Koszul homology and syzygies of Veronese subalgebras

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

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 a parts per thousand currency sign 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) the cth Veronese subring of a polynomial ring has Green-Lazarsfeld index a parts per thousand yen c + 1. The same conclusion also holds for an arbitrary standard graded algebra, provided c >> 0.