Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Decomposing algebraic vector bundles on the projective line

Author: Charles C. Hanna
Journal: Proc. Amer. Math. Soc. 61 (1976), 196-200
MSC: Primary 14F05
MathSciNet review: 0429898
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: $ \mathcal{V}(R)$ denotes the category of algebraic vector bundles on $ {\mathbf{P}}_R^1,\;R$ a commutative, noetherian ring. If $ K$ is a field, it is known that any $ \mathcal{F} \in \mathcal{V}(K)$ is isomorphic to a (unique) direct sum of line bundles. If $ \mathfrak{p} \in \operatorname{Spec} R$ and $ K(\mathfrak{p})$ is the quotient field of $ R/\mathfrak{p}$, any $ \mathcal{F} \in \mathcal{V}(R)$ induces a bundle in $ \mathcal{V}(K(\mathfrak{p}))$, and so a decomposition into line bundles. If the decomposition is the same for each $ \mathfrak{p},\;\mathcal{F}$ is said to be uniform. It is shown that if $ R$ is reduced, uniform vector bundles on $ {\mathbf{P}}_R^1$ are sums of tensor products of (pullbacks of) bundles on $ \operatorname{Spec} R$ with line bundles on $ {\mathbf{P}}_R^1$.

References [Enhancements On Off] (What's this?)

  • [1] H. Bass, Algebraic $ K$-theory, Benjamin, New York, 1968. MR 40 #2736. MR 0249491 (40:2736)
  • [2] G. D. Birkhoff, Singular points of ordinary linear differential equations, Trans. Amer. Math. Soc. 10 (1909), 436-470. MR 1500848
  • [3] A. Grothendieck, Sur la classification des fibres holomorphes sur la sphere de Riemann, Amer. J. Math. 79 (1957), 121-138. MR 19, 315. MR 0087176 (19:315b)
  • [4] G. Horrocks, Projective modules over an extension of a local ring, Proc. London Math. Soc. (3) 14 (1964), 714-718. MR 30 #121. MR 0169878 (30:121)
  • [5] M. P. Murthy, Projective modules, Lecture Notes, Univ. of Chicago (to appear).
  • [6] D. Quillen, Projective modules over polynomial rings (to appear). MR 0427303 (55:337)
  • [7] L. Roberts, Indecomposable vector bundles on the projective line, Canad. J. Math. 24 (1972), 149-154. MR 45 #6828. MR 0297776 (45:6828)
  • [8] J.-P. Serre, Faisceaux algebriques coherents, Ann. of Math. (2) 61 (1955), 197-278. MR 16, 953. MR 0068874 (16:953c)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 14F05

Retrieve articles in all journals with MSC: 14F05

Additional Information

Keywords: Algebraic vector bundle, projective line, commutative noetherian ring, projective module, uniform vector bundle, decomposition of vector bundles
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society