Spectral space

In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topoi.

Definition

Let X be a topological space and let K^$\circ$(X) be the set of all compact open subsets of X. Then X is said to be spectral if it satisfies all of the following conditions:

X is compact and T₀.
K^$\circ$(X) is a basis of open subsets of X.
K^$\circ$(X) is closed under finite intersections.
X is sober, i.e., every nonempty irreducible closed subset of X has a (necessarily unique) generic point.

Equivalent descriptions

Let X be a topological space. Each of the following properties are equivalent to the property of X being spectral:

X is homeomorphic to a projective limit of finite T₀-spaces.
X is homeomorphic to the spectrum of a bounded distributive lattice L. In this case, L is isomorphic (as a bounded lattice) to the lattice K^$\circ$(X) (this is called Stone representation of distributive lattices).
X is homeomorphic to the spectrum of a commutative ring.
X is the topological space determined by a Priestley space.
X is a T₀ space whose frame of open sets is coherent (and every coherent frame comes from a unique spectral space in this way).

Properties

Let X be a spectral space and let K^$\circ$(X) be as before. Then:

K^$\circ$(X) is a bounded sublattice of subsets of X.
Every closed subspace of X is spectral.
An arbitrary intersection of compact and open subsets of X (hence of elements from K^$\circ$(X)) is again spectral.
X is T₀ by definition, but in general not T₁.^[1] In fact a spectral space is T₁ if and only if it is Hausdorff (or T₂) if and only if it is a boolean space if and only if K^$\circ$(X) is a boolean algebra.
X can be seen as a pairwise Stone space.^[2]

Spectral maps

A spectral map f: X → Y between spectral spaces X and Y is a continuous map such that the preimage of every open and compact subset of Y under f is again compact.

The category of spectral spaces, which has spectral maps as morphisms, is dually equivalent to the category of bounded distributive lattices (together with homomorphisms of such lattices).^[3] In this anti-equivalence, a spectral space X corresponds to the lattice K^$\circ$(X).

Citations

^ A.V. Arkhangel'skii, L.S. Pontryagin (Eds.) General Topology I (1990) Springer-Verlag ISBN 3-540-18178-4 (See example 21, section 2.6.)
^ G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, A. Kurz, (2010). "Bitopological duality for distributive lattices and Heyting algebras." Mathematical Structures in Computer Science, 20.
^ Johnstone 1982.

References

M. Hochster (1969). Prime ideal structure in commutative rings. Trans. Amer. Math. Soc., 142 43—60
Johnstone, Peter (1982), "II.3 Coherent locales", Stone Spaces, Cambridge University Press, pp. 62–69, ISBN 978-0-521-33779-3.
Dickmann, Max; Schwartz, Niels; Tressl, Marcus (2019). Spectral Spaces. New Mathematical Monographs. Vol. 35. Cambridge: Cambridge University Press. doi:10.1017/9781316543870. ISBN 9781107146723.

[1] A.V. Arkhangel'skii, L.S. Pontryagin (Eds.) General Topology I (1990) Springer-Verlag ISBN 3-540-18178-4 (See example 21, section 2.6.)

[2] G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, A. Kurz, (2010). "Bitopological duality for distributive lattices and Heyting algebras." Mathematical Structures in Computer Science, 20.

[FOOTNOTEJohnstone1982-3] Johnstone 1982.

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