This exercise shows that Čech cohomology will agree with the usual cohomology whenever the sheaf has no cohomology on any of the open sets. More precisely, let \(X\) be a topological space, \(\mcf\) a sheaf of abelian groups, and \(\mathfrak{U}=(U_i)\) an open cover. Assume for any finite intersection \(V=U_{i_0} \intersect \cdots \intersect U_{i_p}\) of open sets of the covering, and for any \(k>0\), that \(H^k(V, \ro{\mcf}{V})=0\). Then prove that for all \(p \geq 0\), the natural maps \[ \check{H}^p(\mathfrak{U}, \mcf) \to H^p(X, \mcf) \] of (4.4) are isomorphisms. Show also that one can recover (4.5) as a corollary of this more general result.
Solution
Choose an injective resolution \(\mcf\to I^\bullet\) in the category of abelian sheaves on \(X\). Use the double complex \[ B^{p,q}=C^p(\mathfrak U,I^q) =\prod_{i_0<\cdots<i_p}\Gamma(U_{i_0}\cap\cdots\cap U_{i_p},I^q) \qquad(p,q\ge0), \] with the Čech differential horizontally and the resolution differential vertically. Its total differential on bidegree \((p,q)\) is \(d_{\mathrm{Cech}}+(-1)^p d_I\). Each total degree contains only finitely many bidegrees, even when the cover itself is infinite.
1. The augmentation \(\Gamma(X,I^\bullet)\to\operatorname{Tot}B\) is a quasi-isomorphism.
Every injective abelian sheaf \(I^q\) is flasque [@Har10a, Lemma III.2.4]. A flasque sheaf has zero positive-degree Čech cohomology on every open cover, and its degree-zero Čech cohomology is its group of global sections [@Har10a, Lemma III.4.1 and Proposition III.4.3]. Thus the horizontal cohomology of \(B\) is concentrated in column zero, where it is the complex \(\Gamma(X,I^\bullet)\). The first-quadrant double-complex filtration gives the asserted quasi-isomorphism. Equivalently, filter the cone of the augmentation by the resolution degree: its graded complexes are the exact augmented Čech rows, so its total cohomology is zero. The filtration has finitely many terms in each total degree, which is the convergence condition used here. This is the augmented double-complex construction in P-AGH344REFINEMENTH1, step 3.
2. Under the intersection-acyclicity hypothesis, \(C^\bullet(\mathfrak U,\mcf)\to\operatorname{Tot}B\) is a quasi-isomorphism as well.
1.Thus no finiteness of the index set of the cover was required.
3. The natural comparison maps in the statement are isomorphisms in all degrees.
3.Both maps are now isomorphisms, giving\[ \boxed{\check H^p(\mathfrak U,\mcf)\xrightarrow{\cong}H^p(X,\mcf)\qquad(p\ge0).} \]The construction is compatible with morphisms of sheaves and refinements of the cover by the same double-complex maps.
4. The affine-cover comparison theorem (III.4.5) follows.
Let \(X\) be a noetherian separated scheme, let \(\mathfrak U\) be an affine open cover, and let \(\mcf\) be quasi-coherent. Every finite intersection of these opens is affine because \(X\) is separated. The restriction of \(\mcf\) to such an intersection is quasi-coherent, so all its positive-degree cohomology vanishes by T-COHAFF [@Har10a, Theorem III.3.5]. The hypothesis of step 2 is therefore satisfied. Step 3 supplies the Čech comparison isomorphisms in all degrees, which is the assertion of Theorem III.4.5.
5. Q.E.D.
Steps <1>1–<1>3 prove the general acyclic-cover theorem, and step 4 recovers the requested affine-cover corollary.