Koszul homology and syzygies of Veronese subalgebras

TL;DR

This paper investigates the syzygies and Koszul homology of Veronese subalgebras, establishing bounds on their Green-Lazarsfeld index, which measures the complexity of their defining relations.

Contribution

It proves that the c-th Veronese subring of a polynomial ring has a Green-Lazarsfeld index at least c+1, extending to general graded algebras for large c.

Findings

Veronese subrings have high Green-Lazarsfeld index

Index bounds are c+1 for polynomial rings

Results apply to general graded algebras for large c

Abstract

A graded K-algebra R has property N_p if it is generated in degree 1, has relations in degree 2 and the syzygies of order less or equal to p on the relations are linear. The Green-Lazarsfeld index of R is the largest p such that it satisfies the property N_p. Our main results assert that (under a mild assumption on the base field K) the c-th Veronese subring of a polynomial ring has Green-Lazarsfeld index greater or equal to c+1. The same conclusion also holds for an arbitrary standard graded algebra, provided c is sufficiently large.