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arXiv · 1309.6147

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

Marc Hoyois
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Abstract

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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Marc Hoyois. 2014-08-28. A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula. https://doi.org/10.2140/agt.2014.14.3603

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