The Hochschild cohomology ring modulo nilpotence of a stacked monomial algebra

Tom 105 / 2006

Edward L. Green, Nicole Snashall Colloquium Mathematicum 105 (2006), 233-258 MSC: Primary 16E40, 16G20, 16S15. DOI: 10.4064/cm105-2-6


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.


  • Edward L. GreenDepartment of Mathematics
    Virginia Tech
    Blacksburg, VA 24061-0123, U.S.A.
  • Nicole SnashallDepartment of Mathematics
    University of Leicester
    University Road
    Leicester, LE1 7RH, England

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