Abstract
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 Grobner 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.
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References
A. Conca, M. E. Rossi, and G. Valla, “Grobner flags and Gorenstein algebras,” Compositio Math., 129, 95–121 (2001).
A. Conca, N. V. Trung, and G. Valla, “Koszul property for points in projective spaces,” Math. Scand., 89, No.2, 201–216 (2001).
S. Blum, “Initially Koszul algebras,” Beitrage Algebra Geom., 41, No.2, 455–467 (2000).
A. Conca, “Universally Koszul algebras,” Math. Ann., 317, No.2, 329–346 (2000).
A. Conca, “Universally Koszul algebras defined by monomials,” Rend. Sem. Mat. Univ. Padova, 107, 1–5 (2002).
J. Herzog, T. Hibi, and G. Restuccia, “Strongly Koszul algebras,” Math. Scand., 86, No.2, 161–178 (2000).
S. B. Priddy, “Koszul resolutions,” Trans. Amer. Math. Soc., 152, No.1, 39–60 (1970).
L. E. Positselski, “Relation between the Hilbert series of dual quadratic algebras does not imply Koszulity,” Funkts. Anal. Prilozhen., 29, No.3, 83–87 (1995).
J.-E. Roos, “On the characterization of Koszul algebras. Four counter-examples,” C. R. Acad. Sci. Paris, Ser. I, 321, No.1, 15–20 (1995).
D. I. Piontkovskii, “On the Hilbert series of Koszul algebras,” Funkts Anal. Prilozhen., 35, No.2, 64–69 (2001).
A. Polishchuk and L. Positselskii, Quadratic Algebras, Preprint, 2000.
V. E. Govorov, “The dimension of graded algebras,” Mat. Zametki, 12, No.2, 209–216 (1973).
J. Backelin, A Distributiveness Property of Augmented Algebras, and Some Related Homological Results, Ph. D. thesis, Stockholm, 1982.
V. A. Ufnarovskii, “Algebras defined by two quadratic relations. Studies in the theory of rings, algebras, and modules,” Mat. Issled., No. 76, 148–171 (1984).
V. A. Ufnarovskii, “Combinatorial and asymptotic methods in algebra,” In: Itogi Nauki i Tekhniki, Current Problems in Math., Fundamental Directions [in Russian], Vol. 57, VINITI, Moscow, 1990, pp. 5–177; English transl.: Algebra VI, Encyclopedia Math. Sci., Vol. 57, Springer-Verlag, Berlin, 1995, pp. 1–196.
A. A. Davydov, “Totally positive sequences and R-matrix quadratic algebras,” J. Math. Sci. (New York), 100, No.1, 1871–1876 (2000).
M. Wambst, “Complexes de Koszul quantiques,” Ann. Inst. Fourier, Grenoble, 43, No.4, 1089–1156 (1993).
A. Beilinson, V. Ginzburg, and W. Soergel, “Koszul duality patterns in representation theory,” J. Amer. Math. Soc., 9, No.2, 473–527 (1996).
D. Anick, “On the homology of associative algebras,” Trans. Amer. Math. Soc., 296, No.2, 641–659 (1986).
D. Anick, “Generic algebras and CWcomplexes,” In: Proc. of 1983 Conf. on Algebraic Topology and K-Theory in Honor of John Moore, Ann. Math. Study, Vol. 113, Princeton Univ. Press, 1987, pp. 247–331.
J. J. Zhang, “Non-Noetherian regular rings of dimension 2,” Proc. Amer. Mat. Soc., 126, No.6, 1645–1653 (1998).
D. I. Piontkovskii, “Hilbert series and relations in algebras,” Izv. Ross. Akad. Nauk, Ser. Mat., 64, No.6, 205–219 (2000); English transl.: Russian Acad. Sci. Izv. Math., 64, No. 6, 1297–1311 (2000).
D. I. Piontkovskii, “Noncommutative Grobner bases and coherence of the monomial associative algebra,” Fundam. Prikl. Mat., 2, No.2, 501–509 (1996).
N. K. Iyudu, “Algorithmic solvability of the problem of recognition of zero divisors in a class of algebras,” Fundam. Prikl. Mat., 1, No.2, 541–544 (1995).
D. I. Piontkovskii, “Noncommutative Grobner bases, coherence of associative algebras, and divisibility in semigroups,” Fundam. Prikl. Mat., 7, No.2, 495–513 (2001).
V. E. Govorov, “Dimension and multiplicities of graded algebras,” Sib. Mat. J., 14, 1200–1206 (1973).
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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 47–60, 2005
Original Russian Text Copyright © by D. I. Piontkovskii
Supported in part by the Russian Foundation for Basis Research under project 02-01-00468.
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Piontkovskii, D.I. Koszul Algebras and Their Ideals. Funct Anal Its Appl 39, 120–130 (2005). https://doi.org/10.1007/s10688-005-0024-6
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DOI: https://doi.org/10.1007/s10688-005-0024-6