2013/09/30 by Marc Hoyois · 2 citations
Mathematics · #math.AG #math.AT #math.KT #msc:14F42 #msc:11E81
paper · pdf · doi:10.2140/agt.2014.14.3603
published as Algebr. Geom. Topol. 14 (2014) 3603-3658 · Final version (to appear in Alg. Geom. Top.)
arxiv created 2014/08/28 · arxiv updated 2015/05/27
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.