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Algebras of Holomorphic Functions in Ringed Spaces, I

Published online by Cambridge University Press:  20 November 2018

Maxwell E. Shauck*
Affiliation:
Tulane University, New Orleans, Louisiana; Yale University, New Haven, Connecticut
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A pair () is a ringed space if it is a subsheaf of rings with 1 of the sheaf of germs of continuous functions on X. If U is an open subset of X, we denote the set of sections over U relative to by . If , then implies that there exists some open neighbourhood V of u, VU, and some g continuous on V such that the germ of g at u, ug is ϕ(u). Now we define ϕ(u) (u) to be g(u) and in this way we obtain, in a unique fashion, a continuous complex-valued function on U. The collection of all such functions for a given set is denoted by and is called the -holomorphic functions on U.

THEOREM. Let X be a locally connected Hausdorff space and () a ringed space.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Bochner, S. and Martin, W., Several complex variables, Princeton Mathematical Series, Vol. 10 (Princeton Univ. Press, Princeton, N.J., 1948).Google Scholar
2. Godement, R., Théorie des faisceaux (Hermann, Paris, 1958).Google Scholar
3. Hoffman, K., Domains of holomorphy, Mimeographed notes, Massachusetts Institute of Technology, Cambridge, Mass., 1958.Google Scholar
4. Quigley, F., Approximation by algebras of functions, Math. Ann. 135 (1958), 8192.Google Scholar
5. Quigley, F., Lectures on several complex variables, Tulane University, New Orleans, Louisiana, 1964-65, 1965-66.Google Scholar