Characterisation of coherent sheaves on an algebraic variety - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:19:25Z http://mathoverflow.net/feeds/question/25365 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25365/characterisation-of-coherent-sheaves-on-an-algebraic-variety Characterisation of coherent sheaves on an algebraic variety Dan Petersen 2010-05-20T14:15:23Z 2010-05-20T18:19:05Z <p>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 <em>X</em> (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:</p> <p>i) the sheaf $\mathcal{O}_X$ is itself coherent;</p> <p>ii) if, in a short exact sequence of sheaves, two of the sheaves are coherent, then so is the third. </p> <p>I'm skeptical, but I would still like to know if this is true. If so, does anyone know a reference?</p> http://mathoverflow.net/questions/25365/characterisation-of-coherent-sheaves-on-an-algebraic-variety/25401#25401 Answer by MartinG for Characterisation of coherent sheaves on an algebraic variety MartinG 2010-05-20T18:19:05Z 2010-05-20T18:19:05Z <p>This is false. Every sheaf in that class would have zero first Chern class, since $c_1$ is additive over short exact sequences.</p>