New York Journal of Mathematics
Volume 28 (2022), 800-823


Dario Spirito

Isolated points of the Zariski space

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Published: May 16, 2022.
Keywords: Zariski space; constructible topology; Cantor space; isolated points; perfect spaces; extensions of valuations.
Subject: 13F30; 13A15; 13A18; 54D99.

Let D be an integral domain and L be a field containing D. We study the isolated points of the Zariski space Zar(L|D), with respect to the constructible topology. In particular, we completely characterize when L (as a point) is isolated and, under the hypothesis that L is the quotient field of D, when a valuation domain of dimension 1 is isolated; as a consequence, we find all isolated points of Zar(D) when D is a Noetherian domain and, under the hypothesis that D and D' are Noetherian, local and countable, we characterize when Zar(D) and Zar(D') are homeomorphic. We also show that if V is a valuation domain and L is transcendental over V then the set of extensions of V to L has no isolated points.


I would like to thank the referee for pointing out several problems in the first version of the paper and for his or her patience in pointing them out.

Author information

Dario Spirito:
Dipartimento di Matematica "Tullio Levi-Civita"
Universitá degli Studi di Padova, Padova, Italy;
Current address: Dipartimento di Scienze Matematiche, Informatiche e Fisiche
Universitá degli Studi di Udine, Udine, Italy