Topological duality via maximal spectrum functor

Abstract

The Isbell duality tells us a dual equivalence between spatial frames (aka. locales) and sober spaces; it is induced by the prime spectrum functor on frames. In the present paper, we give another dual equivalence induced by the maximal spectrum functor. The Isbell duality subsumes all sober spaces, but not all ${\mathrm{T}}_1$ spaces; the duality shown in this paper subsumes all ${\mathrm{T}}_1$ spaces, but not all sober spaces. Non-sober ${\mathrm{T}}_1$ spaces are particularly important in classical algebraic geometry; they include, inter alia, algebraic varieties in the traditional sense, the points of which can be recovered from their open set frames via the maximal spectrum functor (and cannot via the prime spectrum functor). The duality in this paper is particularly useful for those spaces in algebraic geometry. In addition to the duality induced by maximal spectra, we give a dual adjunction lurking behind it, and an algebraic characterization of having enough points in terms of maximal spectra.

Communicated by Jason P. Bell

Keywords:

• Algebraic varieties
• categorical duality
• classical algebraic geometry
• Isbell duality
• maximal spectrum
• prime spectrum
• Zariski topology

Mathematics Subject Classification:

• 18B30
• 14A10
• 06B15
• 54H10

Acknowledgments

I would like to thank the referee for his/her substantial comments and suggestions for improvement.

Additional information

Funding

This work was supported by JSPS Kakenhi (grant code: 17K14231), JST PRESTO (grant code: JPMJPR17G9), and the JSPS Core-to-Core Program “Mathematical Logic and its Applications” (A. Advanced Research Networks).