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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 Hochschildcohomology to ${\rm Mold}_{n, d}$. The dimension of the tangent space $T_{{\rmMold}_{n, d}/{\Bbb Z}, x}$ of ${\rm Mold}_{n, d}$ over ${\Bbb Z}$ at $x$ can becalculated by the Hochschild cohomology $H^{1}({\mathcal A}(x), {\rmM}_n(k(x))/{\mathcal A}(x))$. We show that $H^{2}({\mathcal A}(x), {\rmM}_n(k(x))/{\mathcal A}(x)) = 0$ is a sufficient condition for the canonicalmorphism ${\rm Mold}_{n, d} \to {\Bbb Z}$ being smooth at $x$. We alsocalculate $H^{i}(A, {\rm M}_n(k)/A)$ for several $R$-subalgebras $A$ of ${\rmM}_n(R)$ over a commutative ring $R$. In particular, we summarize the resultson $H^{i}(A, {\rm M}_n(k)/A)$ for all $k$-subalgebras $A$ of ${\rm M}_n(k)$over an algebraically closed field $k$ in the case $n=2, 3$.

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