Cohomology algebras of a family of cochain DG skew polynomial algebras

Abstract

Let $\mathcal{A}$ be a connected cochain DG algebra such that its underlying graded algebra ${\mathcal{A}}^{\#}$ is the graded skew polynomial algebra $$k⟨x_1,x_2,x_3⟩/\left(\begin{array}{c}{x}_1{x}_2+x_2{x}_1\\ x_2{x}_3+x_3{x}_2\\ x_3{x}_1+x_1{x}_3\end{array}\right),|x_1|=|x_2|=|x_3|=1.$$ From \cite{MWZ} or \cite{MWYZ}, one sees that the differential ${\partial }_{\mathcal{A}}$ is determined by \begin{align*} \left( \begin{array}{c} \partial_{\mathcal{A}}(x_1) \partial_{\mathcal{A}}(x_2) \partial_{\mathcal{A}}(x_3) \end{array} \right)=M\left( \begin{array}{c} x_1^2 x_2^2 x_3^2 \end{array} \right), \end{align*} for some $M\in M_3(k)$. For the case $1\le r(M)\le 3$, we compute $H(\mathcal{A})$ case by case. The computational results in this paper give substantial support for \cite{MWZ}, where the various homological properties of such DG algebras are systematically studied. We find some examples, which indicate that the cohomology graded algebra of a Koszul Calabi-Yau DG algebra may be not left (right) Gorenstein.

arXiv admin note: substantial text overlap with arXiv:2009.03524

References in corpus (1)

• DG algebras with exterior homology