# Climbing Mount Bourbaki

Thoughts on mathematics

August 25, 2011

## Analytic spaces

Posted by Akhil Mathew under functional analysis | Tags: analytic spaces, coherent sheaves, FAC, Oka's theorem |
Consider the space ${\mathbb{C}^n}$ and the sheaf ${\mathcal{O}}$ of holomorphic functions on it. One should think of this as the analog of complex affine space ${\mathbb{C}^n}$, with the Zariski topology, and with the sheaf ${\mathcal{O}_{reg}}$ of regular functions.
In algebraic geometry, if ${I \subset \mathbb{C}[x_1, \dots, x_n]}$ is an ideal, or if ${\mathcal{I} \subset \mathcal{O}_{reg}}$ is a coherent sheaf of ideals, then we can define a closed subset of ${\mathbb{C}[x_1,\dots, x_n]}$ corresponding to the roots of the polynomials in ${I}$. This construction gives the notion of an affine variety, and by gluing these one gets general varieties.
More precisely, here is what an affine variety is. If ${\mathcal{I} \subset \mathcal{O}_{reg}}$ is a coherent sheaf of ideals, then we define a ringed space ${(\mathrm{supp} \mathcal{O}_{reg}/\mathcal{I}, \mathcal{O}_{reg}/\mathcal{I})}$; this gives the associated affine variety. Here the “support” corresponds to taking the common zero locus of the functions in ${\mathcal{I}}$. In this way an affine variety is not just a subset of ${\mathbb{C}^n}$, but a locally ringed space.
Now we want to repeat this construction in the holomorphic category. If ${\mathcal{I} \subset \mathcal{O}}$ is a finitely generated ideal—that is, an ideal which is locally finitely generated—in the sheaf of holomorphic functions on ${\mathbb{C}^n}$, then we define the space cut out by ${\mathcal{I}}$ to be ${(\mathrm{supp} \mathcal{O}/\mathcal{I}, \mathcal{O}_{reg}/\mathcal{I})}$. We can think of these as “affine analytic spaces.”
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