Morse theory from an algebraic viewpoint
HTML articles powered by AMS MathViewer
- by Emil Sköldberg PDF
- Trans. Amer. Math. Soc. 358 (2006), 115-129 Request permission
Abstract:
Forman’s discrete Morse theory is studied from an algebraic viewpoint, and we show how this theory can be extended to chain complexes of modules over arbitrary rings. As applications we compute the homologies of a certain family of nilpotent Lie algebras, and show how the algebraic Morse theory can be used to derive the classical Anick resolution as well as a new two-sided Anick resolution.References
- Grant F. Armstrong, Grant Cairns, and Barry Jessup, Explicit Betti numbers for a family of nilpotent Lie algebras, Proc. Amer. Math. Soc. 125 (1997), no. 2, 381–385. MR 1353371, DOI 10.1090/S0002-9939-97-03607-1
- David J. Anick, On the homology of associative algebras, Trans. Amer. Math. Soc. 296 (1986), no. 2, 641–659. MR 846601, DOI 10.1090/S0002-9947-1986-0846601-5
- Jörgen Backelin, The Gröbner basis calculator Bergman, Available at http://www.math.su.se/bergman/.
- Jörgen Backelin, La série de Poincaré-Betti d’une algèbre graduée de type fini à une relation est rationnelle, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 13, A843–A846 (French, with English summary). MR 551760
- Michael J. Bardzell, The alternating syzygy behavior of monomial algebras, J. Algebra 188 (1997), no. 1, 69–89. MR 1432347, DOI 10.1006/jabr.1996.6813
- Eric Babson, Anders Björner, Svante Linusson, John Shareshian, and Volkmar Welker, Complexes of not $i$-connected graphs, Topology 38 (1999), no. 2, 271–299. MR 1660341, DOI 10.1016/S0040-9383(98)00009-3
- Donald W. Barnes and Larry A. Lambe, A fixed point approach to homological perturbation theory, Proc. Amer. Math. Soc. 112 (1991), no. 3, 881–892. MR 1057939, DOI 10.1090/S0002-9939-1991-1057939-0
- E. Batzies and V. Welker, Discrete Morse theory for cellular resolutions, J. Reine Angew. Math. 543 (2002), 147–168. MR 1887881, DOI 10.1515/crll.2002.012
- Manoj K. Chari, On discrete Morse functions and combinatorial decompositions, Discrete Math. 217 (2000), no. 1-3, 101–113 (English, with English and French summaries). Formal power series and algebraic combinatorics (Vienna, 1997). MR 1766262, DOI 10.1016/S0012-365X(99)00258-7
- Svetlana Cojocaru, Alexander Podoplelov, and Victor Ufnarovski, Non-commutative Gröbner bases and Anick’s resolution, Computational methods for representations of groups and algebras (Essen, 1997) Progr. Math., vol. 173, Birkhäuser, Basel, 1999, pp. 139–159. MR 1714607
- Robin Forman, Morse theory for cell complexes, Adv. Math. 134 (1998), no. 1, 90–145. MR 1612391, DOI 10.1006/aima.1997.1650
- Jakob Jonsson, On the topology of simplicial complexes related to 3-connected and Hamiltonian graphs, J. Combin. Theory Ser. A 104 (2003), no. 1, 169–199. MR 2018427, DOI 10.1016/j.jcta.2003.07.001
- Michael Jöllenbeck and Volkmar Welker, Resolution of the residue class field via algebraic discrete morse theory, arXiv:math.AC/0501179, 2005.
- John Shareshian, Discrete Morse theory for complexes of $2$-connected graphs, Topology 40 (2001), no. 4, 681–701. MR 1851558, DOI 10.1016/S0040-9383(99)00076-2
- V. A. Ufnarovskiĭ, On the use of graphs for calculating the basis, growth and Hilbert series of associative algebras, Mat. Sb. 180 (1989), no. 11, 1548–1560, 1584 (Russian); English transl., Math. USSR-Sb. 68 (1991), no. 2, 417–428. MR 1034428, DOI 10.1070/SM1991v068n02ABEH001373
Additional Information
- Emil Sköldberg
- Affiliation: Department of Mathematics, National University of Ireland, Galway, Ireland
- Email: emil.skoldberg@nuigalway.ie
- Received by editor(s): August 4, 2003
- Published electronically: August 25, 2005
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 115-129
- MSC (2000): Primary 16E05; Secondary 16E40, 17B56
- DOI: https://doi.org/10.1090/S0002-9947-05-04079-1
- MathSciNet review: 2171225