Algebraic & Geometric Topology

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

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 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.

Article information

Algebr. Geom. Topol. Volume 14, Number 6 (2014), 3603-3658.

Received: 1 November 2013
Revised: 13 June 2014
Accepted: 23 June 2014
First available in Project Euclid: 19 December 2017

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Zentralblatt MATH identifier

Primary: 14F42: Motivic cohomology; motivic homotopy theory [See also 19E15]
Secondary: 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30] 11E81: Algebraic theory of quadratic forms; Witt groups and rings [See also 19G12, 19G24]

motivic homotopy theory Grothendieck–Witt group trace formula


Hoyois, Marc. A quadratic refinement of the Grothendieck–Lefschetz–Verdier trace formula. Algebr. Geom. Topol. 14 (2014), no. 6, 3603--3658. doi:10.2140/agt.2014.14.3603.

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