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III.4: Čech Cohomology

V.4.1. #to_work

Let \(f: X \rightarrow Y\) be an affine morphism of noetherian separated schemes (II, Ex. 5.17). Show that for any quasi-coherent sheaf \(\mathcal{F}\) on \(X\), there are natural isomorphisms for all \(i \geqslant 0\), ¹ \begin{align*} H^i(X, \mathcal{F}) \cong H^i\left(Y, f_* \mathcal{F}\right) \end{align*}

V.4.2. #to_work

Prove Chevalley’s theorem: Let \(f: X \rightarrow Y\) be a finite surjective morphism of noetherian separated schemes, with \(X\) affine. Then \(Y\) is affine.

• Let \(f: X \rightarrow Y\) be a finite surjective morphism of integral noetherian schemes. Show that there is a coherent sheaf \(\mathcal{M}\) on \(X\), and a morphism of sheaves \(\alpha: \mathcal{O}_Y^r \rightarrow f_* \mathcal{M}\) for some \(r>0\), such that \(\alpha\) is an isomorphism at the generic point of \(Y\).

• For any coherent sheaf \(\mathcal{F}\) on \(Y\), show that there is a coherent sheaf \(\mathcal{G}\) on \(X\), and a morphism \(\beta: f_* \mathcal{G} \rightarrow \mathcal{F}^r\) which is an isomorphism at the generic point of \(Y\). ²

• Now prove Chevalley’s theorem. First use (Ex. 3.1) and (Ex. 3.2) to reduce to the case \(X\) and \(Y\) integral. Then use (3.7), (Ex. 4.1), consider \(\operatorname{ker} \beta\) and coker \(\beta\), and use noetherian induction on \(Y\).

V.4.3. #to_work

Let \(X=\mathbf{A}_k^2=\operatorname{Spec} k[x, y]\), and let \(U=X-\{(0,0)\}\). Using a suitable cover of \(U\) by open affine subsets, show that \(H^1\left(U, \mathcal{O}_U\right)\) is isomorphic to the \(k\)-vector space spanned by \(\left\{x^i y^j \mathrel{\Big|}i, j<0\right\}\). In particular, it is infinite-dimensional. ³

V.4.4. #to_work

On an arbitrary topological space \(X\) with an arbitrary abelian sheaf \(\mathcal{F}\), Čech cohomology may not give the same result as the derived functor cohomology. But here we show that for \(H^1\), there is an isomorphism if one takes the limit over all coverings.

• Let \(\mathfrak{U}=\left(U_i\right)_{i \in I}\) be an open covering of the topological space \(X\). A refinement of \(\mathfrak{U}\) is a covering \(\mathfrak{B}=\left(V_j\right)_{j \in J}\), together with a map \(\lambda: J \rightarrow I\) of the index sets, such that for each \(j \in J, V_j \subseteq U_{\lambda(j)}\). If \(\mathfrak{B}\) is a refinement of \(\mathfrak{X}\), show that there is a natural induced map on Čech cohomology, for any abelian sheaf \(\mathcal{F}\), and for each \(i\), \begin{align*} \lambda^i: \check{H}^i(\mathfrak{U}, \mathcal{F}) \rightarrow \check{H}^i(\mathfrak{B}, \mathcal{F}) . \end{align*} The coverings of \(X\) form a partially ordered set under refinement, so we can consider the Ceech cohomology in the limit \begin{align*} \colim_{\mathfrak U} \check{H}^i(\mathfrak{U}, \mathcal{F}) . \end{align*}

• For any abelian sheaf \(\mathcal{F}\) on \(X\), show that the natural maps (4.4) for each covering \begin{align*} \check{H}^i(\mathfrak{U}, \mathcal{F}) \rightarrow H^i(X, \mathcal{F}) \end{align*} are compatible with the refinement maps above.

• Now prove the following theorem. Let \(X\) be a topological space, \(\mathcal{F}\) a sheaf of abelian groups. Then the natural map \begin{align*} \colim_{\mathfrak U} \check{H}^1(\mathfrak{U}, \mathcal{F}) \rightarrow H^1(X, \mathcal{F}) \end{align*} is an isomorphism. ⁴

V.4.5. #to_work

For any ringed space \(\left(X, \mathcal{O}_X\right)\), let \(\operatorname{Pic}X\) be the group of isomorphism classes of invertible sheaves (II, §6). Show that \(\operatorname{Pic}X \cong H^1\left(X, \mathcal{O}_X^*\right)\), where \(\mathcal{O}_X^*\) denotes the sheaf whose sections over an open set \(U\) are the units in the ring \(\Gamma\left(U, \mathcal{O}_X\right)\), with multiplication as the group operation. ⁵

V.4.6. #to_work

Let \(\left(X, \mathcal{O}_X\right)\) be a ringed space, let \(\mathcal{I}\) be a sheaf of ideals with \(\mathcal{I}^2=0\), and let \(X_0\) be the ringed space \(\left(X, \mathcal{O}_X / \mathcal{I}\right)\). Show that there is an exact sequence of sheaves of abelian groups on \(X\), \begin{align*} 0 \rightarrow \mathcal{I} \rightarrow \mathcal{O}_X^* \rightarrow \mathcal{O}_{X_0}^* \rightarrow 0, \end{align*} where \(\mathcal{O}_X^*\) (respectively, \(\mathcal{O}_{X_0}^*\) ) denotes the sheaf of (multiplicative) groups of units in the sheaf of rings \(\mathcal{O}_X\) (respectively, \(\mathcal{O}_{X_0}\) ) the map \(\mathcal{I} \rightarrow \mathcal{O}_X^*\) is defined by \(a \mapsto\) \(1+a\), and \(\mathcal{I}\) has its usual (additive) group structure. Conclude there is an exact sequence of abelian groups \begin{align*} \ldots \rightarrow H^1(X, \mathcal{I}) \rightarrow \operatorname{Pic} X \rightarrow \operatorname{Pic} X_0 \rightarrow H^2(X, \mathcal{I}) \rightarrow \ldots . \end{align*}

V.4.7. #to_work

Let \(X\) be a subscheme of \(\mathbf{P}_k^2\) defined by a single homogeneous equation \(f\left(x_0, x_1, x_2\right)=0\) of degree \(d\). (Do not assume \(f\) is irreducible.) Assume that \((1,0,0)\) is not on \(X\). Then show that \(X\) can be covered by the two open affine subsets \(U=X \cap\left\{x_1 \neq 0\right\}\) and \(V=X \cap\left\{x_2 \neq 0\right\}\). Now calculate the Čech complex \begin{align*} \Gamma\left(U, \mathcal{O}_X\right) \oplus \Gamma\left(V, \mathcal{O}_X\right) \rightarrow \Gamma\left(U \cap V, \mathcal{O}_X\right) \end{align*} explicitly, and thus show that \begin{align*} \operatorname{dim} H^0\left(X, \mathcal{O}_X\right)&=1, \\ \operatorname{dim} H^1\left(X, \mathcal{O}_X\right)&=\frac{1}{2}(d-1)(d-2) . \end{align*}

V.4.8. Cohomological Dimension. #to_work

Let \(X\) be a noetherian separated scheme. We define the cohomological dimension of \(X\), denoted \(\operatorname{cd}(X)\), to be the least integer \(n\) such that \(H^i(X, \mathcal{F})=0\) for all quasi-coherent sheaves \(\mathcal{F}\) and all \(i>n\).

Thus for example, Serre’s theorem (3.7) says that \(\operatorname{cd}(X)=0\) if and only if \(X\) is affine. Grothendieck’s theorem (2.7) implies that \(\operatorname{cd}(X) \leqslant \operatorname{dim} X\).

• In the definition of \(\operatorname{cd}(X)\), show that it is sufficient to consider only coherent sheaves on \(X\). Use (II, Ex. 5.15) and (2.9).

• If \(X\) is quasi-projective over a field \(k\), then it is even sufficient to consider only locally free coherent sheaves on \(X\). Use (II, 5.18).

• Suppose \(X\) has a covering by \(r+1\) open affine subsets. Use Čech cohomology to show that \(\operatorname{cd}(X) \leqslant r\).

• * If \(X\) is a quasi-projective scheme of dimension \(r\) over a field \(k\), then \(X\) can be covered by \(r+1\) open affine subsets. Conclude (independently of (2.7)) that \(\operatorname{cd}(X) \leqslant \operatorname{dim} X\).

• Let \(Y\) be a set-theoretic complete intersection (I, Ex. 2.17) of codimension \(r\) in \(X=\mathbf{P}_k^n\). Show that \(\operatorname{cd}(X-Y) \leqslant r-1\).

V.4.9. #to_work

Let \(X=\operatorname{Spec} k\left[x_1, x_2, x_3, x_4\right]\) be affine four-space over a field \(k\). Let \(Y_1\) be the plane \(x_1=x_2=0\) and let \(Y_2\) be the plane \(x_3=x_4=0\). Show that \(Y=Y_1 \cup Y_2\) is not a set-theoretic complete intersection in \(X\). Therefore the projective closure \(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu\) in \(\mathbf{P}_k^4\) is also not a set-theoretic complete intersection. ⁶

V.4.10. #to_work

* Let \(X\) be a nonsingular variety over an algebraically closed field \(k\), and let \(\mathcal{F}\) be a coherent sheaf on \(X\). Show that there is a one-to-one correspondence between the set of infinitesimal extensions of \(X\) by \(\mathcal{F}\) (II, Ex. 8.7) up to isomorphism, and the group \(H^1(X, \mathcal{F} \otimes \mathcal{T})\), where \(\mathcal{T}\) is the tangent sheaf of \(X\), see \((II \S 8)\). ⁷

V.4.11. #to_work

This exercise shows that Cech 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, \(\mathcal{F}\) a sheaf of abelian groups, and \(\mathfrak{U}=\left(U_i\right)\) an open cover. Assume for any finite intersection \(V=U_{i_0} \cap \ldots \cap U_{i_p}\) of open sets of the covering, and for any \(k>0\), that \(H^k\left(V,\left.\mathcal{F}\right|_V\right)=0\). Then prove that for all \(p \geqslant 0\), the natural maps \begin{align*} \check{H}^p(\mathfrak{U}, \mathcal{F}) \rightarrow H^p(X, \mathcal{F}) \end{align*} of (4.4) are isomorphisms. Show also that one can recover (4.5) as a corollary of this more general result.