Mathematical Research Letters

Volume 15, Issue 3, May 2008 pp. 447-458.

A vanishing theorem for oriented intersection multiplicities

Authors: J. Fasel (1) and V. Srinivas (2)
Author institution: École Polytechnique Fédérale de Lausanne (1) and Tata Institute of Fundamental Research (2)

Summary: 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 Soul\'e).

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