Arakelov motivic cohomology I

Holmstrom, Andreas; Scholbach, Jakob

doi:10.1090/jag/648

Authors: Andreas Holmstrom and Jakob Scholbach
Journal: J. Algebraic Geom. 24 (2015), 719-754
DOI: https://doi.org/10.1090/jag/648
Published electronically: April 23, 2015
MathSciNet review: 3383602
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Abstract | References | Additional Information

Abstract:

This paper introduces a new cohomology theory for schemes of finite type over an arithmetic ring. The main motivation for this Arakelov-theoretic version of motivic cohomology is the conjecture on special values of $L$-functions and zeta functions formulated by the second author. Taking advantage of the six functors formalism in motivic stable homotopy theory, we establish a number of formal properties, including pullbacks for arbitrary morphisms, pushforwards for projective morphisms between regular schemes, localization sequences, $h$-descent. We round off the picture with a purity result and a higher arithmetic Riemann-Roch theorem.

In a sequel to this paper, we relate Arakelov motivic cohomology to classical constructions such as arithmetic $K$ and Chow groups and the height pairing.

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Additional Information

Andreas Holmstrom
Affiliation: Institut des Hautes Études Scientifiques Le Bois-Marie, 35 Route de Chartres, F-91440 Bures-sur-Yvette, France
Email: andreas.holmstrom@gmail.com

Jakob Scholbach
Affiliation: Universität Münster, Mathematisches Institut, Einsteinstrasse 62, D-48149 Münster, Germany
Email: jakob.scholbach@uni-muenster.de

Received by editor(s): October 10, 2012
Received by editor(s) in revised form: June 26, 2013
Published electronically: April 23, 2015
Article copyright: © Copyright 2015 University Press, Inc.