The Stacks project

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

\[ ((A_1, P_1), (A_2, P_2), (\text{id}_ A, \varphi )) \]

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. $\square$

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