Uniform bounds on projective dimension and Castelnuovo-Mumford regularity

TL;DR

This paper establishes uniform, effective upper bounds for the projective dimension and Castelnuovo-Mumford regularity of homogeneous ideals in polynomial rings, independent of the number of variables, extending previous results.

Contribution

It extends McCullough's 2012 result to provide bounds on both projective dimension and regularity based on a fraction of syzygies, independent of variables count.

Findings

Derived bounds are independent of the number of variables.

Extended McCullough's result from regularity to projective dimension.

Provided bounds depend on partial data from the resolution.

Abstract

In this article we obtain uniform effective upper bounds for the projective dimension and the Castelnuovo-Mumford regularity of homogeneous ideals inside a standard graded polynomial ring $S$ over a field. Such bounds are independent of the number of variables of $S$, in the spirit of Stillman's conjecture and of the Ananyan-Hochster's theorem, and depend on partial data extracted from the beginning or the end of the resolution. In this direction, we extend a result of McCullough from 2012 regarding a bound on regularity in terms of half the syzygies to a bound on the projective dimension and the regularity of an ideal in terms of a fraction of the syzygies.