A theory of algebraic cocycles
Authors:
Eric M. Friedlander and H. Blaine Lawson
Journal:
Bull. Amer. Math. Soc. 26 (1992), 264268
MSC (2000):
Primary 14C05; Secondary 14C30, 14F35
MathSciNet review:
1118701
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Abstract: We introduce the notion of an algebraic cocycle as the algebraic analogue of a map to an EilenbergMacLane space. Using these cocycles we develop a "cohomology theory" for complex algebraic varieties. The theory is bigraded, functorial, and admits Gysin maps. It carries a natural cup product and a pairing to Lhomology. Chern classes of algebraic bundles are defined in the theory. There is a natural transformation to (singular) integral cohomology theory that preserves cup products. Computations in special cases are carried out. On a smooth variety it is proved that there are algebraic cocycles in each algebraic rational cohomology class.
 [F]
Eric
M. Friedlander, Algebraic cycles, Chow varieties, and Lawson
homology, Compositio Math. 77 (1991), no. 1,
55–93. MR
1091892 (92a:14005)
 [FL]
E. Friedlander and H. B. Lawson, A theory of algebraic cocycles, Ann. of Math. (to appear).
 [FM]
Eric
M. Friedlander and Barry
Mazur, Filtrations on the homology of algebraic varieties,
Mem. Amer. Math. Soc. 110 (1994), no. 529, x+110.
With an appendix by Daniel Quillen. MR 1211371
(95a:14023), http://dx.doi.org/10.1090/memo/0529
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P. LimaFilho, On a homology theory for algebraic varieties, IAS preprint, 1990.
 [LF2]
, Completions and fibrations for topological monoids and excision for Lawson homology, Compositio Math., (1991).
 [L]
H.
Blaine Lawson Jr., Algebraic cycles and homotopy theory, Ann.
of Math. (2) 129 (1989), no. 2, 253–291. MR 986794
(90h:14008), http://dx.doi.org/10.2307/1971448
 [LM]
H.
Blaine Lawson Jr. and MarieLouise
Michelsohn, Algebraic cycles, Bott periodicity, and the Chern
characteristic map, The mathematical heritage of Hermann Weyl (Durham,
NC, 1987) Proc. Sympos. Pure Math., vol. 48, Amer. Math. Soc.,
Providence, RI, 1988, pp. 241–263. MR 974339
(90d:14010), http://dx.doi.org/10.1090/pspum/048/974339
 [F]
 E. Friedlander, Algebraic cycles, Chow varieties, and Lawson homology, Compositio Math. 77 (1991), 5593. MR 1091892 (92a:14005)
 [FL]
 E. Friedlander and H. B. Lawson, A theory of algebraic cocycles, Ann. of Math. (to appear).
 [FM]
 E. Friedlander and B. Mazur, Filtrations on the homology of algebraic varieties (to appear). MR 1211371 (95a:14023)
 [LF1]
 P. LimaFilho, On a homology theory for algebraic varieties, IAS preprint, 1990.
 [LF2]
 , Completions and fibrations for topological monoids and excision for Lawson homology, Compositio Math., (1991).
 [L]
 H. B. Lawson, Jr., Algebraic cycles and homotopy theory, Ann. of Math. (2) 129 (1989), 253291. MR 986794 (90h:14008)
 [LM]
 H. B. Lawson, Jr. and M.L. Michelsohn, Algebraic cycles, Bott periodicity, and the Chern characteristic map, The Mathematical Heritage of Herman Weyl, Amer. Math. Soc., Providence, RI, 1988, pp. 241264. MR 974339 (90d:14010)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S027309791992002690
PII:
S 02730979(1992)002690
Keywords:
Algebraic cycle,
Chow variety,
algebraic cocycle,
cohomology
Article copyright:
© Copyright 1992
American Mathematical Society
