Spectral space

In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring.

Definition

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

• X is quasi-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:

1. X is homeomorphic to a projective limit of finite T₀-spaces.
2. 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).
3. X is homeomorphic to the spectrum of a commutative ring.
4. X is the topological space determined by a Priestley space.
5. X is a coherent space in the sense of topology (this indeed is only another name).

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 quasi-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. 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.
• X can be seen as a Pairwise Stone space.^[1]

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 quasi-compact subset of Y under f is again quasi-compact.

The category of spectral spaces which has spectral maps as morphisms is dually equivalent to the category of bounded distributive lattices (together with morphisms of such lattices).^[2] In this anti-equivalence, a spectral space X corresponds to the lattice K^$\circ$(X).

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.

Footnotes

1. G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, A. Kurz, (2010). Bitopological duality for distributive lattices and Heyting algebras. Mathematical Structures in Computer Science, 20.
2. (Johnstone 1982)