Cohomology algebras of a family of cochain DG skew polynomial algebras

TL;DR

This paper computes the cohomology algebras of a specific family of cochain DG skew polynomial algebras, revealing new insights into their homological properties and providing examples that challenge existing assumptions about Koszul Calabi-Yau DG algebras.

Contribution

It explicitly determines the cohomology algebras for a class of DG skew polynomial algebras and explores their homological characteristics, including Gorenstein properties.

Findings

Cohomology algebras computed for cases with rank 1 to 3.

Examples showing Koszul Calabi-Yau DG algebras may not be Gorenstein.

Supports systematic study of homological properties of DG 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 $$k\langle x_1,x_2, x_3\rangle/\left(\begin{array}{ccc} x_1x_2+x_2x_1\\ x_2x_3+x_3x_2\\ x_3x_1+x_1x_3 \end {array}\right), |x_1|=|x_2|=|x_3|=1.$$ 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.