Available in electronic format
Available in print format		 St.Petersburg Mathematical Journal
St.Petersburg Mathematical Journal
ISSN: 1547-7371(e) ISSN: 1061-0022(p)
Previous issue | Table of contents | Next issue
Recently posted articles | Most recent issue | All issues
Previous Article
     

Orientations and transfers in cohomology of algebraic varieties

Author(s): A. L. Smirnov
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 18 (2006), nomer 2.
Journal: St. Petersburg Math. J. 18 (2007), 305-346.
MSC (2000): Primary 14F99
Posted: March 20, 2007
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: Algebro-geometric cohomology theories are described axiomatically, with a systematic treatment of their orientations. For every oriented theory, transfer mappings are constructed for mappings of smooth varieties that are proper on supports. In some basic cases, transfers are calculated. The presentation is illustrated by motivic cohomology, $ K$-theory, algebraic cobordism, and other examples.

References:

1.
F. Morel and V. Voevodsky, Homotopy category of schemes over a base, Preprint, 1997.

2.
V. Voevodsky, $ \mathbf A^1$-homotopy theory, Proceedings of the International Congress of Mathematicians, Vol. 1 (Berlin, 1998), Doc. Math. 1998, Extra Vol. I, pp. 579-604. MR 1648048 (99j:14018)

3.
I. Panin and A. Smirnov, Push-forwards in oriented cohomology theories of algebraic varieties, $ K$-Theory Preprint Archives, 459, 2000.

4.
J. S. Milne, Étale cohomology, Princeton Math. Ser., vol. 33, Princeton Univ. Press, Princeton, NJ, 1980. MR 559531 (81j:14002)

5.
R. Thomason and T. Trobaugh, Higher algebraic $ K$-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. 3, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247-435. MR 1106918 (92f:19001)

6.
R. Hartshorne, Algebraic geometry, Grad. Texts in Math., No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157 (57:3116)

7.
A. Suslin and V. Voevodsky, Bloch-Kato conjecture and motivic cohomology with finite coefficients, The Arithmetic and Geometry of Algebraic Cycles, NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer Acad. Publ., Dordrecht, 2000, pp. 117-189. MR 1744945 (2001g:14031)

8.
A. Neeman, Triangulated categories, Ann. of Math. Stud., No. 148, Princeton Univ. Press, Princeton, NJ, 2001. MR 1812507 (2001k:18010)

9.
M. Hovey, Model categories, Math. Surveys Monogr., vol. 63, Amer. Math. Soc., Providence, RI, 1999. MR 1650134 (99h:55031)

10.
A. Smirnov, The Leibniz rule in algebraic $ K$-theory, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 319 (2004), 264-292; English transl., J. Math. Sci. (N. Y.) 134 (2006), no. 6, 2582-2597. MR 2117861 (2005j:19001)

11.
D. Quillen, Higher algebraic $ K$-theory. I, Algebraic $ K$-Theory, I: Higher $ K$-Theories (Proc. Conf. Seattle Res. Center, Battelle Memorial Inst., 1972), Lecture Notes in Math., vol. 341, Springer, Berlin, 1973, pp. 85-147. MR 0338129 (49:2895)

12.
F. Morel, Basic properties of the stable homotopy category of smooth schemes, Preprint received approximately in March 2000.

13.
I. Panin, Push-forwards in oriented cohomology theories of algebraic varieties. II, $ K$-Theory Preprint Archives, 619, 2003.

14.
D. Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969), 1293-1298. MR 0253350 (40:6565)

15.
W. Fulton, Intersection theory, Ergeb. Math. Grenzgeb. (3). Ser. Modern Surveys in Math., vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323 (99d:14003)

16.
P. Deligne, Catégories tannakiennes, The Grothendieck Festschrift. Vol. 2, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 111-195. MR 1106898 (92d:14002)

17.
Théorie de topos et cohomologie étale des schémas. Tome 1. Théorie des topos, Séminaire de géométrie algébrique du Bois-Marie (1963-1964) (SGA4). Dirigé par M. Artin, A. Grothendieck et J. L. Verdier, Lecture Notes in Math., vol. 269, Springer-Verlag, Berlin-New York, 1972. MR 0354652 (50:7130)

18.
A. Smirnov, Homotopy properties of algebraic vector bundles, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 319 (2004), 261-263; English transl., J. Math. Sci. (N. Y.) 134 (2006), no. 6, 2580-2581. MR 2117860 (2005i:14025)

19.
J.-P. Serre, Groupes algébriques et corps de classes, Publ. Inst. Math. Univ. Nancago, No. VII, Actualités Sci. Indust., No. 1264, Hermann, Paris, 1975. MR 0466151 (57:6032)

Similar Articles:

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 14F99

Retrieve articles in all Journals with MSC (2000): 14F99

Additional Information:

A. L. Smirnov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: smirnov@pdmi.ras.ru

DOI: 10.1090/S1061-0022-07-00952-1
PII: S 1061-0022(07)00952-1
Keywords: Algebraic variety, motivic cohomology, algebraic cobordism, orientation, transfer, characteristic class, residue
Received by editor(s): 10/JAN/2006
Posted: March 20, 2007
Additional Notes: Partially supported by RFBR (grant no. 03-01-00633a)
Copyright of article: Copyright 2007, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google