The Stacks project

Remark 91.4.8. Let $(X, \mathcal{O}_ X)$, $(X, \mathcal{O}_ X) \to (X'_ i, \mathcal{O}_{X'_ i})$, $\mathcal{I}_ i$, and $h : (X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2})$ be as in Remark 91.4.7. Assume that we are given trivializations $\pi _ i : X'_ i \to X$ such that $\pi _1 = h \circ \pi _2$. In other words, assume $h$ is a morphism of trivialized first order thickening of $(X, \mathcal{O}_ X)$. Let $(\mathcal{K}_ i, c_ i)$, $i = 1, 2$ be a pair consisting of an $\mathcal{O}_ X$-module $\mathcal{K}_ i$ and a map $c_ i : \mathcal{I}_ i \otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{K}_ i$. Assume furthermore given a map of $\mathcal{O}_ X$-modules $\mathcal{K}_2 \to \mathcal{K}_1$ such that

\[ \xymatrix{ \mathcal{I}_2 \otimes _{\mathcal{O}_ X} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes _{\mathcal{O}_ X} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 } \]

is commutative. In this situation the construction of Remark 91.4.6 induces a commutative diagram

\[ \xymatrix{ \{ \mathcal{F}'_2\text{ as in (08L4) with } c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2\} \ar[d] \ar[rr] & & \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathcal{F}, \mathcal{K}_2) \ar[d] \\ \{ \mathcal{F}'_1\text{ as in (08L4) with } c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1\} \ar[rr] & & \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathcal{F}, \mathcal{K}_1) } \]

where the vertical map on the right is given by functoriality of $\mathop{\mathrm{Ext}}\nolimits $ and the map $\mathcal{K}_2 \to \mathcal{K}_1$ and the vertical map on the left is the one from Remark 91.4.7.

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Your email address will not be published.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 08LD. Beware of the difference between the letter 'O' and the digit '0'.



View Remark 91.4.8 as pdf