Abstract
These notes consist of five sections. The aim of these notes is to provide a summary of the theory of noncommutative Gröbner bases and how to apply this theory in representation theory; most notably, in constructing projective resolutions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Reference
W. Adams and P. Loustaunau: An Introduction to Gröbner bases, Graduate St. in Math, AMS 3, 1994.
D. Anick: Noncocomutative graded algebras and their Hilbert series, J. of Algebra 78, (1982), 120–140.
D. Anick: On monomial algebras of finite global dimension, Transactions AMS 291, (1985), 291–310.
D. Anick: Recent progress in Hilbert and Poincaré series, LNM 1318, (1986) Springer-Verlag, 1–25.
D. Anick and E.L. Green: On the homology of path algebras, Comm in Algebra 15, (1985), 641–659.
M. Auslander I. Reiten, and S. Smal∅: Representation Theory of Artin Algebras, Cambridge Studies in Advanced Math. 36, (1995), Cambridge Univ. Press.
J. Backelin, R. Froberg: Koszul algebras, Veronese subrings and rings with linear resolutions, Rev. Roumaine Math. Pures Appl. 30, (1980), 85–97.
M. Bardzell: The alternating syzygy behavior of monomial algebras, J. of Algebra 188, (1997), no. 1, 69–89.
T. Becker and V. Weispfenning: Gröbner bases. A Computational Approach to Commutative Algebra, GTM 141, Springer-Verlag, 1993.
A. Beilinson, V. Ginsburg, & W. Soergel: Koszul Duality Patterns in Representation Theory, J. Amer. Math.Soc.9, (1996) 473–527.
G. Bergman: The diamond Lemma for ring theory,Adv. Math. 29, (1978) 178–218.
A.I. Bondal: Helices, representations of quivers and Koszul algebras, London Math. Soc. Lecture Note Ser. 148,(1990), 75–95.
Buchberger: An algorithm for finding a basis for the residue class ring of a zero-dimensional ideal, Ph.D. Thesis, University of Innsbruck, (1965).
E. Cline, B. Parshall, and L. Scott: Finite dimensional algebras and highest weight categories, J. Reine Angew. Math. 391, (1988), 85–99.
D. Cox, J. Little, and D. O’Shea: Ideals, Varieties, and Algorithms, UTM Series, Springer-Verlag (1992).
D.R. Farkas, C. Feustel, and E.L. Green: Synergy in the theories of Gröbner bases and path algebras, Canad. J. of Mathematics 45, (1993), 727–739.
C. Feustel, E.L. Green, E. Kirkman, and J. Kuzmanovich: Constructing projective resolutions, Comm. in Alg. 21, (1993) 1869–1887.
E.L. Green: Representation theory of tensor algebras, J. Algebra 34, (1975), 136–171.
E.L. Green: Poincaré-Birkhoff-Witt bases and Gröbner bases, preprint.
E.L. Green and R. Huang: Projective resolutions of straightening closed algebras generated by minors, Adv. in Math. 110, (1995), 314–333.
E.L. Green, and R. Martínez Villa: Koszul and Yoneda algebras, Canadian Math. Soc. 18,(1994), 247–298.
E.L. Green, and R. Martinez Villa: Koszul and Yoneda algebras II, in Yoneda algebras II, Canadian Math. Soc., Proceedings of ICRA, Ed. Reiten, Smalö, Solberg, 1998.
E.L. Green and D. Zacharia: The cohomology ring of a monomial algebra, Manuscripta Math. 85, (1994).
R. Hartshone: Residues and Duality, LNM 20, Springer-Verlag, (1966).
C. Löfwall: On the subalgebra generated by the one-dimensional elements in the Yoneda ext-algebra, LNM 1183, Springer-Verlag, (1986), 291–338.
R. Martinez Villa: Applications of Koszul algebras: the preprojective algebra, Canadian Math. Soc. 18, (1994), 487–504.
S. McLane: Homology, Springer-Verlag, 1963.
T. Mora: Gröbner bases for non-commutative polynomial rings, Proc. AAECC3 L.N.C.S. 229, (1986).
S. Priddy: Koszul resolutions, Trans. AMS 152, (1970), 39–60.
J.E. Roos: Relations between the Poincaré-Betti series of loop spaces and of local rings, LNM 740, Springer-Verlag, 285–322.
M. Rosso: Koszul resolutions and quantum groups, Nuclear Phys. B Proc. Suppl. 18b, (1990), 269–276.
V. Ufnarovskii: A growth criterion for graphs and algebras defined by words, Mat. Zemati 31, (1980) 465–472; Math. Notes 37, (1982) 238–241.
Y. Yoshino: Modules with linear resolutions over a polynomial ring in two variables, Nagoya Math. J. 113, (1989), 89–98.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Basel AG
About this paper
Cite this paper
Green, E.L. (1999). Noncommutative Gröbner Bases, and Projective Resolutions. In: Dräxler, P., Ringel, C.M., Michler, G.O. (eds) Computational Methods for Representations of Groups and Algebras. Progress in Mathematics, vol 173. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8716-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8716-8_2
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9740-2
Online ISBN: 978-3-0348-8716-8
eBook Packages: Springer Book Archive