# Tagged Questions

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### Hypercohomology: finding a resolution for the de Rham complex of $\mathbb{CP}^1$

Let $\mathbb{P}^1$ be the complex projective line. Using the standard affine cover, $\mathcal{U} = \lbrace U,U' \rbrace, \ \$ we can define some quasi-coherent sheaves on $\mathbb{P}^1$. We can ...
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### Colimit and push forward of quasi-coherent sheaves in the analytic setting

I'm currently working on my master thesis and have some unresolved questions about quasi-coherent sheaves. Since I'm new to algebraic geometry, they might be rather trivial. I'm working in the ...
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### Exact sequence of sheaves of holomorphic functions

This is from Exercise 2.4.P. June 2013 version of Ravi Vakil's Math 216 notes. The idea is to show $\mathscr{O}_X \xrightarrow{\text{exp}} \mathscr{O}^*_X$ is an epimorphism. It seems ...
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### some ring theory applied to holomorphic functions

I'd like to know if my understanding of this business is correct. Let $U \subset \mathbb{C}^n$ be open and connected. The set $\mathcal{K}(U)$ of meromorÂ­phic functions on $U$ is a field. Is it true ...
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### relation between quotient field of holomorphic functions and meromorphic functions

Let $U \in \mathbb{C}^n$ open connected. Is it true that the quotient field of $\mathcal{O}_{\mathbb{C}^n,z}$ for $z \in U$ is the stalk of the sheaf $\mathcal{K}$ where $\mathcal{K}(U)$ is the field ...
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### Meromorphic functions a constant sheaf?

Is the sheaf of meromorphic functions on a (connected) compact Riemann surface constant? I am refering here to meromorphic functions in the sense of complex analysis and not to those of algebraic ...
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### Resolution of direct image functor

Let $i: X \to Y$ be an embedding of compact complex manifolds (not necessarily projective) and $E\to X$ a holomorphic vector bundle. I've seen it stated that the direct image sheaf $i_* E$ has a ...