Cohomology Algebras of a Family of DG Skew Polynomial Algebras

Xuefeng Mao () and Gui Ren
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Xuefeng Mao: Department of Mathematics, Shanghai University, Shanghai 200444, China
Gui Ren: School of Economics, Shanghai University, Shanghai 200444, China

Mathematics, 2023, vol. 11, issue 7, 1-46

Abstract: Let A be a connected cochain DG algebra such that its underlying graded algebra A # is the graded skew polynomial algebra k ⟨ x 1 , x 2 , x 3 ⟩ / x 1 x 2 + x 2 x 1 x 2 x 3 + x 3 x 2 x 3 x 1 + x 1 x 3 , | x 1 | = | x 2 | = | x 3 | = 1 . Then the differential ∂ A is determined by ∂ A ( x 1 ) ∂ A ( x 2 ) ∂ A ( x 3 ) = M x 1 2 x 2 2 x 3 2 for some M ∈ M 3 ( k ) . When the rank r ( M ) of M belongs to { 1 , 2 , 3 } , we compute H ( A ) case by case. The computational results in this paper give substantial support for the research of the various homological properties of such DG algebras. We find some examples, which indicate that the cohomology graded algebras of such kind of DG algebras may be not left (right) Gorenstein.

Keywords: cochain DG algebra; cohomology algebra; DG skew polynomial algebra; AS-Gorenstein algebra (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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