Homology of algebraic varieties: An introduction to the works of Suslin and Voevodsky
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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
- M. Artin, Grothendieck topologies, Seminar notes, Harvard Univ. Dept of Math., 1962.
- M. F. Atiyah, $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1967. Lecture notes by D. W. Anderson. MR 0224083
- M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, pp. 7–38. MR 0139181
- Spencer Bloch, Algebraic cycles and higher $K$-theory, Adv. in Math. 61 (1986), no. 3, 267–304. MR 852815, DOI 10.1016/0001-8708(86)90081-2
- S. Bloch, The moving lemma for higher Chow groups, J. Algebraic Geom. 3 (1994), no. 3, 537–568. MR 1269719
- S. Bloch and S. Lichtenbaum, A spectral sequence for motivic cohomology, preprint (1995).
- Armand Borel and Jean-Pierre Serre, Le théorème de Riemann-Roch, Bull. Soc. Math. France 86 (1958), 97–136 (French). MR 116022, DOI 10.24033/bsmf.1500
- C. Chevalley, Anneaux de Chow et Applications, Sém. C. Chevalley, 2 Paris, 1958.
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
- Albrecht Dold and René Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. (2) 67 (1958), 239–281 (German). MR 97062, DOI 10.2307/1970005
- A. Duady and J.-L. Verdier, Séminaire de Géométrie Analytique de l’Ecole Normal Sup. Astérisque 36-7, 1976.
- William G. Dwyer and Eric M. Friedlander, Algebraic and etale $K$-theory, Trans. Amer. Math. Soc. 292 (1985), no. 1, 247–280. MR 805962, DOI 10.1090/S0002-9947-1985-0805962-2
- Eric M. Friedlander, Étale $K$-theory. I. Connections with etale cohomology and algebraic vector bundles, Invent. Math. 60 (1980), no. 2, 105–134. MR 586424, DOI 10.1007/BF01405150
- Eric M. Friedlander, Étale $K$-theory. II. Connections with algebraic $K$-theory, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 231–256. MR 683636, DOI 10.24033/asens.1427
- Eric M. Friedlander and Ofer Gabber, Cycle spaces and intersection theory, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 325–370. MR 1215970
- T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
- E. Friedlander and B. Lawson, Moving algebraic cycles of bounded degree, preprint (1994).
- E. M. Friedlander and B. Mazur, Correspondence homomorphisms for singular varieties, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 3, 703–727 (English, with English and French summaries). MR 1303882, DOI 10.5802/aif.1415
- E. Friedlander and V. Voevodsky, Bivariant cycle cohomology, preprint (1995).
- William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
- Ofer Gabber, $K$-theory of Henselian local rings and Henselian pairs, Algebraic $K$-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989) Contemp. Math., vol. 126, Amer. Math. Soc., Providence, RI, 1992, pp. 59–70. MR 1156502, DOI 10.1090/conm/126/00509
- Daniel R. Grayson, Weight filtrations via commuting automorphisms, $K$-Theory 9 (1995), no. 2, 139–172. MR 1340843, DOI 10.1007/BF00961457
- A. Grothendieck, Classes de faisceaux et théorème de Riemann-Roch, Exposé 0, SGA 6, Lecture Notes in Math. 225(1971) 297-364.
- Alexander Grothendieck, La théorie des classes de Chern, Bull. Soc. Math. France 86 (1958), 137–154 (French). MR 116023, DOI 10.24033/bsmf.1501
- Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. MR 0354652
- Henri A. Gillet and Robert W. Thomason, The $K$-theory of strict Hensel local rings and a theorem of Suslin, Proceedings of the Luminy conference on algebraic $K$-theory (Luminy, 1983), 1984, pp. 241–254. MR 772059, DOI 10.1016/0022-4049(84)90037-9
- T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
- T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
- B. Kahn, The Quillen-Lichtenbaum conjecture at the prime 2, preprint (1997).
- Max Karoubi and Orlando Villamayor, $K$-théorie algébrique et $K$-théorie topologique. I, Math. Scand. 28 (1971), 265–307 (1972) (French). MR 313360, DOI 10.7146/math.scand.a-11024
- Marc Levine, Bloch’s higher Chow groups revisited, Astérisque 226 (1994), 10, 235–320. $K$-theory (Strasbourg, 1992). MR 1317122
- M. Levine, Motivic cohomology and algebraic cycles, preprint(1995).
- G. J. Whitrow, On the Lobatchewskian trigonometry of a static substratum, Quart. J. Math. Oxford Ser. 10 (1939), 313–319. MR 756, DOI 10.1093/qmath/os-10.1.313
- James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
- F. Morel, Théorie de l’homotopie et motifs, I, preprint(1995).
- D. Mumford, Rational equivalence of $0$-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195–204. MR 249428, DOI 10.1215/kjm/1250523940
- Daniel Quillen, Cohomology of groups, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 47–51. MR 0488054
- Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129
- Joel Roberts, Chow’s moving lemma, Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer School in Math.), Wolters-Noordhoff, Groningen, 1972, pp. 89–96. Appendix 2 to: “Motives” (Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer School in Math.), pp. 53–82, Wolters-Noordhoff, Groningen, 1972) by Steven L. Kleiman. MR 0382269
- A. A. Roĭtman, Rational equivalence of zero-dimensional cycles, Mat. Sb. (N.S.) 89(131) (1972), 569–585, 671 (Russian). MR 0327767
- A. A. Rojtman, The torsion of the group of $0$-cycles modulo rational equivalence, Ann. of Math. (2) 111 (1980), no. 3, 553–569. MR 577137, DOI 10.2307/1971109
- M. Rost, On the spinor norm and $A_0(X,\mathcal {K}_1)$ for quadrics, preprint (1988).
- M. Rost, Some new results on the Chow groups of quadrics, preprint (1990).
- Francesco Severi, Problèmes résolus et problèmes nouveaux dans la théorie des systèmes d’équivalence, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, Erven P. Noordhoff N. V., Groningen; North-Holland Publishing Co., Amsterdam, 1956, pp. 529–541 (French). MR 0098751
- A. Suslin, On the $K$-theory of algebraically closed fields, Invent. Math. 73 (1983), no. 2, 241–245. MR 714090, DOI 10.1007/BF01394024
- Andrei A. Suslin, On the $K$-theory of local fields, Proceedings of the Luminy conference on algebraic $K$-theory (Luminy, 1983), 1984, pp. 301–318. MR 772065, DOI 10.1016/0022-4049(84)90043-4
- A. Suslin, Higher Chow groups of affine varieties and étale cohomology, preprint (1994).
- T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
- A. Suslin and V. Voevodsky, Relative cycles and Chow sheaves, preprint (1994).
- A. Suslin and V. Voevodsky, Bloch-Kato conjecture and motivic cohomology with finite coefficients, preprint(1995).
- V. Voevodsky, Homology of schemes II, preprint (1994).
- V. Voevodsky, Triangulated categories of motives over a field, preprint (1995).
- V. Voevodsky, The Milnor conjecture, preprint (1996).
- C. Weibel, The two-torsion in the K-theory of the integers, preprint (1996).
- André Weil, Foundations of algebraic geometry, American Mathematical Society, Providence, R.I., 1962. MR 0144898
Additional Information
- Marc Levine
- Affiliation: Department of Mathematics Northeastern University Boston, Massachusetts 02115
- MR Author ID: 113315
- Email: marc@neu.edu
- Received by editor(s): March 4, 1996
- Received by editor(s) in revised form: January 7, 1997
- Additional Notes: Research supported by the NSF
- © Copyright 1997 American Mathematical Society
- 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