Let \(X\) be a noetherian scheme. Show that \(X\) is affine if and only if \(X_{\mathrm{red}}\) (II, Ex. 2.3) is affine.
Hint: Use (3.7), and for any coherent sheaf \(\mcf\) on \(X\), consider the filtration \(\mcf \supseteq \mcn \cdot \mcf \supseteq \mcn^2 \cdot \mcf \supseteq \cdots\), where \(\mcn\) is the sheaf of nilpotent elements on \(X\).
Solution
Let \(\mcn \subseteq \OO_X\) be the sheaf of nilpotent sections, the ideal sheaf of the closed immersion \(X_{\mathrm{red}} \to X\).
1. If \(X = \Spec A\) is affine, then \(X_{\mathrm{red}} = \Spec A/N\) for the nilradical \(N\) of \(A\), so \(X_{\mathrm{red}}\) is affine.
On an affine scheme the reduced induced structure is defined by the nilradical.
2. \(\mcn^n = 0\) for some \(n \geq 1\).
Cover \(X\) by finitely many affine opens \(\Spec A_j\) with \(A_j\) Noetherian; \(\mcn\) restricts on \(\Spec A_j\) to the sheaf of the nilradical \(N_j\). \(N_j\) is generated by finitely many nilpotent elements \(x_1, \ldots, x_s\), with \(x_i^m = 0\) for all \(i\), and every product of \(s(m-1) + 1\) generators contains some \(x_i\) at least \(m\) times, so \(N_j^{s(m-1)+1} = 0\). Take \(n\) to be the largest of these finitely many exponents.
3. If \(X_{\mathrm{red}}\) is affine, then \(H^1(X, \mcf) = 0\) for every coherent sheaf \(\mcf\) on \(X\).
For \(0 \leq i < n\), the coherent sheaf \(\mcn^i \mcf / \mcn^{i+1} \mcf\) is annihilated by \(\mcn\), so it is the pushforward of a coherent sheaf on \(X_{\mathrm{red}}\) along the closed immersion, which is a homeomorphism; its \(H^1\) on \(X\) is its \(H^1\) on the affine Noetherian scheme \(X_{\mathrm{red}}\), which is \(0\). The exact sequences \(0 \to \mcn^{i+1} \mcf \to \mcn^i \mcf \to \mcn^i \mcf / \mcn^{i+1} \mcf \to 0\) give \(H^1(X, \mcn^{i+1} \mcf) = 0 \Rightarrow H^1(X, \mcn^i \mcf) = 0\). Since \(\mcn^n \mcf = 0\) by step 2, descending induction on \(i\) gives \(H^1(X, \mcf) = 0\).
4. Q.E.D.
Step 1 is the forward implication. Conversely, if \(X_{\mathrm{red}}\) is affine, step 3 gives \(H^1(X, \mci) = 0\) for every coherent sheaf of ideals \(\mci\), so \(X\) is affine by Serre’s criterion T-5IOUR.