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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004615&_mathId=si1.gif&_user=111111111&_pii=S0021869315004615&_rdoc=1&_issn=00218693&md5=400d594d4f41414ced20bf6bfeb180f5" title="Click to view the MathML source">D(w,i)class="mathContainer hidden">class="mathCode">$\mathcal{D}(\mathsf{w},i)$ for a unique-up-to-scalar-multiples twisted superpotential class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004615&_mathId=si2.gif&_user=111111111&_pii=S0021869315004615&_rdoc=1&_issn=00218693&md5=a4b9246f19a88e7c71c420350920a73f" title="Click to view the MathML source">wclass="mathContainer hidden">class="mathCode">$\mathsf{w}$. By definition, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004615&_mathId=si1.gif&_user=111111111&_pii=S0021869315004615&_rdoc=1&_issn=00218693&md5=400d594d4f41414ced20bf6bfeb180f5" title="Click to view the MathML source">D(w,i)class="mathContainer hidden">class="mathCode">$\mathcal{D}(\mathsf{w},i)$ is the quotient of the tensor algebra TV  , where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004615&_mathId=si4.gif&_user=111111111&_pii=S0021869315004615&_rdoc=1&_issn=00218693&md5=ee5e264e35160632f5ce984689822227" title="Click to view the MathML source">V=D(w,i)₁class="mathContainer hidden">class="mathCode">$V=\mathcal{D}{(\mathsf{w},i)}_1$, by class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004615&_mathId=si5.gif&_user=111111111&_pii=S0021869315004615&_rdoc=1&_issn=00218693&md5=fa97688de7cf8f181008c7acaa93ec4c" title="Click to view the MathML source">(∂ⁱw)class="mathContainer hidden">class="mathCode">$({\partial }^i\mathsf{w})$, the ideal generated by all i  -th-order left partial derivatives of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004615&_mathId=si2.gif&_user=111111111&_pii=S0021869315004615&_rdoc=1&_issn=00218693&md5=a4b9246f19a88e7c71c420350920a73f" title="Click to view the MathML source">wclass="mathContainer hidden">class="mathCode">$\mathsf{w}$. The restriction map class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004615&_mathId=si6.gif&_user=111111111&_pii=S0021869315004615&_rdoc=1&_issn=00218693&md5=8077effae152651312347de825e9a65e" title="Click to view the MathML source">σ↦σ|_Vclass="mathContainer hidden">class="mathCode">$\sigma \mapsto \sigma {|}_V$ is used to identify the group of graded algebra automorphisms of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004615&_mathId=si1.gif&_user=111111111&_pii=S0021869315004615&_rdoc=1&_issn=00218693&md5=400d594d4f41414ced20bf6bfeb180f5" title="Click to view the MathML source">D(w,i)class="mathContainer hidden">class="mathCode">$\mathcal{D}(\mathsf{w},i)$ with a subgroup of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004615&_mathId=si22.gif&_user=111111111&_pii=S0021869315004615&_rdoc=1&_issn=00218693&md5=a823f9bedacf34187969c8c5c97e82e1" title="Click to view the MathML source">GL(V)class="mathContainer hidden">class="mathCode">$\mathrm{GL}(V)$. We show that the homological determinant of a graded algebra automorphism σ of an m  -Koszul Artin–Schelter regular algebra class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004615&_mathId=si1.gif&_user=111111111&_pii=S0021869315004615&_rdoc=1&_issn=00218693&md5=400d594d4f41414ced20bf6bfeb180f5" title="Click to view the MathML source">D(w,i)class="mathContainer hidden">class="mathCode">$\mathcal{D}(\mathsf{w},i)$ is given by the formula class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315004615&_mathId=si8.gif&_user=111111111&_pii=S0021869315004615&_rdoc=1&_issn=00218693&md5=4b2660925c5e4f86c0a01042b97935ea" title="Click to view the MathML source">hdet(σ)w=σ^⊗(m+i)(w)class="mathContainer hidden">class="mathCode">$\mathrm{hdet}(\sigma )\mathsf{w}={\sigma }^{\otimes (m+i)}(\mathsf{w})$. 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.