Mathematical Research Letters

Volume 15 (2008)

Number 3

A vanishing theorem for oriented intersection multiplicities

Pages: 447 – 458

DOI: http://dx.doi.org/10.4310/MRL.2008.v15.n3.a5

Authors

J. Fasel (École Polytechnique Fédérale de Lausanne)

V. Srinivas (Tata Institute of Fundamental Research)

Abstract

Let $A$ be a regular local ring containing $1/2$, which is either equicharacteristic, or is smooth over a d.v.r. of mixed characteristic. We prove that the product maps on derived Grothendieck-Witt groups of $A$ satisfy the following property: given two elements with supports which do not intersect properly, their product vanishes. This gives an analogue for “oriented intersection multiplicities” of Serre’s vanishing result for intersection multiplicities. It also suggests a Vanishing Conjecture for arbitrary regular local rings containing $1/2$, which is analogous to Serre’s (which was proved independently by Roberts, and Gillet and Soulé).

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