The elementary transformation of vector bundles on regular schemes
HTML articles powered by AMS MathViewer
- by Takuro Abe PDF
- Trans. Amer. Math. Soc. 359 (2007), 4285-4295 Request permission
Abstract:
We give a generalized definition of an elementary transformation of vector bundles on regular schemes by using Maximal Cohen-Macaulay 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 quasi-projective varieties over an algebraically closed field. Moreover, we give an application of this theory to reflexive sheaves.References
- T. Abe, The elementary Transformation of vector bundles on regular schemes, preprint in Kyoto University, 9, 2004.
- T. Abe and M. Yoshinaga, Splitting criterion for reflexive sheaves, arXiv, mathAG/0503710, preprint in RIMS, 1496 (2005).
- I. Dolgachev and M. Kapranov, Arrangements of hyperplanes and vector bundles on $\mathbf P^n$, Duke Math. J. 71 (1993), no.ย 3, 633โ664. MR 1240599, DOI 10.1215/S0012-7094-93-07125-6
- Robin Hartshorne, Local cohomology, Lecture Notes in Mathematics, No. 41, Springer-Verlag, Berlin-New York, 1967. A seminar given by A. Grothendieck, Harvard University, Fall, 1961. MR 0224620
- Robin Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. MR 0282977
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Robin Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no.ย 2, 121โ176. MR 597077, DOI 10.1007/BF01467074
- G. Horrocks and D. Mumford, A rank $2$ vector bundle on $\textbf {P}^{4}$ with $15,000$ symmetries, Topology 12 (1973), 63โ81. MR 382279, DOI 10.1016/0040-9383(73)90022-0
- George R. Kempf, A criterion for the splitting of a vector bundle, Forum Math. 2 (1990), no.ย 5, 477โ480. MR 1067213, DOI 10.1515/form.1990.2.477
- Hing Sun Luk and Stephen S.-T. Yau, Cohomology and splitting criterion for holomorphic vector bundles on $\textbf {C}\textrm {P}^n$, Math. Nachr. 161 (1993), 233โ238. MR 1251020, DOI 10.1002/mana.19931610117
- Masaki Maruyama, On a family of algebraic vector bundles, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp.ย 95โ146. MR 0360587
- Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911
- Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhรคuser, Boston, Mass., 1980. MR 561910
- Kyoji Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no.ย 2, 265โ291. MR 586450
- Chad Schoen, On the geometry of a special determinantal hypersurface associated to the Mumford-Horrocks vector bundle, J. Reine Angew. Math. 364 (1986), 85โ111. MR 817640, DOI 10.1515/crll.1986.364.85
- Hideyasu Sumihiro, A theorem on splitting of algebraic vector bundles and its applications, Hiroshima Math. J. 12 (1982), no.ย 2, 435โ452. MR 665505
- Hideyasu Sumihiro, Elementary transformations of algebraic vector bundles, Algebraic and topological theories (Kinosaki, 1984) Kinokuniya, Tokyo, 1986, pp.ย 305โ327. MR 1102263
- Hideyasu Sumihiro, Elementary transformations of algebraic vector bundles. II, Algebraic geometry and commutative algebra, Vol. II, Kinokuniya, Tokyo, 1988, pp.ย 713โ748. MR 977780
- Hideyasu Sumihiro, Determinantal varieties associated to rank two vector bundles on projective spaces and splitting theorems, Hiroshima Math. J. 29 (1999), no.ย 2, 371โ434. MR 1704256
- Hideyasu Sumihiro and Shigehiro Tagami, A splitting theorem for rank two vector bundles on projective spaces in positive characteristic, Hiroshima Math. J. 31 (2001), no.ย 1, 51โ57. MR 1820694
- Hiroshi Tango, On morphisms from projective space $P^{n}$ to the Grassmann variety $G\textrm {r}(n, d)$, J. Math. Kyoto Univ. 16 (1976), no.ย 1, 201โ207. MR 401787, DOI 10.1215/kjm/1250522969
- Hiroshi Tango, On vector bundles on $\textbf {P}^n$ which have $\sigma$-transition matrices, Tokyo J. Math. 16 (1993), no.ย 1, 1โ29. MR 1223285, DOI 10.3836/tjm/1270128979
Additional Information
- Takuro Abe
- Affiliation: Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, 606-8502, Japan
- Address at time of publication: Department of Mathematics, Hokkaido University, Kita-10, Nishi-8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan
- Email: abetaku@kusm.kyoto-u.ac.jp, abetaku@math.sci.hokudai.ac.jp
- Received by editor(s): July 16, 2004
- Received by editor(s) in revised form: July 23, 2005
- Published electronically: March 20, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 4285-4295
- MSC (2000): Primary 14F05
- DOI: https://doi.org/10.1090/S0002-9947-07-04161-X
- MathSciNet review: 2309185