The big de Rham-Witt forms over fields and motives of non-reduced schemes
Jinhyun Park
TL;DR
This work resolves a long-standing problem on the relative Milnor K-theory of Artin local algebras of embedding dimension 1 by establishing a level-wise, characteristic-free isomorphism with the big de Rham–Witt forms ${\mathbb W}_m\Omega_k^{n-1}$. The author constructs a geometric inverse Bloch map using additive higher Chow groups and Chow groups of vanishing cycles, proving both injectivity and surjectivity through a novel deconcatenation mechanism and cycle-theoretic transfer. As a consequence, the paper identifies the relative Milnor K-groups with vanishing-cycle motivic cohomology and yields a natural extension of $d\log$ to the Artin setting, along with a split exact sequence linking Milnor and de Rham–Witt structures. The results unify and extend perspectives from Bloch–Kato, Rülling–Saito, and Elmanto–Morrow, and have implications for motivic cohomology with modulus and non-reduced schemes. Overall, the work provides a characteristic-free, cycle-theoretic bridge between Milnor K-theory and big de Rham–Witt forms, with broad applications in arithmetic geometry and motivic cohomology.
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.