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A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula. (English) Zbl 1351.14013

Summary: 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.

MSC:

14F42 Motivic cohomology; motivic homotopy theory
11E81 Algebraic theory of quadratic forms; Witt groups and rings
55M20 Fixed points and coincidences in algebraic topology
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