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

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 ${\mathrm{N}}_m(R)$ is a quadratic monomial algebra over $R$. We calculate ${\mathrm{HH}}^{\ast }({\mathrm{N}}_m(R),{\mathrm{M}}_m(R)/{\mathrm{N}}_m(R))$ as $R$-modules. We also determine the $R$-algebra structure of the Hochschild cohomology ring ${\mathrm{HH}}^{\ast }({\mathrm{N}}_m(R),{\mathrm{N}}_m(R))$. For $m\ge 3$, ${\mathrm{HH}}^{\ast }({\mathrm{N}}_m(R),{\mathrm{N}}_m(R))$ is an infinitely generated algebra over $R$ and has no Batalin-Vilkovisky algebra structure giving the Gerstenhaber bracket.

54 pages