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The Wikipedia article on coherent sheaves makes the following claim (without any reference), which I had trouble proving or finding a reference for: on an algebraic variety X (or I guess possibly even on a locally noetherian scheme), the coherent sheaves can be defined as the smallest class of sheaves of $\mathcal{O}_X$-modules with the following two properties:

i) the sheaf $\mathcal{O}_X$ is itself coherent;

ii) if, in a short exact sequence of sheaves, two of the sheaves are coherent, then so is the third.

I'm skeptical, but I would still like to know if this is true. If so, does anyone know a reference?

flag	asked May 20 2010 at 14:15 Dan Petersen 10.4k●1●22●46

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Wouldn't the first Chern class of every sheaf in that class be zero, since $c_1$ is additive over short exact sequences? – MartinG May 20 2010 at 14:25

Oh wow, I feel silly. You should post that as an answer so I can accept it. – Dan Petersen May 20 2010 at 14:43

It isn't silly, after all it was claimed at Wikipedia, and you were skeptical.. – MartinG May 20 2010 at 18:21

I think if you replace (ii) with the requirement that the kernel and cokernel of any morphism between coherent sheaves is coherent, then it works. – Sándor Kovács Jun 19 2011 at 2:34

MartinG · Accepted Answer · 2010-05-20 18:19:05Z UTC

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This is false. Every sheaf in that class would have zero first Chern class, since $c_1$ is additive over short exact sequences.

answered May 20 2010 at 18:19

MartinG
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Perhaps the original Wikipedia article meant to have "$\mathcal{O}_{Z}$ is coherent for every subvariety $Z\subseteq X$" in place of (i). Then the fact that the smallest class of sheaves satisfying (i) and (ii) is the category of coherent sheaves is an application of the usual dévissage-type argument. – Mike Roth May 26 2010 at 3:20

Characterisation of coherent sheaves on an algebraic variety

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