Algebraic & Geometric Topology

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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513716051

Digital Object Identifier
doi:10.2140/agt.2014.14.3603

Zentralblatt MATH identifier
1351.14013

Subjects
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]

Keywords
motivic homotopy theory Grothendieck–Witt group trace formula

Citation

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. https://projecteuclid.org/euclid.agt/1513716051


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