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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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/

[1] Conca A., Rossi M. E., Valla G., “Groebner flags and Gorenstein algebras”, Compositio Math., 129 (2001), 95–121 | DOI | MR | Zbl

[2] Conca A., Trung N. V., Valla G., “Koszul property for points in projective spaces”, Math. Scand., 89:2 (2001), 201–216 | DOI | MR | Zbl

[3] Blum S., “Initially Koszul algebras”, Beiträge Algebra Geom., 41:2 (2000), 455–467 | MR | Zbl

[4] Conca A., “Universally Koszul algebras”, Math. Ann., 317:2 (2000), 329–346 | DOI | MR | Zbl

[5] Conca A., “Universally Koszul algebras defined by monomials”, Rend. Sem. Mat. Univ. Padova, 107 (2002), 1–5 | MR

[6] Herzog J., Hibi T., Restuccia G., “Strongly Koszul algebras”, Math. Scand., 86:2 (2000), 161–178 | DOI | MR | Zbl

[7] Priddy S. B., “Koszul resolutions”, Trans. Amer. Math. Soc., 152:1 (1970), 39–60 | DOI | MR | Zbl

[8] Positselskii L. E., “Sootnoshenie mezhdu ryadami Grëbnera kvadratichno dvoistvennykh algebr ne vlechet ikh koshulevosti”, Funkts. analiz i ego pril., 29:3 (1995), 83–87 | MR

[9] Roos J.–E., “On the characterization of Koszul algebras. Four counter-examples”, C. R. Acad. Sci. Paris, Sér. I, 321:1 (1995), 15–20 | MR | Zbl

[10] Piontkovskii D. I., “O ryadakh Gilberta kozyulevykh algebr”, Funkts. analiz i ego pril., 35:2 (2001), 64–69 | DOI | MR | Zbl

[11] Polishchuk A., Positselski L., Qudratic algebras, Preprint, 2000 | MR

[12] Govorov V. E., “O razmernosti graduirovannykh algebr”, Matem. zametki, 14:2 (1973), 209–216 | MR | Zbl

[13] Backelin J., A distributiveness property of augmented algebras, and some related homological results, Ph. D. thesis, Stockholm, 1982

[14] Ufnarovskii V. A., “Algebry, zadannye dvumya kvadratichnymi sootnosheniyami”, Issledovaniya po teorii kolets, algebr i modulei, Matem. issled., 76, 1984, 148–172 | MR

[15] Ufnarovskii V. A., “Kombinatornye i asimptoticheskie metody v algebre”, Itogi nauki i tekhniki, Sovremennye problemy matematiki, Fundamentalnye napravleniya, 57, VINITI, M., 1990, 5–177 | MR

[16] Davydov A. A., “Totally positive sequences and R-matrix quadratic algebras”, J. Math. Sci. (New York), 100:1 (2000), 1871–1876 | DOI | MR | Zbl

[17] Wambst M., “Complexes de Koszul quantiques”, Ann. Inst. Fourier, Grenoble, 43:4 (1993), 1089–1156 | DOI | MR | Zbl

[18] Beilinson A., Ginzburg V., Soergel W., “Koszul duality patterns in representation theory”, J. Amer. Math. Soc., 9:2 (1996), 473–527 | DOI | MR | Zbl

[19] Anick D., “On the homology of associative algebras”, Trans. Amer. Math. Soc., 296:2 (1986), 641–659 | DOI | MR | Zbl

[20] Anick D., “Generic algebras and CW-complexes”, Proc. of 1983 Conf. on algebra, topology and K-theory in honor of John Moore, Princeton Univ., 1988, 247–331 | MR

[21] Zhang J. J., “Non-Noetherian regular rings of dimension 2”, Proc. Amer. Mat. Soc., 126:6 (1998), 1645–1653 | DOI | MR | Zbl

[22] Piontkovskii D. I., “Ryady Gilberta i sootnosheniya v algebrakh”, Izv. RAN, ser. matem., 64:6 (2000), 205–219 | DOI | MR | Zbl

[23] Piontkovskii D. I., “Bazisy Grëbnera i kogerentnost monomialnoi assotsiativnoi algebry”, Fundam. prikl. matem., 2:2 (1996), 501–509 | MR | Zbl

[24] Iyudu N. K., “Razreshimost problemy raspoznavaniya delitelei nulya v odnom klasse algebr”, Fundam. prikl. matem., 1:2 (1995), 541–544 | MR | Zbl

[25] Piontkovskii D. I., “Nekommutativnye bazisy Grëbnera, kogerentnost assotsiativnykh algebr i delimost v polugruppakh”, Fundam. prikl. matem., 7:2 (2001), 495–513 | MR | Zbl

[26] Govorov V. E., “Razmernost i kratnost graduirovannykh algebr”, Sib. matem. zh., 14 (1973), 1200–1206 | MR | Zbl