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

Abstract

Let Moldn,d be the moduli of rank d subalgebras of Mn over Z. For x∈Moldn,d, let A(x)⊆Mn(k(x)) be the subalgebra of Mn corresponding to x, where k(x) is the residue field of x. In this article, we apply Hochschild cohomology to Moldn,d. The dimension of the tangent space TMoldn,d/Z,x of Moldn,d over Z at x can be calculated by the Hochschild cohomology H1(A(x),Mn(k(x))/A(x)). We show that H2(A(x),Mn(k(x))/A(x))=0 is a sufficient condition for the canonical morphism Moldn,d→Z being smooth at x. We also calculate Hi(A,Mn(k)/A) for several R-subalgebras A of Mn(R) over a commutative ring R. In particular, we summarize the results on Hi(A,Mn(k)/A) for all k-subalgebras A of Mn(k) over an algebraically closed field k in the case n=2,3.