303x

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. ¹

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 \(A\).

• Show that \(\Gamma_{\mathrm{{\mathfrak{a}}}}(\cdot)\) (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 \(H_{{\mathfrak{a}}}^i(\cdot)\).

• Now let \(X=\operatorname{Spec} A, Y=V(\mathfrak{a})\). Show that for any \(A\)-module \(M\), \begin{align*} H_{{\mathfrak{a}}}^i(M)=H_Y^i(X, \tilde{M}), \end{align*} where \(H_Y^i(X, \cdot)\) denotes cohomology with supports in \(Y(\) Ex. 2.3).

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

V.3.4. Cohomological Interpretation of Depth. #to_work

If \(A\) is a ring, \({\mathfrak{a}}\) an ideal, and \(M\) an \(A\) module, then \(\operatorname{depth}_{\mathfrak{a}}M\) 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 \((II, \S 8)\).

• Assume that \(A\) is noetherian. Show that if \(\operatorname{depth}_{\mathfrak{a}}M \geqslant 1\), then \(\Gamma_{\mathfrak{a}}(M)=0\), and the converse is true if \(M\) is finitely generated. ²

• Show inductively, for \(M\) finitely generated, that for any \(n \geqslant 0\), the following conditions are equivalent:

  • \(\operatorname{depth}_{\mathfrak{a}} M \geqslant n\);
  • \(H_{\mathfrak{a}}^i(M)=0\) for all \(i<n\).

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:

• \(\operatorname{depth}\mathcal{O}_P \geqslant 2\);
• 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.

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

• * Show that any injective object of \({\mathsf{QCoh}}(X)\) is flasque. ³

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

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 \(X-V(\mathfrak{a})\).

• For any \(A\)-module \(M\), establish the following formula of Deligne: \begin{align*} \Gamma(U, \tilde{M}) \cong \colim_n \mathop{\mathrm{Hom}}_A\left({\mathfrak{a}}^n, M\right) . \end{align*}

• 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.