Spectral space

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

Definition[edit]

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:

Equivalent descriptions[edit]

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[edit]

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[edit]

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[edit]

  • M. Hochster (1969). Prime ideal structure in commutative rings. Trans. Amer. Math. Soc., 142 43—60

Footnotes[edit]

  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)