## Arkiv för Matematik

• Ark. Mat.
• Volume 49, Number 1 (2011), 129-148.

### Homomorphisms of infinitely generated analytic sheaves

Vakhid Masagutov

#### Abstract

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.

#### Note

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.

#### Article information

Source
Ark. Mat., Volume 49, Number 1 (2011), 129-148.

Dates
Revised: 17 March 2010
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485907133

Digital Object Identifier
doi:10.1007/s11512-010-0129-x

Mathematical Reviews number (MathSciNet)
MR2784261

Zentralblatt MATH identifier
1216.32003

Rights

#### Citation

Masagutov, Vakhid. Homomorphisms of infinitely generated analytic sheaves. Ark. Mat. 49 (2011), no. 1, 129--148. doi:10.1007/s11512-010-0129-x. https://projecteuclid.org/euclid.afm/1485907133

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