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Suppose I have a trivial vector bundle $V\cong \mathcal{O}_C^{\oplus s} \rightarrow C$ on an algebraic variety $C$, and suppose furthermore that I have an action $\mu$ of a cyclic finte group $G$ on the vector bundle $V$ that gives a decomposition of $V$ into eigenbundles, indexed by the characters of $G$. Are the eigen-bundles trivial as well? Why?

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No. Let $C$ be an open affine part of an elliptic curve over the complex numbers and $L$ a non-trivial line bundle on $C$. Now, $L\oplus L^{-1}$ is trivial, so let $G=\mathbf Z/2\mathbf Z$ operate by $1$ on $L$ and by $-1$ on $L^{-1}$.

For complete varieties (for simplicity, say having a rational point), all endomorphisms of a free bundle are given by constant matrices, so all direct summands are free again.

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so, if I understand properly, my claim is correct for complete variety, right? (I need this in order to thick your answer!! ;-) ) –  IMeasy Jun 22 '12 at 12:20
    
Right. . –  user2035 Jun 22 '12 at 15:19

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