The big de Rham-Witt forms over fields and motives of non-reduced schemes

Abstract

Using algebraic cycles as a medium, we prove that the groups of the big (Hesselholt-Madsen) de Rham-Witt forms over arbitrary fields are isomorphic to the relative improved (Gabber-Kerz) Milnor $K$-groups of Artin local algebras of embedding dimension $1$. This answers an old problem on the relative Milnor $K$-groups studied since 1970s, especially in ${\rm char} (k) = p>0$. Applications include an interpretation of the big de Rham-Witt forms precisely as the vanishing cycles of the Elmanto-Morrow motivic cohomology of non-reduced schemes, as well as a construction of an extended logarithmic derivative map $d\log$ on the Milnor $K$-theory of some Artin rings to the de Rham-Witt forms.