Lemma 92.7.2. Example 92.7.1 satisfies the Rim-Schlessinger condition (RS). In particular, $\mathcal{D}\! \mathit{ef}_ P$ is a deformation category for any graded $k$-algebra $P$.

**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 89.16.4 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.

Consider an object

of the category $\mathcal{F}(A_1) \times _{\mathcal{F}(A)} \mathcal{F}(A_2)$. Then we consider $P_1 \times _\varphi P_2$. Since $\varphi : P_1 \otimes _{A_1} A \to P_2 \otimes _{A_2} A$ is an isomorphism of graded algebras, we see that the graded pieces of $P_1 \times _\varphi P_2$ are finite projective $A_1 \times _ A A_2$-modules, see proof of Lemma 92.4.2. Thus $P_1 \times _\varphi P_2$ is an object of $\mathcal{F}(A_1 \times _ A A_2)$. This construction determines a quasi-inverse to our functor and the proof is complete. $\square$

## 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Your email address will not be published.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.

## Comments (0)