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.



Augmented algebras over semisimple ring
This is the set-up of *Beilinson, Alexander; Ginzburg, Victor; Soergel, Wolfgang*, [**Koszul duality patterns in representation theory**](http://dx.doi.org/10.1090/S0894-0347-96-00192-0), J. Am. Math. Soc. 9, No. 2, 473-527 (1996). [ZBL0864.17006](https://zbmath.org/?q=an:0864.17006).


It has been cited 888 times according to Google Scholar, so it should be considered a classic.


