Arakelov motivic cohomology I

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.

final version, to appear in Journal of Algebraic Geometry

References in corpus (10)

• Triangulated categories of mixed motives
• Derived Algebraic Geometry III: Commutative Algebra
• The Abel-Jacobi Map for Higher Chow Groups, II
• Algebraic K-theory, A^1-homotopy and Riemann-Roch theorems
• Arakelov motivic cohomology I
• Generalized holomorphic analytic torsion
• Special L-values of geometric motives
• Preorientations of the derived motivic multiplicative group
• Zeta functions of regular arithmetic schemes at s=0
• On Goncharov's regulator and higher arithmetic Chow groups

Cited by in corpus (14)

• Orientation theory in arithmetic geometry
• Arakelov motivic cohomology I
• On higher regulators of Siegel threefolds II: the connection to the special value
• The rigid syntomic ring spectrum
• Symmetric operads in abstract symmetric spectra
• Regulators and cycle maps in higher-dimensional differential algebraic K-theory
• The Beilinson regulator is a map of ring spectra
• Overconvergent global analytic geometry
• Homotopy theory of smooth compactifications of algebraic varieties
• The flat Grothendieck-Riemann-Roch theorem without adiabatic techniques
• On the Riemann-Roch formula without projective hypothesis
• Deligne-Beilinson cycle maps for Lichtenbaum cohomology
• Localization theorem for higher arithmetic K-theory
• Cycle maps on cohomology theories for dg-categories and their applications