Applications of Hochschild cohomology to the moduli of subalgebras of the full matrix ring

Abstract

Let ${\mathrm{Mold}}_{n,d}$ be the moduli of rank $d$ subalgebras of ${\mathrm{M}}_n$ over $\mathbb{Z}$. For $x\in {\mathrm{Mold}}_{n,d}$, let $\mathcal{A}(x)\subseteq {\mathrm{M}}_n(k(x))$ be the subalgebra of ${\mathrm{M}}_n$ corresponding to $x$, where $k(x)$ is the residue field of $x$. In this article, we apply Hochschild cohomology to ${\mathrm{Mold}}_{n,d}$. The dimension of the tangent space $T_{{\mathrm{Mold}}_{n,d}/\mathbb{Z},x}$ of ${\mathrm{Mold}}_{n,d}$ over $\mathbb{Z}$ at $x$ can be calculated by the Hochschild cohomology $H^1(\mathcal{A}(x),{\mathrm{M}}_n(k(x))/\mathcal{A}(x))$. We show that $H^2(\mathcal{A}(x),{\mathrm{M}}_n(k(x))/\mathcal{A}(x))=0$ is a sufficient condition for the canonical morphism ${\mathrm{Mold}}_{n,d}\to \mathbb{Z}$ being smooth at $x$. We also calculate $H^i(A,{\mathrm{M}}_n(k)/A)$ for several $R$-subalgebras $A$ of ${\mathrm{M}}_n(R)$ over a commutative ring $R$. In particular, we summarize the results on $H^i(A,{\mathrm{M}}_n(k)/A)$ for all $k$-subalgebras $A$ of ${\mathrm{M}}_n(k)$ over an algebraically closed field $k$ in the case $n=2,3$.

51 pages