# Simple sheaves, definition and characterization

Hi,

I have found the (unexplained) notion of a simple coherent sheaf $F$. Is it right that one defines it as a coherent sheaf where every nonzero homomorphism $F\rightarrow F$ is invertible?

And is this equivalent to: $F$ has no nontrivial (coherent?) subsheaves?

Thanks a lot and greetings!

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To begin with this is mostly used for proper varieties. In nay case as a definition it seems OK (though I guess some people might make some restrictions on $F$ such as being torsion free). It is definitely not equivalent to having no non-trivial subsheaves. Consider for instance a proper geometrically integral scheme over a field and the structure sheaf, it is simple in this sense but there are lots of coherent ideals (unless the scheme is zero-dimensional). –  Torsten Ekedahl Aug 15 '11 at 12:34