In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a spectral space is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
that is
homeomorphic to the
spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to
coherent topos.
Definition
Let ''X'' be a topological space and let ''K''
(''X'') be the set of all
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
open subsets
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a Set (mathematics), set along with a metric (mathematics), distance defined between any two points), open sets are the sets that, with every ...
of ''X''. Then ''X'' is said to be ''spectral'' if it satisfies all of the following conditions:
*''X'' is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
and
T0.
* ''K''
(''X'') is a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
of open subsets of ''X''.
* ''K''
(''X'') is
closed under
In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but n ...
finite intersections.
* ''X'' is
sober
In cryptography, SOBER is a family of stream ciphers initially designed by Greg Rose of QUALCOMM Australia starting in 1997. The name is a contrived acronym for ''S''eventeen ''O''ctet ''B''yte ''E''nabled ''R''egister. Initially the cipher wa ...
, i.e., every nonempty
irreducible
In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole.
Emergence ...
closed subset of ''X'' has a (necessarily unique)
generic point
In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point.
In classical algebraic g ...
.
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
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits c ...
of finite
T0-spaces.
#''X'' is homeomorphic to the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
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
In mathematics, a Priestley space is an ordered topological space with special properties. Priestley spaces are named after Hilary Priestley who introduced and investigated them. Priestley spaces play a fundamental role in the study of distributiv ...
.
#''X'' is a T
0 space whose
frame
A frame is often a structural system that supports other components of a physical construction and/or steel frame that limits the construction's extent.
Frame and FRAME may also refer to:
Physical objects
In building construction
*Framing (con ...
of open sets is coherent (and every coherent frame comes from a unique spectral space in this way).
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 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 T
1 if and only if it is
Hausdorff (or T
2) if and only if it is a
boolean space
In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first h ...
if and only if ''K''
(''X'') is a
boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
.
*''X'' can be seen as a
pairwise Stone space
In mathematics and particularly in topology, pairwise Stone space is a bitopological space \scriptstyle (X,\tau_1,\tau_2) which is pairwise compact, pairwise Hausdorff, and pairwise zero-dimensional.
Pairwise Stone spaces are a bitopological ...
.
[G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, A. Kurz, (2010). "Bitopological duality for distributive lattices and Heyting algebras." ''Mathematical Structures in Computer Science'', 20.]
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 morphisms of such lattices). In this anti-equivalence, a spectral space ''X'' corresponds to the lattice ''K''
(''X'').
Citations
References
*
M. Hochster (1969). Prime ideal structure in commutative rings. ''
Trans. Amer. Math. Soc.'', 142 43—60
*.
*
{{DEFAULTSORT:Spectral Space
General topology
Algebraic geometry
Lattice theory