2003/12/30 by Patricia Hersh, Hersh, Patricia, Volkmar Welker +1
Computer Science · Mathematics · #05E25 #06A07 #13D02 #13P10 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Topological and Geometric Data Analysis #math.AC #math.CO #msc:05E25 #msc:06A07 #msc:13D02 #msc:13P10
paper · pdf · doi:10.48550/arxiv.math/0312505
arxiv created 2003/12/30 · openalex publication_date 2003/12/30 · arxiv updated 2009/12/01 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
The purpose of this paper is twofold. 1. We give combinatorial bounds on the ranks of the groups \TorR_\bullet(k,k)_\bullet in the case where R = k[Λ] is an affine semi-group ring, and in the process provide combinatorial proofs for bounds by Eisenbud, Reeves and Totaro on which Tor groups vanish. In addition, we show that if the bounds hold for a field k then they hold for \field[Λ] and any field \field. Moreover, we provide a combinatorial construction for a free resolution of \field over \field[Λ] which achieves these bounds. 2. We extend the lexicographic discrete Morse function construction of Babson and Hersh for the determination of the homotopy type and homology of order complexes of posets to a larger class of facet orderings that includes orders induced by monomial term orders. Since it is known that the order complexes of finite intervals in the poset of monomials in k[Λ] ordered by divisibility in k[Λ] govern the \Tor-groups, the newly developed tools are applicable and serve as the main ingredients for the proof of the bounds and the construction of the resolution.