Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-03T01:05:01.076Z Has data issue: false hasContentIssue false

Roitman's theorem for singular projective varieties in arbitrary characteristic

Published online by Cambridge University Press:  14 November 2008

Vivek Mohan Mallick
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, INDIA, [email protected].
Get access

Abstract

In this paper, we prove Roitman's theorem regarding torsion 0-cycles for singular projective varieties over algebraically closed fields of arbitrary characteristic, for torsion which is of exponent prime to the characteristic. This generalizes earlier results for complex projective varieties. Our proof even in that case is different from the earlier ones.

Type
Research Article
Copyright
Copyright © ISOPP 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Barbieri-Viale, Luca, Pedrini, Claudio, and Weibel, Charles. Roitman's theorem for singular complex projective surfaces. Duke Mathematical Journal 84 (1):155190, 07 1996.CrossRefGoogle Scholar
2.Barbieri-Viale, Luca and Srinivas, Vasudevan. Albanese and Picard 1-motives. Mém. Soc. Math. Fr. (N.S.), 2001.CrossRefGoogle Scholar
3.Biswas, Jishnu and Srinivas, Vasudevan. Roitman's theorem for singular projective varieties. Compositio Mathematica 119:213237, 1999.CrossRefGoogle Scholar
4.Biswas, Jishnu Gupta. Topics in Algebraic Cycles: Analogs for singular varieties of Lefschetz and Roitman. PhD thesis, Faculty of Sciences, University of Mumbai, 1997.Google Scholar
5.Bloch, Spencer. Lectures on Algebraic Cycles. Mathematics Department Duke University, 1980.Google Scholar
6.Collino, A.. Torsion in the chow group of codimension 2: The case of varieties with isolated singularities. Journal of Pure and Applied Algebra 34:147153, 1984.CrossRefGoogle Scholar
7.Deligne, Pierre. Théorie de Hodge. III. Inst. Hautes Études Sci. Publ. Math. 44:577, 1974.CrossRefGoogle Scholar
8.Esnault, Hélène, Srinivas, Vasudevan, and Viehweg, Eckart. The universal regular quotient of the chow group of points on projective varieties. Inventiones Mathematicae 135 (3):595664, 1999.CrossRefGoogle Scholar
9.Faltings, G. and Wüstholz, G.. Einbettungen kommutativer algebraischer Gruppen und einige ihrer Eigenschaften. J. Reine Angew. Math. 354:175205, 1984.Google Scholar
10.Fulton, William. Intersection Theory. Springer, 2nd edition, 1998.CrossRefGoogle Scholar
11.Geisser, Thomas. Duality via cycle complexes. arxiv:math.AG/0608456 v2 21 09 2006.Google Scholar
12.Hartshorne, Robin. Algebraic Geometry. Springer, 1977.CrossRefGoogle Scholar
13.Hoobler, Raymond T.. The Merkuriev-Suslin theorem for any semi-local ring. J. Pure Appl. Algebra 207 (3):537552, 2006.CrossRefGoogle Scholar
14.Kleiman, Steven L. and Altman, Allen B.. Bertini theorems for hypersurface sections containing a subscheme. Communications in Algebra 7 (8):775790, 1979.CrossRefGoogle Scholar
15.Krishna, Amalendu and Srinivas, Vasudevan. Zero-cycles and k-theory on normal surfaces. Annals of Mathematics 156:155195, 2002.CrossRefGoogle Scholar
16.Lang, Serge. Abelian Varieties. Interscience Publishers, Inc., 1959.Google Scholar
17.Levine, Marc. Torsion zero cycles on singular varieties. American Journal of Mathematics 107:737757, 1985.CrossRefGoogle Scholar
18.Levine, Marc and Weibel, Chuck. Zero cycles and complete intersections on singular varieties. J. Reine Angew. Math. 359:106120, 1985.Google Scholar
19.Milne, James S.. Étale Cohomology. Princeton University Press, 1980.Google Scholar
20.Milne, James S.. Zero cycles in algebraic varieties in nonzero characteristic: Rojtman's theorem. Compositio Math. 47 (3):271287, 1982.Google Scholar
21.Oort, F.. A contruction of generalized Jacobian varieties by group extensions. Mathematische Annalen 147:277286, 1962.CrossRefGoogle Scholar
22.Ramachandran, Niranjan. Duality of Albanese and Picard 1-motives. K-Theory 22 (3):271301, 2001.CrossRefGoogle Scholar
23.Roitman, A. A.. The torsion group of zero cycles modulo rational equivalence. Annals of Mathematics 111:553570, 1980.CrossRefGoogle Scholar
24.Russell, Henrik. Generalized albanese and its dual. arXiv:math.AG/0606250v1 10 06 2006.Google Scholar
25.Serre, Jean-Pierre. Morphisms universels et differéntielles de troisième espèce. Seminaire Chevalley, 19581959.Google Scholar
26.Swan, Richard G.. On seminormality. Journal of Algebra 67 (1):210229, 1980.CrossRefGoogle Scholar
27.Weil, A.. Sur les critéres d'équivalence en géométrie algébrique. Mathematische Annalen 128:95127, 1954.CrossRefGoogle Scholar
28.Zhang, Bin. Théorèmes du type bertini en caractéristique positive. Achiv der Mathematik 64:209215, 1995.CrossRefGoogle Scholar
29.Zhang, Bin. Sur les jacobiennes des courbes à singularités ordinaires. Manuscripta Mathematica 92:112, 1997.CrossRefGoogle Scholar