equivalence of Grothendieck-style versus Cech-style sheaf cohomology

Question

Given a topological space $X$, we can define the sheaf cohomology of $X$ in

I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$)

or

II. the Čech style (first by defining the Čech cohomology groups subordinate to an open cover, and then taking the direct limit of these groups over all covers).

When exactly are these two definitions equivalent? I'm unhappy with the explanation given by Hartshorne. Are they the same for any paracompact Hausdorff space? Or a locally contractible space?

And what is the relationship between these two sheaf cohomologies and singular cohomology?

Any elaboration on this circle of ideas related to the relationship between all the different cohomology theories would be appreciated.

Answer 1

Dear Victoria: here is a summary of the main comparison results I know of between Grothendieck cohomology (which is usually just called cohomology and written $\newcommand{\F}{\mathcal F}H^i(X,\F)$ ) and other cohomologies.

1) If $X$ is locally contractible then the cohomology of a constant sheaf coincides with singular cohomology. [This is Eric's answer, but there is no need for his hypothesis that open subsets be acyclic]

2) Cartan's theorem: Given a topological space $X$ and a sheaf $\F$, assume there exists a basis of open sets $\mathcal{U}$, stable under finite intersections, such that the CECH cohomology groups for the sheaf $\F$ are trivial (in positive dimension) for every open $U$ in the basis: $H^i(U,\F)=0$ Then the Cech cohomology of $\F$ on $X$ coincides with (Grothendieck) cohomology

3) Leray's Theorem: Given a topological space $X$ and a sheaf $\F$, assume that for some covering $(U_i)$ of $X$ we know that the (Grothendieck!) cohomology in positive dimensions of $\F$ vanishes on every finite intersection of the $U_i$'s. Then the cohomology of $\F$ is already calculated by the Cech cohomology OF THE COVERING $(U_i)$: no need to pass to the inductive limit on all covers. This contains Dinakar's favourite example of a quasi-coherent sheaf on a separated scheme covered by affines.

4) If $X$ is paracompact and Hausdorff, Cech cohomology coincides with Grothendieck cohomology for ALL SHEAVES
If you think this is too nice to be true, you can check Théorème 5.10.1 in Godement's book cited below [So Eric's remark that no matter how nice the space is, Cech cohomology would probably not coincide with derived functor cohomology for arbitrary sheaves turns out to be too pessimistic]

5) Cohomology can be calculated by taking sections of any acyclic resolution of the studied sheaf: you don't need to take an injective resolution. This contains De Rham's theorem that singular cohomology can be calculated with differential forms on manifolds.

6) If you study sheaves of non-abelian groups, Cech cohomology is convenient: for example vector bundles on $X$ ( a topological space or manifold or scheme or...) are parametrized by $H^1(X, GL_r)$. I don't know if there is a description of sheaf cohomology for non-abelian sheaves in the derived functor style.

Good references are

a) A classic: Godement, Théorie des faisceaux (in French, alas)

b) S.Ramanan, Global Calculus,AMS graduate Studies in Mahematics, volume 65. (An amazingly lucid book, in the best Indian tradition.)

c) Torsten Wedhorn's quite detailed on-line notes, which prove 1) above (Theorem 9.16, p.92) and much, much more.
By the way, @Wedhorn is one of the two authors of a great book on algebraic geometry.

d) Ciboratu, Proposition 2.1 and Voisin's Hodge Theory and Complex Algebraic Geometry I, Theorem 4.47, page 109 , which both also prove 1) above.

Answer 2

I like to say that there is only a single abstract definition of cohomology: in any $(\infty,1)$-topos $\mathbf{H}$ given objects $X$ and $A$, the cohomology of $X$ with coefficients in $A$ is the connected components of the hom-$\infty$-groupoid $H(X,A) := \pi_0 \mathbf{H}(X,A)$.

Everything else one sees described as "cohomology" is, i claim, a special case and a special realization of this situation.

More on this point of view is at cohomology

In particular, ordinary abelian sheaf cohomology for sheaves on a cite $C$ is the cohomology in this sense of the $(\infty,1)$-topos of $\infty$-stacks on C where the coefficient objects are, moreover, restricted to be objectwise in the image of the Dold-Kan map (are "maximally abelian $\infty$-stacks").

From this perspective the relation betwen Cech-cohomology and other means to compute sheaf-cohomology become conceptually evident: all of these are just models to model the $(\infty,1)$-cateorical hom-space $\mathbf{H}(X,A)$: Cech cohomology does so by finding cofibrant versions of $X$ (namely Cech nerves of Cech covers), derived-functor-style sheaf cohomology usually does so by finding fibrant versions of $A$ (namely injective resolutions of sheaves).

That this is the relation between the two is of course implicitly the old Verdier hypercovering theorem. A particularly clear-sighted description of this is the remarkable old article by Kenneth Brown, Abstract homotopy theory and generalized sheaf cohomology.

A summary of that in the light of the above comments is at nlab:abelian sheaf cohomology.

Technical details are also at Cech cohomology.

Answer 3

I don't know if it's bad form to reply to something this old, but I stumbled on this question because I've wondering about the negative result for a couple of days now. That is, for an explicit example where Cech cohomology differs from (derived functor) sheaf cohomology. In case anyone is also curious about this, I did find an example buried in pages 177-179 of Grothendieck's classic "Tohoku" paper "Sur quelque points...". Perhaps I can say a few words about it since it is surprisingly simple. Take X to be the affine plane over a field, and let $Y\subset X$ be the union of two irreducible curves meeting at two distinct points. Let K be the kernel of the restriction map $Z_X\to Z_Y$ of the Z-valued constant sheaves on the Zariski topology. Then he shows that $H^2(X,K)=Z$ but that the Cech group $\check{H}^2(X,K)= 0$. (I wrote this backwards previously, sorry about that.)

Answer 4

The problem with Cech cohomology is that even if things are acyclic on open sets of your Cech cover, they may not be when you restrict to intersections of those open sets. The usual fix is to make the cover finer so you don't have that problem. Unfortunately there are topological spaces where no cover will be good enough. That's the bad news.

The good news is that for a lot of spaces and categories of sheaves you're interested in, there will be such a cover. My favorite example is the category of quasi-coherent sheaves on a (noetherian) separated scheme. Then Cech cohomology computed on any affine cover will compute the derived functor cohomology.

The even better news is that there is a way to fix Cech cohomology so that it will work for all situations. This is Verdier's theory of hypercovers, and it computes derived functor cohomology for any category with a Grothendieck topology. I must admit I have not played around much with this, but here is a link to a paper that talks about this circle of ideas.

Answer 5

Let $X$ be a topological space, and $T$ its category of open sets with the usual Grothendieck topology. Let $T'$ be any sieve of $T$ (a subcategory of $T$ such that if $U$ is in $T'$ then any subset of $U$ is also in $T'$). For example, $T'$ might be the collection of open subsets subordinate to the open subsets in a cover $\mathcal{U}$. Any sheaf on $T$ induces a functor on $T'$ which can be viewed as a sheaf on $T'$ if $T'$ is given the minimal topology (the only covers are the identity maps). This determines a morphism of topoi $f : T \rightarrow T'$, hence a spectral sequence

$H^p(T', R^q f_\ast F) \Rightarrow H^{p+q}(T, F)$ .

(One could surely also convince oneself that such a spectral sequence exists without any reference to topoi.)

The Cech cohomology of $F$ with respect to some covering family $\mathcal{U}$ is

$H^p(\mathcal{U}, F) = H^p(T', f_\ast F)$

where $T' = T'(U)$ is the sieve associated to the cover $\mathcal{U}$. The Cech cohomology is then the filtered colimit

$\check{H}^p(T, F) = \varinjlim_{(T',f)} H^p(T', f_\ast F)$

taken over the projections $f : T \rightarrow T'$ associated as above to covering families $\mathcal{U}$.

One evidently has edge homomorphisms

$\check{H}^p(T, F) \rightarrow H^p(T, F)$

from the spectral sequence, and the question is when these induce an isomorphism. If we could somehow eliminate the $R^p f_\ast F$, $p > 0$, by passing to a "small enough" cover we would have equality. This condition already holds in many cases; the following condition is more general (but I haven't checked carefully that it actually works!):

For every cover $\mathcal{U}$ of $X$, every $U_1, \ldots, U_n \in \mathcal{U}$, and every class in $\alpha \in H^p(U_1 \mathop{\times}_X \cdots \mathop{\times}_X U_n, F)$, $p > 0$, there exists a refinement $\mathcal{U}'$ of $\mathcal{U}$ such that the restriction of $\alpha$ under the map

$H^p(U_1 \mathop{\times}_X \cdots \mathop{\times}_X U_n, F) \rightarrow H^p(U'_1 \mathop{\times}_X \cdots \mathop{\times}_X U'_n, F)$

is zero.

To make sense of this, one must use some convention for the covers $\mathcal{U}$ and $\mathcal{U}'$ to ensure there is a map as above. For example, one could work only with covers indexed by the points of $X$ (a cover is then a collection of neighborhoods of each point of $X$).

A more refined version of the above condition would say that Cech cohomology equals cohomology in degrees at most $q$ if the above condition holds for $p \leq q$. Since it always holds for $p = 0,1$ this implies that

$\check{H}^1(T, F) = H^1(T, F)$

in general.

Edit in response to David's comment:

The Cech complex always computes cohomology correctly in a presheaf category (i.e., when the topology is "chaotic": an object has no covers by anything except itself). Trying to compute cohomology in an arbitrary site using the Cech complex is (heuristically) something like trying to approximate the site by a presheaf category.

Here is how Cech cohomology computes cohomology of presheaves. Consider any category $T'$. If $F$ is a presheaf of groups on $T'$ then the sheaf cohomology groups of $F$ are the derived functors of the inverse limit for diagrams of shape $T'$. They are also computed as

$Ext(\mathbf{Z}, F)$

where $\mathbf{Z}$ is the constant sheaf associated to the integers. Remarkably, in a presheaf category, $\mathbf{Z}$ has a canonical projective resolution associated to any cover of the final presheaf. A cover of the final presheaf is a collection of objects $U$ of $T'$ such that every object of $T'$ has a map to at least one object of $U$. The $i$-th term of this complex is the direct sum, over all choices of $i$ elements $U_1, ..., U_i$ of $U$, of the groups $\mathbf{Z}_{U_1 \times \cdots \times U_i}$. (You can check this is projective by noting it is the extension by $0$ of $\mathbf{Z}$ from the slice category $T' / U_1 \times \cdots \times U_i$ and extension by $0$ preserves projectives (since it has an exact right adjoint) and $\mathbf{Z}$ is projective on the slice category since all higher cohomology of all sheaves vanishes (since it has a final object). It's also easy to check by a direct calculation.)

Denote this complex by $K$. Since this is a projective resolution of $\mathbf{Z}$, $\mathrm{Hom}(K, F)$ computes the cohomology of $F$. But it is also easy to see that this is just the Cech complex of $F$.

Answer 6

This isn't entirely complete, but here are some results. Exercise III.4.11 in Hartshorne gives that whenever a sheaf is acyclic on any intersection of sets in a cover, Cech cohomology agrees with the derived functors. In particular, if every open cover has a refinement with this property, the derived functors will agree with Cech cohomology defined as the limit over open covers. I don't think you can say any general result involving just the space; you need to know something about the sheaf too. If a sheaf is nowhere locally acyclic, no matter how nice the space is Cech cohomology is probably not going to agree with the derived functors.

These are related to singular cohomology by looking only at constant sheaves. On a locally contractible space every open subset of which is paracompact, singular cohomology with coefficients in A is the same as derived functor cohomology of the constant sheaf A (you may be able to get slightly better hypotheses than this; this is what I found I needed when I tried to prove this a while ago). Basically, this is because the (sheafification of the pre)sheaf of singular cochains forms an acyclic resolution of the constant sheaf. On the other hand, for any space having the homotopy type of a CW-complex, Cech cohomology of constant sheaves agrees with singular cohomology (because it satisfies the Eilenberg-Steenrod axioms). It follows, for example, that for any CW-complex or any manifold, singular cohomology agrees with both Cech and derived cohomology of constant sheaves.

Answer 7

Two small remarks:

1. The edge morphism $\check{\mathrm{H}}^1(X,\mathscr{F}) \to \mathrm{H}^1(X,\mathscr{F})$ is always an isomorphism. This follows from the 5-term exact sequence associated to the Čech-to-derived-functor spectral sequence https://en.wikipedia.org/wiki/%C4%8Cech-to-derived_functor_spectral_sequence. (This spectral sequence can also be used to prove the result 3) mentioned by Georges Elencwajg above, which allows practical computation of abelian sheaf cohomology in certain cases, e.g. quasi-coherent cohomology of separated schemes.)

2. If one uses hypercoverings instead of Čech cohomology, one always gets derived functor cohomology (see e.g. Verdier's hypercovering theorem in https://ncatlab.org/nlab/show/hypercover or Stacks Project https://stacks.math.columbia.edu/tag/01H0; there is also an spectral sequence for hypercoverings as in 1.).

This is also not only for topological spaces, but also for Grothendieck sites and abelian sheaf cohomology. Someone told me once that Čech cohomology is the "wrong" cohomology since it is defined on presheaves instead of sheaves.

Answer 8

I suggest in addition to the other answers checking out Brian Conrad's notes on cohomological descent. They're a little more to the point for applications to geometry.

Answer 9

Here are some positive results and counterexamples for etale cohomology.

Definition: Let $X$ be a scheme. Say that $X$ has property "$AF_{n}$" if for every collection $x_{1},\dotsc,x_{n} \in X$ of $n$ points of $X$ there exists an affine open subscheme $U \subseteq X$ containing all the $x_{i}$. Let $$ a(X) $$ denote the supremum of the positive integers $n$ for which $X$ has $AF_{n}$. We say that $X$ is an "AF-scheme" (or "FA-scheme") if $a(X) = \infty$.

See Gross [4], Section 2.

The relevant theorem is:

Theorem (Artin 1971 [1]): Let $X$ be a quasi-compact AF-scheme. Then for any abelian sheaf $\mathscr{F}$ on the etale site of $X$, the map $$ \check{\mathrm{H}}{}^{i}(X,\mathscr{F}) \to \mathrm{H}_{\mathrm{et}}^{i}(X,\mathscr{F}) $$ is an isomorphism for all $i$.

Milne [6, III, Theorem 2.17] also discusses Artin's proof. This is more a claim about the etale topology on $X$, namely that if $U \to X$ is an etale cover, then any etale cover $V \to U \times_{X} \dotsb \times_{X} U$ may be refined by an etale cover of the form $U' \times_{X} \dotsb \times_{X} U'$ for $U' \to X$ which refines $U \to X$.

Later Schröer refined Artin's result as follows:

Theorem (Schröer 2003 [7]): Let $X$ be a Noetherian scheme and let $n$ be an integer such that $n \le a(X)$. Then for any abelian sheaf $\mathscr{F}$ on the etale site of $X$, the map $$ \check{\mathrm{H}}{}^{i}(X,\mathscr{F}) \to \mathrm{H}_{\mathrm{et}}^{i}(X,\mathscr{F}) $$ is an isomorphism for $i \le n$ and an injection for $i = n+1$.

Example: For an example of a scheme $X$ and an abelian sheaf $\mathscr{F}$ on the etale site of $X$ for which the map $$ \check{\mathrm{H}}{}^{2}(X,\mathscr{F}) \to \mathrm{H}_{\mathrm{et}}^{2}(X,\mathscr{F}) $$ is not an isomorphism, see the answers to this question. The current answers discuss the cases (1) $\mathscr{F}$ is a constant sheaf and (2) $\mathscr{F} = \mathcal{O}_{X}$.

(Gabber) For an example of a scheme $X$ for which the map $$ \check{\mathrm{H}}{}^{2}(X,\mathbb{G}_{m}) \to \mathrm{H}_{\mathrm{et}}^{2}(X,\mathbb{G}_{m}) $$ is not an isomorphism, let $R$ be a normal noetherian strictly henselian local ring of dimension $\ge 2$ whose punctured spectrum $U$ has nonzero Picard group (see e.g. this and this for examples of such $R$), and let $X$ be the gluing of two copies of $\operatorname{Spec} R$ along $U$. This is a local version of the counterexample to $\operatorname{Br} = \operatorname{Br}'$ due to Edidin, Hassett, Kresch, Vistoli [9].

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Here are some conditions relevant to the AF-property:

Lemma (e.g. [8, 01ZY]): Let $X$ be a quasi-compact scheme admitting an ample line bundle. Then $X$ is an AF-scheme.

This is essentially the graded prime avoidance lemma.

The Chevalley conjecture, proved by Kleiman, states that for smooth proper varieties the converse is true:

Theorem (Kleiman 1966 [5]): Let $k$ be an algebraically closed field and let $X$ be a smooth proper $k$-scheme. If $X$ is an AF-scheme, then $X$ is projective over $k$.

Benoist proved the following generalization of Kleiman's result:

Theorem (Benoist 2013 [2]): Let $k$ be an algebraically closed field and let $X$ be a normal, finite type $k$-scheme. If $X$ is an AF-scheme, then $X$ is quasi-projective over $k$.

Theorem (Farnik 2013 [3, Theorem 2.2]) For every integer $n \ge 2$ there is a smooth proper variety $X$ with $a(X) = n$.

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References:

[1] M. Artin, "On the joins of Hensel rings", Advances in Mathematics 7 (1971) pp 282–296, doi:10.1016/S0001-8708(71)80007-5, core.ac.uk.

[2] O. Benoist, "Quasi-projectivity of normal varieties", International Mathematics Research Notices, vol 2013, no 17 (2012) pp 3878–3885, doi:10.1093/imrn/rns163, arXiv:1112.0975.

[3] M. Farnik, "On strengthening of the Kleiman-Chevalley criterion", Proceedings of the AMS 141 no 11 (2013) pp 4005-4013, doi:10.1090/S0002-9939-2013-11695-3.

[4] P. Gross, "Tensor generators on schemes and stacks", Algebraic Geometry 4 (4) (2017) pp 501–522, doi:10.14231/ag-2017-026, arXiv:1306.5418.

[5] S. Kleiman, "Toward a numerical theory of ampleness", Annals of Mathematics 84 No. 3 (1966) pp 293–344, doi:10.2307/1970447.

[6] J.S. Milne, Etale Cohomology, Princeton University Press (1980) JSTOR (subscription needed).

[7] S. Schröer, "The bigger Brauer group is really big", Journal of Algebra 262 (2003) pp 210–225, doi:10.1016/S0021-8693(03)00026-7, arXiv:math/0108135.

[8] Stacks Project, https://stacks.math.columbia.edu/.

[9] Edidin, Hassett, Kresch, Vistoli, "Brauer groups and quotient stacks", American Journal of Mathematics 123 No. 4 (2001), doi:10.1353/ajm.2001.0024, JSTOR, arXiv:math/9905049.

Answer 10

Let $f \colon X\to S$ be a scheme morphism having a direct image functor $f_*$ (for instance, $f_*$ is the global section functor). Since $X$ is quasicompact, there exists a finite affine cover $U$ of $X$ such that $f |_U$ is an affine morphism for any $U$ belongs to a cover. Then the standard complex $C(U; f_*)$ corresponding to the cover U is a resolution of the functor $f_*$. Therefore it can be used for computing higher direct images (= derived functors) of $f_*$.

If the localizations at different open sets of the cover $U$ commute,the complex $C(U; f_*)$ is homotopically equivalent to the Cech complex, $C(U; f_*)$ of the cover $U$. One can show that the following conditions are equivalent:

(a) For any affine cover $U$ of a scheme $X$, If the localizations at different open sets of the cover $U$ commute

(b) The scheme $X$ is separated.

In other words, the Cech complex is equivalent to the standard complex for any affine cover only if the scheme is separated. If the scheme $X$ is not separated, the higher cohomology of the Cech complex $C(U; f_*)$ are not isomorphic, for a general affine cover $U$, to the corresponding derived functors of $f_*$.