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In algebraic geometry it is well-known (see Hartshorne Exercise II.5.16 for example) that there is a 1-1 correspondence between rank $n$ (geometric) vector bundles $\pi\colon Y\to X$ on a scheme $X$ and locally free sheaves of $\mathcal{O}_X$-modules of rank $n$.

As I would imagine is well-known to experts in the subject, there is an analogous result in rigid analytic geometry (in the sense of Tate). Here the analogue of a trivial geometric vector bundle of rank $n$ on a rigid $K$-analytic space $X$ is the fibre product $X\times_K \mathbb{A}^{n,an}$ together with its natural projection onto $X$. Here $\mathbb{A}^{n,an}$ denotes the rigid analytic space obtained by gluing together polydiscs of larger and larger radius as in Example of Non-Archimedean analysis by Bosch, Güntzer and Remmert. One can verify that the sections of the projection map $X\times_K \mathbb{A}^{n,an}\to X$ are naturally a free $\mathcal{O}_X(X)$-module of rank $n$. Given this it is not difficult to make a definition of a general geometric vector bundle on rank $n$ on a rigid $K$-analytic space in such a way that the sections of such a bundle form a locally free sheaf of $\mathcal{O}_X$-modules (that is a vector bundle on $X$ in the only sense I can find in the literature). Moreover, as in the algebraic setting this actually defines a 1-1 correspondence between the two notions of vector bundle.

My question is whether anyone has written this up in a form that can be easily cited.

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I thought this correspondence works the same way for most kinds of ringed spaces. Are there extra difficulties in your setting? –  Piotr Achinger Feb 19 '13 at 17:30
No, I believe that there are no extra difficulties once you have proved that the sections of the trivial geometric vector bundle naturally form a free module over the global sections. A citeable reference to a general statement for ringed spaces (when the base space is equipped with a Grothendieck rather than usual topology) would also be appreciated if one does not exist for this specific case --- the fewer things I would have to explicitly check the better. –  Simon Wadsley Feb 19 '13 at 18:10

1 Answer 1

My guess is that in chapter 4, specially section 4.7, of

Rigid Analytic Geometry and Its Applications, by Jean Fresnel and Marius Van Der Put, Progress in Mathematics, vol. 218, Birkhäuser, Boston, 2004.

you could find the result you are looking for.

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Sadly not. They do talk about locally free sheaves and call them vector bundles but they don't discuss what I call geometric vector bundles in my question. –  Simon Wadsley Feb 21 '13 at 8:57
Looking again I guess you're meaning Proposition 4.7.2(1) which does point in the right direction in that it explains the relationship between the locally free sheaf and the gluing data of the trivial geometric sub-bundles. I'd like something more explicit though. –  Simon Wadsley Feb 21 '13 at 9:03

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