The Partition Complex: an invitation to combinatorial commutative algebra

TL;DR

This paper introduces the partition complex as a new, self-contained foundation for combinatorial commutative algebra and Stanley-Reisner theory, providing novel proofs and generalizations of key theorems.

Contribution

It develops new techniques using the partition complex, including self-contained proofs of Reisner's theorem, Poincaré duality, and a master-theorem for Lefschetz properties.

Findings

Self-contained proof of Reisner's theorem and Schenzel's formula

Generalized Poincaré duality for face rings of manifolds

A master-theorem for Lefschetz theorems on subdivisions

Abstract

We provide a new foundation for combinatorial commutative algebra and Stanley-Reisner theory using the partition complex introduced in [Adi18]. One of the main advantages is that it is entirely self-contained, using only a minimal knowledge of algebra and topology. On the other hand, we also develop new techniques and results using this approach. In particular, we provide - A novel, self-contained method of establishing Reisner's theorem and Schenzel's formula for Buchsbaum complexes. - A simple new way to establish Poincar\'e duality for face rings of manifolds, in much greater generality and precision than previous treatments. - A "master-theorem" to generalize several previous results concerning the Lefschetz theorem on subdivisions. - Proof for a conjecture of K\"uhnel concerning triangulated manifolds with boundary.