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I don't need this statement to be true globally - even if it's locally true I'm fine with that. I'm on a noncompact Riemann surface but that doesn't matter because I only want this to hold locally anyway, so we can just look at little subsets of $\mathbb{C}$. Say $h$ is an arbitrary holomorphic function. Say $f,g$ are holomorphic functions with no common zero. Then I want to show that there exist $\phi_1, \phi_2$ holomorphic such that $h = f\phi_1 + g\phi_2$ locally (so on some neighbourhood of some fixed point). I'm happy to just work with Taylor series of all these functions (which is what I was doing) because as I said, I'm fine with this holding just locally. But I can't quite figure out how it works. I'd appreciate some help, please!

José Carlos Santos
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Acton
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  • I may be missing something stupid, but: since you just want this locally around a point, say $z_0$, then if $f(z_0) \neq 0$ you can take $h = h/f \cdot f + 0 \cdot g$ and if $g(z_0) \neq 0$ you can take $h = 0 \cdot f + h/g \cdot g$...? Where you've shrunk the neighborhood enough so that either $f$ or $g$ does not vanish. – Najib Idrissi May 08 '18 at 11:54
  • Thank you! While I'm happy for the equality to hold locally, I do want all my functions (including $\phi_1, \phi_2$) to be globally defined. – Acton May 08 '18 at 12:02
  • As in, globally defined and holomorphic – Acton May 08 '18 at 12:03
  • Actually no maybe I can't expect that; maybe that's asking for too much. I'm not sure. – Acton May 08 '18 at 12:06

2 Answers2

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Here is an advanced level but very general answer.
Let $X$ be a complex manifold (or even complex space with singularities ) and let $f_1,\cdots, f_n\in \mathcal O(X)$ be globally defined holomorphic functions .
Then we have a an $\mathcal O_X$-linear map of coherent sheaves $$f=(f_1,\cdots, f_n): \mathcal O_X^n\to \mathcal O_X $$ sending $(\phi_1,\cdots, \phi_n)$ to $\sum f_i\phi_i$.
The hypothesis that the $f_i$'s have no common zero at any point $x\in X$ implies that this morphism of sheaves is surjective, so that denoting by $\mathcal K$ its kernel we get the exact sequence of coherent sheaves $$0\to\mathcal K \to\ \mathcal O_X^n\to \mathcal O_X \to 0 $$ The long exact sequence of cohomology then yields the fragment $$\cdots \to \mathcal O_X(X)^n\to \mathcal O_X(X) \to H^1(X,\mathcal K)\to \cdots $$ If $X$ is Stein, as is for example every non-compact Riemann surface, Cartan-Serre's Théorème B for coherent sheaves implies that $H^1(X,\mathcal K)=0$, so that $$\mathcal O_X(X)^n\to \mathcal O_X(X)\to 0$$ is surjective.
This means exactly that every $h\in \mathcal O_X(X)$ can be written as $$h=\sum f_i\Phi_i$$ for some suitable global holomorphic functions $\Phi_i\in \mathcal O_X(X)$.

pshmath0
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Georges Elencwajg
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Hint: In Rudin Real and Complex Analysis there's a proof that while $H(\Omega)$ is not a PID, nonetheless every finitely generated ideal is principal.

David C. Ullrich
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