Zero cycles with modulus and zero cycles on singular varieties. (9th October 2017)

• Record Type: Journal Article
• Authors: Binda, Federico; Krishna, Amalendu
• Is Part Of: Compositio mathematica. Volume 154:Number 1(2018)
• Page Start: 120
• Page End: 187
• Publication Date: 2017-10-09
• Subjects: 14C25 (primary), -- 14F30, -- 13F35, -- 19E15 (secondary); algebraic cycles, -- Chow groups, -- singular schemes, -- cycles with modulus; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X17007503
• Languages: English
• ISSNs: 0010-437X

Abstract

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$\text{CH}_{0}(X|D)$ .