Geodesic
Koszul Algebras and Their Ideals
Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 2, pp. 47-60

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We study associative graded algebras that have a “complete flag” of cyclic modules with linear free resolutions, i.e., algebras over which there exist cyclic Koszul modules with any possible number of relations (from zero to the number of generators of the algebra). Commutative algebras with this property were studied in several papers by Conca and others. Here we present a noncommutative version of their construction. We introduce and study the notion of Koszul filtration in a noncommutative algebra and examine its connections with Koszul algebras and algebras with quadratic Gröbner bases. We consider several examples, including monomial algebras, initially Koszul algebras, generic algebras, and algebras with one quadratic relation. It is shown that every algebra with a Koszul filtration has a rational Hilbert series.
Mots-clés : Koszul filtration, Koszul algebra
Keywords: coherent algebra, Hilbert series. 
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     author = {D. I. Piontkovskii},
     title = {Koszul {Algebras} and {Their} {Ideals}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
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     number = {2},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2005_39_2_a3/}
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D. I. Piontkovskii. Koszul Algebras and Their Ideals. Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 2, pp. 47-60. http://geodesic.mathdoc.fr/item/FAA_2005_39_2_a3/