---
tag: 5FD8V6MQNX
kind: statement
role: theorem
name: Cartan's theorem A
msc: ["32E10", 32C35]
uses: [Z4AMJS5KA6, DSJET02WPR, R7VKRV8AGQ]
about: [Z4AMJS5KA6, DSJET02WPR]
sources:
  - url: https://en.wikipedia.org/wiki/Cartan%27s_theorems_A_and_B
---

## Statement

Let $X$ be a [[Stein space|Z4AMJS5KA6]] and let $\mathcal{F}$ be a [[coherent analytic sheaf|DSJET02WPR]] on $X$.
Then $\mathcal{F}$ is generated by its global sections.
That is, for every $x \in X$, the germs at $x$ of the elements of $\Gamma(X, \mathcal{F})$ generate the stalk $\mathcal{F}_x$ as a module over $\mathcal{O}_{X,x}$.

## Remarks

Henri Cartan proved theorems A and B in his seminar of 1951–52, building on the work of Oka.
In the usual proof, $X$ is exhausted by compact holomorphically convex sets.
On each of them, $\mathcal{F}$ is generated by finitely many sections, by a local argument.
Cartan's theorem B on these pieces and a Runge-type approximation theorem then pass to the limit.
Example: let $\mathcal{I}$ be the ideal sheaf of a closed complex submanifold $Y$ of a Stein manifold $X$.
Theorem A says that near each point, $Y$ is cut out by global holomorphic functions on $X$ that vanish on $Y$.
For $X = \C^n$ and $\mathcal{F} = \mathcal{O}$ the theorem is trivial, because the constant function $1$ generates every stalk.
The Stein hypothesis is needed: on the projective line $\mathbb{P}^1$, the sheaf $\mathcal{O}(-1)$ is coherent but has no nonzero global sections, so it is not generated by them.

## Explanations

### L1 human

A coherent sheaf describes local data, such as the holomorphic functions that vanish on a given analytic set near each point.
Theorem A says that on a Stein space all this local data comes from global data.
Everything one sees near a point is produced by objects defined on the whole space.
It is the complex-analytic counterpart of the fact that every quasi-coherent sheaf on an affine scheme is generated by its global sections.

### L2 human

Theorem A is usually proved together with [[Cartan's theorem B|R7VKRV8AGQ]], and each is used in the proof of the other on exhausting compact sets.
A typical consequence: global holomorphic functions give local coordinates at every point of a Stein manifold.
Another one: every analytic subset of a Stein space is the common zero set of global holomorphic functions.