 
Summary: FREE RESOLUTIONS OVER SHORT LOCAL RINGS
LUCHEZAR L. AVRAMOV, SRIKANTH B. IYENGAR, AND LIANA M. SžEGA
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(m2) < rankk(m/m2)
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 nonKoszul modules are classified. Structure theorems are established for
the graded kalgebra ExtR(k, k) and its graded module ExtR(M, k).
Introduction
This paper is concerned with the structure of minimal free resolutions of finite
(that is, finitely generated) modules over a commutative noetherian local ring R,
whose maximal ideal m satisfies m3
= 0. Over the last 30 years this special class
has emerged as a testing ground for properties of infinite free resolutions. For a
finite module M such properties are often stated in terms of its Betti numbers
R
n (M) = rankk Extn
R(M, k), where k = R/m, or in terms of its PoincarŽe series
PR
