Jacobian criteria for complete intersections. The graded case

Abstract

LetP be a positively graded polynomial ring over a fieldk of characteristic zero, letI be a homogeneous ideal ofP, and setR=P/I. The paper investigates the homological properties of someR-modules canonically associated withR, among them the module Ω R/k of Kahler differentials and the conormal moduleI/I 2. It is shown that a subexponential bound on the Betti numbers of any of these modules implies thatI is generated by aP-regular sequence. In particular, the finiteness of the projective dimension of the conormal module impliesR is a complete intersection. Similarly, the finiteness of the projective dimension of the differential module impliesR is a reduced complete intersection. This provides strong converses to some well-known properties of complete intersections, and establishes special cases of conjectures of Vasconcelos. The proofs of these results make extensive use of differential graded homological algebra. The crucial step is to show that any homomorphism of complexes from the minimal cotangent complexL R/k of Andre and Quillen into the minimal free resolution of the irrelevant maximal ideal m ofR, which extends the Euler map Ω R/k →, is a split embedding of gradedR-modules.