.

ψ is injective. Let c_{1} ∈ H^{1}(S_{α_{1}}, G_{α_{1}}) and c_{2} ∈ H^{1}(S_{α_{2}}, G_{α_{2}}) be such that ψ(c_{1}) = ψ(c_{2}). We must show that c_{1} and c_{2} have the same image under the respective canonical maps

(2.2)  H^{1}(S_{α_{n}}, G_{α_{n}}) → \underrightarrow{lim}_{λ≥α_{n}} H^{1}(S_{λ}, G_{λ}),

where n = 1, 2. Since Λ is directed, we may assume with no loss of generality that α_{1} = α_{2} = α for some α ∈ Λ. We may also assume, as explained above and after taking a common refinement, that c_{n} corresponds to a cocyle z_{n}^{α} ∈ Z^{1}(U^{α}, G_{α}) for some covering U^{α} = (U_{i}^{α} → S_{α})_{i∈I} of S_{α} with I finite and U_{i}^{α} affine. Since ψ(c_{1}) = ψ(c_{2}), there exists a refinement V = (V_{j} → S)_{j∈J} of the cover U^{α} ×_{S_{α}} S = (U_{i}^{α} ×_{S_{α}} S → S)_{i∈I} where the images of z_{1}^{α} and z_{2}^{α} become cohomologous. We may again assume J to be finite and the V_{j} to be affine.

Let Λ′ = {λ ∈ Λ : λ ≥ α}. Assume i ∈ I and j ∈ J are such that a morphism V_{j} → U_{i}^{α} ×_{S_{α}} S is part of our refinement. The same reasoning used at the end of the proof of Lemma 2.2 shows that this morphism is of finite presentation.^{1}

For λ ∈ Λ′ define S′_{λ} = U_{i}^{α} ×_{S_{α}} S_{λ} and S′ = U_{i}^{α} ×_{S_{α}} S. Then S′ = \underleftarrow{lim}S′_{λ} where the limit is taken over Λ′. By Proposition 2.3 applied to S′ , there exists λ ∈ Λ′ such that our (flat, étale...) morphism V_{j} → S′ comes from a (flat, étale...) morphism V_{j}^{λ} → S′_{λ} by the base change S′ → S′_{λ} arising from u_{λ}. In fact, since I and J are finite and Λ′ is directed, there exists β ≥ α such that our entire refinement V of U^{α} comes from a refinement V^{β} of the covering U^{α} ×_{S_{α}} S_{β} by the base change S → S_{β}. Replacing the z_{n}^{α} by their respective images z_{n}^{β} ∈ Z^{1}(U^{α} ×_{S_{α}} S_{β}, G_{β}), and then U^{α} ×_{S_{α}} S_{β} by its refinement V^{β} does not change our c_{n}. This allows us to reduce to the case where our original cocycles z_{1}^{α} and z_{2}^{α} are such that their images z_{1} and z_{2} in Z^{1}(U^{α} ×_{S_{α}} S, G) are cohomologous. Accordingly, there exists elements g_{i} ∈ G(U_{i}^{α} ×_{S_{α}} S) such that

1To see that U_{i}^{α} ×_{S_{α}} S is quasiseparated observe that because U_{i}^{α} is affine, it is quasiseparated over S_{α}. Thus U_{i}^{α} ×_{S_{α}} S is quasiseparated over S, and we can now conclude from the fact that S is quasiseparated.

(2.3)  g_{i}(z_{1})_{ij}g_{j}^{−1} = (z_{2})_{ij} for all i, j ∈ I

(where in (2.3) the g_{i}’s are restricted to the U_{i}^{α} ×_{S_{α}} U_{j}^{α} ×_{S_{α}} S as usual). Because G is locally of finite presentation the g_{i}’s may be assumed to come from some elements g_{i}^{γ} ∈ G_{γ}(U_{i}^{α} ×_{S_{α}} S_{γ}) for some γ ≥ α. Replacing z_{1}^{α} and z_{2}^{α} by their images z_{1}^{γ} and z_{2}^{γ} under the base change u_{αγ} : S_{γ} → S_{α} we see that g_{i}^{γ}(z_{1}^{γ})_{ij}(g_{j}^{γ})^{−1} and (z_{2}^{γ})_{ij} have the same image under the base change u_{γ} : S → S_{γ}. Again since G is of finite presentation, we obtain that g_{i}^{δ}(z_{1}^{δ})_{ij}(g_{j}^{δ})^{−1} = (z_{2}^{δ})_{ij} after a base change u_{γδ} : S_{δ} → S_{γ} with δ ≥ γ suitably chosen.

Remark 2.4. We assume throughout that S_{0} and the S_{λ} are quasicompact and quasiseparated.

(a) If G_{0} is flat, affine and locally of finite presentation over S_{0}, then by descent theory the sheaf torsors whose isomorphism classes are measured by H_{fppf}^{1}(S_{λ}, G_{λ}) are representable. Similarly for H_{fppf}^{1}(S, G). The surjectivity of the map \underrightarrow{lim}H_{fppf}^{1}(S_{λ}, G_{λ}) → H_{fppf}^{1}(S, G) is as in (10.16) of [SGA3] VI_{B} (where the case when the S_{λ} are affine and the G_{λ} are of finite presentation is studied). Groups which are locally of finite presentation but not finitely presented (which are covered by our result) arise naturally in the classification of reductive groups, in particular of tori.

(b) Let G be a finitely presented group scheme over S. By Proposition 2.3 the group G is obtained from a finitely presented group over S_{α} by base change. Replacing Λ by Λ_{α} = {λ ∈ Λ : λ ≥ α} puts us back within the assumptions of Theorem 2.1. Thus, H^{1}(S, G) can be computed in terms of direct limits.

(c) If G is a flat affine and finitely presented group scheme over S, and if Y is a torsor over S under G, then the twisted S-group _{Y} G is also finitely presented and the considerations of (b) above apply.

(d) In Theorem 2.1 the assumption that G_{0} be representable is not crucial. The proof goes through as long as G_{0} is a group functor on S_{0} which is locally of finite presentation.^{2} An important example is the case when G_{0} is the group of automorphisms Aut_{S_{0}}(X_{0}) of a finitely presented scheme X_{0} over S_{0} (see [A] for details).

References

[A] M. Artin, Algebraic approximation of structures over complete local rings, I.H.É.S. Publ. Math 36 (1969), 23–58.

2I am thankful to B. Conrad for bringing this point to my attention.

[CTO] J.–L. Colliot–Thélène, M. Ojanguren, Espaces principaux homogènes localement triviaux, I.H.É.S. Publ. Math. 75 (1992), 97–122.

[GP1] P. Gille and A. Pianzola, Isotriviality of torsors over Laurent polynomials rings, C. R. Acad. Sci. Paris, Ser. I 340 (2005) 725–729.

[GP2] P. Gille and A. Pianzola, Galois cohomology and forms of algebras over Laurent polynomial rings, Math. Annalen (in press).

[GP3] P. Gille and A. Pianzola, Isotriviality and étale cohomology of Laurent polynomial rings, preprint (2007).

[Gi] J. Giraud, Cohomologie non abélienne, Die Grundlehren der mathematischen Wissenschaften 179 (1971), Springer-Verlag.

[M] J.S. Milne, Étale cohomology, Princeton University Press (1980).

[EGAIV] A. Grothendieck (avec la collaboration de J. Dieudonne) Eléments de Géométrie Algébrique IV, Public. de l’I.H.É.S. no 20, 24, 28 and 32 (1964 - 1967).

[SGA3] Séminaire de Géométrie algébrique de l’I.H.É.S., 1963-1964, Schémas en groupes, dirigé par M. Demazure et A. Grothendieck, Lecture Notes in Math. 151-153. Springer (1970).

[SGA4] Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964, Théorie des topos et cohomologie étale des schémas. Tome 2, dirigé par M. Artin, A. Grothendieck et J. L. Verdier, Lecture Notes in Mathematics 270 (1972), Springer-Verlag.

Benedictus Margaux Laboratoire de Recherche “Princess Stephanie” Monte Carlo 51840, Monaco benedictus.margaux@gmail.com