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

TL;DR

This paper establishes an isomorphism between big de Rham-Witt forms over fields and relative Milnor K-groups of Artin local algebras, solving a longstanding problem and linking to motivic cohomology.

Contribution

It proves a new isomorphism connecting de Rham-Witt forms with Milnor K-groups over fields, addressing a problem from the 1970s in positive characteristic.

Findings

Isomorphism between big de Rham-Witt forms and Milnor K-groups.

Interpretation of de Rham-Witt forms as vanishing cycles in motivic cohomology.

Construction of an extended logarithmic derivative map for Artin rings.

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.