 
Summary: JACOBIAN CRITERIA FOR COMPLETE INTERSECTIONS.
THE GRADED CASE
Luchezar L. Avramov and J urgen Herzog
Dedicated to Professor Ernst Kunz on his sixtieth birthday
Abstract. Let P be a positively graded polynomial ring over a eld k of characteristic zero,
let I be a homogeneous ideal of P , and set R = P=I. The paper investigates the homological
properties of some R{modules canonically associated with R, among them the
module
Rjk
of Kahler dierentials and the conormal module I=I 2 .
It is shown that a subexponential bound on the Betti numbers of any of these modules
implies that I is generated by a P {regular sequence. In particular, the niteness of the pro
jective dimension of the conormal module implies R is a complete intersection. Similarly, the
niteness of the projective dimension of the dierential module implies R is a reduced com
plete intersection. This provides strong converses to some wellknown properties of complete
intersections, and establishes special cases of conjectures of Vasconcelos.
The proofs of these results make extensive use of dierential graded homological algebra.
The crucial step is to show that the morphism of complexes from the minimal cotangent
complex L Rjk of Andre and Quillen into the minimal free resolution of the irrelevant maximal
ideal m of R, which extends the Euler
