TY - JOUR

T1 - APPLICATIONS OF HOCHSCHILD COHOMOLOGY TO THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING

AU - Nakamoto, Kazunori

AU - Torii, Takeshi

N1 - Publisher Copyright:
Copyright © 2020, The Authors. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/6/14

Y1 - 2020/6/14

N2 - 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.MSC Codes 16E40 (Primary) 14D22, 16S50, 16S80 (Secondary)

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

KW - And phrases. Hochschild cohomology

KW - Matrix ring

KW - Moduli of molds

KW - Subalgebra

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M3 - Article

AN - SCOPUS:85095207648

JO - [No source information available]

JF - [No source information available]

SN - 0402-1215

ER -