#### Vol. 3, No. 4, 2018

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The slice spectral sequence for singular schemes and applications

### Amalendu Krishna and Pablo Pelaez

Vol. 3 (2018), No. 4, 657–708
##### 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.

##### Keywords
algebraic cobordism, Milnor K-theory, motivic homotopy theory, motivic spectral sequence, K-theory, slice filtration, singular schemes
##### Mathematical Subject Classification 2010
Primary: 14C25, 14C35, 14F42, 19E08, 19E15
##### Milestones
Revised: 10 May 2018
Accepted: 31 May 2018
Published: 18 December 2018
##### Authors
 Amalendu Krishna School of Mathematics Tata Institute of Fundamental Research Mumbai India Pablo Pelaez Instituto de Matemáticas Ciudad Universitaria, UNAM México