A commutative P^1-spectrum representing motivic cohomology over Dedekind domains

By Markus Spitzweck. 2012

93 citations recorded by the source corpus.

We construct a motivic Eilenberg-MacLane spectrum with a highly structured multiplication over smooth schemes over Dedekind domains which represents Levine's motivic cohomology. The latter is defined via Bloch's cycle complexes. Our method is by gluing p-completed and rational parts along an arithmetic square. Hereby the finite coefficient spectra are obtained by truncated \'etale sheaves (relying on the now proven Bloch-Kato conjecture) and a variant of Geisser's version of syntomic cohomology, and the rational spectra are the ones which represent Beilinson motivic cohomology. As an application the arithmetic motivic cohomology groups can be realized as Ext-groups in a triangulated category of Tate sheaves with integral coefficients. These can be modelled as representations of derived fundamental groups. Our spectrum is compatible with base change giving rise to a formalism of six functors for triangulated categories of motivic sheaves over general base schemes including the localization triangle. Further applications include a generalization of the Hopkins-Morel isomorphism and a structure result for the dual motivic Steenrod algebra in the case where the coefficient characteristic is invertible on the base scheme.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c3962b7f99357a5b6786609913f41fad3eb6fbad

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Slices of hermitian $K$–theory and Milnor's conjecture on quadratic forms (2013, background): “(24) Here MGL denotes Voevodsky’s algebraic cobordism spectrum, xn denotes the canonical image of Lazard’s generator in MGL∗,∗, MZ/2 denotes Spitzweck’s motivic cohomology spectrum with F2-coeﬃcients [51], and the maps are induced by the respective canonical orientations. All maps in diagram (24) in”
Integral mixed motives in equal characteristic (2014, background): “However, at this moment, there is no proof that Spitzweck’s motivic cohomology satisfies the absolute purity theorem, and we do not know how to compare Spitzweck’s construction with triangulated categories of motives constructed in the language of algebraic correspondences (except for fields).”
Modules over algebraic cobordism (2019, background): “f. The sheaf A S has canonical nite locally free transfers [BH18, Proposition 13.13], and in particular nite syntomic transfers. Let HZ S 2SH(S) be the motivic cohomology spectrum dened by Spitzweck [Spi18], which is an E 1-algebra in Mod MGL(SH(S)) [Spi18, Remark 10.2]. The following theorem is a r”