  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 $n=2, 3$.
