Foundations of the theory of Klein surfaces
Foundations of the theory of Klein surfaces
复制标题
克莱因曲面理论的基础
DOI：
10.1007/bfb0060987
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发表时间：
1971
影响因子：
0.8
通讯作者：
N. Greenleaf
中科院分区：
文献类型：
--
作者：
N. L. Alling;N. Greenleaf
众所周知，紧黎曼曲面范畴S和非常数解析映射范畴C与复代数函数场范畴C和复变函数范畴C通过两个逆变函数是余等价的，从而将解析理论和代数理论联系在一起。在研究具有非空边界ǝX([A]，[A₂]，[A])的紧黎曼曲面上的几个Banach代数时，第一作者提出了以下问题：什么是可以与X相联系的最简单的代数对象，并且可以从中恢复*？答案似乎是这样的：设E(*)是f(ǝX)CRU{∞}上的所有函数f亚纯的域。这个域是实数上的一元代数函数域。然后很自然地会问相反的问题：给定这样一个域E，是否存在一个紧致的Riemann曲面(可能有边界)使得EE()？有趣的是，这个问题的答案是否定的。代数几何学家长期熟知的下列领域为2 2这样的猜想提供了一个反例：设E=R(x，y)，其中X+y=-1。目前的合作就是在这一关头开始的。设(R)是所有实代数函数域和所有实线性同构的范畴。给定这样一个域E，代数几何学家早就知道如何将曲线X与E联系起来；例如，设X={0：0是E在R上的赋值环。放置在这样的曲线上的常见拓扑是中的Zariski拓扑
It has long been known that the category S of compact Riemann surfaces and non-constant analytic maps, and the category C of complex-algebraic function fields and complex isomorphisms are, via two contravariant functions, coequivalent; thus an analytic theory and an algebraic theory are tied together. While investigating several Banach algebras on compact Riemann surfaces with non-empty boundary ǝX ([A],[A₂],[A]), the first author posed the following question for himself: what is the simplest algebraic object which can be associated with X, from which* can be recovered? The answer seems to be the following: let E (*) be the field of all functions f meromorphic on that f (ǝX) CRU {∞}. This field is an algebraic function field in one variable over the reals. It is natural then to ask the converse question: given such a field E, is there a compact Riemann surface (possibly with boundary) such that EE ()? The such answer to this question, interestingly, is no. The following field, long known to algebraic geometers, supplies a counter-example to 2 2 such a conjecture: let E= R (x, y), where X+ y=-1. The present collaboration began at this juncture. Let (R be the category of all real-algebraic function fields and all real-linear isomorphisms. Given such a field E, the algebraic geometers have long known how to associate a curve X with E; for example let X={0: 0 a valuation ring of E over R}. The usual topology put on such a curve is the Zariski topology in