Supriya Ghosh (Editor)

Spectral space

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In mathematics, a spectral space (sometimes called a coherent space) is a topological space that is homeomorphic to the spectrum of a commutative ring.

Contents

Definition

Let X be a topological space and let K ∘ (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 ∘ (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

    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 ∘ (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.

    Properties

    Let 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 quasi-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. 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.

    • 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). In this anti-equivalence, a spectral space X corresponds to the lattice K ∘ (X).

    References

    Spectral space Wikipedia
    (Text) CC BY-SA