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Hochschild cohomology of the quadratic monomial algebra ${\rm N}_m$

Tomohiro Itagaki, Kazunori Nakamoto, Takeshi Torii

Abstract

Let ${\rm N}_m(R) = \{ (a_{ij}) \in {\rm M}_m(R) \mid a_{11} = a_{22} = \cdots = a_{mm} \mbox{ and } a_{ij} = 0 \mbox{ for any } i > j \}$ for a commutative ring $R$. Then ${\rm N}_m(R)$ is a quadratic monomial algebra over $R$. We calculate ${\rm HH}^{\ast}({\rm N}_m(R), {\rm M}_m(R)/{\rm N}_m(R))$ as $R$-modules. We also determine the $R$-algebra structure of the Hochschild cohomology ring ${\rm HH}^{\ast}({\rm N}_m(R), {\rm N}_m(R))$. For $m \ge 3$, ${\rm HH}^{\ast}({\rm N}_m(R), {\rm N}_m(R))$ is an infinitely generated algebra over $R$ and has no Batalin-Vilkovisky algebra structure giving the Gerstenhaber bracket.

Hochschild cohomology of the quadratic monomial algebra ${\rm N}_m$

Abstract

Let for a commutative ring . Then is a quadratic monomial algebra over . We calculate as -modules. We also determine the -algebra structure of the Hochschild cohomology ring . For , is an infinitely generated algebra over and has no Batalin-Vilkovisky algebra structure giving the Gerstenhaber bracket.
Paper Structure (12 sections, 28 theorems, 139 equations)

This paper contains 12 sections, 28 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Hochschild cohomology
  3. Spectral sequences
  4. Review on the construction of spectral sequences associated to filtered cochain complexes
  5. Filtrations and spectral sequences on Hochschild cochain complexes
  6. Gradings on spectral sequences
  7. ${\rm HH}^{\ast}({\rm N}_m(R), R)$
  8. Quadratic monomial algebras
  9. The $R$-algebra structure of ${\rm HH}^{\ast}({\rm N}_m(R), R)$
  10. Several results on $\varphi(n)$
  11. ${\rm HH}^{\ast}({\rm N}_m(R), {\rm M}_m(R)/{\rm N}_m(R))$
  12. Spectral sequences for subquotients of ${\rm M}_m(R)$

Key Result

Theorem 1.1

Let $m \ge 3$. The Hochschild cohomology ${\rm HH}^{n}({\rm N}_m(R), {\rm M}_m(R)/{\rm N}_m(R))$ is a free $R$-module for $n\ge 0$. The rank of ${\rm HH}^{n}({\rm N}_m(R), {\rm M}_m(R)/{\rm N}_m(R))$ for $n\ge 0$ is given by

Theorems & Definitions (37)

  • Theorem 1.1: Theorem \ref{['th:HHMmNm']} and Corollary \ref{['cor:HHMmNm']}
  • Theorem 1.2: Theorem \ref{['th:mainthm']}
  • Theorem 1.3: Theorems \ref{['th:productzerom=3']} and \ref{['th:productzerom>=4']} and Corollary \ref{['cor:infinitelygenerated']}
  • Theorem 1.4: Theorem \ref{['th:noBVstructure']}
  • Theorem 1.5: Theorem \ref{['th:dimtangentspace']}
  • Definition 2.1
  • Remark 2.2
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • ...and 27 more