Let \(X\) be a noetherian scheme, and let \(P\) be a closed point of \(X\). Show that the following conditions are equivalent:
\(\depth \mco_P \geq 2\);
if \(U\) is any open neighborhood of \(P\), then every section of \(\mco_X\) over \(U-P\) extends uniquely to a section of \(\mco_X\) over \(U\).
This generalizes (I, Ex. 3.20), in view of (II, 8.22A).
Solution
Put \(R=\OO_{X,P}\), let \(\mathfrak m\) be its maximal ideal, and let \(S=X_P\) be the local space of P-AGH325LOCALSPACE. Write \(H_P^i\) for cohomology with supports in the closed singleton \(\{P\}\).
1. The space \(S\), equipped with the inverse image of \(\OO_X\), is the scheme \(\Spec R\).
Choose an affine neighborhood \(V=\Spec A\) of \(P\) and let \(\mathfrak p\subseteq A\) correspond to \(P\). Every generization of \(P\) lies in every open neighborhood of \(P\), hence lies in \(V\). Inside \(V\), these generizations are exactly the prime ideals \(\mathfrak q\subseteq\mathfrak p\). They are the image of the localization map \(\Spec A_{\mathfrak p}\to\Spec A\). This map is a homeomorphism onto that subspace: its distinguished opens are the intersections of the subspace with the opens \(D(a)\) for \(a\in A\). At the point associated to \(\mathfrak q\), the map on local rings is \[ A_{\mathfrak q}\xrightarrow{\cong}(A_{\mathfrak p})_{\mathfrak qA_{\mathfrak p}}. \] Thus the natural map from the inverse-image structure sheaf to the structure sheaf of \(\Spec A_{\mathfrak p}\) is an isomorphism on every stalk and hence an isomorphism. Since \(A_{\mathfrak p}=R\), this proves the assertion. The point \(P\) corresponds to \(\mathfrak m\).
2. For every open neighborhood \(U\) of \(P\) and every \(i\ge0\), there is a natural isomorphism\[ H_P^i(U,\OO_U)\cong H_{\mathfrak m}^i(R). \]
The scheme \(U\) is noetherian, so its underlying space is a Zariski space by P-AGH2317ZARISKISPACE, part (a). Its local space at \(P\) is the same \(S\), since all the generizations of \(P\) lie in \(U\). Apply P-AGH325LOCALSPACE to \(U\) and its structure sheaf. By step 1, the result is \[ H_P^i(U,\OO_U)\cong H_{\{\mathfrak m\}}^i(\Spec R,\OO_{\Spec R}). \] The comparison in P-AGH333LOCALCOH, part (b), identifies the right side with \(H_{\mathfrak m}^i(R)\), since \(V(\mathfrak m)=\{\mathfrak m\}\). This proves the stated comparison for arbitrary \(U\), not only for affine neighborhoods.
3. Condition (i) implies condition (ii).
The ring \(R\) is a nonzero noetherian local ring and a finite module over itself. By P-AGH334DEPTHCOH, part (b), the inequality \(\depth R\ge2\) gives \[ H_{\mathfrak m}^0(R)=H_{\mathfrak m}^1(R)=0. \] For any neighborhood \(U\) of \(P\), step 2 therefore makes both \(H_P^0(U,\OO_U)\) and \(H_P^1(U,\OO_U)\) zero. The supported-cohomology long exact sequence from P-AGH323SUPPORTS, part (e), begins \[ 0\to H_P^0(U,\OO_U)\to\Gamma(U,\OO_U) \to\Gamma(U\setminus\{P\},\OO_U) \to H_P^1(U,\OO_U). \] The two outer groups vanish, so the restriction map is an isomorphism. Surjectivity gives existence of every extension and injectivity gives its uniqueness, as required in (ii).
4. Condition (ii) implies condition (i).
Take an affine neighborhood \(V\) of \(P\). Condition (ii) makes \(\Gamma(V,\OO_V)\to\Gamma(V\setminus\{P\},\OO_V)\) an isomorphism. The exact sequence used in step 3 first gives \(H_P^0(V,\OO_V)=0\). Its continuation has the exact segment \[ \Gamma(V,\OO_V)\to\Gamma(V\setminus\{P\},\OO_V) \to H_P^1(V,\OO_V)\to H^1(V,\OO_V). \] The first arrow is surjective, and the last term is zero by affine vanishing [@Har10a, Theorem III.3.5]. Hence \(H_P^1(V,\OO_V)=0\) as well. Step 2 gives \(H_{\mathfrak m}^0(R)=H_{\mathfrak m}^1(R)=0\). The converse in P-AGH334DEPTHCOH, part (b), now yields \(\depth R\ge2\).
5. Q.E.D.
Steps 3 and 4 prove the two implications, using the local-ring comparison in steps <1>1–<1>2.