---
tag: DSJET02WPR
kind: concept
name: Coherent analytic sheaf
msc: [32C35]
sources:
  - url: https://en.wikipedia.org/wiki/Coherent_sheaf
---

## Summary

A coherent analytic sheaf on a [[complex analytic space|HXSZA44J5J]] is a [[sheaf of modules|S33B2H3MYD]] over the structure sheaf that is locally finitely generated.
Moreover, the relations among finitely many local sections must again be locally finitely generated.
Coherent sheaves are the right class of sheaves for complex analytic geometry.
They include the structure sheaf, ideal sheaves of analytic subsets and sheaves of sections of holomorphic vector bundles.
Kernels, cokernels and images of morphisms between coherent sheaves are coherent.
They are the analytic counterpart of [[coherent sheaves|56RXEJ8Z6S]] on schemes.

## Explanations

### L1 human

A coherent sheaf attaches to each small region of a space a module of functions or sections.
Locally everything is described by finitely many generators and finitely many relations.
The model is a system of linear equations with holomorphic coefficients: its solutions form a coherent sheaf.
For example, on $\C^2$ the germs $(g_1, g_2)$ with $g_1 z_1 + g_2 z_2 = 0$ form a coherent sheaf, generated by the single relation $(z_2, -z_1)$.