Download Introduction to spectral spaces

Introduction to spectral spaces
Marcus Tressl
University of Manchester
MODNET Workshop July 14-18, 2008
1. Definition of spectral spaces
A spectral space is a topological space X satisfying:
S1: X is quasi-compact and T0 . compact
S2: The
set
◦
K(X ) := {O ⊆ X |O is open and quasi-compact} is
◦
an open basis for X . voidK(X ) =clopens are a basis
◦
S3: K(X ) is closed under finite intersections. voidtrivial
S4: Every nonempty, closed and irreducible subset of X
has a generic point. trivialtrivial
A subset S of X is called irreducible if for all closed sets B, C
with S ⊆ B ∪ C we have S ⊆ B or S ⊆ C .
A generic point of S is a point x ∈ S with S = {x}.
Initial examples:
X finite and T0
X Hausdorff boolean
1. Definition
We write x
y for y ∈ {x} and say y is a specialisation of x
Warning: In general, X is not Hausdorff and singletons are not
closed.
Theorem (Reformulations)
The following are equivalent for every topological space X .
1
X is spectral.
2
X is profinite T0 , i.e. X = lim Fi with finite T0 -spaces Fi .
3
X is homeomorphic to the Zariski-spectrum of some
commutative ring (Hochster).
4
X is homeomorphic to some space of partial types (in the
sense of model theory).
←
2. Stone duality
Definition (distributive lattice)
A lattice L is a poset L = (L, ≤) such that for all a, b ∈ L the
supremum x ∨ y and the infimum x ∧ y exists in L. We shall
always assume that L is bounded (i.e. L has a smallest element 0
and a largest element 1) and that L is distributive (i.e.
a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)).
Definition (prime filter)
A filter F of L is a nonempty subset of L satisfying
a, b ∈ F ⇒ a ∧ b ∈ and c ≥ a ∈ F ⇒ c ∈ F .
A filter F is called prime if F 6= L and
a ∨ b ∈ F ⇒ a ∈ F or b ∈ F .
2. Stone duality
Let
Spec L = {P ⊆ L|P prime filter}.
For a ∈ L let
VL (a) := {P ∈ Spec L | a ∈ P}.
Theorem (Stone duality)
∪
For a, b ∈ L we have VL (a ∨
∧ b) = VL (a) ∩ VL (b), and the map
VL : L −→ {VL (a)|a ∈ L}
is an isomorphism of lattices.
The proof uses mainly the following
Proposition (Krull)
If F is a filter of L and a ∈ L, a 6∈ F then there is a prime filter P
of L containing F with a ∈
6 P.
3. Spectral topology on Spec L
By Stone-duality, the sets VL (a) form a basis of closed sets of a
topology on Spec L and this basis is closed under finite
intersections.
Theorem (3.1)
Spec L is a spectral space and a subset S of Spec L is of the form
VL (a) if and only if X \ S is open and quasi-compact. In other
words
◦
K(Spec L) = {Spec L \ VL (a)|a ∈ L}.
3. Spectral topology on Spec L
Conversely, if X is a spectral space, let
◦
K(X ) := {X \ O | O ∈ K(X )}.
Theorem (Inverse Stone)
K(X ) is a bounded lattice of subsets of Spec L and the map
X
−→ Spec K(X )
x
7−→ {V ∈ K(X ) | x ∈ V }
is an homeomorphism.
4. The category of spectral spaces
A spectral map f : X −→ X 0 between spectral spaces is a
continuous map which satisfies
O 0 ⊆ X 0 open and quasi-compact ⇒ f −1 (O 0 ) ⊆ X quasi-compact.
In other words, taking preimages induces a map K(X 0 ) −→ K(X ).
I Every homomorphism ϕ : L0 −→ L of bounded distributive
lattices induces a spectral map Spec ϕ : Spec L −→ Spec L0 via
P 7→ ϕ−1 (P).
Summary
Spec is an anti-equivalence between the category of bounded
distributive lattices and the category of spectral spaces.
The quasi-inverse is given by X 7→ K(X )
If “lattices” is replaced by “rings”, this statement becomes very
false.
5. A real example
Let L be the lattice of all finite unions of finite intersections of sets
of the form {f ≥ 0}, where f ∈ R[x], x = (x1 , ..., xn ). We have a
map
τ : Sn (R) −→ Spec L; τ (p) = {{f ≥ 0}|f (x) ≥ 0 ∈ p}.
By quantifier elimination, τ is bijective.
Sn (R)
τ
cGG
GG
GG
G
Rn
/ Spec L
w;
ww
w
w
ww r 7→{C |r ∈C }
The map Rn −→ Spec L is an homeomorphism onto its image!
I Moreover Spec L is connected - even if R is replaced by an
arbitrary real closed field.
5. A real example
Exercise∗ : Suppose C ⊆ Rn is closed and definable. Is C ∈ L ?
We approach the question as follows: Let L0 be the lattice of all
closed and definable subsets of Rn . So L ⊆ L0 . We want to show
L = L0 .
Sn (R)
O
/ Spec L0
O
/ Spec L
O
?
?
/ VL0 (C )
/ D?
hC i
It remains to show that the image D of VL0 (C ) under
Spec L0 −→ Spec L is closed. This in turn translates into a
very local problem:
If P 0 , Q 0 ∈ Spec L0 and (P 0 ∩ L)
(Q 0 ∩ L), show that
0
0
P
Q . This can be solved using valuation theory.
6. Spectral spaces attached to structures and theories
I Let L0 be a first order language, let L be an extension of L0 by
relation symbols and let E be a another binary relation symbol.
I Let T0 be an L0 -theory and let T be an L-theory.
I Let A |= T0 .
I For every L0 -homomorphism f : A −→ M |= T , let A(f ) be the
natural expansion of A to an L(E )-structure obtained from pulling
back the structure of M, i.e.
E A(f ) = f −1 (diagonalM ) ⊆ A2 and R A(f ) = f −1 (R M ) ⊆ Aarity of R .
Define
T − Spec A := {A(f ) | f : A −→ M |= T }.
T − Spec A is a spectral space. Basic closed sets: pick a positive
quantifier free sentence δ(a) in the language L(E ) with parameters
a ⊆ A and take V (δ(a)) = {A(f )|A(f ) |= δ(a)}.
6. Spectral spaces attached to structures and theories
More formally, T - Spec A can be seen as (is homeomorphic to) the
spectrum of the lattice
qf+ - Sen[L(E )(a|a ∈ A)] modulo T ∪ qf+ - diag(A)
Theorem
There is an L(E )-theory, denoted by T - Spec T0 , such that for
every A |= T0 we have
T - Spec A := {A(f ) | A(f ) |= T - Spec T0 }.
Moreover T - Spec T0 is universal with respect to T0 , i.e.
T0 ∪ (T - Spec T0 )∀ ` T - Spec T0 .
References
R. Berr, Spectral Spaces and First Order Theories.
Available in the DDG, 1999
M. Hochster, Prime ideal structure in commutative rings.
Trans. Amer. Math. Soc. 142 (1969) 43-60
G. Grätzer, General lattice theory. Birkhäuser, Basel, 1998.
P.T. Johnstone, Stone Spaces. Cambridge Studies in
Advanced Mathematics, 3. Cambridge University Press,
Cambridge, 1986.
R. O. Robson, Model theory and spectra. J. Pure Appl.
Algebra 63 (1990), no. 3, 301–327
S. Vickers, Topology via Logic. Cambridge Tracts in
Theoretical Computer Science, 5. Cambridge University Press,
Cambridge, 1989.