Holomorphic convexity of compact sets in analytic spaces and the structure of algebras of holomorphic germs
Author:
William R. Zame
Journal:
Trans. Amer. Math. Soc. 222 (1976), 107-127
MSC:
Primary 32E25; Secondary 32C35
DOI:
https://doi.org/10.1090/S0002-9947-1976-0425179-8
MathSciNet review:
0425179
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Let $(X,{\mathcal {O}_X})$ be a reduced analytic space and let K be a compact, holomorphically convex subset of X. It is shown that analogs of Cartan’s Theorems A and B are valid for coherent analytic sheaves on K. This result is applied to the study of the algebra of germs on K of functions holomorphic near K. In particular, characterizations of finitely generated ideals, prime ideals and homomorphisms are obtained.
- G. R. Allan, An extension of the Silov-Arens-Calderón theorem, J. London Math. Soc. 44 (1969), 595–601. MR 243356, DOI https://doi.org/10.1112/jlms/s1-44.1.595
- G. R. Allan, On lifting analytic relations in commutative Banach algebras, J. Functional Analysis 5 (1970), 37–43. MR 0262832, DOI https://doi.org/10.1016/0022-1236%2870%2990035-2
- Richard Arens and A. P. Calderón, Analytic functions of several Banach algebra elements, Ann. of Math. (2) 62 (1955), 204–216. MR 71735, DOI https://doi.org/10.2307/1969675
- Joseph Becker, Parametrizations of analytic varieties, Trans. Amer. Math. Soc. 183 (1973), 265–292. MR 344513, DOI https://doi.org/10.1090/S0002-9947-1973-0344513-8
- Jan-Erik Björk, Holomorphic convexity and analytic structures in Banach algebras, Ark. Mat. 9 (1971), 39–54. MR 385170, DOI https://doi.org/10.1007/BF02383636
- Glen E. Bredon, Sheaf theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. MR 0221500 H. Cartan, Séminaire 1951/52, École Normale Supérieure, 1952. MR 16, 233.
- Henri Cartan, Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France 85 (1957), 77–99 (French). MR 94830
- R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR 0221256
- Jacques Frisch, Points de platitude d’un morphisme d’espaces analytiques complexes, Invent. Math. 4 (1967), 118–138 (French). MR 222336, DOI https://doi.org/10.1007/BF01425245
- Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. MR 0410387
- Hans Grauert, Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Inst. Hautes Études Sci. Publ. Math. 5 (1960), 64 (German). MR 121814
- Hans Grauert and Reinhold Remmert, Bilder und Urbilder analytischer Garben, Ann. of Math. (2) 68 (1958), 393–443 (German). MR 102612, DOI https://doi.org/10.2307/1970254
- Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
- Reese Harvey and R. O. Wells Jr., Compact holomorphically convex subsets of a Stein manifold, Trans. Amer. Math. Soc. 136 (1969), 509–516. MR 235158, DOI https://doi.org/10.1090/S0002-9947-1969-0235158-8
- Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR 0155856
- Raghavan Narasimhan, Introduction to the theory of analytic spaces, Lecture Notes in Mathematics, No. 25, Springer-Verlag, Berlin-New York, 1966. MR 0217337
- Yum-tong Siu, Noetherianness of rings of holomorphic functions on Stein compact series, Proc. Amer. Math. Soc. 21 (1969), 483–489. MR 247135, DOI https://doi.org/10.1090/S0002-9939-1969-0247135-7
- Edgar Lee Stout, The theory of uniform algebras, Bogden & Quigley, Inc., Tarrytown-on-Hudson, N. Y., 1971. MR 0423083 M. A. van Kuilenberg, Convexity and approximation in Stein spaces, Dissertation, Tulane University, 1972.
- William R. Zame, Algebras of analytic germs, Trans. Amer. Math. Soc. 174 (1972), 275–288. MR 313545, DOI https://doi.org/10.1090/S0002-9947-1972-0313545-7
- William R. Zame, Homomorphisms of rings of germs of analytic functions, Proc. Amer. Math. Soc. 33 (1972), 410–414. MR 289808, DOI https://doi.org/10.1090/S0002-9939-1972-0289808-6
- William Zame, Induced homomorphisms of algebras of analytic germs, Rice Univ. Stud. 59 (1973), no. 2, 157–163. MR 330509 ---, Ideals in rings of holomorphic germs, Preliminary report, Notices Amer. Math. Soc. 20 (1973), A-114. Abstract #701-32-2. ---, Holomorphic convexity of compact sets and cohomology vanishing theorems, Preliminary report, Notices Amer. Math. Soc. 20 (1973), A-636. Abstract #73T-B309.
- G. R. Allan, H. G. Dales, and J. P. McClure, Pseudo-Banach algebras, Studia Math. 40 (1971), 55–69. MR 333737, DOI https://doi.org/10.4064/sm-40-1-55-69
Retrieve articles in Transactions of the American Mathematical Society with MSC: 32E25, 32C35
Retrieve articles in all journals with MSC: 32E25, 32C35
Additional Information
Keywords:
Holomorphic convexity,
holomorphic germs,
analytic spaces,
Theorem B
Article copyright:
© Copyright 1976
American Mathematical Society