Definition:
Haz analítico: a sheaf $\mathcal{F}$ tal que para cada abierto $U$, $\mathcal{F}(U)$ es un $\mathcal{O}(U)$-módulo.
Haz coherente: haz analítico $\mathcal{F}$ tal que para cada punto $p$ existe un entorno abierto $U_p$ y una sucesión exacta
$$ \bigoplus_s \mathcal{O}(U_p) \longrightarrow \bigoplus_r \mathcal{O}(U_p) \longrightarrow \mathcal{F}(U_p) \longrightarrow 0 $$Idea: In general, vector bundles are locally free sheaves. The kernel and image of vector bundle morphisms are not vector bundles (unless they are constant rank morphisms) but they are, at least, sheaves. Those sheaves are therefore not necessarily locally free ones but coherent sheaves. Coherent sheaves constitute the class of sheaves such that their image and kernel are still sheaves of the same class. You can understand them as a generalization of vector bundles.
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Author of the notes: Antonio J. Pan-Collantes
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