APPLICATIONS OF HOCHSCHILD COHOMOLOGY TO THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING

Let Mold_n,d be the moduli of rank d subalgebras of Mn over Z. For x ∈ Mold_n,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 Mold_n,d. The dimension of the tangent space T_Moldn,d/Z,x of Mold_n,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 Mold_n,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.

MSC Codes 16E40 (Primary) 14D22, 16S50, 16S80 (Secondary)