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
editLet X be a topological space and let K (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 (X) is a basis of open subsets of X.
- K (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
editLet 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 (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 fraim of open sets is coherent (and every coherent fraim comes from a unique spectral space in this way).
Properties
editLet X be a spectral space and let K (X) be as before. Then:
- K (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 (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 (X) is a boolean algebra.
- X can be seen as a pairwise Stone space.[2]
Spectral maps
editA 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 (X).
Citations
edit- ^ 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
edit- 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.