The Stacks project

Lemma 93.4.2. Example 93.4.1 satisfies the Rim-Schlessinger condition (RS). In particular, $\mathcal{D}\! \mathit{ef}_ V$ is a deformation category for any finite dimensional vector space $V$ over $k$.

Proof. Let $A_1 \to A$ and $A_2 \to A$ be morphisms of $\mathcal{C}_\Lambda $. Assume $A_2 \to A$ is surjective. According to Formal Deformation Theory, Lemma 90.16.4 it suffices to show that the functor $\mathcal{F}(A_1 \times _ A A_2) \to \mathcal{F}(A_1) \times _{\mathcal{F}(A)} \mathcal{F}(A_2)$ is an equivalence of categories.

Thus we have to show that the category of finite projective modules over $A_1 \times _ A A_2$ is equivalent to the fibre product of the categories of finite projective modules over $A_1$ and $A_2$ over the category of finite projective modules over $A$. This is a special case of More on Algebra, Lemma 15.6.9. We recall that the inverse functor sends the triple $(M_1, M_2, \varphi )$ where $M_1$ is a finite projective $A_1$-module, $M_2$ is a finite projective $A_2$-module, and $\varphi : M_1 \otimes _{A_1} A \to M_2 \otimes _{A_2} A$ is an isomorphism of $A$-module, to the finite projective $A_1 \times _ A A_2$-module $M_1 \times _\varphi M_2$. $\square$

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Your email address will not be published.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