Zero cycles with modulus and zero cycles on singular varieties

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 ${\mathrm{CH}}_0(X)$ and the Chow group of 0-cycles with modulus ${\mathrm{CH}}_0(X|D)$ on $X$. When $X$ is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of ${\mathrm{CH}}_0(X|D)$. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that ${\mathrm{CH}}_0(X|D)$ is torsion-free and there is an injective cycle class map ${\mathrm{CH}}_0(X|D)\hookrightarrow K_0(X,D)$ if $X$ is affine. For a smooth affine surface $X$, this is strengthened to show that $K_0(X,D)$ is an extension of ${\mathrm{CH}}_1(X|D)$ by ${\mathrm{CH}}_0(X|D)$.

62 pages. Final version to appear in Compositio Math

References in corpus (3)

• Relative cycles with moduli and regulator maps
• A moving lemma for algebraic cycles with modulus and contravariance
• Higher Chow groups with modulus and relative Milnor K-theory

Cited by in corpus (15)

• Relative cycles with moduli and regulator maps
• A moving lemma for algebraic cycles with modulus and contravariance
• K-theory and 0-cycles on schemes
• The slice spectral sequence for singular schemes and applications
• Relative $K_0$ and relative cycle class map
• Bloch's formula for 0-cycles with modulus and higher dimensional Class Field Theory
• Zero-cycles on normal varieties
• A motivic homotopy theory without ${\mathbb{A}}^1$ invariance
• A remark on the tensor product of SC-reciprocity sheaves
• Chern classes with modulus
• Derived Log Albanese Sheaves
• Tensor structures in the theory of modulus presheaves with transfers
• Semi-purity for cycles with modulus
• Torsion and divisibility for reciprocity sheaves and 0-cycles with modulus
• ${\mathbb{A}}^1$-equivalence of zero cycles on surfaces II