Free resolutions over short local rings

To the memory of our friend and colleague Anders Frankild. Abstract. The structure of minimal free resolutions of finite modules M over commutative local rings (R,m, k) with m3 = 0 and rankk(m 2) < rankk(m/m 2) is studied. It is proved that over generic R every M has a Koszul syzygy mod-ule. Explicit families of Koszul modules are identified. When R is Gorenstein the non-Koszul modules are classified. Structure theorems are established for the graded k-algebra ExtR(k, k) and its graded module ExtR(M,k)