Cohomology Algebras of a Family of DG Skew Polynomial Algebras - Xuefeng Mao; Gui Ren (Mathematics, 2023)
Let <semantics>A</semantics> be a connected cochain DG algebra such that its underlying graded algebra <semantics>A#</semantics> is the graded skew polynomial algebra <semantics><mfenced separators="" open="" close="">k⟨x1,x2,x3⟩/<mfenced separators="" open="(" close=")"><mtable><mtr><mtd>x1x2+x2x1</mtd></mtr><mtr><mtd>x2x3+x3x2</mtd></mtr><mtr><mtd>x3x1+x1x3</mtd></mtr></mtable></mfenced></mfenced>,|x1|=|x2|=|x3|=1.</semantics> Then the differential <semantics>∂A</semantics> is determined by <semantics><mfenced separators="" open="(" close=")"><mtable><mtr><mtd>∂A(x1)</mtd></mtr><mtr><mtd>∂A(x2)</mtd></mtr><mtr><mtd>∂A(x3)</mtd></mtr></mtable></mfenced>=M<mfenced separators="" open="(" close=")"><mtable><mtr><mtd><msubsup>x12</msubsup></mtd></mtr><mtr><mtd><msubsup>x22</msubsup></mtd></mtr><mtr><mtd><msubsup>x32</msubsup></mtd></mtr></mtable></mfenced></semantics> for some <semantics>M∈M3(k)</semantics>. When the rank <semantics>r(M)</semantics> of M belongs to <semantics>{1,2,3}</semantics>, we compute <semantics>H(A)</semantics> 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.