A quadratic refinement of the Grothendieck–Lefschetz–Verdier trace formula

We prove a trace formula in stable motivic homotopy theory over a general base scheme, equating the trace of an endomorphism of a smooth proper scheme with the “Euler characteristic integral” of a certain cohomotopy class over its scheme of fixed points. When the base is a field and the fixed points are étale, we compute this integral in terms of Morel’s identification of the ring of endomorphisms of the motivic sphere spectrum with the Grothendieck–Witt ring. In particular, we show that the Euler characteristic of an étale algebra corresponds to the class of its trace form in the Grothendieck–Witt ring.

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Keywords

motivic homotopy theory, Grothendieck–Witt group, trace formula

Mathematical Subject Classification 2010

Primary: 14F42

Secondary: 47H10, 11E81

References

Publication

Received: 1 November 2013

Revised: 13 June 2014

Accepted: 23 June 2014

Published: 15 January 2015

Authors

Marc Hoyois
Department of Mathematics Northwestern University 2033 Sheridan Road Evanston, IL 60208 USA
http://math.northwestern.edu/~hoyois/

Volume 14, issue 6 (2014)

Marc Hoyois

Abstract