I will briefly recall Milnor cycle modules over a field as defined by Rost (1996) and their significance and properties. Recently, 'modules' over Milnor-Witt K-theory or alternatively Milnor-Witt cycle modules over field have been formalized by N. Feld (2020).
I will talk about recent joint work with Chetan Balwe and Amit Hogadi, where we considered the Milnor-Witt cycle modules over excellent DVR and studied a subclass of these that satisfy certain lifting conditions on residue maps associated with horizontal valuations. As an important example, Milnor-Witt K-theory of fields belongs to this subclass. Moreover, this condition is sufficient to deduce the local acyclicity property and A^1-homotopy invariance of the associated Gersten complex.
