The Multiplicity Conjecture for Barycentric Subdivisions

TL;DR

This paper investigates how barycentric subdivision affects algebraic invariants of Stanley-Reisner rings, confirming a conjecture relating multiplicity to shifts in free resolutions.

Contribution

It proves a conjecture by verifying the relationship between multiplicity and shifts for Stanley-Reisner rings of barycentric subdivisions, developing new theoretical insights.

Findings

Confirmed the conjecture relating multiplicity to shifts in free resolutions.

Developed new results on algebraic invariants under barycentric subdivision.

Analyzed behavior of dimension, Hilbert series, and cohomology in this context.

Abstract

For a simplicial complex $\Delta$ we study the effect of barycentric subdivision on ring theoretic invariants of its Stanley-Reisner ring. In particular, for Stanley-Reisner rings of barycentric subdivisions we verify a conjecture by Huneke and Herzog & Srinivasan, that relates the multiplicity of a standard graded $k$-algebra to the product of the maximal and minimal shifts in its minimal free resolution up to the height. On the way to proving the conjecture we develop new and list well known results on behavior of dimension, Hilbert series, multiplicity, local cohomology, depth and regularity when passing from the Stanley-Reisner ring of $\Delta$ to the one of its barycentric subdivision.