Spectral space

{{Short description|Space homeomorphic to some ring spectrum}} 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''^{<math>\circ</math>}(''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''^{<math>\circ</math>}(''X'') is a basis of open subsets of ''X''. * ''K''^{<math>\circ</math>}(''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:

#''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''^{<math>\circ</math>}(''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 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''^{<math>\circ</math>}(''X'') be as before. Then: *''K''^{<math>\circ</math>}(''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''^{<math>\circ</math>}(''X'')) is again spectral. *''X'' is T₀ by definition, but in general not T₁.<ref>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.)''</ref> 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''^{<math>\circ</math>}(''X'') is a boolean algebra. *''X'' can be seen as a pairwise Stone space.<ref>G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, A. Kurz, (2010). "Bitopological duality for distributive lattices and Heyting algebras." ''Mathematical Structures in Computer Science'', 20.</ref>

==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 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).{{sfn|Johnstone|1982}} In this anti-equivalence, a spectral space ''X'' corresponds to the lattice ''K''^{<math>\circ</math>}(''X'').

==References== {{reflist}}

==Further reading== {{refbegin}} *M. Hochster (1969). Prime ideal structure in commutative rings. ''Trans. Amer. Math. Soc.'', 142 43—60 *{{citation | last = Johnstone | first = Peter | author-link = Peter Johnstone (mathematician) | isbn = 978-0-521-33779-3 | publisher = Cambridge University Press | title = Stone Spaces | contribution = II.3 Coherent locales | pages = 62–69 | year = 1982}}. * {{cite book | last1=Dickmann | first1=Max | last2=Schwartz | first2= Niels | last3=Tressl | first3= Marcus | title=Spectral Spaces| doi=10.1017/9781316543870 | year=2019 | publisher=Cambridge University Press | series=New Mathematical Monographs | volume=35 | location=Cambridge | isbn=9781107146723 }} {{refend}}

{{DEFAULTSORT:Spectral Space}} Category:General topology Category:Algebraic geometry Category:Lattice theory