Zero cycles with modulus and zero cycles on singular varieties. (9th October 2017) - Binda, Federico; Krishna, Amalendu
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$\text{CH}_{0}(X|D)$ .