The slice spectral sequence for singular schemes and applications

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 a large class of projective schemes. We also deduce applications to the torsion in the motivic cohomology of singular schemes.