nLab
spectral topological space

Contents

Contents

Idea

A coherent space (alias spectral topological space) is a topological space which is homeomorphic to the spectrum of a commutative ring, hence to the topological space underlying an affine scheme.

Equivalently, it is a compact sober topological space whose collection of compact open subsets is closed under finite intersections and forms a topological base.

Morphisms of coherent spaces are continuous maps such that preimages of compact open subsets are again compact.

Stone duality for coherent spaces

Passing from a coherent space to its lattice of compact open subsets establishes a contravariant equivalence from the category of coherent spaces to the category of (bounded) distributive lattices.

A coherent space is Hausdorff if and only if it is a Stone space. Under Stone duality for coherent spaces, this corresponds to the fact that in a distributive lattice LL every element has a complement if and only if LL is a Boolean algebra.

In particular, restricting the Stone duality equivalence between coherent spaces and distributive lattices to Stone spaces and Boolean algebras recovers the classical Stone duality.

Coherent locales

A locale is coherent if its compact elements are closed under finite meets and any element is a join of compact elements.

A morphism of coherent locales is a morphism ff of locales such that f *f^* preserves compact elements.

Assuming the axiom of choice, coherent locales are automatically spatial.

The category of coherent locales is contravariantly equivalent to the category of distributive lattices. This statement does not depend on the axiom of choice, unlike the point-set version above.

References

  • Peter Johnstone, Stone Spaces.

  • Max Dickmann?, Niels Schwartz?, Marcus Tressl?, Spectral Spaces. New Mathematical Monographs 35 (2019). Cambridge: Cambridge University Press. ISBN 9781107146723.

  • Wikipedia, Spectral space

Last revised on May 8, 2019 at 19:58:35. See the history of this page for a list of all contributions to it.