Cohomology algebras of a family of cochain DG skew polynomial algebras
Xuefeng Mao, Huan Wang, Gui Ren

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 be a connected cochain DG algebra such that its underlying graded algebra is the graded skew polynomial algebra From \cite{MWZ} or \cite{MWYZ}, one sees that the differential 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 . For the case , we compute 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…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology · Nonlinear Waves and Solitons
