Download Projective limits of topological vector spaces

Projective limits of topological vector spaces
Jordan Bell
[email protected]
Department of Mathematics, University of Toronto
April 3, 2014
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Definitions
Let (I, ) be a directed poset: if i ∈ I then i i; if i, j, k ∈ I, i j, and j k,
then i k: if i, j ∈ I, i j, and j i, then i = j; if i, i ∈ I, then there is some
k ∈ I such that i k and j k.1
A projective system in a category C is a set {Xi : i ∈ I} of objects in C and
a set {φij : Xi → Xj : i, j ∈ I, i j} of morphisms, such that if i ∈ I then
φii = idXi and such that if i, j, k ∈ I and k j i then
φik
Xi
φij
Xk
φjk
Xj
commutes. If Y is an object in C and for each i ∈ I, ψi : Y → Xi is a
morphism, we say that the morphisms ψi are compatible with the projective
system {Xi , φij , I} if φij ◦ ψi = ψj whenever j i, that is,
Xi
φij
ψi
Xj
ψj
Y
commutes.
Suppose that X is an object in C and the morphisms φi : X → Xi are
compatible with the projective system {Xi , φij , I}. We say that X is a projective
limit of the projective system if whenever Y is an object in C and there are
1 I am using the first chapter of L. Ribes and Z. Zalesskii, Profinite Groups, which has a
clean presentation of projective and direct limits, and also Paul Garrett’s notes Functions
on circles and Basic categorical constructions, which are on his homepage. I have not used
them, but J. L. Taylor, Notes on locally convex topological vector spaces looks readable and
comprehensive.
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morphisms ψi : Y → Xi that are compatible with the projective system, there
is one and only one morphism ψ : Y → X such that
φi ◦ ψ = ψi
for all i ∈ I. This is a definition of a projective limit by a universal property, but
it is not clear which projective systems have projective limits; but any object
and morphisms that satisfy this property will be called a projective limit.
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Direct products of topological vector spaces
A topological vector space is a vector space V over C that is a Hausdorff topological space, and such that addition V × V → V is continuous and scalar
multiplication C × V → V is continuous.
Let I be a set and let Vi be topological vector spaces, where αi : Vi ×Vi → Vi
is the addition map and µi : C × Vi → Vi is the scalar multiplication map. Let
n
o
Y
[
V =
Vi = v : I →
Vi : if i ∈ I then vi ∈ Vi ,
i∈I
i∈I
the Cartesian product of the sets Vi , and let V have the product topology. V is
a Hausdorff topological space. For each i ∈ I, define pi : V → Vi by pi (v) = vi ,
the projection from V to Vi ; pi is continuous.
Define α : V × V → V by α(v, w)i = αi (vi , wi ), and define µ : C × V → V by
µ(λ, v)i = µi (λ, vi ). V is a vector space with addition α and scalar multiplication
−1
µ. Let Ui be an open set in Vi and let U = p−1
iQ (Ui ). Let Wj = αi (Ui ) for
−1
j = i and Wj = Vj × Vj for j 6= i, and let W = j∈I Wj : W = α (U ) and W
is an open set in V × V . Since α−1 (U ) is an open set in V × V for each subbasic
open set U in V , it follows that α is continuous. Moreover, let Ui be an open
−1
−1
set in Vi and let U = pQ
i (Ui ). Let Wj = αi (Ui ) for j = i and Wj = C × Vj
−1
for j 6= i, and let W = j∈I Wj : W = µi (U ) and W is an open set in C × V .
It follows that µ is continuous. Therefore V is a topological vector space.2
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Projective limits of topological vector spaces
Let C be the category of topological vector spaces: an object of C is a topological
vector space, and a morphism V → W is a linear map that is continuous. Let
{Vi , φij , I} be a projective system in C. Let
Y
X=
Vj ,
j∈I
and let pi : X → Vi be the projection map. Define
n
o
V = v ∈ X : if i, j ∈ I and j i then φij (pi (v)) = pj (v) .
2 Paul Garrett in his notes gives a diagrammatic proof that the direct product of topological
vector spaces is a topological vector space.
2
V is a Hausdorff topological space with the subspace topology. Since both pi and
φij are linear, it follows that V is a vector subspace of X. Let α : X × X → X
be addition and µ : C × X → X be scalar multiplication, and let ι : V ,→ X be
the inclusion map. ι is continuous, so ι × ι : V × V → X × X is continuous, and
hence α ◦ (ι × ι) : V × V → V and µ ◦ (ι × ι) : V × V → V are continuous; that
their codomains are V follows from V being a vector subspace of X. Thus V is
a topological vector space.
For i ∈ I, define φi : V → Vi by φi = pi ◦ ι, which is a morphism. If j i
then φij ◦ φi = φj , which follows from the definition of V , so φi : V → Vi
are compatible with the projective system {Vi , φij , I}. Suppose that Y is a
topological vector space and ψi : Y → Vi are compatible with the projective
system {Vi , φij , I}. Define ψ : Y → V by ψ(y)i = ψi (y); that the codomain of
ψ is V follows from ψi : Y → Vi being compatible with the projective system.
Since each ψi is linear, ψ is linear. Let Ui be open in Vi . We have
−1
ψ −1 (V ∩ p−1
i (Ui )) = ψi (Ui ),
hence ψ −1 (U ) for each subbasic open set U in V , and so ψ is continuous. Let
y ∈Y.
φi (ψ(y)) = pi (ι(ψ(y))) = ψ(y)i = ψi (y),
so φi ◦ ψ = ψi for each in i ∈ I. Suppose that χ : Y → V is a morphism such
that
φi ◦ χ = ψi
for all i ∈ I. On the one hand φi (χ(y)) = χ(y)i , and on the other hand
ψi (y) = ψ(y)i , so φi ◦ χ = ψ for all i ∈ I. Therefore V is a projective limit of
the projective system {Vi , φij , I}.
In any category: Let {Xi , φij , I} be a projective system. If (X, φi ) and
(Y, ψi ) are projective limits, then there is a unique isomorphism φ : X → Y
such that ψi ◦ φ = φi for all i. This can be cleanly proved in a straightforward
way; a proof is written down in Ribes and Zalesskii for the category of topological
spaces, but it doesn’t use anything that is special to that category.3
Thus in the category of topological vector spaces, a projective limit of a
projective system exists and is unique up to unique isomorphism. We denote
the projective limit of a projective system {Vi , φij , I} of topological vector spaces
by
lim Vi .
←−
i∈I
Q
Let v ∈ i∈I Vi \ limi∈I Vi . Since v 6∈ limi∈I Vi , there are some r, s ∈ I, r s
←−
←−
such that φrs (vr ) 6= vs . As φrs (vr ) and vs are distinct points in the Hausdorff
space Vs , there are disjoint open sets U1 , U2 in Vs with φrs (vr ) ∈ U1 and vs ∈ U2 .
3 Ribes and Zalesskii’s book is about profinite groups, which are projective limits of finite
discrete topological groups; indeed there exist projective limits in the category of topological
groups. An example of an infinite profinite group is the p-adic integers Zp , for p prime, which
is the projective limit of the groups Z/pn , where the map Z/pn → Z/pm , n ≥ m, is taking
the remainder modulo pm .
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0
Let U10 = φ−1
rs (U1 ); thisQis an open set in Vr .QDefine Wr = U1 , W
Qs = U2 , and
Wi =QVi if i 6= r, s.
W
is
open
in
V
,
and
w
∈
i
i inI
i∈I i
i∈I Wi . Let
w ∈ i∈I Wi . φrs (wr ) ∈ U1 and ws ∈ U2 , and U1 and U2 are disjoint, so
φrs (wr ) 6= ws . Thus w 6∈ limi∈I Vi . Therefore, limi∈I Vi is a closed subset of
←−
←−
Q
i∈I Vi .
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Locally convex topological vector spaces
We say that a topological vector space V is locally convex if for every open
neighborhood U of 0, there is some convex open 0 ∈ U 0 ⊆ U . Let (Vj , φij , I) be
a projective system of locally convex topological vector spaces and let V be the
projective limit in the category of topological vector spaces. We shall show that
V is locally convex. Let J ⊂ I be finite and let Uj be open in Vj for each j ∈ J.
Since Vj is locally convex, for each j ∈ J there is some convex open Uj0 ⊆ Uj
such thatQ0 ∈ Uj0 . Define Wj = Uj0 for j ∈ J and Wi = Vi for
Q i ∈ I \ J, and
let W = i∈I Wi . W is open and convex. This shows that i∈I Vi is locally
convex. It is apparent that a vector subspace of a locally convex topological
vector space is itself a locally convex topological vector space with the product
topology. Thus V is locally convex. It follows that a projective system in the
category of locally convex topological vector spaces has a projective limit that
is unique up to unique isomorphism. (A morphism of locally convex topological
vector spaces is the same as a morphism of topological vector spaces.)
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Cofinality
Let (I, ) be a directed poset, and suppose that J ⊆ I is also directed with .4
We say that J is cofinal in I if for all i ∈ I there is some j ∈ J such that i j.
Let C be a category in which there exists a projective limit of any projective
system; we have shown above that the category of topological vector spaces
and the category of locally convex topological vector spaces are such categories.
This projective limit will be unique up to unique isomorphism.
Let (Xi , φij , I) be a projective system and let J be cofinal in I. Then
(Xi , φij , J) is a projective system. Let (XI , φi , I) be a projective limit of the
first system and let (XJ , φ0i , J) be a projective limit of the second system. Let
i ∈ I, take j ∈ J with j i, and define ψi : XJ → Xi by ψi = φji ◦ φ0j ; if
j, j 0 i then φj 0 i ◦ φ0j 0 = φji ◦ φ0j . Let i0 , i ∈ I with i0 i, and let j ∈ J with
i j. Because (Xi , φij , I) is a projective system, φii0 ◦ φji = φji0 , and hence
φii0 ◦ ψi = φii0 ◦ φji ◦ φ0j = φji0 ◦ φ0j = ψi0 ,
showing that the morphisms ψi : XJ → Xi are compatible with the projective
system (Xi , φij , I). As (XI , φi , I) is a projective limit of (Xi , φij , I), it follows
that there is a unique morphism ψ : XJ → XI such that φi ◦ ψ = ψi for all
i ∈ I.
4 This
section follows Paul Garrett’s notes Basic categorical constructions.
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Since φi : XI → Xi , i ∈ I, are compatible with the projective system
(Xi , φij , I), it follows that φj : XI → Xj , j ∈ J, are compatible with (Xi , φij , J).
As (XJ , φ0i , J) is a projective limit of (Xi , φij , I), there is a unique morphism
φ : XI → XJ such that φ0j ◦ φ = φj for all j ∈ J.
On the one hand, φ ◦ ψ : XJ → XJ is a morphism such that φ0j ◦ (φ ◦ ψ) = φ0j
for all j ∈ J. On the other hand, idXJ : XJ → XJ is a morphism such that
φ0j ◦ idXJ = φ0j for all j ∈ J. By the definition of projective limit, we get
φ ◦ ψ = idXJ . Likewise, ψ ◦ φ = idXI . Thus ψ : XJ → XI and φ : XI → XJ
are isomorphisms. Since XJ and XI are isomorphic, (XJ , ψi , I) is a projective
limit of (Xi , φij , I). This is similar to how the limit of a convergent sequence is
the same as the limit of any infinite subsequence.
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