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Zero cycles with modulus and zero cycles on singular varieties

Part of: $K$-theory in geometry Arithmetic rings and other special rings Cycles and subschemes (Co)homology theory

Published online by Cambridge University Press:  09 October 2017

Federico Binda  and
Amalendu Krishna
Federico Binda
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040, Regensburg, Germany email [email protected]
Amalendu Krishna
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Colaba, Mumbai, India email [email protected]
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Abstract

Given a smooth variety $X$ and an effective Cartier divisor $D\subset X$, we show that the cohomological Chow group of 0-cycles on the double of $X$ along $D$ has a canonical decomposition in terms of the Chow group of 0-cycles $\text{CH}_{0}(X)$ and the Chow group of 0-cycles with modulus $\text{CH}_{0}(X|D)$ on $X$. When $X$ is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of $\text{CH}_{0}(X|D)$. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that $\text{CH}_{0}(X|D)$ is torsion-free and there is an injective cycle class map $\text{CH}_{0}(X|D){\hookrightarrow}K_{0}(X,D)$ if $X$ is affine. For a smooth affine surface $X$, this is strengthened to show that $K_{0}(X,D)$ is an extension of $\text{CH}_{1}(X|D)$ by $\text{CH}_{0}(X|D)$.

Keywords

algebraic cyclesChow groupssingular schemescycles with modulus

MSC classification

Primary: 14C25: Algebraic cycles
Secondary: 14F30: $p$-adic cohomology, crystalline cohomology 13F35: Witt vectors and related rings 19E15: Algebraic cycles and motivic cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 1 , January 2018 , pp. 120 - 187
Copyright
© The Authors 2017 

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