Let \(\mcf\) be a sheaf of abelian groups on a topological space \(X\). For an open set \(U\subseteq X\) and a section \(s\in\mcf(U)\), the support of the section is \[ \supp(s)\coloneqq\ts{x\in U\st s_x\ne0\text{ in }\mcf_x}. \] The support of the sheaf is \[ \supp(\mcf)\coloneqq\ts{x\in X\st\mcf_x\ne0}. \] Here \(s_x\) is the germ of \(s\) in the stalk \(\mcf_x\).
For every open set \(U\subseteq X\) and \(s\in\mcf(U)\), the subset \(\supp(s)\) is closed in \(U\).
If \(s_x=0\), the definition of a germ gives an open neighborhood \(x\in V\subseteq U\) with \(s|_V=0\). Every point of \(V\) therefore lies outside \(\supp(s)\). Thus \(U\setminus\supp(s)\) is open.
(A sheaf with nonclosed support)
Let \(p\) be a prime number and put \(X=\Spec\ZZ_{(p)}\). Its points are the generic point \(\eta\) and the closed point \(x=(p)\), and its open sets are \(\varnothing\), \(\{\eta\}\), and \(X\). Define a sheaf of abelian groups \(\mcf\) by \[ \mcf(X)=0,\qquad\mcf(\{\eta\})=\ZZ,\qquad\mcf(\varnothing)=0, \] with zero restriction maps for proper inclusions. Every open cover of a nonempty open set contains that open set itself, so the sheaf axioms hold. The only neighborhood of \(x\) is \(X\), whereas \(\{\eta\}\) is a neighborhood of \(\eta\). Consequently \(\mcf_x=0\) and \(\mcf_\eta\cong\ZZ\), giving \(\supp(\mcf)=\{\eta\}\), which is not closed in \(X\). For the open inclusion \(j:\{\eta\}\hookrightarrow X\), this sheaf is the extension by zero \(j_!\ul{\ZZ}\).
Let \(X\) be a scheme and \(\mcf\) a quasi-coherent \(\OO_X\)-module of finite type. Then \(\supp(\mcf)\) is closed. For an affine open \(U=\Spec A\) with \(\mcf|_U\cong\widetilde M\), \[ \supp(\mcf)\cap U=V(\Ann_A M). \] For \(M\) generated by \(m_1,\ldots,m_r\), a prime \(\mfp\) has \(M_\mfp=0\) if and only if some \(a\notin\mfp\) annihilates every \(m_i\), that is, if and only if \(\mfp\not\supseteq\Ann_AM\) (P-AGH256SUPPORT).
Let \(i:Z\to X\) be a closed immersion of schemes, and let \(\mcf\) be a sheaf of abelian groups on \(Z\). Then \[ \supp(i_*\mcf)=i(\supp\mcf). \] In particular, \(\supp(i_*\mcf)\) is closed in \(X\) if and only if \(\supp\mcf\) is closed in \(Z\).
For \(x\notin i(Z)\), the open set \(X\setminus i(Z)\) is a neighborhood on which \(i_*\mcf\) is zero, so \((i_*\mcf)_x=0\). For \(x=i(z)\), the sets \(i^{-1}(U)\), with \(U\) an open neighborhood of \(x\) in \(X\), form a neighborhood basis at \(z\) in \(Z\) because \(i\) is a homeomorphism onto its image. Taking the colimit of \((i_*\mcf)(U)=\mcf(i^{-1}(U))\) gives \((i_*\mcf)_{i(z)}\cong\mcf_z\). The support equality follows. The closedness equivalence follows because \(i\) identifies \(Z\) with a closed subspace of \(X\).
The identity \(i:X\to X\) is a closed immersion. For \(X=\Spec\ZZ_{(p)}\) and the sheaf with \(\mcf_\eta\cong\ZZ\) and \(\mcf_x=0\), its direct image has support \(\{\eta\}\), which is not closed.
Dependencies
D-0QSI0Stalks and germsuses