The Stacks project

Lemma 6.27.3. Let $X$ be a topological space, and let $x \in X$ a point. The functors $\mathcal{F} \mapsto \mathcal{F}_ x$ and $A \mapsto i_{x, *}A$ are adjoint. In a formula

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Sets}}(\mathcal{F}_ x, A) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (X)}(\mathcal{F}, i_{x, *}A). \]

A similar statement holds for the case of abelian groups, algebraic structures. In the case of sheaves of modules we have

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X, x}}(\mathcal{F}_ x, A) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, i_{x, *}A). \]

Proof. Omitted. Hint: The stalk functor can be seen as the pullback functor for the morphism $i_ x : \{ x\} \to X$. Then the adjointness follows from adjointness of $i_ x^{-1}$ and $i_{x, *}$ (resp. $i_ x^*$ and $i_{x, *}$ in the case of sheaves of modules). $\square$

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Your email address will not be published.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 009C. Beware of the difference between the letter 'O' and the digit '0'.