## III.3: Cohomology of a Noetherian Affine Scheme 

### V.3.1.  #to-work

Let $X$ be a noetherian scheme. Show that $X$ is affine if and only if $X_{\text {red }}$ (II, Ex. 2.3) is affine.[^hint.3.3.1]

[^hint.3.3.1]: 
Hint: Use (3.7), and for any coherent sheaf $\mathcal{F}$ on $X$, consider the filtration $\mathcal{F} \supseteq \mathcal{N} \cdot \mathcal{F} \supseteq \mathcal{N}^2 \cdot \mathcal{F} \supseteq \ldots$, where $\mathcal{N}$ is the sheaf of nilpotent elements on $X$.


### V.3.2.  #to-work

Let $X$ be a reduced noetherian scheme. Show that $X$ is affine if and only if each irreducible component is affine.


### V.3.3.  #to-work

Let $A$ be a noetherian ring, and let $\mathfrak{a}$ be an ideal of $A$.

a. Show that $\Gamma_{\mathrm{\mfa}}(\cdot)$ (II, Ex. 5.6) is a left-exact functor from the category of $A$-modules to itself. We denote its right derived functors, calculated in $\mathsf{Mod}(A)$, by $H_{\mfa}^i(\cdot)$.

b. Now let $X=\operatorname{Spec} A, Y=V(\mathfrak{a})$. Show that for any $A$-module $M$,
\[
H_{\mfa }^i(M)=H_Y^i(X, \tilde{M}),
\]
where $H_Y^i(X, \cdot)$ denotes cohomology with supports in $Y($ Ex. 2.3).

c. For any $i$, show that $\Gamma_{\mfa}\left(H_{\mathfrak{a}}^i(M)\right)=H_{\mathfrak{a}}^i(M)$.

### V.3.4. Cohomological Interpretation of Depth. #to-work

If $A$ is a ring, $\mfa$ an ideal, and $M$ an $A$ module, then $\depth_\mfa M$ is the maximum length of an $M$-regular sequence $x_1, \ldots, x_r$, with all $x_i \in \mathfrak{a}$. This generalizes the notion of depth introduced in $(II, \S 8)$.

a. Assume that $A$ is noetherian. Show that if $\depth_\mfa M \geqslant 1$, then $\Gamma_{\mathfrak{a}}(M)=0$, and the converse is true if $M$ is finitely generated.[^hint.3.3.4.a]


b. Show inductively, for $M$ finitely generated, that for any $n \geqslant 0$, the following conditions are equivalent:
    (i) $\depth_{\mathfrak{a}} M \geqslant n$;
    (ii) $H_\mfa^i(M)=0$ for all $i<n$.

[^hint.3.3.4.a]: 
Hint: When $M$ is finitely generated, both conditions are equivalent to saying that $\mathfrak{a}$ is not contained in any associated prime of $M$.


### V.3.5.  #to-work

Let $X$ be a noetherian scheme, and let $P$ be a closed point of $X$. Show that the following conditions are equivalent:

(i) $\depth \mathcal{O}_P \geqslant 2$;
(ii) if $U$ is any open neighborhood of $P$, then every section of $\mathcal{O}_X$ over $U-P$ extends uniquely to a section of $\mathcal{O}_X$ over $U$.

This generalizes (I, Ex. 3.20), in view of (II, 8.22A).

### V.3.6.  #to-work

Let $X$ be a noetherian scheme.

a. Show that the sheaf $\mathcal{G}$ constructed in the proof of (3.6) is an injective object in the category $\QCoh(X)$ of quasi-coherent sheaves on $X$. Thus $\QCoh(X)$ has enough injectives.

b. \* Show that any injective object of $\QCoh(X)$ is flasque.[^hint.3.3.6.b]

c. Conclude that one can compute cohomology as the derived functors of $\Gamma(X, \cdot)$, considered as a functor from $\QCoh(X)$ to $\Ab$.


[^hint.3.3.6.b]: 
Hints: The method of proof of (2.4) will not work, because $\mathcal{O}_U$ is not quasi-coherent on $X$ in general. Instead, use (II, Ex. 5.15) to show that if $\mathcal{I} \in \mathbb{Q} \mathrm{co}(X)$ is injective, and if $U \subseteq X$ is an open subset, then $\left.\mathcal{I}\right|_U$ is an injective object of $\operatorname{Qco}(U)$. Then cover $X$ with open affines $\cdots$.


### V.3.7.  #to-work

Let $A$ be a noetherian ring, let $X=\operatorname{Spec} A$, let $\mathfrak{a} \subseteq A$ be an ideal, and let $U \subseteq X$ be the open set $X-V(\mathfrak{a})$.

a. For any $A$-module $M$, establish the following formula of Deligne:
\[
\Gamma(U, \tilde{M}) \cong \colim_n \Hom_A\left(\mfa^n, M\right) .
\]

b. Apply this in the case of an injective $A$-module $I$, to give another proof of (3.4).


### V.3.8.  #to-work

Without the noetherian hypothesis, (3.3) and (3.4) are false. Let $A=k\left[x_0, x_1, x_2, \ldots\right]$ with the relations $x_0^n x_n=0$ for $n=1,2, \ldots$. Let $I$ be an injective $A$-module containing $A$. Show that $I \rightarrow I_{x_0}$ is not surjective.