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