The Stacks project

Lemma 66.29.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module. Let $\overline{x}$ be a geometric point of $X$ and let $\overline{y} = f \circ \overline{x}$ be the image in $Y$. Then there is a canonical isomorphism

\[ (f^*\mathcal{G})_{\overline{x}} = \mathcal{G}_{\overline{y}} \otimes _{\mathcal{O}_{Y, \overline{y}}} \mathcal{O}_{X, \overline{x}} \]

of the stalk of the pullback with the tensor product of the stalk with the local ring of $X$ at $\overline{x}$.

Proof. Since $f^*\mathcal{G} = f_{small}^{-1}\mathcal{G} \otimes _{f_{small}^{-1}\mathcal{O}_ Y} \mathcal{O}_ X$ this follows from the description of stalks of pullbacks in Lemma 66.19.9 and the fact that taking stalks commutes with tensor products. A more direct way to see this is as follows. Choose a commutative diagram

\[ \xymatrix{ U \ar[d]_ p \ar[r]_\alpha & V \ar[d]^ q \\ X \ar[r]^ a & Y } \]

where $U$ and $V$ are schemes, and $p$ and $q$ are surjective ├ętale. By Lemma 66.19.4 we can choose a geometric point $\overline{u}$ of $U$ such that $\overline{x} = p \circ \overline{u}$. Set $\overline{v} = \alpha \circ \overline{u}$. Then we see that

\begin{align*} (f^*\mathcal{G})_{\overline{x}} & = (p^*f^*\mathcal{G})_ u \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} \\ & = (\alpha ^*q^*\mathcal{G})_ u \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} \\ & = (q^*\mathcal{G})_ v \otimes _{\mathcal{O}_{V, v}} \mathcal{O}_{U, u} \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} \\ & = (q^*\mathcal{G})_ v \otimes _{\mathcal{O}_{V, v}} \mathcal{O}_{X, \overline{x}} \\ & = (q^*\mathcal{G})_ v \otimes _{\mathcal{O}_{V, v}} \mathcal{O}_{Y, \overline{y}} \otimes _{\mathcal{O}_{Y, \overline{y}}} \mathcal{O}_{X, \overline{x}} \\ & = \mathcal{G}_{\overline{y}} \otimes _{\mathcal{O}_{Y, \overline{y}}} \mathcal{O}_{X, \overline{x}} \end{align*}

Here we have used Lemma 66.29.4 (twice) and the corresponding result for pullbacks of quasi-coherent sheaves on schemes, see Sheaves, Lemma 6.26.4. $\square$


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Your email address will not be published.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 05VQ. Beware of the difference between the letter 'O' and the digit '0'.