Triviality of finite free deformations of semisimple algebras over the power series ring

Determinant criterion: let $\phi :R^N\to R^N$ be $R$-linear with reduction $\bar{\phi }$ invertible, and let $D:=\det (\phi )$. Reducing the identity $D=\det (\phi )$ modulo $t$ gives $D<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>t=\det (\bar{\phi })\mathrm{\ne }0$, so $D=c(1+tw)$ with $c\in {\mathbb{C}}^{\times }$; here $c$ is a unit of $R$ and $1+tw\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}t)$ is a unit by [L1], so $D$ is a unit and $\phi$ is an isomorphism by [L2]. This is the determinant unit criterion of the Statement.

The algebra $A$ is complete and separated for the $t$-adic topology: it is a finite free $R$-module, so the $t$-adic filtration on $A=t^{0}A\supseteq tA\supseteq t^{2}A\supseteq \cdots$ is obtained from that on $R$ by taking a finite direct sum, and follows from [F1] componentwise. The quotient $A\mathrm{/}tA=\bar{A}$ is the product of matrix algebras given in the Statement, of $\mathbb{C}$-dimension $N={\sum }_i{d}_i^2$, so ${\mathrm{rank}}_{R}A=N$.

The idempotents ${\bar{e}}_1,\dots ,{\bar{e}}_r$ of $A\mathrm{/}tA$ are pairwise orthogonal, so by [F2] applied with the ideal $tA$ there are pairwise orthogonal idempotents $e_1,\dots ,e_r\in A$ with $e_i\equiv {\bar{e}}_i(\mathrm{m}\mathrm{o}\mathrm{d}tA)$ for each $i$.

Fix $i$ and put $V_i:=A{e}_i$, $K_i:=A(1-e_i)$ and ${\rho }_i(a):=a{e}_i$. The map ${\rho }_i$ is an $R$-linear idempotent endomorphism of the free module $A$ with image $V_i$ and kernel $K_i$, so $A=V_i\oplus K_i$; it preserves both $tA$ and the filtration, hence induces an idempotent endomorphism of $\bar{A}$ with image $\bar{A}{\bar{e}}_i$, the $i$-th column module, of $\mathbb{C}$-dimension $d_i$, and kernel $\bar{A}(1-{\bar{e}}_i)$, of dimension $N-d_i$. Choose elements $x_1,\dots ,x_{d_i}\in V_i$ and $y_1,\dots ,y_{N-d_i}\in K_i$ whose images modulo $tA$ are bases of $V_i\mathrm{/}t{V}_i$ and $K_i\mathrm{/}t{K}_i$ respectively; the union $(x,y)$, read in an $R$-basis of $A$, has a coordinate matrix $\mathrm{\Gamma }$ whose reduction modulo $t$ is invertible, because the images of the $x$'s and $y$'s together form a basis of $A\mathrm{/}tA=V_i\mathrm{/}t{V}_i\oplus K_i\mathrm{/}t{K}_i$. By step 1.1 the matrix $\mathrm{\Gamma }$ is invertible over $R$, so $x_1,\dots ,x_{d_i},y_1,\dots ,y_{N-d_i}$ is an $R$-basis of $A$ on which ${\rho }_i$ is diagonal with $d_i$ entries $1$ and $N-d_i$ entries $0$. In particular $V_i$ is free of rank $d_i$ over $R$, and $K_i$ is free of rank $N-d_i$.

By step 3.1 each $V_i$ is a free $R$-module of rank $d_i$, so ${\mathrm{End}}_R(V_i)\cong {\mathrm{M}}_{d_i}(R)$, and the left multiplication action of $A$ on the left $A$-modules $V_i=A{e}_i$ gives an $R$-algebra homomorphism $\phi :A⟶{\prod }_{i=1}^r{\mathrm{End}}_R(V_i)\cong {\prod }_{i=1}^r{\mathrm{M}}_{d_i}(R),\phi (a):=(v\mapsto av)\mathrm{.}$

The reduction $\bar{\phi }$ of $\phi$ modulo $t$ is the action of $\bar{A}={\prod }_i{\mathrm{M}}_{d_i}(\mathbb{C})$ on ${⨁}_i{V}_i\mathrm{/}t{V}_i\cong {⨁}_i{\mathbb{C}}^{d_i}$; the $i$-th factor acts on the $i$-th summand through the isomorphism ${\mathrm{M}}_{d_i}(\mathbb{C})\stackrel{\sim }{\to }{\mathrm{End}}_{\mathbb{C}}({\mathbb{C}}^{d_i})$, and all other factors act as $0$, so $\bar{\phi }$ is an isomorphism ${\prod }_i{\mathrm{M}}_{d_i}(\mathbb{C})\to {\prod }_i{\mathrm{M}}_{d_i}(\mathbb{C})$. Both $A$ and ${\prod }_i{\mathrm{M}}_{d_i}(R)$ are free of rank ${\sum }_i{d}_i^2=N$ over $R$, so $\phi$ is a finite-rank $R$-linear map whose reduction is invertible; the determinant criterion of step 1.1 makes $\phi$ an isomorphism. Hence the $R$-algebra $A$ is isomorphic to ${\prod }_{i=1}^r{\mathrm{M}}_{d_i}(R)$. Every object was produced by the explicit liftings of step 2.1 and the finite bases of step 3.1 and no selection of a family is required, so no choice principle is used.