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

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 \'etale, 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 \'etale algebra corresponds to the class of its trace form in the Grothendieck-Witt ring.