Recall the notions of support of a section of a sheaf, support of a sheaf, and subsheaf with supports from (Ex. 1.14) and (Ex. 1.20).
Let \(A\) be a ring, let \(M\) be an \(A\)-module, let \(X = \Spec A\), and let \(\mcf = \widetilde M\). For any \(m \in M = \Gamma(X,\mcf)\), show that \(\supp m = V(\Ann m)\), where \(\Ann m = \ts{a \in A \st am = 0}\).
Now suppose that \(A\) is noetherian and \(M\) finitely generated. Show that \(\supp \mcf = V(\Ann M)\).
The support of a coherent sheaf on a noetherian scheme is closed.
For any ideal \(\mfa \subseteq A\), define a submodule \(\Gamma_\mfa(M)\) of \(M\) by \[ \Gamma_\mfa(M) = \ts{m \in M \st \mfa^n m = 0 \text{ for some } n > 0}. \] Assume that \(A\) is noetherian, and \(M\) any \(A\)-module. Show that \(\Gamma_\mfa(M)^\sim \cong \mch^0_Z(\mcf)\), where \(Z = V(\mfa)\) and \(\mcf = \widetilde M\).
Let \(X\) be a noetherian scheme, and let \(Z\) be a closed subset. If \(\mcf\) is a quasi-coherent, respectively coherent, \(\OO_X\)-module, then \(\mch^0_Z(\mcf)\) is also quasi-coherent, respectively coherent.
Hint
For part (d), use (Ex. 1.20) and (5.8) to show a priori that \(\mch^0_Z(\mcf)\) is quasi-coherent, then show that \(\Gamma_\mfa(M) \cong \Gamma_Z(\mcf)\).
Solution
Use the support conventions of P-AGH2114SUPPORT and the subsheaf of sections with support in a closed set. For \(X=\Spec A\) and \(\mcf=\widetilde M\), the stalk at \(\mathfrak p\) is \(M_{\mathfrak p}\) and the sections on \(D(f)\) are \(M_f\) [@Har10a, Proposition II.5.1].
1. For part (a), \(\supp m=V(\Ann m)\).
For a prime ideal \(\mathfrak p\subseteq A\), the germ \(m/1\) is zero in \(M_{\mathfrak p}\) exactly when there exists \(s\in A\setminus\mathfrak p\) with \(sm=0\). Equivalently, \(\Ann m\) is not contained in \(\mathfrak p\). Thus the germ is nonzero exactly at the primes containing \(\Ann m\), proving the equality.
2. For part (b), \(\supp\mcf=V(\Ann M)\).
Choose generators \(m_1,\ldots,m_r\) of \(M\). If \(M_{\mathfrak p}=0\), for each \(j\) there is \(s_j\notin\mathfrak p\) with \(s_jm_j=0\). The product \(s=\prod_j s_j\) is outside \(\mathfrak p\) and annihilates all generators, hence all of \(M\). Therefore \(\Ann M\) is not contained in \(\mathfrak p\). Conversely, an element of \(\Ann M\setminus\mathfrak p\) becomes a unit after localization and annihilates \(M_{\mathfrak p}\), forcing that module to be zero. Consequently \(M_{\mathfrak p}\ne0\) exactly when \(\Ann M\subseteq\mathfrak p\). For \(M=0\) the empty product is \(1\), and the same argument gives empty support.
3. The support in part (c) is closed.
On each affine open \(U=\Spec A\subseteq X\), the coherent sheaf \(\mcf\) restricts to \(\widetilde M\) for a finite \(A\)-module \(M\) [@Har10a, Proposition II.5.4]. Step 2 shows that \(\supp\mcf\cap U=V(\Ann M)\) is closed in \(U\). Thus \(U\setminus\supp\mcf\) is open in \(U\), hence in \(X\). The complement \(X\setminus\supp\mcf\) is the union of these open subsets as \(U\) runs over an affine cover, so it is open.
4. For part (d), the inclusion \(\Gamma_\mfa(M)\subseteq M\) induces an isomorphism\[ \widetilde{\Gamma_\mfa(M)}\cong\mch_Z^0(\widetilde M). \]
4.1. The sections of \(\widetilde M\) supported in \(Z\) are precisely \(\Gamma_\mfa(M)\).
The set \(\Gamma_\mfa(M)\) is a submodule: a common power of \(\mfa\) annihilates the sum of two elements annihilated by powers of \(\mfa\), and scalar multiplication preserves this property. By step 1 and the ideal description of closed subsets of an affine spectrum, \[ \supp m\subseteq V(\mfa) \quad\Longleftrightarrow\quad V(\Ann m)\subseteq V(\mfa) \quad\Longleftrightarrow\quad \mfa\subseteq\sqrt{\Ann m}. \] If \(\mfa^n m=0\), the last containment holds. Conversely, choose generators \(a_1,\ldots,a_r\) of \(\mfa\) and positive integers \(n_i\) such that \(a_i^{n_i}m=0\). For \(N=1+\sum_i(n_i-1)\), every monomial of total degree \(N\) in the \(a_i\) has some exponent at least \(n_i\). These monomials generate \(\mfa^N\), so \(\mfa^N m=0\). If \(\mfa=0\), its first power already annihilates every element. This proves the desired equality of section modules.
4.2. For every \(f\in A\), localization identifies\[ \Gamma_\mfa(M)_f=\Gamma_{\mfa A_f}(M_f) \]as submodules of \(M_f\).
Localization of the inclusion \(\Gamma_\mfa(M)\subseteq M\) is injective. An element of its source is still annihilated by a power of \(\mfa A_f\), giving one containment. For the other, let \(m/f^q\) be annihilated by \((\mfa A_f)^n\). Choose finitely many generators \(b_1,\ldots,b_s\) of \(\mfa^n\). For every \(j\), the equality \(b_jm/f^q=0\) gives an integer \(e_j\ge0\) with \(f^{e_j}b_jm=0\). Choose \(e\) at least all the \(e_j\). Then \(\mfa^n(f^em)=0\), and \[ m/f^q=(f^em)/f^{q+e}\in\Gamma_\mfa(M)_f. \] If \(\mfa^n=0\), one may take its generating list to consist of the single element zero and take \(e=0\).
4.3. Q.E.D.
Apply step 1 over the noetherian ring \(A_f\), with module \(M_f\) and ideal \(\mfa A_f\). The sections of \(\mch_Z^0(\widetilde M)\) on \(D(f)\) are exactly \(\Gamma_{\mfa A_f}(M_f)\). By step 2, these are also the sections of \(\widetilde{\Gamma_\mfa(M)}\). All identifications are inclusions into \(M_f\) and commute with restriction. They therefore give the asserted isomorphism on a basis, hence on \(X\).
5. Part (e) follows from the affine description in step
4.
For an affine open \(U=\Spec A\) in \(X\), write \(Z\cap U=V(\mfa)\) and \(\mcf|_U\cong\widetilde M\). The definition of the supported subsheaf commutes with restriction to \(U\). Step 4 therefore gives \[ \mch_Z^0(\mcf)|_U\cong\widetilde{\Gamma_\mfa(M)}. \] These descriptions make \(\mch_Z^0(\mcf)\) quasi-coherent. If \(\mcf\) is coherent, then \(M\) is finitely generated over the noetherian ring \(A\). Its submodule \(\Gamma_\mfa(M)\) is consequently finitely generated, so the displayed associated sheaf is coherent. This proves the second assertion on an affine cover of \(X\).
6. Q.E.D.
Steps 1, 2, and 3 prove parts (a), (b), and (c). Step 4 proves part (d), and step 5 proves part (e).