On the coniveau filtration on algebraic $K$-theory of singular schemes

TL;DR

This paper introduces two new functorial filtrations on algebraic K-theory for schemes with singularities, extending classical filtrations from smooth schemes and connecting to motivic homotopy theory.

Contribution

It constructs the coniveau and motivic coniveau filtrations on algebraic K-theory for singular schemes, generalizing classical filtrations and linking to motivic homotopy theory.

Findings

The coniveau filtration matches Quillen's classical filtration on smooth schemes.

The motivic coniveau filtration aligns with Levine's homotopy coniveau filtration.

Both filtrations are functorial and applicable to schemes with arbitrary singularities.

Abstract

We construct two functorial filtrations on the algebraic $K$-theory of schemes of finite type over a field $k$ that may admit arbitrary singularities and may be non-reduced, one called the coniveau filtration, and the other called the motivic coniveau filtration. Restricting to the subcategory of smooth $k$-schemes, our coniveau filtration coincides with the classical coniveau (also known as the topological) filtration on algebraic $K$-theory of D. Quillen, whereas our motivic coniveau filtration coincides with the homotopy coniveau filtration for algebraic $K$-theory of M. Levine.