do stalks determine a sheaf?
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From: Alex on 21 Jan 2010 07:31 Let F and G be sheaves on a toplogical space X such that there is an isomorphism Fx=Gx on every stalk with x in X. Is F necessarily isomorphic to G? Or do I have to assume that there is a morphism of sheaves from F to G inducing all the isomorphisms on the stalks?
From: victor_meldrew_666 on 21 Jan 2010 14:27
On 21 Jan, 12:31, Alex <mynameisrab...(a)hotmail.com> wrote: > Let F and G be sheaves on a toplogical space X such that there is an > isomorphism Fx=Gx on every stalk with x in X. Is F necessarily > isomorphic to G? No. > Or do I have to assume that there is a morphism of > sheaves from F to G inducing all the isomorphisms on the stalks? Yes. On the circle there's a sheaf whose stalks are all Z but each non-zero component of its espace etale winds around twice. It's not isomorphic to the constant sheaf but it has isomorphic stalks. |