# Zero-cycle groups on algebraic varieties

@article{Binda2021ZerocycleGO, title={Zero-cycle groups on algebraic varieties}, author={Federico Binda and Amalendu Krishna}, journal={Journal de l’{\'E}cole polytechnique — Math{\'e}matiques}, year={2021} }

We compare various groups of 0-cycles on quasi-projective varieties over a field. As applications, we show that for certain singular projective varieties, the Levine-Weibel Chow group of 0-cycles coincides with the corresponding Friedlander-Voevodsky motivic cohomology. We also show that over an algebraically closed field of positive characteristic, the Chow group of 0-cycles with modulus on a smooth projective variety with respect to a reduced divisor coincides with the Suslin homology of the…

## 6 Citations

### Zero-cycles on normal varieties

- Mathematics
- 2020

We prove an extension of the Kato-Saito unramified class field theory for smooth projective schemes over a finite field to a class of normal projective schemes. As an application, we obtain Bloch’s…

### ZERO-CYCLES ON NORMAL PROJECTIVE VARIETIES

- MathematicsJournal of the Institute of Mathematics of Jussieu
- 2022

We prove an extension of the Kato–Saito unramified class field theory for smooth projective schemes over a finite field to a class of normal projective schemes. As an application, we obtain Bloch’s…

### Zero-cycles in families of rationally connected varieties

- Mathematics
- 2022

We study zero-cycles in families of rationally connected varieties. We show that for a smooth projective scheme over a henselian discrete valuation ring the restriction of relative zero cycles to the…

### A decomposition theorem for 0-cycles and applications to class field theory

- MathematicsANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
- 2022

. We prove a decomposition theorem for the cohomological Chow group of 0-cycles on the double of a quasi-projective R 1 -scheme over a ﬁeld along a closed subscheme, in terms of the Chow groups, with…

### Suslin homology via cycles with modulus and applications

- Mathematics
- 2022

We show that for a smooth projective variety X over a field k and a reduced effective Cartier divisor D ⊂ X, the Chow group of 0-cycles with modulus CH0(X ∣D) coincides with the Suslin homology H S 0…

### Bloch’s formula for 0-cycles with modulus and higher-dimensional class field theory

- Computer ScienceJournal of Algebraic Geometry
- 2022

Bloch’s formula for the Chow group of 0-cycles with modulus on a smooth quasi-projective surface over a field is proved to give a simple proof of the rank one case of a conjecture of Deligne and Drinfeld on lisse.

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We prove an extension of the Kato-Saito unramified class field theory for smooth projective schemes over a finite field to a class of normal projective schemes. As an application, we obtain Bloch’s…

### A module structure and a vanishing theorem for cycles with modulus

- Mathematics
- 2014

We show that the higher Chow groups with modulus of Binda-Kerz-Saito for a smooth quasi-projective scheme $X$ is a module over the Chow ring of $X$. From this, we deduce certain pull-backs, the…

### On the cycle class map for zero-cycles over local fields

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We study the Chow group of zero-cycles of smooth projective varieties over local and strictly local fields. We prove in particular the injectivity of the cycle class map to integral l-adic cohomology…

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Let X be a smooth proper variety over a perfect field k of arbitrary characteristic. Let D be an effective divisor on X with multiplicity. We introduce an Albanese variety Alb(X, D) of X of modulus D…

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. We prove a decomposition theorem for the cohomological Chow group of 0-cycles on the double of a quasi-projective R 1 -scheme over a ﬁeld along a closed subscheme, in terms of the Chow groups, with…

### Chow group of $0$-cycles with modulus and higher-dimensional class field theory

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One of the main results of this paper is a proof of the rank one case of an existence conjecture on lisse l-adic sheaves on a smooth variety over a finite field due to Deligne and Drinfeld. The…

### Rigidity for relative 0-cycles

- MathematicsANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
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We present a relation between the classical Chow group of relative $0$-cycles on a regular scheme $\mathcal{X}$, projective and flat over an excellent Henselian discrete valuation ring, and the…

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Let R be a henselian discrete valuation ring. Let X be a regular projective flat scheme over Spec(R) with generalized semistable reduction. We prove a bijectivity theorem for etale cycle class maps…

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Abstract We prove that for a smooth projective variety X of dimension d defined over a finite field k, the structure map σ : X → Spec k induces an isomorphism σ∗ : CH d+1(X, 1) ≅ CH 1(k, 1) = k*. We…