Milnor-Witt cycle modules over an excellent DVR

Abstract

The definition of Milnor-Witt cycle modules in [5] can easily be adapted over general regular base schemes. However, there are simple examples (see (2.9)) 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 A ¹-invariance hold. Moreover R5 is vacuously satisfied when the base is a field and it is also satisfied by K^MWover any base. As a corollary, we obtain the strict A ¹-invariance and the exactness of Gersten complex K^MWfor over an excellent DVR.