Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 14, Number 6 (2014), 3603-3658.
A quadratic refinement of the Grothendieck–Lefschetz–Verdier trace formula
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.
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
Permanent link to this document
Digital Object Identifier
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]
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