Let $A$ be a non-negatively graded algebra such that $A_0 = k$. We say that $A$ is Koszul if $k$ has a projective resolution by projective modules such that the i-th piece is generated in degree $i$. Let us now replace the condition $A_0 = k$ with $S: = A_0$ being semi-simple. Then, $A$ can be considered as an augmented graded algebra over $S$. What can be said in such situation? Has a theory of such algebras been developed? I know there is a big theory of Koszul algebras/Koszul duality, but I couldn't find anything about this (seemingly easy) generalisation. In particular, I am interested in an eventual analogue of Koszul duality and $Ext^{\bullet}_{A}(S,S)$. Can you give me any references? Thanks.
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asked Dec 17, 2019 at 17:21
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1 Answer
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This is the set-up of Beilinson, Alexander; Ginzburg, Victor; Soergel, Wolfgang, Koszul duality patterns in representation theory, J. Am. Math. Soc. 9, No. 2, 473-527 (1996). ZBL0864.17006.
It has been cited 888 times according to Google Scholar, so it should be considered a classic.
answered Dec 17, 2019 at 19:19
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$\begingroup$ My bad, it seems I didn't search enough. Thanks! $\endgroup$ Commented Dec 18, 2019 at 14:34