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A Note on Analytically Irreducible Domains

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Abstract

If D is a one-dimensional, Noetherian, local domain, it is well known that D is analytically irreducible if and only if D is unibranched and the integral closure D of D is finitely generated as D-module. However, the proof of this result is split into pieces and spread over the literature. This paper collects the pieces and assembles them to a complete proof. Next to several results on integral extensions and completions of modules, we use Cohen’s structure theorem for complete, Noetherian, local domains to prove the main result. The purpose of this survey is to make this characterization of analytically irreducible domains more accessible.

Keywords

  • Unibranched
  • Analytically irreducible
  • Noetherian
  • Local
  • One-dimensional

MSC 2010

  • 13-02; 13B22
  • 13B35 13J05
  • 3J10
  • 13H05

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  • DOI: 10.1007/978-3-319-65874-2_17
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Acknowledgements

R. Rissner is supported by the Austrian Science Fund (FWF): P 27816-N26.

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Correspondence to Roswitha Rissner .

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Rissner, R. (2017). A Note on Analytically Irreducible Domains. In: Fontana, M., Frisch, S., Glaz, S., Tartarone, F., Zanardo, P. (eds) Rings, Polynomials, and Modules. Springer, Cham. https://doi.org/10.1007/978-3-319-65874-2_17

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