The Stacks project

Lemma 92.4.2. Example 92.4.1 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 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.

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 $M_1 \times _\varphi M_2$. $\square$


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