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# Mathematical Research Letters

## Volume 15 (2008)

### Number 6

### On deep Frobenius descent and flat bundles

Pages: 1101 – 1115

DOI: https://dx.doi.org/10.4310/MRL.2008.v15.n6.a3

#### Authors

#### Abstract

Let $R$ be an integral domain of finite type over $\ZZ$ and let $f:\shX \ra \Spec R$ be a smooth projective morphism of relative dimension $d \geq 1$. We investigate, for a vector bundle $\shE$ on the total space $\shX$, under what arithmetical properties of a sequence $(\fop_n, e_n)_{n \in \NN}$, consisting of closed points $\fop_n$ in $\Spec R$ and Frobenius descent data $\shE_{\fop_n} \cong {F^{e_n}}^*(\shF)$ on the closed fibers $\shX_{\fop_n}$, the bundle $\shE_0$ on the generic fiber $\shX_0$ is semistable.

Published 1 January 2008