The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

17.6 Closed immersions and abelian sheaves

Recall that we think of an abelian sheaf on a topological space $X$ as a sheaf of $\underline{\mathbf{Z}}_ X$-modules. Thus we may apply any results, definitions for sheaves of modules to abelian sheaves.

Lemma 17.6.1. Let $X$ be a topological space. Let $Z \subset X$ be a closed subset. Denote $i : Z \to X$ the inclusion map. The functor

\[ i_* : \textit{Ab}(Z) \longrightarrow \textit{Ab}(X) \]

is exact, fully faithful, with essential image exactly those abelian sheaves whose support is contained in $Z$. The functor $i^{-1}$ is a left inverse to $i_*$.

Proof. Exactness follows from the description of stalks in Sheaves, Lemma 6.32.1 and Lemma 17.3.1. The rest was shown in Sheaves, Lemma 6.32.3. $\square$

Let $\mathcal{F}$ be a sheaf on $X$. There is a canonical subsheaf of $\mathcal{F}$ which consists of exactly those sections whose support is contained in $Z$. Here is the exact statement.

Lemma 17.6.2. Let $X$ be a topological space. Let $Z \subset X$ be a closed subset. Let $\mathcal{F}$ be a sheaf on $X$. For $U \subset X$ open set

\[ \Gamma (U, \mathcal{H}_ Z(\mathcal{F})) = \{ s \in \mathcal{F}(U) \mid \text{ the support of }s\text{ is contained in }Z \cap U\} \]

Then $\mathcal{H}_ Z(\mathcal{F})$ is an abelian subsheaf of $\mathcal{F}$. It is the largest abelian subsheaf of $\mathcal{F}$ whose support is contained in $Z$. The construction $\mathcal{F} \mapsto \mathcal{H}_ Z(\mathcal{F})$ is functorial in the abelian sheaf $\mathcal{F}$.

Proof. This follows from Lemma 17.5.2. $\square$

This seems like a good opportunity to show that the functor $i_*$ has a right adjoint on abelian sheaves.

Lemma 17.6.3. Let $i : Z \to X$ be the inclusion of a closed subset into the topological space $X$. Denote1 $i^! : \textit{Ab}(X) \to \textit{Ab}(Z)$ the functor $\mathcal{F} \mapsto i^{-1}\mathcal{H}_ Z(\mathcal{F})$. Then $i^!$ is a right adjoint to $i_*$, in a formula

\[ \mathop{Mor}\nolimits _{\textit{Ab}(X)}(i_*\mathcal{G}, \mathcal{F}) = \mathop{Mor}\nolimits _{\textit{Ab}(Z)}(\mathcal{G}, i^!\mathcal{F}). \]

In particular $i_*$ commutes with arbitrary colimits.

Proof. Note that $i_*i^!\mathcal{F} = \mathcal{H}_ Z(\mathcal{F})$. Since $i_*$ is fully faithful we are reduced to showing that

\[ \mathop{Mor}\nolimits _{\textit{Ab}(X)}(i_*\mathcal{G}, \mathcal{F}) = \mathop{Mor}\nolimits _{\textit{Ab}(X)}(i_*\mathcal{G}, \mathcal{H}_ Z(\mathcal{F})). \]

This follows since the support of the image via any homomorphism of a section of $i_*\mathcal{G}$ is contained in $Z$, see Lemma 17.5.2. $\square$

Remark 17.6.4. In Sheaves, Remark 6.32.5 we showed that $i_*$ as a functor on the categories of sheaves of sets does not have a right adjoint simply because it is not exact. However, it is very close to being true, in fact, the functor $i_*$ is exact on sheaves of pointed sets, sections with support in $Z$ can be defined for sheaves of pointed sets, and $i^!$ makes sense and is a right adjoint to $i_*$.

[1] This is likely nonstandard notation.

Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01AW. Beware of the difference between the letter 'O' and the digit '0'.