Milnor-Witt cycle modules over an excellent DVR

TL;DR

This paper extends the theory of Milnor-Witt cycle modules to excellent DVRs, establishing conditions under which Gersten complex exactness and $ ext{A}^1$-invariance hold, especially for $K^{MW}$.

Contribution

It introduces an extra axiom R5 for Milnor-Witt cycle modules over excellent DVRs, ensuring key properties like Gersten complex exactness and $ ext{A}^1$-invariance.

Findings

Gersten complex is exact under R5 for certain Milnor-Witt modules.

$K^{MW}$ satisfies R5 over any base, ensuring invariance.

Results apply to excellent DVRs, generalizing previous field-based theories.

Abstract

The definition of Milnor-Witt cycle modules in [Feld, N., Milnor-Witt cycle modules, Journal of Pure and Applied Algebra 224 (2020) 106298] can easily be adapted over general regular base schemes. However, there are simple examples to show that Gersten complex fails to be exact for cycle modules in general if the base is not a field. The goal of this article is to show that, for a restricted class of Milnor-Witt cycle modules over an excellent DVR satisfying an extra axiom, called here as R5, the expected properties of exactness of Gersten complex and $\mathbb{A}^1$-invariance hold. Moreover R5 is vacuously satisfied when the base is a perfect field and it is also satisfied by $K^{MW}$ over any base. As a corollary, we obtain the strict $\mathbb{A}^1$-invariance and the exactness of Gersten complex for $K^{MW}$ over an excellent DVR.