Subadditivity of syzygies of Koszul algebras

TL;DR

This paper investigates the growth of degrees of minimal syzygies in quotient algebras of polynomial rings, showing they increase slowly, especially under certain conditions, with implications for Koszul algebras.

Contribution

It establishes bounds on the degree growth of syzygies for a broad class of algebras, including Koszul algebras, and refines these bounds under additional conditions.

Findings

Syzygy degrees increase by at most 2 between consecutive modules.

Slower growth of syzygies is proven under the N_q condition with q > 1.

Results apply to Koszul algebras in almost all characteristics.

Abstract

Estimates are obtained for the degrees of minimal syzygies of quotient algebras of polynomial rings. For a class that includes Koszul algebra in almost all characteristics, these degrees are shown to increase by at most 2 from one syzygy module to the next one. Even slower growth is proved if, in addition, the algebra satisfies Green and Lazarsfeld's condition N_q with q > 1.