Norms in motivic homotopy theory
By Tom Bachmann, Marc Hoyois. Astérisque, 2017
139 citations recorded by the source corpus.
If $f : S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor $f_\otimes : \mathcal{H}_{\bullet}(S')\to \mathcal{H}_{\bullet}(S)$, where $\mathcal{H}_\bullet(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite étale, we show that it stabilizes to a functor $f_\otimes : \mathcal{S}\mathcal{H}(S') \to \mathcal{S}\mathcal{H}(S)$, where $\mathcal{S}\mathcal{H}(S)$ is the $\mathbb{P}^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic $E_\infty$-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum $H\mathbb{Z}$, the homotopy $K$-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $H\mathbb{Z}$ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/866c1139430b6661255b72ccf881d374e692c4ad
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In Mathematical Physics
Citations in the corpus
- Fundamental classes in motivic homotopy theory (2018, unknown): “(3) 2, Springer, Berlin (1998) Zbl 0885.”
- EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS (2020, background): “) is the K-theory of oriented (respectively symplectic) vector bundles in the usual sense, and MSL (respectively MSp) is the Thom spectrum as defined in [PW10a]. Proposition 4.10 (see Example 16.20 in [BH17]). The Thom spectra MGL,MSL,MSp are stable under base change. Now let ξ∈KG(X). Then there is a”