The slice spectral sequence for singular schemes and applications

TL;DR

This paper develops a spectral sequence approach for studying the motivic cobordism and cohomology of singular schemes, establishing isomorphisms and injectivity results that deepen understanding of their algebraic and geometric properties.

Contribution

It introduces a new spectral sequence for singular schemes using the slice filtration, linking motivic cobordism and cohomology, and proves injectivity of the cycle class map for projective schemes.

Findings

Spectral sequence for motivic cobordism of singular schemes is constructed.

Isomorphism between geometric parts of motivic cobordism and cohomology is established.

Cycle class map from motivic cohomology to homotopy K-theory is injective for projective schemes.

Abstract

We examine the slice spectral sequence for the cohomology of singular schemes with respect to various motivic T-spectra, especially the motivic cobordism spectrum. When the base field k admits resolution of singularities and X is a scheme of finite type over k, we show that Voevodsky's slice filtration leads to a spectral sequence for MGL(X) whose terms are the motivic cohomology groups of X defined using the cdh-hypercohomology. As a consequence, we establish an isomorphism between certain geometric parts of the motivic cobordism and motivic cohomology of X. A similar spectral sequence for the connective K-theory leads to a cycle class map from the motivic cohomology to the homotopy invariant K-theory of X. We show that this cycle class map is injective for projective schemes. We also deduce applications to the torsion in the motivic cohomology of singular schemes.