Several complex variables and analytic spaces Stein manifolds and coherent sheaves Theorem

Cartan's theorem A

Builds on Stein space Coherent analytic sheaf Cartan's theorem B

Statement

Let $X$ be a Stein space and let $\mathcal{F}$ be a coherent analytic sheaf 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.

Going deeper

For a student

Theorem A is usually proved together with Cartan's theorem B, 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.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Tag	`5FD8V6MQNX`
URI	`m2://5FD8V6MQNX`
Kind	statement, theorem
Writing	W0 stub
MSC	32E10, 32C35
Status	This copy of the site has no build data. Run `m2 status 5FD8V6MQNX`.
About	Stein space, Coherent analytic sheaf
Uses	Stein space, Coherent analytic sheaf, Cartan's theorem B