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Register a new userA quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula
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Authors
Marc Hoyois
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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 etale, we compute this integral in terms of Morels 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 etale algebra corresponds to the class of its trace form in the Grothendieck-Witt ring.
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