Jacobian criteria for complete intersections. The graded case

Summary 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 _R/k of Kähler 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 André and Quillen into the minimal free resolution of the irrelevant maximal ideal m ofR, which extends the Euler map _R/k , is a split embedding of gradedR-modules.Oblatum 14-IV-1993 & 9-IX-1993Dedicated to Professor Ernst Kunz on his sixtieth birthdayThe first author was partly supported by a grant from NSF. During the peparation of this paper the second author was supported by Purdue University, whose hospitality he wishes to acknowledge