Let \(\mcf\) be a sheaf on \(X\) and let \(s \in \mcf(U)\) be a section over an open set \(U\). The support of \(s\), denoted \(\supp s\), is defined to be \(\ts{P \in U \st s_P \neq 0}\), where \(s_P\) denotes the germ of \(s\) in the stalk \(\mcf_P\). Show that \(\supp s\) is a closed subset of \(U\).
Define the support of \(\mcf\) to be \(\supp \mcf \da \ts{P \in X \st \mcf_P \neq 0}\). Show that this need not be a closed subset.
Solution
1. The complement of the support of a section is open:\[ U\setminus\operatorname{supp}s =\{P\in U:s_P=0\}. \]
Let \(P\in U\) satisfy \[ s_P=0 \] in the stalk \(\mcf_P\). By the definition of a stalk, equality of the germ \(s_P\) with the zero germ means that there exists an open neighborhood \[ P\in V\subseteq U \] such that \[ s|_V=0. \] For every \(Q\in V\), therefore, \[ s_Q=0. \] Thus \[ V\subseteq U\setminus\operatorname{supp}s. \] Every point of the complement has an open neighborhood contained in the complement, so the complement is open.
2. Hence\[ \boxed{\operatorname{supp}s\text{ is closed in }U.} \]
This is exactly the conclusion of 1.
3. The support of a sheaf need not be closed.
Take \[ X=\mathbb A^1_{\mathbb C} =\operatorname{Spec}\mathbb C[x]. \] For each positive integer \(n\), let \[ i_n:\{n\}\hookrightarrow X \] be the closed-point inclusion and let \[ \mathcal K_n=i_{n*}\mathbb C \] be the corresponding skyscraper sheaf. Define \[ \mcf=\bigoplus_{n\ge1}\mathcal K_n \] in the category of sheaves of abelian groups.
The stalk functor commutes with direct sums, so for every \(P\in X\), \[ \mcf_P \cong \bigoplus_{n\ge1}(\mathcal K_n)_P. \] For a skyscraper sheaf, \[ (\mathcal K_n)_P \cong \begin{cases} \mathbb C,&P=n,\\ 0,&P\ne n. \end{cases} \] Therefore \[ \operatorname{supp}\mcf =\{1,2,3,\ldots\} \subseteq\mathbb A^1_{\mathbb C}. \]
This subset is infinite. Every proper Zariski-closed subset of \(\mathbb A^1_{\mathbb C}\) is finite, because it is the zero set of a nonzero polynomial in one variable. Hence \[ \{1,2,3,\ldots\} \] is Zariski dense but is not all of \(X\). It is therefore not closed.
4. Thus there is a genuine distinction:\[ \boxed{ \operatorname{supp}s\text{ is always closed, whereas } \operatorname{supp}\mcf\text{ need not be closed}. } \]
The first assertion is 2 and the second is the example in 3.
5. Q.E.D.
Step 4 is exactly the pair of assertions requested.