## 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 $i \geqslant 0$,[^hint.3.4.1]
\[
H^i(X, \mathcal{F}) \cong H^i\left(Y, f_* \mathcal{F}\right)
\]

[^hint.3.4.1]: 
Hint: Use (II, 5.8).

### 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.

a. 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 $Y$.


b. 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 $Y$.[^hint.3.4.2.b]


c. 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 $Y$.

[^hint.3.4.2.b]: 
Hint: Apply $\sheafhom(\cdot, \mathcal{F})$ to $\alpha$ and use (II, Ex. 5.17e).


### 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 $\left\{x^i y^j \mid i, j<0\right\}$. In particular, it is infinite-dimensional.[^rmk.3.4.3]

[^rmk.3.4.3]: 
Using (3.5), this provides another proof that $U$ is not affine-cf. (I, Ex. 3.6).

### 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.

a. 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 $i$,
\[
\lambda^i: \check{H}^i(\mathfrak{U}, \mathcal{F}) \rightarrow \check{H}^i(\mathfrak{B}, \mathcal{F}) .
\]
The coverings of $X$ form a partially ordered set under refinement, so we can consider the Ceech cohomology in the limit
\[
\colim_{\mathfrak U} \check{H}^i(\mathfrak{U}, \mathcal{F}) .
\]

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

c. Now prove the following theorem. Let $X$ be a topological space, $\mathcal{F}$ a sheaf of abelian groups. Then the natural map
\[
\colim_{\mathfrak U} \check{H}^1(\mathfrak{U}, \mathcal{F}) \rightarrow H^1(X, \mathcal{F})
\]
is an isomorphism.[^hint.3.4.qqqqqqq]

[^hint.3.4.qqqqqqq]: 
Hint: Embed $\mathcal{F}$ in a flasque sheaf $\mathcal{G}$, and let $\mathcal{R}=\mathcal{G}/\mathcal{F}$, so that we have an exact sequence
\[
0 \rightarrow \mathcal{F} \rightarrow \mathcal{G} \rightarrow \mathcal{R} \rightarrow 0 .
\]
Define a complex $D^{\cdot}(\mathfrak{U})$ by
\[
0 \rightarrow C^{\cdot}(\mathfrak{U}, \mathcal{F}) \rightarrow C^{\cdot}(\mathfrak{U}, \mathcal{G}) \rightarrow D^{\prime}(\mathfrak{U}) \rightarrow 0 .
\]
Then use the exact cohomology sequence of this sequence of complexes, and the natural map of complexes
\[
D^*(\mathfrak{U}) \rightarrow C^*(\mathfrak{U}, \mathcal{R}),
\]
and see what happens under refinement.


### V.4.5.  #to-work

For any ringed space $\left(X, \mathcal{O}_X\right)$, let $\Pic X$ be the group of isomorphism classes of invertible sheaves (II, §6). Show that $\Pic X \cong H^1\left(X, \mathcal{O}_X^*\right)$, 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.[^hint.3.4.5]

[^hint.3.4.5]: 
Hint: For any invertible sheaf $\mathcal{L}$ on $X$, cover $X$ by open sets $U_i$ on which $\mathcal{L}$ is free, and fix isomorphisms $\phi_i: \OO_{U_i} \iso \ro{\mcl}{U_i}$. 
Then on $U_i \cap U_j$, we get an isomorphism $\varphi_i^{-1} \circ \varphi_j$ of $\mathcal{O}_{U_i \cap U_j}$ with itself. These isomorphisms give an element of $\check{H}^1\left(\mathfrak{U}, \mathcal{O}_X^*\right)$. Now use (Ex. 4.4).


### 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 $X$,
\[
0 \rightarrow \mathcal{I} \rightarrow \mathcal{O}_X^* \rightarrow \mathcal{O}_{X_0}^* \rightarrow 0,
\]
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
\[
\ldots \rightarrow H^1(X, \mathcal{I}) \rightarrow \operatorname{Pic} X \rightarrow \operatorname{Pic} X_0 \rightarrow H^2(X, \mathcal{I}) \rightarrow \ldots .
\]

### 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
\[
\Gamma\left(U, \mathcal{O}_X\right) \oplus \Gamma\left(V, \mathcal{O}_X\right) \rightarrow \Gamma\left(U \cap V, \mathcal{O}_X\right)
\]
explicitly, and thus show that
\[
\operatorname{dim} H^0\left(X, \mathcal{O}_X\right)&=1, \\
\operatorname{dim} H^1\left(X, \mathcal{O}_X\right)&=\frac{1}{2}(d-1)(d-2) .
\]

### 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 $\operatorname{cd}(X) \leqslant \operatorname{dim} X$.

a. 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).

b. 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).

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

d. \* 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 $\operatorname{cd}(X) \leqslant \operatorname{dim} X$.

e. 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 $\bar{Y}$ in $\mathbf{P}_k^4$ is also not a set-theoretic complete intersection.[^hint.3.4.9]

[^hint.3.4.9]: 
Hints: Use an affine analogue of (Ex. 4.8e). Then show that $H^2\left(X-Y, \mathcal{O}_X\right) \neq 0$, by using (Ex. 2.3) and (Ex. 2.4). If $P=Y_1 \cap Y_2$, imitate (Ex. 4.3) to show $H^3\left(X-P, \mathcal{O}_X\right) \neq 0$.

### 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 $(II \S 8)$.[^hint.3.4.10]

[^hint.3.4.10]: 
Hint: Use (II, Ex. 8.6) and (4.5).


### 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
\[
\check{H}^p(\mathfrak{U}, \mathcal{F}) \rightarrow H^p(X, \mathcal{F})
\]
of (4.4) are isomorphisms. Show also that one can recover (4.5) as a corollary of this more general result.