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Koszul differential graded algebras and BGG correspondence. II. (English) Zbl 1194.16010

Summary: The concept of Koszul differential graded (DG for short) algebra is introduced in part I [J. Algebra 320, No. 7, 2934-2962 (2008; Zbl 1193.16012)]. Let \(A\) be a Koszul DG algebra. If the Ext-algebra of \(A\) is finite-dimensional, i.e., the trivial module \(_Ak\) is a compact object in the derived category of DG \(A\)-modules, then it is shown in part I [loc. cit.] that \(A\) has many nice properties. However, if the Ext-algebra is infinite-dimensional, little is known about \(A\). As shown in [X.-F. Mao, Q.-S. Wu, Sci. China, Ser. A 52, No. 4, 648-676 (2009; Zbl 1197.16011)] (see also Proposition 2.2), \(_Ak\) is not compact if \(H(A)\) is finite-dimensional. In this paper, it is proved that the Koszul duality theorem also holds when \(H(A)\) is finite-dimensional by using Foxby duality. A DG version of the BGG correspondence is deduced from the Koszul duality theorem.

MSC:

16E45 Differential graded algebras and applications (associative algebraic aspects)
16S37 Quadratic and Koszul algebras
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
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References:

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