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

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