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.
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References
Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley, Reading, MA (1969)
Bourbaki, N.: Commutative Algebra, Chapters 1–7. Springer, Berlin (1989)
Cahen, P.-J., Chabert, J.-L.: Integer-Valued Polynomials. Mathematical Surveys and Monographs, vol. 48. American Mathematical Society, Providence (1997)
Huneke, C., Swanson, I.: Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006)
Lang, S.: Algebra. Graduate Texts in Mathematics, vol. 211. Springer, New York (2002)
Matsumura, H.: Commutative Ring Theory. Cambridge University Press, Cambridge (1986)
Nagata, M.: Local Rings. Interscience Tracts in Pure and Applied Mathematics, vol. 13. Wiley, New York (1962)
Northcott, D.G.: A general theory of one-dimensional local rings. Proc. Glasgow Math. Assoc. 2, 159–169 (1956)
Olberding, B.: One-dimensional bad Noetherian domains. Trans. Am. Math. Soc. 366(8), 4067–4095 (2014)
Acknowledgements
R. Rissner is supported by the Austrian Science Fund (FWF): P 27816-N26.
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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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DOI: https://doi.org/10.1007/978-3-319-65874-2_17
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