Coherent sheaves

The concept of coherent sheaf is of great interest.  The definition is subtle but it rests on very basic ideas. Consider a stalk $F_x$ of a sheaf $F$. Then each element $s\in F_x$ has a representative $s_U$ for some $U$ containing $x$.  But we cannot in general find a $U$ which does for all such $s$. Consider a countable set of points having the origin of the complex plane as an accumulation point. And consider the sheaf of holomorphic functions having zeros at the origin and on the points of the set. Then the stalk at the origin is isomorphic to 0 but locally around 0 our sheaf is not isomorphic to the null sheaf. Thus a morphism of sheaves which is an isomorphism on the stalk at at a point $x$ is not necessarily locally an isomorphism of sheaves around $x$. Note the similarity to uniform continuity. Coherence expresses a coincidence between locality and infinitesimal locality which "coheres" - one of the characteristics of holomorphic functions.

An excellent discussion and motivation is given in Greuel and Pfister's Local Analytic Geometry.