Lemma 20.16.2. Let $X$ be a Hausdorff and quasi-compact topological space. Let $\mathcal{F}$ be an abelian sheaf on $X$. Then the map $\check{H}^ n(X, \mathcal{F}) \to H^ n(X, \mathcal{F})$ defined in (20.15.0.1) is an isomorphism for all $n$.

Proof. We already know that $\check{H}^ n(X, -) \to H^ p(X, -)$ is an isomorphism of functors for $n = 0, 1$, see Lemma 20.16.1. The functors $H^ n(X, -)$ form a universal $\delta$-functor, see Derived Categories, Lemma 13.20.4. If we show that $\check{H}^ n(X, -)$ forms a universal $\delta$-functor and that $\check{H}^ n(X, -) \to H^ n(X, -)$ is compatible with boundary maps, then the map will automatically be an isomorphism by uniqueness of universal $\delta$-functors, see Homology, Lemma 12.12.5.

Let $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ be a short exact sequence of abelian sheaves on $X$. Let $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ be an open covering. This gives a complex of complexes

$0 \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{G}) \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{H}) \to 0$

which is in general not exact on the right. The sequence defines the maps

$\check{H}^ n(\mathcal{U}, \mathcal{F}) \to \check{H}^ n(\mathcal{U}, \mathcal{G}) \to \check{H}^ n(\mathcal{U}, \mathcal{H})$

but isn't good enough to define a boundary operator $\delta : \check{H}^ n(\mathcal{U}, \mathcal{H}) \to \check{H}^{n + 1}(\mathcal{U}, \mathcal{F})$. Indeed such a thing will not exist in general. However, given an element $\overline{h} \in \check{H}^ n(\mathcal{U}, \mathcal{H})$ which is the cohomology class of a cocycle $h = (h_{i_0 \ldots i_ n})$ we can choose open coverings

$U_{i_0 \ldots i_ n} = \bigcup W_{i_0 \ldots i_ n, k}$

such that $h_{i_0 \ldots i_ n}|_{W_{i_0 \ldots i_ n, k}}$ lifts to a section of $\mathcal{G}$ over $W_{i_0 \ldots i_ n, k}$. By Topology, Lemma 5.13.5 (this is where we use the assumption that $X$ is hausdorff and quasi-compact) we can choose an open covering $\mathcal{V} : X = \bigcup _{j \in J} V_ j$ and $\alpha : J \to I$ such that $V_ j \subset U_{\alpha (j)}$ (it is a refinement) and such that for all $j_0, \ldots , j_ n \in J$ there is a $k$ such that $V_{j_0 \ldots j_ n} \subset W_{\alpha (j_0) \ldots \alpha (j_ n), k}$. We obtain maps of complexes

$\xymatrix{ 0 \ar[r] & \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \ar[d] \ar[r] & \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{G}) \ar[d] \ar[r] & \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{H}) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F}) \ar[r] & \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G}) \ar[r] & \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{H}) \ar[r] & 0 }$

In fact, the vertical arrows are the maps of complexes used to define the transition maps between the Čech cohomology groups. Our choice of refinement shows that we may choose

$g_{j_0 \ldots j_ n} \in \mathcal{G}(V_{j_0 \ldots j_ n}),\quad g_{j_0 \ldots j_ n} \longmapsto h_{\alpha (j_0) \ldots \alpha (j_ n)}|_{V_{j_0 \ldots j_ n}}$

The cochain $g = (g_{j_0 \ldots j_ n})$ is not a cocycle in general but we know that its Čech boundary $\text{d}(g)$ maps to zero in $\check{\mathcal{C}}^{n + 1}(\mathcal{V}, \mathcal{H})$ (by the commutative diagram above and the fact that $h$ is a cocycle). Hence $\text{d}(g)$ is a cocycle in $\check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F})$. This allows us to define

$\delta (\overline{h}) = \text{class of }\text{d}(g)\text{ in } \check{H}^{n + 1}(\mathcal{V}, \mathcal{F})$

Now, given an element $\xi \in \check{H}^ n(X, \mathcal{G})$ we choose an open covering $\mathcal{U}$ and an element $\overline{h} \in \check{H}^ n(\mathcal{U}, \mathcal{G})$ mapping to $\xi$ in the colimit defining Čech cohomology. Then we choose $\mathcal{V}$ and $g$ as above and set $\delta (\xi )$ equal to the image of $\delta (\overline{h})$ in $\check{H}^ n(X, \mathcal{F})$. At this point a lot of properties have to be checked, all of which are straightforward. For example, we need to check that our construction is independent of the choice of $\mathcal{U}, \overline{h}, \mathcal{V}, \alpha : J \to I, g$. The class of $\text{d}(g)$ is independent of the choice of the lifts $g_{i_0 \ldots i_ n}$ because the difference will be a coboundary. Independence of $\alpha$ holds1 because a different choice of $\alpha$ determines homotopic vertical maps of complexes in the diagram above, see Section 20.15. For the other choices we use that given a finite collection of coverings of $X$ we can always find a covering refining all of them. We also need to check additivity which is shown in the same manner. Finally, we need to check that the maps $\check{H}^ n(X, -) \to H^ n(X, -)$ are compatible with boundary maps. To do this we choose injective resolutions

$\xymatrix{ 0 \ar[r] & \mathcal{F} \ar[r] \ar[d] & \mathcal{G} \ar[r] \ar[d] & \mathcal{H} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{I}_1^\bullet \ar[r] & \mathcal{I}_2^\bullet \ar[r] & \mathcal{I}_3^\bullet \ar[r] & 0 }$

as in Derived Categories, Lemma 13.18.9. This will give a commutative diagram

$\xymatrix{ 0 \ar[r] & \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \ar[r] \ar[d] & \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \ar[r] \ar[d] & \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}_1^\bullet )) \ar[r] & \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}_2^\bullet )) \ar[r] & \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}_3^\bullet )) \ar[r] & 0 }$

Here $\mathcal{U}$ is an open covering as above and the vertical maps are those used to define the maps $\check{H}^ n(\mathcal{U}, -) \to H^ n(X, -)$, see Lemma 20.11.2. The bottom complex is exact as the sequence of complexes of injectives is termwise split exact. Hence the boundary map in cohomology is computed by the usual procedure for this lower exact sequence, see Homology, Lemma 12.13.12. The same will be true after passing to the refinement $\mathcal{V}$ where the boundary map for Čech cohomology was defined. Hence the boundary maps agree because they use the same construction (whenever the first one is defined on an element in Čech cohomology on a given covering). This finishes our discussion of the construction of the structure of a $\delta$-functor on Čech cohomology and why this structure is compatible with the given $\delta$-functor structure on usual cohomology.

Finally, we may apply Lemma 20.11.1 to see that higher Čech cohomology is trivial on injective sheaves. Hence we see that Čech cohomology is a universal $\delta$-functor by Homology, Lemma 12.12.4. $\square$

 This is an important check because the nonuniqueness of $\alpha$ is the only thing preventing us from taking the colimit of Čech complexes over all open coverings of $X$ to get a short exact sequence of complexes computing Čech cohomology.

Comment #4856 by Kiran Kedlaya on

Maybe make it more clear in the text that the Hausdorff assumption is being used in the call to 09UW? It would also be helpful to point to Grothendieck's example (from section 3.8 of "Sur quelques points") that shows why this hypothesis is needed. (There is also a variant by Schroer which is Hausdorff but not paracompact; see arXiv:1309.2524.)

Comment #4862 by MAO Zhouhang on

Let me have a try to explain conceptually how separatedness and compactness are used. In fact, Hausdorff + paracompactness implies existence of (locally finite) star refinements. A reference is http://at.yorku.ca/p/a/c/a/02.pdf. Let me explain the definition of star refinement: Given a cover $\mathcal V$ of a space $X$. A cover $\mathcal U$ is called a star refinement of the cover $\mathcal V$ if for all $U\in\mathcal U$, the star $\operatorname{st}(U,\mathcal U):=\{U'\in\mathcal U\colon U\cap U'\neq\emptyset\}$ of $U$ in $\mathcal U$ is a subset of some $V\in\mathcal V$. Note that this condition is "point-less", that is, we can migrate it to any locale. This implies that, for a paracompact space $X$ and any open cover $\mathcal U$ of $X$, if we are given an open cover for all $p+1$-intersections, then we can refine $\mathcal U$ such that all $p+1$-intersections constitute a refinement of open covers on $p+1$-intersections of $\mathcal U$ in some fashion (precisely, Lemma 7.2.3.5 in Lurie's Higher Algebra). As a consequence, we deduce Čech-derived comparison in question. This is essentially the proof in Godement's book, which is modernizes in Theorem 1.3.13 in Brylinski's Loop Spaces, Characteristic Classes and Geometric Quantization.

A side note: it seems that the concept of star refinements could be generalized to sites but I don't know whether it is useful.

Comment #5145 by on

@#4856: OK, I added that here.

@#4862. Thanks for the explanation.

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