TITLE: Krull-Schmidt-Remak for vector bundles
QUESTION [5 upvotes]: I'm reading Nori's paper The fundamental group scheme, and I have some problems in certain passages of the proofs. This one is from chapter 1, 2.3. 
Let $X$ be a complete connected reduced $k$-scheme. We have that $\operatorname{H}^0(X, \operatorname{end}V)$ is of finite dimension for all finite vector bundles $V$, and Nori states that this implies that Krull-Schmidt-Remak holds: for all finite vector bundles there is a unique decomposition (up to reorder blah blah blah...) in a direct sum of indecomposable subbundles ($V$ is indecomposable if $V=V_1\oplus V_2\Rightarrow V=V_1\vee V=V_2$). Why?

REPLY [3 votes]: For future readers, I answer to this old question of mine. The proof can be found in Atiyah's paper On the Krull-Schmidt theorem with application to sheaves.