The elementary transformation of vector bundles on regular schemes
Author:
Takuro Abe
Journal:
Trans. Amer. Math. Soc. 359 (2007), 42854295
MSC (2000):
Primary 14F05
Published electronically:
March 20, 2007
MathSciNet review:
2309185
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Abstract: We give a generalized definition of an elementary transformation of vector bundles on regular schemes by using Maximal CohenMacaulay sheaves on divisors. This definition is a natural extension of that given by Maruyama, and has a connection with that given by Sumihiro. By this elementary transformation, we can construct, up to tensoring line bundles, all vector bundles from trivial bundles on nonsingular quasiprojective varieties over an algebraically closed field. Moreover, we give an application of this theory to reflexive sheaves.
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 [A]
 T. Abe, The elementary Transformation of vector bundles on regular schemes, preprint in Kyoto University, 9, 2004.
 [AY]
 T. Abe and M. Yoshinaga, Splitting criterion for reflexive sheaves, arXiv, mathAG/0503710, preprint in RIMS, 1496 (2005).
 [DK]
 I. Dolgachev and M. Kapranov, Arrangements of Hyperplanes and Vector Bundles on , Duke. Math. J., 71 (1993), 633664. MR 1240599 (95e:14029)
 [G]
 A. Grothendieck, Local cohomology, Lecture Notes in Math., 41, 1967. MR 0224620 (37:219)
 [H1]
 R. Hartshorne, Ample Subvarieties of Algebraic Varieties, Lecture Notes in Math., 156, SpringerVerlag, 1970. MR 0282977 (44:211)
 [H2]
 R. Hartshorne, Algebraic Geometry, Graduated Texts in Mathematics, SpringerVerlag, 1977. MR 0463157 (57:3116)
 [H3]
 R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), 121176. MR 0597077 (82b:14011)
 [HN]
 G. Horrocks and D. Mumford, A rank 2 vector bundle on with 15,000 symmetries, Topology. 12 (1973), 6381. MR 0382279 (52:3164)
 [K]
 G. Kempf, A criterion for the splitting of a vector bundle, Forum Math. 2 (1990), 477480. MR 1067213 (91j:14036)
 [LY]
 H. S. Luk and S. T. Yau, Cohomology and splitting criterion for holomorphic vector bundles on , Math. Nachr. 161 (1993), 233238. MR 1251020 (94k:14037)
 [Mar]
 M. Maruyama, On a family of algebraic vector bundles, Number Theory, Algebraic Geometry and Commutative Algebra (1973), 95149, Kinokuniya. MR 0360587 (50:13035)
 [Mat]
 H. Matsumura, Commutative Algebra, W.A. Benjamin Co., New York (1970). MR 0266911 (42:1813)
 [OSS]
 C. Okonek, M. Schneider and H. Spindler, Vector Bundles on Complex Projective Spaces. 3 (1980), Birkhäuser. MR 0561910 (81b:14001)
 [Sa]
 K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math 27 (1980), no. 2, 265291. MR 0586450 (83h:32023)
 [Sch]
 C. Schoen, On the geometry of a special determinantal hypersurface associated to the MumfordHorrocks vector bundle, J. reine. angew. Math. 364 (1986), 85111. MR 0817640 (87e:14039)
 [Su1]
 H. Sumihiro, A theorem on splitting of algebraic vector bundles and its applications, Hiroshima Math J. 12 (1982), 435452. MR 0665505 (83m:14010)
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 H. Sumihiro, Elementary Transformations of Algebraic Vector Bundles, Algebraic and Topological Theories (1985), 305327, Kinokuniya. MR 1102263 (92e:14012)
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 [ST]
 H. Sumihiro and S. Tagami, A splitting theorem for rank two vector bundles on projective spaces in positive characteristic, Hiroshima Math J. 31 (2001), 5157. MR 1820694 (2001m:14063)
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 H. Tango, On morphisms from projective space to the Grassmann variety , J. Math. Kyoto Univ. 161 (1976), 201207 MR 0401787 (53:5614)
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Additional Information
Takuro Abe
Affiliation:
Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, 6068502, Japan
Address at time of publication:
Department of Mathematics, Hokkaido University, Kita10, Nishi8, KitaKu, Sapporo, Hokkaido, 0600810, Japan
Email:
abetaku@kusm.kyotou.ac.jp, abetaku@math.sci.hokudai.ac.jp
DOI:
http://dx.doi.org/10.1090/S000299470704161X
PII:
S 00029947(07)04161X
Received by editor(s):
July 16, 2004
Received by editor(s) in revised form:
July 23, 2005
Published electronically:
March 20, 2007
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
