Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Homology of algebraic varieties: An introduction to the works of Suslin and Voevodsky


Author: Marc Levine
Journal: Bull. Amer. Math. Soc. 34 (1997), 293-312
MSC (1991): Primary 19-02, 19E15, 14C25; Secondary 19E08, 19E20, 14F20, 18F10
DOI: https://doi.org/10.1090/S0273-0979-97-00723-4
MathSciNet review: 1432056
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give an overview of the ideas Suslin and Voevodsky have introduced in their works on algebraic cycles and their relation to the mod-$n$ homology of algebraic varieties.


References [Enhancements On Off] (What's this?)

  • 1. M. Artin, Grothendieck topologies, Seminar notes, Harvard Univ. Dept of Math., 1962.
  • 2. M. Atiyah, K-theory, Benjamin 1967. MR 36:7130
  • 3. M. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, Amer. Math. Soc. Symp. in Pure Math. III(1961) 7-38. MR 25:2617
  • 4. S. Bloch, Algebraic cycles and higher $K$-theory, Adv. in Math. 61 No. 3(1986) 267-304. MR 88f:18010
  • 5. S. Bloch, The moving lemma for the higher Chow groups, J. Alg. Geom. 3(1994) 537-568. MR 96c:14007
  • 6. S. Bloch and S. Lichtenbaum, A spectral sequence for motivic cohomology, preprint (1995).
  • 7. A. Borel and J.-P. Serre, Le théorem de Riemann-Roch, Bull. Soc. Math. France 86(1958) 97-136. MR 22:6817
  • 8. C. Chevalley, Anneaux de Chow et Applications, Sém. C. Chevalley, 2 Paris, 1958.
  • 9. W.-L. Chow, On the equivalence classes of cycles in an algebraic variety, Ann. of Math. 64(1956) 450-479. MR 18:509a
  • 10. P. Deligne et al., Cohomologie Etale, Lecture Notes in Math. 569, Springer Verlag 1977. MR 57:3132
  • 11. A. Dold and R. Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. 67(1958) 239-281. MR 20:3542
  • 12. A. Duady and J.-L. Verdier, Séminaire de Géométrie Analytique de l'Ecole Normal Sup. Astérisque 36-7, 1976.
  • 13. W. Dwyer and E. Friedlander, Algebraic and étale K-theory, Trans. Amer. Math. Soc. 292(1985) 247-280. MR 87h:18013
  • 14. E. Friedlander, Etale K-theory I: Connections with Etale Cohomology and Algebraic Vector Bundles, Invent. Math. 60(1980) 105-134. MR 82e:14029
  • 15. E. Friedlander, Etale K-theory II: Connections with algebraic K-theory, Ann. Sci. Ec. Norm. Sup. $4^{\text e}$ sér. 15 no. 2(1982) 231-256. MR 85c:14014
  • 16. E. Friedlander and O. Gabber, Cycle spaces and intersection theory, in Topological methods in modern mathematics, Publish or Perish, Houston, TX (1993) 325-370. MR 94j:14010
  • 17. E. Friedlander and B. Lawson, Duality relating spaces of algebraic cocycles and cycles, Topology 36 (1997) 533-565. MR 1:415 605
  • 18. E. Friedlander and B. Lawson, Moving algebraic cycles of bounded degree, preprint (1994).
  • 19. E. Friedlander and B. Mazur, Correspondence homomorphisms for singular varieties, Ann. Inst. Fourier (Grenoble) 44 (1994) 703-727. MR 95j:14009
  • 20. E. Friedlander and V. Voevodsky, Bivariant cycle cohomology, preprint (1995).
  • 21. W. Fulton, Intersection Theory, Springer-Verlag, New York 1984. MR 85k:14004
  • 22. O. Gabber, K-theory of henselian local rings and henselian pairs, in Algebraic K-theory, Commutative Algebra, and Algebraic Geometry, Contemp. Math. 126, Amer. Math Soc., Providence, RI (1992) 59-70. MR 93c:19005
  • 23. D. Grayson, Weight filtrations via commuting automorphisms, $K$-Theory 9 (1995) 139-172. MR 96h:19001
  • 24. A. Grothendieck, Classes de faisceaux et théorème de Riemann-Roch, Exposé 0, SGA 6, Lecture Notes in Math. 225(1971) 297-364.
  • 25. A. Grothendieck, La théorie des classes des Chern, Bull. Soc. Math. France 86(1958) 137-154. MR 22:6818
  • 26. A. Grothendieck et al., Théorie de Topos et Cohomologie Etale des Schémas, I, Lecture Notes in Math. 269, Springer Verlag 1972. MR 50:7130
  • 27. H. Gillet and R. Thomason, On the K-theory of strict hensel local rings and a theorem of Suslin, JPAA 34(1984) 241-254. MR 86e:18014
  • 28. M. Hanamura, Mixed motives and algebraic cycles, I, Math. Res. Lett. 2 (1995) 811-821; II, preprint (1995). MR 1:362 972
  • 29. A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996) 51-93. MR 1:423 020
  • 30. B. Kahn, The Quillen-Lichtenbaum conjecture at the prime 2, preprint (1997).
  • 31. M. Karoubi and O. Villamayor, $K$-théorie algébrique et $K$-théorie topologique, I. Math. Scand. 28(1971) 265-307. MR 47:1915
  • 32. M. Levine, Bloch's higher Chow groups revisited, in K-theory, Strasbourg, 1992, Asterisque 226(1994)235-320. MR 96c:14008
  • 33. M. Levine, Motivic cohomology and algebraic cycles, preprint(1995).
  • 34. S. Lichtenbaum, Values of zeta-functions at non-negative integers, in Number Theory, Lecture Notes in Math. 1068, Springer, Berlin (1984) 127-138. MR 756:089
  • 35. J.S. Milne, Etale Cohomology, Princeton Math. Ser. 33, Princeton Univ. Press 1980. MR 81j:14002
  • 36. F. Morel, Théorie de l'homotopie et motifs, I, preprint(1995).
  • 37. D. Mumford, Rational equivalence of zero-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195-204. MR 40:2673
  • 38. D. Quillen, Cohomology of groups, International Congress of Mathematics, Nice 1970. MR 58:7627a
  • 39. D. Quillen, Higher algebraic K-theory I, in Algebraic K-theory I: Higher K-theories, Lecture Notes in Math. 341, Springer Verlag(1973) 85-147. MR 49:2895
  • 40. J. Roberts, Chow's moving lemma, in Algebraic Geometry, Oslo 1979, F. Oort editor. MR 52:3154
  • 41. A.A. Roitman, Rational equivalence of zero-dimensional cycles, Math. of the USSR Sbornik 18(1972) 571-588. MR 48:6109
  • 42. A.A. Roitman, The torsion of the group of $0$-cycles modulo rational equivalence, Ann. of Math. 111(1980) 553-569. MR 81g:14003
  • 43. M. Rost, On the spinor norm and $A_0(X,{\cal K}_1)$ for quadrics, preprint (1988).
  • 44. M. Rost, Some new results on the Chow groups of quadrics, preprint (1990).
  • 45. F. Severi, Problèmes résolus et problèmes nouveaux dans la théories des systémes d'équivalence, Proc. Intern. Cong. Math. 3(1954), 529-541. MR 20:5206
  • 46. A. Suslin, On the K-theory of algebraically closed fields, Invent. Math. 73(1983) 241-245. MR 85h:18008a
  • 47. A. Suslin, On the K-theory of local fields, JPAA 34(1984) 301-318.MR 86d:18010
  • 48. A. Suslin, Higher Chow groups of affine varieties and étale cohomology, preprint (1994).
  • 49. A. Suslin and V. Voevodsky, Singular homology of abstract algebraic varieties, Inv. Math. 123 Fasc. 1(1996) 61-94. MR 1:376 246
  • 50. A. Suslin and V. Voevodsky, Relative cycles and Chow sheaves, preprint (1994).
  • 51. A. Suslin and V. Voevodsky, Bloch-Kato conjecture and motivic cohomology with finite coefficients, preprint(1995).
  • 52. V. Voevodsky, Homology of schemes II, preprint (1994).
  • 53. V. Voevodsky, Triangulated categories of motives over a field, preprint (1995).
  • 54. V. Voevodsky, The Milnor conjecture, preprint (1996).
  • 55. C. Weibel, The two-torsion in the K-theory of the integers, preprint (1996).
  • 56. A. Weil, Foundations of Algebraic Geometry, AMS Colloquium publ. 29, AMS 1962. MR 26:2439

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1991): 19-02, 19E15, 14C25, 19E08, 19E20, 14F20, 18F10

Retrieve articles in all journals with MSC (1991): 19-02, 19E15, 14C25, 19E08, 19E20, 14F20, 18F10


Additional Information

Marc Levine
Affiliation: Department of Mathematics Northeastern University Boston, Massachusetts 02115
Email: marc@neu.edu

DOI: https://doi.org/10.1090/S0273-0979-97-00723-4
Keywords: Motives, cycles
Received by editor(s): March 4, 1996
Received by editor(s) in revised form: January 7, 1997
Additional Notes: Research supported by the NSF
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society