Resolution of the residue class field via algebraic discrete Morse theory

2005/01/12 by Michael Joellenbeck, Joellenbeck, Michael, Volkmar Welker +1
Computer Science · Mathematics · #05E99 #13D02 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.math/0501179

openalex publication_date 2005/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Forman's Discrete Morse theory is studied from an algebraic viewpoint. Analogous to independent work of Emil Skoeldberg we show that this theory can be extended to chain complexes of free modules over a ring. We provide three applications of this theory: We construct new resolutions of the residue class field k over A, where A is the quotient of a (i) commutative polynomial ring or (ii) non-commutaitve polynomial ring by a (twosided) ideal and (iii) we construct a new resolution of A as an A ⊗ A^op-module in the situation (ii). In either case we prove minimality of the resolution for certain classes of algebras A.

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