Koszul algebras and their syzygies

摘要

These are the notes of the lectures of the author at the 2013 CIME/CIRMsummer school on Combinatorial Algebraic Geometry. Koszul algebras, introducedby Priddy, are positively graded K-algebras R whose residue field K has alinear free resolution as an R-module. The first part of the notes is devotedto the introduction of Koszul algebras and their characterization in terms ofCastelnuovo-Mumford regularity. In the second part we discuss recernt resultson the syzygies of Koszul algebras. Finally in the last part we discuss theKoszul property of Veronese algebras and of algebras associated withcollections of hyperspaces.