Arkiv för Matematik

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

Homomorphisms of infinitely generated analytic sheaves

Vakhid Masagutov

Full-text: Open access

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

Permanent link to this document
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
2010 © Institut Mittag-Leffler

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


Export citation

References

  • Eisenbud, D., Commutative Algebra, Graduate Texts in Mathematics, 150, Springer, New York, 1995.
  • Grauert, H. and Remmert, R., Coherent Analytic Sheaves, Grundlehren der Mathematischen Wissenschaften 265, Springer, Berlin, 1984.
  • Leiterer, J., Banach coherent analytic Fréchet sheaves, Math. Nachr. 85 (1978), 91–109.
  • Lempert, L., Coherent sheaves and cohesive sheaves, in Complex Analysis, Trends in Mathematics, pp. 227–244, Birkhäuser, Basel, 2010.
  • Lempert, L. and Patyi, I., Analytic sheaves in Banach spaces, Ann. Sci. École Norm. Sup. 40 (2007), 453–486.
  • Matsumura, H., Commutative Algebra, 2nd ed., Mathematics Lecture Note Series 56, Benjamin/Cummings, Reading, MA, 1980.
  • Mujica, J., Complex Analysis in Banach Spaces, Holomorphic Functions and Domains of Holomorphy in Finite and Infinite Dimensions, North-Holland Mathematics Studies 120, North-Holland, Amsterdam, 1986.
  • Serre, J. P., Faisceaux algébriques cohérents, Ann. of Math. 61 (1955), 197–278.