The Milnor-Witt motivic ring spectrum and its associated theories

Abstract

We build a ring spectrum representing Milnor-Witt motivic cohomology, as well as its étale local version and show how to deduce out of it three other theories: Borel-Moore homology, cohomology with compact support and homology. These theories, as well as the usual cohomology, are defined for singular schemes and satisfy the properties of their motivic analog (and more), up to considering more general twists. In fact, the whole formalism of these four theories can be functorially attached to any ring spectrum, giving finally maps between the Milnor-Witt motivic ones to the classical motivic ones.

arXiv comment28 pages. Comments welcome

References in corpus (4)

• Triangulated categories of mixed motives
• Orientation theory in arithmetic geometry
• MW-motivic complexes
• Bivariant theories in motivic stable homotopy

Cited by in corpus (6)

• Fundamental classes in motivic homotopy theory
• On the effectivity of spectra representing motivic cohomology theories
• Chow-Witt rings of classifying spaces for symplectic and special linear groups
• Cancellation theorem for Grothendieck-Witt-correspondences and Witt-correspondences
• General Motivic Cohomology and Symplectic Orientation
• Zariski-local framed ${\mathbb{A}}^1$-homotopy theory