The Stacks project

Remark 90.17.6. The action of Lemma 90.17.5 is functorial. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of deformation categories. Let $A' \to A$ be a surjective ring map whose kernel $I$ is annihilated by $\mathfrak m_{A'}$. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(A))$. In this situation $\varphi $ induces the vertical arrows in the following commutative diagram

\[ \xymatrix{ \text{Lift}(x, A') \times (T\mathcal{F} \otimes _ k I) \ar[d]_{(\varphi , d\varphi \otimes \text{id}_ I)} \ar[r] & \text{Lift}(x, A') \ar[d]^\varphi \\ \text{Lift}(\varphi (x), A') \times (T\mathcal{G} \otimes _ k I) \ar[r] & \text{Lift}(\varphi (x), A') } \]

The commutativity follows as each of the maps (90.17.5.2), (90.17.5.1), and (90.17.5.3) of the proof of Lemma 90.17.5 gives rise to a similar commutative diagram.


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Your email address will not be published.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 06JJ. Beware of the difference between the letter 'O' and the digit '0'.