Riemann Surfaces

Adrien Douady () and Régine Douady
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Adrien Douady: Université Paris-Sud Orsay
Régine Douady: Université Paris Denis-Diderot

Chapter 6 in Algebra and Galois Theories, 2020, pp 305-405 from Springer

Abstract: Abstract We begin this chapter with the definition of a Riemann surface. Let B be a compact connected Riemann surface, and $${{\mathscr {M}}}(B)$$ the field of meromorphic functions on B. If X is a compact connected Riemann surface over B (i.e. equipped with a non constant morphism $$\pi :X\rightarrow B$$ ), then the field $${{\mathscr {M}}}(X)$$ is a finite extension of $${{\mathscr {M}}}(B)$$ . Moreover, there is a finite subset $$\Delta $$ of B such that $$X'=X- \pi ^{-1}(\Delta )$$ is a connected finite cover of $$B-\Delta $$ . The functors $$X\mapsto {{\mathscr {M}}}(X)$$ and $$X\mapsto X'$$ give an equivalence between the category $${{\mathscr {V}}}^1_B$$ $${{\mathscr {V}}}^1_B$$ of compact connected Riemann surfaces over B, and respectively, the opposite category of the category of finite extensions of $${{\mathscr {M}}}(B)$$ , and the direct limit category of categories of connected covers of $$B-\Delta $$ , where $$\Delta \subset B$$ is finite.

Date: 2020
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DOI: 10.1007/978-3-030-32796-5_6

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