Stillman's Question for Exterior Algebras and Herzog's Conjecture on Betti Numbers of Syzygy Modules

TL;DR

This paper constructs examples of ideals in exterior algebras with unbounded regularity, providing counterexamples to Stillman's Question in this setting and addressing Herzog's conjecture on Betti numbers of syzygy modules.

Contribution

It demonstrates that the analogue of Stillman's Question fails for exterior algebras and offers counterexamples to Herzog's conjecture on Betti numbers.

Findings

Existence of ideals with unbounded Castelnuovo-Mumford regularity in exterior algebras

Counterexamples to Stillman's Question for exterior algebras

Counterexamples to Herzog's conjecture on Betti numbers

Abstract

Let K be a field of characteristic 0 and consider exterior algebras of finite dimensional K-vector spaces. In this short paper we exhibit principal quadric ideals in a family whose Castelnuovo-Mumford regularity is unbounded. This negatively answers the analogue of Stillman's Question for exterior algebras posed by I. Peeva. We show that these examples are dual to modules over polynomial rings that yield counterexamples to a conjecture of J. Herzog on the Betti numbers in the linear strand of syzygy modules.