Holmstrom Chow groups

Chow groups

Following Levine’s chapter in K-theory handbook. We consider a field $k$ and a $k$-scheme $X$ of finite type. We define groups $CH_p(X,n)$, and for $X$ locally equidimensional over $k$, also groups $CH^p(X,n)$, for $n$ a natural number. The classical Chow groups are $CH_p(X,0)$.

Functoriality for $CH_p$: (Might be an error here, compare with another source.) Pushforwards for proper map, and pullbacks for equidimensional l.c.i. map increasing degree by $d$, where $d$ is the fiber dimension.

Possibly, to get nice functorial properties for $CH^p(-, n)$ we might have to restrict attention to $Sm_k$.

On page 442, there is a discussion on more general domain cats: schemes essentially of finite type over $k$, schemes of finite type over a regular base of Krull dimensions one.

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Chow groups

The Milnor-Chow homomorphism revisited, by Moritz Kerz and Stefan Mueller-Stach: http://www.math.uiuc.edu/K-theory/0767

Grothendieck defined, for a nonsingular variety, Chern classes of vector bundles $c_P(E) \in CH^p(X)$. Composing these with the cycle class map, I would guess that one obtains Chern classes in any cohomology theory with a cycle class map.

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Chow groups

Gillet: K-theory and Intersection theory (looks excellent)

Something might be in here.

A very intereting paper by Joshua on motivic DGA.

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Chow groups

There is also a brief intro in Lectures on Arakelov geometry, in Chapter I, including Chow groups with supports, explicit def of rational equivalence, and relation to K-theory.

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Chow groups

MathSciNet

Google Scholar

Google

arXiv: Experimental full text search

arXiv: Abstract search

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Chow groups

AG (Algebraic geometry)

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Chow groups

Homotopy invariance and localization. See Levine (page 443). Also Mayer-Vietoris long exact seq. Products. Projective bundle formula.

The Kunneth formula fails for Chow groups, at least over the complex numbers. Counterexample: See Totaro’s Algebraic cycles exam, there one shows that the natural homomorphism $CH^*(E) \otimes_{\mathbb{Z}} CH^*(E) \to CH^*(E \times E)$ is not surjective.

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Chow groups

Levine in K-theory handbook.

My own notes from Totaro’s course, spring 2008.

Seminaire Chevalley: Anneaux de Chow et applications (1958)

Soulé et al: Lectures on Arakelov Geometry. First chapter treats intersection theory for an arbitrary regular noetherian finite-dimensional scheme.

What about Bloch: An elementary presentation… (in Motives vol)?

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Chow groups

Mixed, Pure

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Chow groups

Steenrod operations

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The Bloch-Beilinson filtration. Some interesting material in Nagel and Peters: Algebraic cycles and motives, volume II.

Shuji Saito on filtration

Jannsen: Motivic sheaves and filtrations on Chow groups (Motives vol)

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Chow groups

See also Algebraic cycles, Higher Chow groups, Chow homology, Chow groups with coefficients, Chow cohomology

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Chow groups

Asakura and Saito on surfaces over p-adic fields with very infinite Chow groups of 0-cycles. No Cam subscription to Algebra and N Th. review. The review and abstract mentions related finiteness problems, for example Srinivas and Rosenschon.

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Chow groups

The Chow groups and the motive of the Hilbert scheme of points on a surface, by Mark Andrea de Cataldo and Luca Migliorini: http://www.math.uiuc.edu/K-theory/0415

Saito: On the bijectivity of some cycle maps (in Motives vol)

Pedrini and Weibel: K-theory and Chow groups on singular varieties (1986)

Roberts: Chow’s moving lemma. (Appendix to Kleiman’s Motives article in the Oslo volume)

Saito et al: We study the higher Chow groups CH^2(X,1) and CH^3(X,2) of smooth, projective algebraic surfaces over a field of char 0.

The oriented Chow ring, by J. Fasel

Chow rings of excellent quadrics, by Nobuaki Yagita: http://www.math.uiuc.edu/K-theory/0787

MR1780429 (2001m:11106) Otsubo, Noriyuki(J-TOKYO) Selmer groups and zero-cycles on the Fermat quartic surface.

Soulé: Groupes de Chow et K-théorie de varietes sur un corps fini (1984)

Biswat and Srinivas: The Chow ring of a singular surface

Kresch: Canonical rational equivalence of intersections of divisors (1999)

Roberts: Recent developments on Serre’s multiplicity conjectures: Gabber’s proof of the nonnegativity conjecture (1998)

Complex varieties for which the Chow group mod n is not finite (2002)

Laterveer: Algebraic varieties with small Chow groups. Discusses things related to “representability of Chow groups”, and the Hodge conjecture.

Green, Griffiths: Formal Deformation of Chow Groups. In Abel volume.

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Chow groups

http://mathoverflow.net/questions/17634/definition-of-chow-groups-over-spec-z

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Chow groups

Several papers by Akhtar on K-theory archive.

http://mathoverflow.net/questions/99897/examples-of-chow-rings-of-surfaces mentions surfaces over finite fields

http://mathoverflow.net/questions/11343/algorithm-for-calculating-the-chow-groups-of-a-variety-over-a-finite-field

arXiv:1207.4703 On the Chow groups of certain geometrically rational 5-folds fra arXiv Front: math.NT av Ambrus Pal We give an explicit regular model for the quadric fibration studied in Pirutka (2011). As an application we show that this construction furnishes a counterexample for the integral Tate conjecture in any odd characteristic for some sufficiently large finite field. We study the étale cohomology of this regular model, and as a consequence we derive that these counterexamples are not torsion.

arXiv:1206.2704 On the torsion of Chow groups of Severi-Brauer varieties fra arXiv Front: math.AG av Sanghoon Baek For a large class of central simple algebras we provide upper bounds for the annihilators of the torsion subgroups of the Chow groups of the corresponding Severi-Brauer varieties.

nLab page on Chow groups