Koszul Algebras and Their Syzygies

Koszul algebras, introduced by Priddy in [P], are positively graded Kalgebras R whose residue field K has a linear free resolution as an R-module. Here linear means that the non-zero entries of the matrices describing the maps in the R-free resolution of K have degree 1. For example, the polynomial ring S = K[x1, . . . ,xn] over a field K (i.e the symmetric algebra SymK(V ) of a n-dimensional K-vector space V ), then the residue field K is resolved by the Koszul complex which is linear. Analougously, over the exterior algebra E = ∧ K V the residue field K is resolved by the so-called Cartan complex which is also linear. In these lectures we deal mainly with standard graded commutative K-algebras, that is, quotient rings of the polynomial ring S by homogeneous ideals. In the first lecture we present various characterizations of Koszul algebras and strong versions of Koszulness.