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Vector bundles with a Frobenius structure on the punctured unit disc. (English) Zbl 1074.14028
The paper under review deals with the classification of $$\sigma$$-vector bundles on the punctured unit disc $$\dot{D} \subset \mathbb{C}$$, where $$\mathbb{C}$$ is a complete non-archimedian valued algebraically closed field of positive characteristic $$p$$. A $$\sigma$$-bundle on $$\dot{D}$$ is a vector bundle together with a fixed isomorphism with its pull-back under the arithmetic Frobenius. For $$r$$ and $$d$$ relatively prime, there exist $$\sigma$$-bundles denoted $${\mathcal F}_{d,r}$$ of rank $$r$$, which are quite easy to construct.
The main result of the paper (Theorem 11.1) states that every $$\sigma$$-bundle is isomorphic to the direct sum of such indecomposable bundles. This result resembles the classification of vector bundle over an elliptic curve by Atiyah, but the $$\sigma$$-line bundles here are just given by $${\mathcal F}_{d,1}$$, so isomorphic with $$\mathbb{Z}$$.

##### MSC:
 14H60 Vector bundles on curves and their moduli 14G22 Rigid analytic geometry 13A35 Characteristic $$p$$ methods (Frobenius endomorphism) and reduction to characteristic $$p$$; tight closure
Dieudonne theory
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