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April 2011 Homomorphisms of infinitely generated analytic sheaves
Vakhid Masagutov
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Ark. Mat. 49(1): 129-148 (April 2011). DOI: 10.1007/s11512-010-0129-x


We prove that every homomorphism $\mathcal{O}^{E}_{\zeta}\rightarrow\mathcal{O}^{F}_{\zeta}$, with E and F Banach spaces and ζ∈ℂm, is induced by a $\mathop{\mathrm{Hom}}(E,F)$-valued holomorphic germ, provided that 1≤m<∞. A similar structure theorem is obtained for the homomorphisms of type $\mathcal{O}^{E}_{\zeta}\rightarrow\mathcal{S}_{\zeta}$, where $\mathcal{S}_{\zeta}$ is a stalk of a coherent sheaf of positive depth. We later extend these results to sheaf homomorphisms, obtaining a condition on coherent sheaves which guarantees the sheaf to be equipped with a unique analytic structure in the sense of Lempert–Patyi.

Funding Statement

Research partially supported by NSF grant DMS0700281 and the Mittag-Leffler Institute, Stockholm. I am grateful to both organizations, and I, particularly, would like to express my gratitude to the Mittag-Leffler Institute for their hospitality during my research leading to this paper. I am indebted to Professor Lempert for his guidance and for proposing questions that motivated this work. I am especially grateful for his suggestions and critical remarks that were invaluable at the research and writing phases. Lastly, I would like to thank the anonymous referee for devoting the time and effort to thoroughly review the manuscript and for suggesting numerous improvements.


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Vakhid Masagutov. "Homomorphisms of infinitely generated analytic sheaves." Ark. Mat. 49 (1) 129 - 148, April 2011.


Received: 27 April 2009; Revised: 17 March 2010; Published: April 2011
First available in Project Euclid: 31 January 2017

zbMATH: 1216.32003
MathSciNet: MR2784261
Digital Object Identifier: 10.1007/s11512-010-0129-x

Rights: 2010 © Institut Mittag-Leffler


Vol.49 • No. 1 • April 2011
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