On extension of the motivic cohomology beyond smooth schemes

TL;DR

This paper extends motivic cohomology to include schemes with singularities and non-reduced schemes, providing a functorial algebraic-cycle based model that generalizes existing theories and detects nilpotence.

Contribution

It introduces a new motivic cohomology model applicable to all schemes of finite type over a field, including singular and non-reduced schemes, aligning with classical theory on smooth schemes.

Findings

Model is functorial on the category of schemes.

Detects nilpotence in schemes.

Agrees with higher Chow theory on smooth schemes.

Abstract

We construct an algebraic-cycle based model for the motivic cohomology on the category of schemes of finite type over a field, where schemes may admit arbitrary singularities and may be non-reduced. We show that our theory is functorial on the category, that it detects nilpotence, and that its restriction to the subcategory of smooth schemes agrees with the pre-existing motivic cohomology theory, which is the higher Chow theory of S. Bloch (Adv. Math., 1986). A few structures and applications are discussed.