m-Koszul Artin-Schelter regular algebras

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• 作者：Izuru Mori^a ; ¹ ; ^{simouri@ipc.shizuoka.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; S. Paul Smith^b ; ^{smith@math.washington.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：16E65 ; 16W50
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 January 2016
• 年：2016
• 卷：446
• 期：Complete
• 页码：373-399
• 全文大小：481 K

文摘

This paper studies the homological determinants and Nakayama automorphisms of not-necessarily-noetherian m-Koszul twisted Calabi–Yau or, equivalently, m  -Koszul Artin–Schelter regular, algebras. Dubois-Violette showed that such an algebra is isomorphic to a derivation quotient algebra D(w,i) for a unique-up-to-scalar-multiples twisted superpotential w. By definition, D(w,i) is the quotient of the tensor algebra TV  , where V=D(w,i)₁, by (∂ⁱw), the ideal generated by all i  -th-order left partial derivatives of w. The restriction map σ↦σ|_V$\sigma \mapsto \sigma {|}_V$ is used to identify the group of graded algebra automorphisms of D(w,i) with a subgroup of GL(V)$\mathrm{GL}(V)$. We show that the homological determinant of a graded algebra automorphism σ of an m  -Koszul Artin–Schelter regular algebra D(w,i) is given by the formula hdet(σ)w=σ^⊗(m+i)(w). It follows from this that the homological determinant of the Nakayama automorphism of an m-Koszul Artin–Schelter regular algebra is 1. As an application, we prove that the homological determinant and the usual determinant coincide for most quadratic noetherian Artin–Schelter regular algebras of dimension 3.