Let \(X\) be a Zariski space (II, Ex. 3.17). Let \(P \in X\) be a closed point, and let \(X_P\) be the subset of \(X\) consisting of all points \(Q \in X\) such that \(P \in \bar{\ts{Q}}\). We call \(X_P\) the local space of \(X\) at \(P\), and give it the induced topology.
Let \(j: X_P \to X\) be the inclusion, and for any sheaf \(\mcf\) of abelian groups on \(X\), let \(\mcf_P=j^* \mcf\). Show that for all \(i\ge0\) and \(\mcf\), we have \[ H_P^i(X, \mcf)=H_P^i(X_P, \mcf_P). \]
Solution
Put \(S=X_P\) and write \(j^{-1}\) for the inverse-image functor on abelian sheaves, denoted \(j^*\) in the statement. Thus \(\mcf_P\) denotes an entire sheaf on \(S\), not the stalk of \(\mcf\) at \(P\). Write \(\Gamma_P\) for sections supported in the closed singleton \(\{P\}\), as in P-AGH323SUPPORTS. A Zariski space is noetherian and has a unique generic point for every nonempty irreducible closed subset.
1. Every open neighborhood of \(P\) contains \(S\).If \(V\subseteq X\) is open and \(C\subseteq V\) is relatively closed with \(C\cap S=\varnothing\), there is an open neighborhood \(U\) of \(P\) such that \(U\cap V\cap C=\varnothing\).
If \(Q\in S\), every open neighborhood of the specialization \(P\) contains \(Q\). This proves the first assertion.
The closed subset \(C\) of the noetherian space \(V\) has finitely many irreducible components \(C_1,\ldots,C_t\). Each has a generic point \(\eta_a\in C_a\). Indeed, the closure of \(C_a\) in \(X\) has a generic point, and its intersection with \(V\) is the nonempty open subset \(C_a\); the generic point belongs to this intersection. If \(P\) belonged to \(\overline{C_a}\), it would belong to \(\overline{\{\eta_a\}}\), giving \(\eta_a\in S\cap C\), a contradiction. Consequently \[ U=X\setminus\bigcup_{a=1}^t\overline{C_a} \] is a neighborhood of \(P\) disjoint from \(C\). This includes \(C=\varnothing\), when \(U=X\).
Equivalently, if \(W\subseteq V\) is open and contains \(S\cap V\), then some neighborhood \(U\) of \(P\) satisfies \(U\cap V\subseteq W\). We will use this form both to cover a neighborhood and to make finitely many local sections agree.
2. For every abelian sheaf \(A\) on \(X\) and open \(V\subseteq X\), the natural map\[ \varinjlim_{U\ni P}\Gamma(U\cap V,A) \longrightarrow\Gamma(S\cap V,j^{-1}A) \]is an isomorphism, where the transition maps are restriction to smaller neighborhoods \(U\) of \(P\).
Every subspace of a noetherian space is noetherian, so \(S\cap V\) is quasi-compact. For example, this follows by extending a cover of the subspace to opens in \(X\) and using quasi-compactness of their union, an open subset of a noetherian space.
Take a section of \(j^{-1}A\) on \(S\cap V\). By the construction of inverse image, it is locally represented by sections \(a_i\in A(V_i)\) for opens \(V_i\subseteq V\). Choose finitely many of these opens covering \(S\cap V\). On \(V_i\cap V_h\), the germs of \(a_i-a_h\) are zero at every point of \(S\) in this overlap. The zero-germ locus of a section is open, so step 1 gives a neighborhood \(U_{ih}\) of \(P\) on whose intersection with \(V_i\cap V_h\) the two sections agree. The same step gives a neighborhood \(U_0\) with \(U_0\cap V\subseteq\bigcup_iV_i\). Intersect these finitely many neighborhoods and call the result \(U\). The \(a_i\) now glue on \(U\cap V\) and represent the given section on \(S\cap V\). This proves surjectivity.
For injectivity, let \(a\in A(U\cap V)\) restrict to zero on \(S\cap V\). Its zero-germ locus is an open subset of \(U\cap V\) containing \(S\cap V\), since \(S\subseteq U\). Step 1, applied inside \(U\cap V\), gives a smaller neighborhood \(U'\) of \(P\) on which \(a|_{U'\cap U\cap V}=0\). Thus \(a\) is already zero in the indicated direct limit. All the constructions use the natural restriction map, so the isomorphism is natural in \(A\) and compatible with restriction in \(V\).
3. If \(A\) is flasque on \(X\), then \(j^{-1}A\) is flasque on \(S\).
Let \(T\subseteq T'\) be open subsets of \(S\) and choose an open \(V\subseteq X\) with \(T=S\cap V\). A section on \(T\) is represented, by step 2, by a section of \(A\) on \(U\cap V\) for some neighborhood \(U\) of \(P\). Flasqueness extends this section to all of \(X\). Its inverse-image section on \(S\), restricted to \(T'\), extends the original section on \(T\). Thus every restriction map on \(j^{-1}A\) is surjective. This conclusion has been proved for the present inclusion; it is not an assertion that arbitrary inverse images preserve flasqueness.
4. For every abelian sheaf \(A\), restriction gives a natural isomorphism\[ \Gamma_P(X,A)\xrightarrow{\cong}\Gamma_P(S,j^{-1}A). \]
A section supported at \(P\) restricts to one with that support on \(S\). If this restriction is zero, its germ at \(P\) is zero, because \((j^{-1}A)_P\cong A_P\). All its other germs were already zero by the support condition, so the original section is zero. This proves injectivity.
Conversely, represent a section on \(S\) supported at \(P\) by \(a\in A(U)\) using step 2 with \(V=X\). Its restriction to \(S\setminus\{P\}\) is zero. Apply the injectivity in step 2 with the open set \(V=X\setminus\{P\}\). After shrinking the neighborhood \(U\) of \(P\), it follows that \(a|_{U\setminus\{P\}}=0\). Glue \(a\) on \(U\) with zero on \(X\setminus\{P\}\). These opens cover \(X\), so this gives a global section supported at \(P\) with the prescribed restriction. The construction is inverse to restriction and hence natural.
5. The supported cohomology groups are naturally isomorphic in every degree.
Choose an injective resolution \(\mcf\to I^\bullet\) on \(X\). Its terms are flasque [@Har10a, Lemma III.2.4]. Inverse image of abelian sheaves is exact, as follows from its stalk formula \((j^{-1}A)_x=A_x\) for \(x\in S\) and stalkwise exactness. Thus \(j^{-1}I^\bullet\) resolves \(\mcf_P\). By step 3 all its terms are flasque, and hence acyclic for sections supported at \(P\) by P-AGH323SUPPORTS, part (c). It therefore computes \(H_P^i(S,\mcf_P)\).
Apply the natural isomorphism of step 4 to every \(I^q\). It commutes with the differentials and identifies the complexes \[ \Gamma_P(X,I^\bullet)\cong\Gamma_P(S,j^{-1}I^\bullet). \] The first computes \(H_P^i(X,\mcf)\) by definition; the second computes the groups just identified. Taking cohomology gives \[ \boxed{H_P^i(X,\mcf)\xrightarrow{\cong}H_P^i(X_P,\mcf_P)\qquad(i\ge0).} \] Naturality follows from the natural maps on sections and comparison of resolutions [@Har10a, Chapter III, §1].
6. Q.E.D.
Steps <1>1–<1>4 establish the restriction and flasqueness comparisons for the local space, and step 5 proves the required isomorphism in every degree.