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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Decomposing algebraic vector bundles on the projective line
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by Charles C. Hanna PDF
Proc. Amer. Math. Soc. 61 (1976), 196-200 Request permission

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 ${\mathbf {P}}_R^1$.
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 61 (1976), 196-200
  • MSC: Primary 14F05
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0429898-4
  • MathSciNet review: 0429898