A criterion for Gorenstein algebras to be regular. (English) Zbl 1251.16013
Let \(A\) be a left Gorenstein connected graded algebra with trivial module \(_Ak\). Assume that \(k\) admits a resolution by finitely generated free modules. The main result of the paper says that \(A\) is Artin-Schelter regular if and only if the Gorenstein index of \(A\) equals \(-\inf\{i\mid\text{Ext}_A^{\text{depth}_AA}(k,k)_i\neq 0\}\). The Gorenstein index is defined as follows. Recall that \(\text{Ext}_A^i(k,A)\) is nonzero if and only if \(i=\text{depth}_AA\), and in that case it is isomorphic to a copy of \(k\). The Gorenstein index is then defined as the internal degree of that copy.
Reviewer: Alex Martsinkovsky (Boston)
MSC:
| 16E65 | Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) |
| 16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
| 16W50 | Graded rings and modules (associative rings and algebras) |
| 16E50 | von Neumann regular rings and generalizations (associative algebraic aspects) |
Keywords:
Gorenstein algebras; AS-Gorenstein algebras; AS-regular algebras; Artin-Schelter regular algebras; connected graded algebras; Gorenstein indicesReferences:
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