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 MGLX 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
Received: 10 November 2017
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