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Zero cycles with modulus and zero cycles on singular varieties

  • Federico Binda (a1) and Amalendu Krishna (a2)
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 $\text{CH}_{0}(X|D)$ .

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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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