The Hochschild cohomology ring modulo nilpotence of a stacked monomial algebra

Edward L. Green¹ ; Nicole Snashall²

¹Department of Mathematics Virginia Tech Blacksburg, VA 24061-0123, U.S.A.
²Department of Mathematics University of Leicester University Road Leicester, LE1 7RH, England

Colloquium Mathematicum, Tome 105 (2006) no. 2, pp. 233-258

Résumé

This paper studies the Hochschild cohomology of finite-dimensional monomial algebras. If ${\mit\Lambda} = K{\mathcal Q}/I$ with $I$ an admissible monomial ideal, then we give sufficient conditions for the existence of an embedding of $K[x_1, \ldots , x_r]/\langle x_ax_b \hbox{ for } a \neq b\rangle$ into the Hochschild cohomology ring $\mathop{\rm HH}^*({\mit\Lambda})$. We also introduce stacked algebras, a new class of monomial algebras which includes Koszul and $D$-Koszul monomial algebras. If ${\mit\Lambda}$ is a stacked algebra, we prove that $\mathop{\rm HH}^*({\mit\Lambda})/{\cal N} \cong K[x_1, \ldots , x_r]/\langle x_ax_b \hbox{ for } a \neq b\rangle$, where ${\cal N}$ is the ideal in $\mathop{\rm HH}^*({\mit\Lambda})$ generated by the homogeneous nilpotent elements. In particular, this shows that the Hochschild cohomology ring of ${\mit\Lambda}$ modulo nilpotence is finitely generated as an algebra.

Détail

DOI : 10.4064/cm105-2-6

Keywords: paper studies hochschild cohomology finite dimensional monomial algebras mit lambda mathcal admissible monomial ideal sufficient conditions existence embedding ldots langle hbox neq rangle hochschild cohomology ring mathop * mit lambda introduce stacked algebras class monomial algebras which includes koszul d koszul monomial algebras mit lambda stacked algebra prove mathop * mit lambda cal cong ldots langle hbox neq rangle where cal ideal mathop * mit lambda generated homogeneous nilpotent elements particular shows hochschild cohomology ring mit lambda modulo nilpotence finitely generated algebra