A New Proof of the New Intersection Theorem

Abstract

In 1987 Roberts completed the proof of the New Intersection Theorem (NIT) by settling the mixed characteristic case using local Chern characters, as developed by Fulton and also by Roberts. His proof has been the only one recorded of the NIT in mixed characteristic. This paper gives a new proof of this theorem, one which mostly parallels Roberts’ original proof, but avoids the use of local Chern characters. Instead, the proof here uses Adams operations on Ktheory with supports as developed by Gillet-Soulé. New Intersection Theorem. Let A be a (commutative, Noetherian) local ring. If the complex of finite rank free A-modules 0 → Fn → · · · → F1 → F0 → 0 has non-zero homology of finite length, then n ≥ dim(A). In 1973, Peskine-Szpiro [7] proved the New Intersection Theorem (NIT) in prime characteristic p > 0 using the Frobenius map. Their work ushered characteristic p methods to the forefront of commutative algebra; by 1975, Hochster’s work [3, 4] established a reduction to characteristic p > 0 from equicharacteristic zero to give a proof of the NIT in all equicharacteristic rings. In 1987, Roberts [9, 10] proved this theorem for mixed characteristic rings using local Chern characters. In this paper, we give a new proof of the NIT in the mixed characteristic case. This proof parallels Roberts’ original proof in many respects, but differs in that it entirely avoids using local Chern theory. Instead, we use Adams operations on K-theory with supports, as developed by Gillet-Soulé [2]. The difference between this proof 2000 Mathematics Subject Classification. 13D22, 13D15, 19A99.

Cite this paper

@inproceedings{PIEPMEYER2008ANP, title={A New Proof of the New Intersection Theorem}, author={GREG PIEPMEYER and Mark E. Walker and Paul Alfred Roberts}, year={2008} }