Homological Milnor-Witt modules and Chow-Witt groups over general bases

TL;DR

This paper develops a comprehensive theory of homological Milnor-Witt cycle modules over general bases, extending existing theories and establishing duality, with applications to Chow-Witt groups and invariants over various schemes.

Contribution

It introduces a unified framework for homological and cohomological Milnor-Witt modules over general bases, extending prior theories and establishing duality and new applications.

Findings

Extended Chow-Witt groups to schemes over general bases.

Proved generalized Bloch formulas and representability.

Computed graded Chow-Witt groups over Dedekind schemes.

Abstract

We introduce a general theory of homological Milnor-Witt cycle modules over an excellent base scheme equipped with a dimension function, extending both Rost's cycle modules and Feld's theory over fields. To any such module we associate a Rost-Schmid type complex whose homology defines a Borel-Moore intersection theory with quadratic coefficients, satisfying homotopy invariance, localization, proper pushforwards, smooth pullbacks, and Gysin morphisms for essentially smoothable lci morphisms. Using duality data induced by pinning structures, we define cohomological Milnor-Witt modules and establish a duality equivalence between homological and cohomological theories. As applications, we extend Chow-Witt groups to schemes over general (possibly singular or arithmetic) bases, prove generalized Bloch formulas and representability results, and compute graded Chow-Witt groups over Dedekind schemes of finite type over the integers. In particular, we obtain finiteness results for Chow-Witt and related Milnor-Witt invariants in dimension at most one.