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Let $R=R_0 \oplus R_1 \oplus R_2 ...$ be a graded not necessarily commutative algebra over field $k$ and $R$ is generated by $R_1$, $R_0=k$. In commutative situation if one wants free resolution of $R/I$ one can start with Koszul (or Koszul-Tate) complex. Is there any noncommutative analogs of such explicit free resolution of $R/I$?

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You can write down the Koszul complex regardless of whether or not R is commutative (see Priddy's original Koszul paper, or the Polishchuk and Positselski book Quadratic Algebras). It's not guaranteed to be a projective resolution in any case of course (because it may fail to be exact). The bar resolution en.wikipedia.org/wiki/Bar_resolution always gives a free resolution of any module you like (it works for arbitrary algebras, no grading needed), but it is very large. –  Matthew Towers Feb 21 '12 at 13:54
    
@mt But this seems only for quadratic algebras ? Generalization to n-homogenous algebras seems to be new ideas: e.g. arxiv.org/abs/math/0301172 Koszulity for nonquadratic algebras II Roland Berger –  Alexander Chervov Feb 21 '12 at 18:38
    
The Annick resolution (q.v.) is a general construction that does what you want in an explicit way. –  Mariano Suárez-Alvarez Feb 21 '12 at 22:00
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Thank you, Marino. I found some answers in Annick's "Noncommutative graded algebras and their Hilbert series". –  Sasha Pavlov Feb 22 '12 at 14:02

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