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^{${\displaystyle \circ }$}(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.
• K^{${\displaystyle \circ }$}(X) is a basis of open subsets of X.
• K^{${\displaystyle \circ }$}(X) is closed under finite intersections.
• X is sober, i.e., every nonempty irreducible closed subset of X has a unique generic point.

From that X is sober it follows that X is T₀. Indeed the definition of a spectral space can be equivalently reformulated through explicitly assuming that X is T₀ and weaking the assumption that X is sober to only require it to be quasi-sober, i.e. every irreducible closed subspace possesses a (not nececssarily unique) generic point. This is the way the definition is formulated in Hochster's 1967 thesis.

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^{${\displaystyle \circ }$}(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 T₀ space whose locale of open sets is coherent (and every coherent locale comes from a unique spectral space in this way).

Properties

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

• K^{${\displaystyle \circ }$}(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^{${\displaystyle \circ }$}(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 (i.e. T₂) if and only if it is a boolean space if and only if K^{${\displaystyle \circ }$}(X) is a boolean algebra.
• X can be seen as a pairwise Stone space. ^[2]

Spectral maps

A spectral mapf: 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^{${\displaystyle \circ }$}(X).

References

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.
3. Johnstone 1982.

Further reading

• 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.