Traces in oriented homology theories of algebraic varieties

Author:
K. Pimenov

Translated by:
the author

Original publication:
Algebra i Analiz, tom **19** (2007), nomer 5.

Journal:
St. Petersburg Math. J. **19** (2008), 805-828

MSC (2000):
Primary 14F43

DOI:
https://doi.org/10.1090/S1061-0022-08-01022-4

Published electronically:
June 27, 2008

MathSciNet review:
2381946

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper is an attempt to axiomatize the notion of oriented homology theory. Under the axioms in question, a Chern structure determines an orientation and *vice versa*. The main result of the paper is the projective bundle theorem in §2.

**[Ne]**Alexander Nenashev,*Projective bundle theorem in homology theories with Chern structure*, Doc. Math.**9**(2004), 487–497. MR**2117424****[P1]**I. Panin,*Oriented cohomology theories of algebraic varieties*, 𝐾-Theory**30**(2003), no. 3, 265–314. Special issue in honor of Hyman Bass on his seventieth birthday. Part III. MR**2064242**, https://doi.org/10.1023/B:KTHE.0000019788.33790.cb**[PS]**I. Panin and A. Smirnov,*Push-forwards in oriented cohomology theories of algebraic varieties*, (2000),`http://www.math.uiuc.edu/K-theory/0459`.**[Pa]**I. Panin,*Riemann-Roch theorems for oriented cohomology*, Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131, Kluwer Acad. Publ., Dordrecht, 2004, pp. 261–333. MR**2061857**, https://doi.org/10.1007/978-94-007-0948-5_8**[Pi]**K. Pimenov,*Traces in oriented homology theories*, (2005),`http://www.math.uiuc.edu/K-theory/ 0724`.**[PY]**I. Panin and S. Yagunov,*Poincaré duality for algebraic varieties*, (2002),`http://www.math.uiuc. edu/K-theory/0576`.**[S]**A. Solynin,*Chern and Thom elements in the representable cohomology theories*, Preprint POMI-03/2004;`www.pdmi.ras.ru/preprint`.**[SV]**Andrei Suslin and Vladimir Voevodsky,*Bloch-Kato conjecture and motivic cohomology with finite coefficients*, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer Acad. Publ., Dordrecht, 2000, pp. 117–189. MR**1744945****[V]**V. Voevodsky,*The Milnor conjecture*, (1996),`http://www.math.uiuc.edu/K-theory/0170`.**[V1]**Vladimir Voevodsky,*𝐀¹-homotopy theory*, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998, pp. 579–604. MR**1648048****[V2]**-,*Cancellation theorem*, (2002)`http://www.math.uiuc.edu/K-theory/0541`.

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Additional Information

**K. Pimenov**

Affiliation:
St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

Email:
kip302002@yahoo.com

DOI:
https://doi.org/10.1090/S1061-0022-08-01022-4

Keywords:
Chern class,
Thom isomorphism,
orientation,
homology theory

Received by editor(s):
April 10, 2007

Published electronically:
June 27, 2008

Additional Notes:
Supported by the Russian Ministry of Education (grant no. PD02-1.1-368) and by INTAS (grant no. 05-1000008-8118)

Dedicated:
Dedicated to the centenary of the birth of Dmitriĭ Konstantinovich Faddeev

Article copyright:
© Copyright 2008
American Mathematical Society