Groebner basis degree bounds on $\Tor^{k[Λ]}_\bullet(k,k)_\bullet$ and discrete Morse theory for posets

Abstract

The purpose of this paper is twofold. 1. We give combinatorial bounds on the ranks of the groups $\Tor^{R}_\bullet(k,k)_\bullet$ 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.