aa r X i v : . [ m a t h . A C ] A ug Topological rigidity as a monoidal equivalence
Laurent Poinsot ∗ LIPN - UMR CNRS 7030University Paris 13, Sorbonne Paris Cité93140 Villetaneuse, [email protected]
Abstract
A topological commutative ring is said to be rigid when for everyset X , the topological dual of the X -fold topological product of thering is isomorphic to the free module over X . Examples are fields witha ring topology, discrete rings, and normed algebras. Rigidity trans-lates into a dual equivalence between categories of free modules andof “topologically-free” modules and, with a suitable topological tensorproduct for the latter, one proves that it lifts to an equivalence be-tween monoids in this category (some suitably generalized topologicalalgebras) and coalgebras. In particular, we provide a description ofits relationship with the standard duality between algebras and coal-gebras, namely finite duality. Keywords:
Topological dual space, topological basis, coalgebras, fi-nite duality.
MSC classification:
The main result of [14] states that given a (Hausdorff ) topological field ( k , τ ) , for every set X , the topological dual (( k , τ ) X ) ′ of the X -fold topo-logical product ( k , τ ) X is isomorphic to the vector space k ( X ) of finitely-supported k -valued maps defined on X (i.e., those maps X f Ð→ k such thatfor all but finitely many members x of X , f ( x ) = ). ∗ Second address: CREA, French Air Force Academy, Base aérienne 701, 13661 Salon-de-Provence, France. All topologies will be assumed separated. rigidity is shared by more generaltopological (commutative unital) rings than only topological fields (a factnot noticed in [14]). For instance any discrete ring is rigid in the abovesense (see Lemma 18). And even if not all topological rings are rigid (seeSection 4.3 for a counter-example), many of them still are (e.g., every realor complex normed commutative algebra).It is our intention to study in more details some consequences of theproperty of rigidity for arbitrary commutative rings in particular for some oftheir topological algebras . So far, for a topological ring ( R , τ ) , rigidity readsas (( R, τ ) X ) ′ ≃ R ( X ) (here, and everywhere else, R stands for the canonicalleft R -module structure on the underlying abelian group of R ) for each set X . Suitably topologized (see Section 3.1), the algebraic dual ( R ( X ) ) ∗ turnsout to be isomorphic to ( R, τ ) X .More appropriately the above correspondence may be upgraded into adual equivalence of categories between free and topologically-free modules,i.e., those topological modules isomorphic to some ( R, τ ) X (Theorem 48)under the algebraic and topological dual functors. (This extends a similarinterpretation from [14] to the realm of arbitrary commutative rigid rings.)Under the rigidity assumption, the aforementioned dual equivalence en-ables to provide a topological tensor product ⍟ ( R ,τ ) for topologically-free ( R , τ ) -modules by transporting the algebraic tensor product ⊗ R along thedual equivalence. It turns out that ⍟ ( R ,τ ) is (coherently) associative, com-mutative and unital, i.e., makes monoidal the category of topologically-freemodules (Proposition 60). Not too surprisingly the above dual equivalenceremains well-behaved, i.e., monoidal, with respect to the (algebraic and topo-logical) tensor products (Theorem 63). According to the theory of monoidalcategories, this in turn provides a dual equivalence between monoids inthe tensor category of topologically-free modules (some suitably generalizedtopological algebras) and coalgebras (Corollary 65). So there are two con-structions: a topological dual coalgebra of a monoid (in the tensor categoryof topologically-free modules) and an algebraic dual monoid of a coalgebra,and these constructions are inverse one from the other (up to isomorphism).There already exists a standard duality theory between algebras and coal-gebras, over a field, known as finite duality but contrary to our “topologicalduality” it is merely an adjunction, not an equivalence. One discusses howthese dualities interact (see Section 7) and in particular one proves that the In this contribution, every ring is assumed commutative and unital (see Section 2.1). The results of the present contribution also serve in a subsequent paper under prepa-ration about topological semisimplicity of commutative topological algebras. A dual equivalence is an equivalence between a category and the opposite of another. Except as otherwise stipulated, all topologies are Hausdorff, and every ringis assumed unital and commutative .For a ring R , R denotes both its underlying set and the canonical left R -module structure on its underlying additive group. Likewise if A is an R -algebra, then A is both its underlying set and its underlying R -module.The unit of a ring R (resp., unital algebra A ) is either denoted by R (resp. A ). A ring map (or morphism of rings) is assumed to preserve the units.A product of topological spaces always has the product topology. Whenfor each x ∈ X , all ( E x , τ x ) ’s are equal to the same topological space ( E, τ ) ,then the X -fold topological product ∏ x ∈ X ( E x , τ x ) is canonically identifiedwith the set E X of all maps from X to E equipped with the topology ofsimple convergence, and is denoted by ( E, τ ) X . Under this identification,the canonical projections ( E, τ ) X π x Ð→ (
E, τ ) are given by π x ( f ) = f ( x ) , x ∈ X , f ∈ E X . Let R be a ring. A (Hausdorff, following our conventions)topology τ of (the carrier set of ) the ring is called a ring topology whenaddition, multiplication and additive inversion of the ring are continuous.By topological ring ( R , τ ) is meant a ring together with a ring topology τ onit . By a field with a ring topology , denoted ( k , τ ) , is meant a topologicalring ( k , τ ) with k a field. A ring R with the discrete topology d is a topological ring. To the contrary an algebra over a ring won’t be assumed commutative, but associativeand unital. In view of Section 2.1, the multiplication of a topological ring is jointly continuous. ( R , τ ) be a topological ring. A pair ( M, σ ) consisting of a (leftand unital ) R -module M and a topology σ on M which makes continuousthe addition, opposite and scalar multiplication R × M → M , is called a topological ( R , τ ) -module . Such a topology is referred to as a ( R , τ ) -moduletopology . In particular, when R is a field k , then this provides topologi-cal ( k , τ ) -vector spaces . Given topological ( R , τ ) -modules ( M, σ ) , ( N, γ ) , a continuous ( R , τ ) -linear map ( M, σ ) f Ð→ (
N, γ ) is a R -linear map M f Ð→ N which is continuous. Topological ( R , τ ) -modules and these morphisms forma category TopMod ( R ,τ ) , which is denoted TopVect ( k ,τ ) , when k is a field.A pair ( A , σ ) , with A a unital R -algebra, and σ a topology on A , is a topological ( R , τ ) -algebra , when σ is a module topology for the underlying R -module A , and the multiplication of A is a bilinear (jointly) continuous map.Given topological ( R , τ ) -algebras ( A , σ ) , ( B , γ ) , a continuous ( R , τ ) -algebramap ( A , σ ) f Ð→ ( B , γ ) is a unit-preserving R -algebra map A f Ð→ B which isalso continuous. Topological ( R , τ ) -algebras with these morphisms form acategory TopAlg ( R ,τ ) . One also has the full subcategory ,c TopAlg ( R ,τ ) of unital and commutative topological algebras. ( R, τ ) is a topological ( R , τ ) -module and the topological ring ( R , τ ) with the previous module structure, is a topological ( R , τ ) -algebra. A R -module (resp. R -algebra) with the discrete topology d is atopological ( R , d ) -module (resp. ( R , d ) -algebra). X -fold product and finitely-supported maps The opposite category C op of C has the same objects and morphisms as C butwith opposite composition. In other words one has C op ( D, C ) = C ( C, D ) . f op denotes the C -morphism f considered as a C op -morphism. Let C F Ð→ D be a functor. Then, C op F op ÐÐ→ D op with for a C -morphism f , F op ( f op ) = Unital means that the scalar action of the unit of R is the identity on the module. One does not worry about size issues and in this presentation “category” means alocally small category while “set” loosely means both small and large set. One assumesthat the reader is familiar with basic notions from category theory among subcategories,(full, faithful) functors, natural transformations, natural isomorphisms, equivalence ofcategories, categorical product, terminal object, left/right adjoints, unit and counit ofan adjunction. However some of them will be recalled in the text, essentially throughfootnotes. Of course [13] is a fundamental reference for this subject. By a full subcategory is meant a subcategory D of C such that D ( C, D ) = C ( C, D ) foreach D -objects C, D , where as usually, C ( C, D ) denotes the hom-set of all C -morphismswith domain C and codomain D . ( f ) op , is called the opposite of F . Each natural transformation α ∶ F ⇒ G ∶ C → D has an opposite natural transformation α op ∶ G op ⇒ F op ∶ C op → D op with ( α op ) C = α op C for each C -object C .Let R be a ring, and let X be a set. The R -module R X of all maps from X to R , equivalently defined as the X -fold power of R in the category Mod R ,is the object component of a functor Set op P R Ð→ Mod R whose action onmaps is as follows: given X f Ð→ Y and g ∈ R Y , P R ( f )( g ) = g ○ f . R X is merelynot just a R -module but, under point-wise multiplication R X × R X M X ÐÐ→ R X ,a commutative R -algebra, the usual function algebra on X , denoted A R ( X ) ,with unit A R ( X ) ∶ = ∑ x ∈ X δ R x , where δ R x , or simply δ x , is the member of R X with δ R x ( x ) = R , x ∈ R , and for y ∈ X , y / = x , δ R x ( y ) = . This actuallyprovides a functor Set op A R Ð→ ,c Alg R .Let f ∈ R X . The support of f is the set supp ( f ) ∶ = { x ∈ X ∶ f ( x ) / = } .Let R ( X ) be the sub- R -module of R X consisting of all finitely-supported maps (or maps with finite support ), i.e., the maps f such that supp ( f ) is finite. R ( X ) is actually the free R -module over X , and a basis is given by { δ R x ∶ x ∈ X } . Observe that the map X δ R X Ð→ R X , x ↦ δ x , is one-to-one if, and only if, R is non-trivial. Of course one has the free module functor
Set F R Ð→ Mod R whichis a left adjoint of the usual forgetful functor Mod R ∣⋅∣ Ð→ Set ; F R ( X ) ∶ = R ( X ) ,and for X f Ð→ Y , p ∈ R ( X ) , F R ( f )( p ) ∶ = ∑ y ∈ Y (∑ x ∈ f − ({ y }) p ( x )) δ R y , i.e., F R ( f )( δ R x ) = δ R f ( x ) , x ∈ X . The map X δ R X Ð→ ∣ R ( X ) ∣ is the component at X ofthe unit of the adjunction F R ⊣ ∣ ⋅ ∣ ∶ Set → Mod R . Let ( R , τ ) be a topological ring and let X be a set. Since for a map The notation α ∶ F ⇒ G ∶ C → D means that α is a natural transformation between twofunctors C F,G
ÐÐ→ D . Given a C -object C , α C ∈ D ( F ( C ) , G ( C )) denotes the componentat C of α . Thus α = ( α C ) C . For each functor C F Ð→ D , there is an identity at F , id F ∶ F ⇒ F ∶ C → D with ( id F ) C ∶= id F ( C ) , also denoted simply id . Mod R is the category of unital left- R -modules with R -linear maps. When R is a field k one uses Vect k instead. Set is the category of sets with set-theoretic maps. Alg R the category of (associative) unital R -algebras with unit-preserving algebramaps. (The multiplication m A of an algebra A thus is a R -bilinear map A × A m A ÐÐ→ A .) ,c Alg R is the category of unital and commutative algebras. In this presentation, by F ⊣ G ∶ C → D is meant an adjunction with left adjoint C F Ð→ D and right adjoint D G Ð→ C . f Ð→ Y , π x ○ P ( R ,τ ) ( f ) = π f ( x ) , x ∈ X , ( R, τ ) Y P R ( f ) ÐÐÐ→ (
R, τ ) X is continuous,and thus one has a topological power functor Set op P ( R ,τ ) ÐÐÐ→
TopMod ( R ,τ ) . ( R, τ ) X × ( R, τ ) X M X ÐÐ→ (
R, τ ) X is continuous. Proof: M X is separately continuous because for each x ∈ X , π x ○ M X = m R ○ ( π x × π x ) . Let A ⊆ X be finite, and for each x ∈ A , let U x bean open neighborhood zero in ( R, τ ) . Continuity of m R at zero impliesthe existence of neighborhoods V x , W x of zero such that m R ( V x , W x ) ⊆ U x . M X (⋂ x ∈ A π − x ( V x ) , ⋂ x ∈ A π − x ( W x )) ⊆ ⋂ x ∈ A π − x ( U x ) ensures continuity at zeroof M X , and thus continuity by [19, Theorem 2.14, p. 16]. ◻ A ( R ,τ ) ∶ X ↦ (( R, τ ) X , M X , A R ( X ) ) is a functor too and the diagram be-low commutes, with the forgetful functors unnamed. ,c TopAlg ( R ,τ ) / / (cid:15) (cid:15) ,c Alg R (cid:15) (cid:15) Set op A ( R ,τ ) j j ❯❯❯❯❯❯❯ P ( R ,τ ) t t ✐✐✐✐✐✐✐✐ A R ❧❧❧❧❧❧ P R ) ) ❘❘❘❘❘❘❘ TopMod ( R ,τ ) / / Mod R (1) The underlying topological ring of A ( R ,τ ) ( X ) is denoted ( R , τ ) X (of course, it is the X -fold product of ( R , τ ) in the category of rings). Let R be a ring. Let M be a R -module. Let M ∗ ∶ = Mod R ( M, R ) be the algebraic (or linear ) dual of M . This is readily a R -module on its own.When ( R , τ ) is a topological ring, then M ∗ may be topologized withthe initial topology ([6, 19]) w ∗( R ,τ ) , called the weak- ∗ topology , inducedby the family ( M ∗ Λ M ( v ) ÐÐÐÐ→ R ) v ∈ M of evaluations at some points, where ( Λ M ( v ))( ℓ ) ∶ = ℓ ( v ) . This provides a structure of topological ( R , τ ) -moduleon M ∗ , which is even Hausdorff (since if ℓ ( v ) = for all v ∈ M , then ℓ = ). The initial topology on X induced by a family F of maps all with domain X , isHausdorff when F separates the points of X , i.e., when for each x /= y in X , there is a map f ∈ F such that f ( x ) /= f ( y ) . M f Ð→ N , N ∗ f ∗ Ð→ M ∗ , ℓ ↦ f ∗ ( ℓ ) ∶ = ℓ ○ f , iscontinuous for the above topologies. Consequently, this provides a functor Mod opR
Alg ( R ,τ ) ÐÐÐÐ→
TopMod ( R ,τ ) called the algebraic dual functor . M ∗ or f ∗ stand for Alg ( R ,τ ) ( M ) or Alg ( R ,τ ) ( f ) . Up to isomorphism, one recovers the module of all R -valued maps on aset X , with its product topology, as the algebraic dual of the module of allfinitely-supported maps duly topologized as above. For each set X , ( R, τ ) X ≃ Alg ( R ,τ ) ( R ( X ) ) (in TopMod ( R ,τ ) )under the map ρ X ∶ ( R, τ ) X → (( R ( X ) ) ∗ , w ∗( R ,τ ) ) given by ( ρ X ( f )( p )) ∶ = ∑ x ∈ X p ( x ) f ( x ) , f ∈ R X , p ∈ R ( X ) . Proof:
Let ℓ ∈ ( R ( X ) ) ∗ . Let us define X ˆ ℓ Ð→ R by ˆ ℓ ( x ) ∶ = ℓ ( δ x ) , x ∈ X .That the two constructions are linear and inverse one from the other is clear.It remains to make sure that there are also continuous. Let ℓ ∈ ( R ( X ) ) ∗ ,and let x ∈ X . Then, π x ( ˆ ℓ ) = ˆ ℓ ( x ) = ℓ ( δ x ) = ( Λ R ( X ) ( δ x ))( ℓ ) , which ensurescontinuity of (( R ( X ) ) ∗ , w ∗( R ,τ ) ) ρ − X ÐÐ→ (
R, τ ) X . Let f ∈ R X , and p ∈ R ( X ) .As ( Λ R ( X ) ( p ))( ρ X ( f )) = ( ρ X ( f ))( p ) = ∑ x ∈ X p ( x ) f ( x ) = ∑ x ∈ X π x ( p ) f ( x ) = ∑ x ∈ X π x ( p ) π x ( f ) , Λ R ( X ) ( p ) ○ ρ X is a finite linear combination of projections,whence is continuous for the product topology, so is ρ X . ◻ Let M be a free R -module. Let B be a basis of M . This defines a familyof R -linear maps, the coefficient maps ( M b ∗ Ð→ R ) b ∈ B such that each v ∈ M is uniquely represented as a finite linear combination v = ∑ b ∈ B b ∗ ( v ) b . Onedenotes FreeMod R the full subcategory of Mod R spanned by the freemodules. When k is a field, FreeMod k is just Vect k itself.
10 Example
For each set X , p x = δ ∗ x , where p x ∶ = R ( X ) ↪ R X π x Ð→ R , x ∈ X ,with R ( X ) ↪ R X the canonical inclusion.
11 Remark b ∗ ( d ) = δ b ( d ) , b, d ∈ B . So B (−) ∗ ÐÐ→ B ∗ ∶ = { b ∗ ∶ b ∈ B } is abijection. Each (even large) subset X of the objects of some category C determines uniquely a fullsubcategory of C called the full subcategory of C spanned by X , namely the subcategory C X whose set of objects is X and C X ( C, D ) = C ( C, D ) , C, D ∈ X . R -module, any choice of a basis B provides the initial topol-ogy Π τB on M ∗ induced by ( Λ M ( b )) b ∈ B . (Of course, Π τB ⊆ w ∗( R ,τ ) .)
12 Lemma
Let M be a free R -module. The topology Π τB is independent ofthe choice of the basis B of M since it is equal to w ∗( R ,τ ) . Moreover, for eachbasis B of M , ( M ∗ , w ∗( R ,τ ) ) ≃ ( R, τ ) B (in TopMod ( R ,τ ) ). Proof:
For ℓ ∈ M ∗ , ( Λ M ( v ))( ℓ ) = ∑ b ∈ B b ∗ ( v ) ℓ ( b ) = ∑ b ∈ B b ∗ ( v )( Λ M ( b ))( ℓ ) , v ∈ M , thus Λ M ( v ) is a finite linear combination of some Λ M ( b ) ’s, whenceis continuous for Π τB , and so w ∗( R ,τ ) ⊆ Π τB . The last assertion is clear. ◻ Let ( R , τ ) be a topological ring, and let ( M, σ ) be a topological ( R , τ ) -module. Let ( M, σ ) ′ ∶ = TopMod ( R ,τ ) (( M, σ ) , ( R, τ )) be the topological dual of ( M, σ ) , which is a R -submodule of M ∗ . Let ( M, σ ) f Ð→ (
N, γ ) be a contin-uous ( R , τ ) -linear map between topological modules. Let ( N, γ ) ′ f ′ Ð→ (
M, σ ) ′ be the R -linear map given by f ′ ( ℓ ) ∶ = ℓ ○ f . All of this evidently forms afunctor TopMod op ( R ,τ ) T op ( R ,τ ) ÐÐÐÐ→
Mod R .Let ( R , τ ) be a topological ring, and let X be a set. Let R ( X ) λ X ÐÐ→ ( R X ) ∗ be given by ( λ X ( p ))( f ) ∶ = ∑ x ∈ X p ( x ) f ( x ) , p ∈ R ( X ) , f ∈ R X .Let ρ X be the map from Lemma 9. Then, for each p ∈ R ( X ) , λ X ( p ) = Λ R ( X ) ( p ) ○ ρ X , which ensures continuity of λ X ( p ) , i.e., λ X ( p ) ∈ (( R, τ ) X ) ′ .Next lemma follows from the equality p ( x ) = ( λ X ( p ))( δ x ) , p ∈ R ( X ) , x ∈ X .
13 Lemma R ( X ) λ X ÐÐ→ ((
R, τ ) X ) ′ is one-to-one. The notion of rigidity , recalled at the beginning of the Introduction, wasoriginally but only implicitly introduced in [14, Theorem 5, p. 156] as themain result therein and the possibility that its conclusion could remain validfor more general topological rings than topological division rings was notnoticed. Since a large part of this presentation is given for arbitrary rigidrings (Definition 15 below), one here provides a stock of basic examples.As [14, Lemma 13, p. 158], one has the following fundamental lemma.8
Let ( R , τ ) be a topological ring, and let X be a set. For each f ∈ R X , ( f ( x ) δ x ) x ∈ X is summable in ( R, τ ) X with sum f .
15 Definition
Let ( R , τ ) be a topological ring. It is said to be rigid whenfor each set X , R ( X ) λ X ÐÐ→ ((
R, τ ) X ) ′ is an isomorphism in Mod R , i.e., λ X is onto. In this situation, one sometimes also called rigid a ring topology τ such that ( R , τ ) is rigid.
16 Lemma
Let ℓ ∈ (( R, τ ) X ) ′ . ℓ ∈ im ( λ X ) if, and only if, ˆ ℓ ∶ X → R givenby ˆ ℓ ( x ) ∶ = ℓ ( δ x ) , belongs to R ( X ) . Moreover, im ( λ X ) ˆ (−) ÐÐ→ R ( X ) , ℓ ↦ ˆ ℓ = ∑ x ∈ X ℓ ( δ x ) δ x , is the inverse of λ X . Of course, the trivial ring is rigid (under the (in)discrete topology!).The first assertion of the following result is a slight generalization of themain theorem in [14], which is precisely the second assertion below, sincethe proof of [14, Theorem 5, p. 156] does not use continuity of the inversion.
17 Lemma
Let ( k , τ ) be a field with a ring topology (see Definition 1).Then, ( k , τ ) is rigid. In particular, any topological field is rigid.
18 Lemma
For each ring R , the discretely topologized ring ( R , d ) is rigid. Proof:
Let ℓ ∈ (( R, d ) X ) ′ . As a consequence of Lemma 14, ( ℓ ( δ x )) x issummable in ( R, d ) , with sum ℓ ( A R ( X ) ) . Since { } is an open neighborhoodof zero in ( R, d ) , ℓ ( δ x ) = for all but finitely many x ∈ X ([19, Theorem 10.5,p. 73]). The conclusion follows by Lemma 16. ◻ Every normed, complex or real, commutative and unital algebra (e.g.,Banach or C ∗ -algebra) is rigid.
19 Lemma
Let k = R , C . Let ( A , ∥ ⋅ ∥) be a commutative normed k -algebra with a unit. Then, as a topological ring under the topology induced by thenorm, it is rigid. A topological field is a field with a ring topology ( k , τ ) such that the inversion α ↦ α − is continuous from k ∖ { } to itself with the subspace topology. In a normed algebra ( A , ∥ ⋅ ∥) , unital or not, commutative or not, the norm is assumed sub-multiplicative , i.e., ∥ xy ∥ ≤ ∥ x ∥∥ y ∥ , which ensures that the multiplication of A is jointlycontinuous with respect to the topology induced by the norm. roof: Let τ ∥−∥ be the topology on A induced by the norm of A , where A is the underlying k -vector space of A . Let X be a set. Let ℓ ∈ (( A, τ ∥−∥ ) X ) ′ .Let f ∈ A X be given by f ( x ) = ∥ ℓ ( δ x )∥ A if x ∈ supp ( ˆ ℓ ) and f ( x ) = for x / ∈ supp ( ˆ ℓ ) . Since by Lemma 14, ( f ( x ) δ x ) x ∈ X is summable with sum f , ( f ( x ) ℓ ( δ x )) x ∈ X is summable in ( A, τ ∥−∥ ) with sum ℓ ( f ) . So according to [19,Theorem 10.5, p. 73], for > ǫ > , there exists a finite set F ǫ ⊆ X such that ∥ f ( x ) ℓ ( δ x )∥ < ǫ for all x ∈ X ∖ F ǫ . But = ∥ f ( x ) ℓ ( δ x )∥ for all x ∈ supp ( ˆ ℓ ) sothat supp ( ˆ ℓ ) is finite, and λ X is onto by Lemma 16. ◻ A ring is said to be von Neumann regular if for each x ∈ R , there exists y ∈ R such that x = xyx [10, Theorem 4.23, p. 65].Let us assume that R is a (commutative) von Neumann regular ring. Foreach x ∈ R , there is a unique x † ∈ R , called the weak inverse of x , such that x = xx † x and x † = x † xx † .
20 Example
A field is a von Neumann regular with x † ∶ = x − , x / = , and † = . More generally, let ( k i ) i ∈ I be a family of fields. Let R be a ring, andlet ∶ R ↪ ∏ i ∈ I k i be a one-to-one ring map. Assume that for each x ∈ R , ( x ) † ∈ im ( ) , where for ( x i ) i ∈ I ∈ ∏ i ∈ I k i , ( x i ) † i ∈ I ∶ = ( x † i ) i ∈ I . Then, R is vonNeumann regular.
21 Remark
Let R be a von Neumann regular ring. For each x ∈ R , x / = if,and only if, xx † / = . Moreover, xx † belongs to the set E ( R ) of all idempotents( e = e ) of R .
22 Proposition
Let ( R , τ ) be a topological ring such that R is von Neumannregular. If / ∈ E ( R ) ∖ { } , then ( R , τ ) is rigid. In particular, if E ( R ) isfinite, then ( R , τ ) is rigid. Proof:
That the second assertion follows from the first is immediate. Let X be a set. Let us assume that / ∈ E ( R ) ∖ { } . Let V ∈ V ( R,τ ) ( ) suchthat V ∩ ( E ( R ) ∖ { }) = ∅ . Let ℓ ∈ (( R, τ ) X ) ′ . Let f ∈ R X be given by Assumed commutative and unital as in Section 2.1. Given y ∈ R with x = xyx , then z ∶ = yxy meets the requirements to be a “weak inverse”of x , and if y, z are two candidates, then one has z = z x = z x y = ( x z ) zy = xzy =( x y ) zy = ( x z ) y = xy = y . Given a topological space ( E, τ ) and x ∈ E , V ( E,τ ) ( x ) is the set of all neighborhoodsof x . ( x ) ∶ = ℓ ( δ x ) † for each x ∈ X . Since ( f ( x ) ℓ ( δ x )) x ∈ X is summable in ( R, τ ) with sum ℓ ( f ) , by Cauchy’s condition [19, Definition 10.3, p. 72], there existsa finite set A f,V ⊆ X such that for all x / ∈ A f,V , f ( x ) ℓ ( δ x ) ∈ V . But for x ∈ X , f ( x ) ℓ ( δ x ) = ℓ ( δ x ) † ℓ ( δ x ) ∈ E ( R ) . Whence, in view of Remark 21, for all butfinitely many x ’s, f ( x ) ℓ ( δ x ) = , i.e., ℓ ( δ x ) = . ◻
23 Remark
Lemma 17 is a consequence of Proposition 22 since for a field k , E ( k ) = { , k } . Now, let ( E i , τ i ) i ∈ I be a family of topological spaces. On ∏ i ∈ I E i is definedthe box topology [9, p. 107] a basis of open sets of which is given by the “box” ∏ i ∈ I V i , where each V i ∈ τ i , i ∈ I . The product ∏ i ∈ I E i together with the boxtopology is denoted by ⊓ i ∈ I ( E i , τ i ) . (This topology is Hausdorff as soon asall the ( E i , τ i ) ’s are.)It is not difficult to see that given a family ( R i , τ i ) i ∈ I of topological rings,then ⊓ i ∈ I ( R i , τ i ) still is a topological ring (under component-wise opera-tions).
24 Proposition
Let ( k i ) i ∈ I be a family of fields, and for each i ∈ I , let τ i be a ring topology on k i . Let R be a ring with a one-to-one ring map ∶ R ↪ ∏ i ∈ I k i . Let us assume that for each x ∈ R , ( x ) † ∈ im ( ) ( ( x i ) † i asin Example 20). Let R be topologized with the subspace topology τ inheritedfrom ⊓ i ∈ I ( k i , τ i ) . Then, ( R , τ ) is rigid. Proof:
Naturally ( x i ) i ∈ I ∈ E ( ∏ i ∈ I k i ) if, and only if, x i ∈ { , k i } for each i ∈ I . Now, for each i ∈ I , let U i be an open neighborhood of zero in ( k i , τ i ) such that k i / ∈ U i . Then, ∏ i U i is an open neighborhood of zero in ⊓ i ∈ I ( k i , τ i ) whose only idempotent member is . Therefore, / ∈ E ( ∏ i ∈ I k i ) ∖ { } .Under the assumptions of the statement, an application of Example 20states that R is a (commutative) von Neumann regular ring. It is also ofcourse a topological ring under τ (since is a one-to-one ring map). It isalso clear that E ( R ) ≃ E ( ( R )) ⊆ E ( ∏ i k i ) . Furthermore, ( E ( R ) ∖ { }) = E ( ( R )) ∖ { } ∩ ( R ) ⊆ E ( ∏ i k i ) ∖ { } , and thus / ∈ E ( R ) ∖ { } accordingto the above discussion. Therefore, by Proposition 22, ( R , τ ) is rigid. ◻ Let ( R , τ ) be a topological ring, and let us consider the topological ( R , τ ) X -module (( R, τ ) X ) X for a given set X . To avoid confusion one denotes by ( R X ) X Π x Ð→ R X the canonical projections.11et us define a linear map ( R X ) X ℓ
Ð→ (
R, τ ) X by setting ℓ ( f ) ∶ x ↦( f ( x ))( x ) , f ∈ ( R X ) X . ℓ is continuous, and thus belongs to ((( R, τ ) X ) X ) ′ ,since for each x ∈ X , π x ○ ℓ = π x ○ Π x . Now, for each x ∈ X , ( ℓ ( δ R X x ))( x ) = δ R X x ( x ) = R X , so that supp ( ˆ ℓ ) = X . Consequently one obtains
25 Proposition
Let ( R , τ ) be topological ring, and let X be a set. If X isinfinite, then ( R , τ ) X is not rigid. However the above negative result may balanced by the following.
26 Proposition
Let ( R , τ ) be a rigid ring. If I is finite, then ( R , τ ) I isrigid too. Proof:
Let ( R , τ ) be a topological ring. For a set I , one recalls that ( R , τ ) I is the underlying ring ( R , τ ) I (Notation 7) of A ( R ,τ ) ( I ) . Any topo-logical ( R , τ ) I -module is also a topological ( R , τ ) -module under restriction ofscalars along the unit map ( R , τ ) η I Ð→ ( R , τ ) I , η I ( R ) = R I , which of courseis a ring map, and is continuous (because η I ( α ) = m R I ( η I ( α ) , R I ) , α ∈ R .)Let X be a set, and let ℓ ∈ ((( R, τ ) I ) X ) ′ , i.e., (( R, τ ) I ) X ℓ
Ð→ (
R, τ ) I iscontinuous and ( R , τ ) I -linear, and by restriction of scalar along η I it is alsoa continuous ( R , τ ) -linear. Therefore for each i ∈ I , (( R, τ ) I ) X ℓ
Ð→ (
R, τ ) I π i Ð→(
R, τ ) belongs to the topological dual space of (( R, τ ) I ) X seen as a ( R , τ ) -module.Let us assume that ( R , τ ) is rigid. Then, by Lemma 16, supp ( ̂ π i ○ ℓ ) isfinite for each i ∈ I . One also has supp ( ˆ ℓ ) = ⋃ i ∈ I supp ( ̂ π i ○ ℓ ) , with X ˆ ℓ Ð→ R I , ˆ ℓ ( x ) ∶ = ℓ ( δ R I x ) , x ∈ X . Whence if I is finite, then supp ( ˆ ℓ ) is finite too. ◻ Let ( R , τ ) and ( S , σ ) be topological rings, and let ( R , τ ) f Ð→ ( S , σ ) be a continuousring map. It may be used to transform a topological ( S , σ ) -module into a topological ( R , τ ) -module by restriction of scalars along f . In details, let ( M, γ ) be a topological ( S , σ ) -module. There is a scalar action of R on M given by α ⋅ v ∶ = f ( α ) v , α ∈ R , v ∈ M (where by juxtaposition is denoted the scalar action S × M → M ). Furthermore this actionis again continuous (by composition of continuous maps). Let f ∗ ( M, γ ) be the topological ( R , τ ) -module just obtained. At present let ( M, γ ) g Ð→ ( N, π ) be a continuous ( S , σ ) -linearmap. g is also R -linear because of g ( α ⋅ v ) = g ( f ( α ) v ) = f ( α ) g ( v ) = α ⋅ g ( v ) , and thusprovides a continuous ( R , τ ) -linear map f ∗ ( M, γ ) g Ð→ f ∗ ( N, π ) . All this results in a functor TopMod ( S ,σ ) f ∗ Ð→ TopMod ( R ,τ ) of restriction of scalars along f . Rigidity as an equivalence of categories
The main result of this section is Theorem 48 which provides a translation ofthe rigidity condition on a topological ring into a dual equivalence betweenthe category of free modules and that of topologically-free modules (see be-low), provided by the topological dual functor with equivalence inverse the(opposite of the) algebraic dual functor, with both functors convenientlyco-restricted. The purpose of this section thus is to prove this result.
Topologically-free modules.
Let ( R , τ ) be a topological ring. Let ( M, σ ) be a topological ( R, τ ) -module. It is said to be a topologically-free ( R , τ ) -module if ( M, σ ) ≃ ( R, τ ) X , in TopMod ( R ,τ ) , for some set X . Such topolog-ical modules span the full subcategory TopFreeMod ( R ,τ ) of TopMod ( R ,τ ) .For a field ( k , τ ) with a ring topology, one defines correspondingly the cat-egory TopFreeVect ( k ,τ ) ↪ TopVect ( k ,τ ) of topologically-free ( k , τ ) -vectorspaces .
27 Remark
The topological power functor
Set op P ( R ,τ ) ÐÐÐ→
TopMod ( R ,τ ) fac-tors as indicated below (the co-restriction obtained is also called P ( R ,τ ) ). Set op P ( R ,τ ) / / P ( R ,τ ) * * ❚❚❚❚❚❚❚❚❚❚❚ TopMod ( R ,τ ) TopFreeMod ( R ,τ ) ?(cid:31) O O (2)Topologically-free modules are characterized by the fact of possessing“topological bases” (see Corollary 31 below) which makes easier a number ofcalculations and proofs, once such a basis is chosen.
28 Definition
Let ( M, σ ) be a topological ( R , τ ) -module. Let B ⊆ M . It issaid to be a topological basis of ( M, σ ) if the following hold.1. For each v ∈ M , there exists a unique family ( b ′ ( v )) b ∈ B , with b ′ ( v ) ∈ R for each b ∈ B , such that ( b ′ ( v ) b ) b is summable in ( M, σ ) with sum v . b ′ ( v ) is referred to as the coefficient of v at b ∈ B .2. For each family ( α b ) b ∈ B of elements of R , there is a member v of M such that b ′ ( v ) = α b , b ∈ B . (By the above point such v is unique.)3. σ is equal to the initial topology induced by the (topological) coefficientmaps ( M b ′ Ð→ (
R, τ )) b ∈ B . (According to the two above points, each b ′ is R -linear.) It is an immediate consequence of the definition that for a topo-logical basis B of some topological module, / ∈ B and b ′ ( d ) = δ b ( d ) , b, d ∈ B (since ∑ b ∈ B δ b ( d ) b = d = ∑ b ∈ B b ′ ( d ) b ). In particular, B (−) ′ ÐÐ→ B ′ ∶ = { b ′ ∶ b ∈ B } is a bijection.
30 Lemma
Let ( M, σ ) and ( N, γ ) be isomorphic topological ( R , τ ) -modules.Let Θ ∶ ( M, σ ) ≃ ( N, γ ) be an isomorphism (in TopMod ( R ,τ ) ). Let B be atopological basis of ( M, σ ) . Then, Θ ( B ) = { Θ ( b ) ∶ b ∈ B } is a topologicalbasis of ( N, γ ) .
31 Corollary
Let ( M, σ ) be a (Hausdorff ) topological ( R , τ ) -module. It ad-mits a topological basis if, and only if, it is topologically-free.
32 Example
Let ( R , τ ) be a topological ring. For each set X , { δ x ∶ x ∈ X } is a topological basis of ( R, τ ) X . Moreover π x = δ ′ x , x ∈ X . Let us now take the time to establish a certain number of quite usefulproperties of topological bases.
33 Lemma
Let ( M, σ ) be a topologically-free ( R , τ ) -module with topologicalbasis B . Then, B is R -linearly independent and the linear span ⟨ B ⟩ of B isdense in ( M, σ ) . Proof:
Concerning the assertion of independence, it suffices to note that may be written as ∑ b ∈ B b , and conclude by the uniqueness of the decompo-sition in a topological basis. Let u ∈ M and let V ∶ = { v ∈ M ∶ b ′ ( v ) ∈ U b , b ∈ A } ∈ V ( M,σ ) ( ) , where A is a finite subset of B and U b ∈ V ( R ,τ ) ( ) , b ∈ A .Let α b ∈ U b , b ∈ A , and v ∶ = ∑ b ∈ A α b b − ∑ b ∈ B ∖ A b ′ ( u ) b ∈ V . So u + v ∈ ⟨ B ⟩ .Thus, u + V meets ⟨ B ⟩ and ⟨ B ⟩ is dense in ( M, σ ) . ◻
34 Corollary
Let ( M, σ ) be a topologically-free ( R , τ ) -module, and let ( N, γ ) be a topological ( R , τ ) -module. Let ( M, σ ) f,g Ð→ (
N, γ ) be two continuous ( R , τ ) -linear maps. f = g if, and only if, for any topological basis B of ( M, σ ) , f ( b ) = g ( b ) for each b ∈ B . Topologically-free modules allow for the definition of changes of topolog-ical bases (see Proposition 51 for a more general construction).
35 Lemma
Let ( M, σ ) and ( N, γ ) be two topologically-free ( R , τ ) -modules,and let B, D be respective topological bases. Let f ∶ B → D be a bijection.Then, there is a unique isomorphism g in TopMod ( R ,τ ) such that g ( b ) = f ( b ) , b ∈ B . roof: The question of uniqueness is settled by Corollary 34. If suchan isomorphism g exists, then g ( v ) = g ( ∑ b ∈ B b ′ ( v ) b ) = ∑ b ∈ B b ′ ( v ) g ( b ) = ∑ b ∈ B b ′ ( v ) f ( b ) = ∑ d ∈ D ( f − ( d )) ′ ( v ) d , v ∈ M . One observes that g as de-fined by the right hand-side of the last equality, is R -linear, and it is alsocontinuous since for each d ∈ D , d ′ ○ g = ( f − ( d )) ′ . ◻
36 Lemma
Let M be a free module with basis B . Then, ( M ∗ , w ∗( R ,τ ) ) isa topologically-free module with topological basis B ∗ ∶ = { b ∗ ∶ b ∈ B } (see Re-mark 11). Proof:
According to Lemma 12, ( M ∗ , w ∗( R ,τ ) ) is a topologically-free mod-ule. Let M θ B Ð→ R ( B ) be the isomorphism given by θ B ( b ) = δ b , b ∈ B .Thus, θ ∗ B ∶ ( R ( B ) ) ∗ ≃ M ∗ , and θ ∗ B ○ ρ B ∶ R B ≃ ( R ( B ) ) ∗ ≃ M ∗ is given by θ ∗ B ( ρ B ( δ R b )) = ρ B ( δ R b ) ○ θ B = p b ○ θ B = b ∗ for b ∈ B (see Example 10 forthe definition of p b ). Now, { δ b ∶ b ∈ B } being a topological basis of R B , byLemma 30, this shows that B ∗ is a topological basis of ( M ∗ , w ∗( R ,τ ) ) . ◻
37 Example { p x ∶ x ∈ X } is a topological basis of ( R ( X ) ) ∗ (Example 10).
38 Remark If B is a basis of a free module M , then B ≃ B ∗ under b ↦ b ∗ ,because for each b, d ∈ B , b ∗ ( d ) = δ b ( d ) .
39 Corollary
Let ( R , τ ) be a topological ring. The algebraic dual functors Mod opR
Alg ( R ,τ ) ÐÐÐÐ→
TopMod ( R ,τ ) factors as illustrated in the diagram below .Moreover the resulting co-restriction of Alg ( R ,τ ) (the bottom arrow of the When k is a field with a ring topology τ , then one has the corresponding factorizationof Vect op k Alg ( k ,τ ) ÐÐÐÐ→
TopVect ( k ,τ ) . Vect op k Alg ( k ,τ ) / / ) ) ❚❚❚❚❚❚❚❚❚❚ TopVect ( k ,τ ) TopFreeVect ( k ,τ ) ?(cid:31) O O (3) iagram) is essentially surjective . Mod opR
Alg ( R ,τ ) / / TopMod ( R ,τ ) FreeMod opR ?(cid:31) O O / / TopFreeMod ( R ,τ ) ?(cid:31) O O (4) Proof:
The first assertion is merely Lemma 36. Regarding the secondassertion, let ( M, σ ) be a topologically-free module. So, for some set X , ( M, σ ) ≃ ( R, τ ) X . By Lemma 9, ( R, τ ) X ≃ Alg ( R ,τ ) ( R ( X ) ) . ◻
40 Lemma
Let ( R , τ ) be a rigid ring. Let ( M, σ ) be a topologically-free ( R , τ ) -module with topological basis B . Then, ( M, σ ) ′ is free with basis B ′ ∶ = { b ′ ∶ b ∈ B } . Proof:
Let Θ B ∶ ( M, σ ) ≃ ( R, τ ) B be given by Θ B ( b ) = δ b . Therefore Θ ′ B ∶ (( R, τ ) B ) ′ ≃ ( M, σ ) ′ , and thus one has an isomorphism Θ ′ B ○ λ B ∶ R ( B ) ≃ ( M, σ ) ′ . Since a module isomorphic to a free module is free, ( M, σ ) ′ is free.The previous isomorphism acts as: Θ ′ B ( λ B ( δ b )) = π b ○ Θ B = b ′ for b ∈ B . Itfollows from Lemma 30 that B ′ is a basis of ( M, σ ) ′ . ◻
41 Example
Let ( R , τ ) be a rigid ring. Let ( M, σ ) = ( R, τ ) X . By Exam-ple 32, { δ ′ x ∶ x ∈ X } = { π x ∶ x ∈ X } is a linear basis of (( R, τ ) X ) ′ .
42 Corollary
Let ( R , τ ) be a rigid ring. The functor TopMod op ( R ,τ ) T op ( R ,τ ) ÐÐÐÐ→
Mod R factors as indicated by the diagram below. TopMod op ( R ,τ ) T op ( R ,τ ) / / Mod R TopFreeMod op ( R ,τ ) ?(cid:31) O O / / FreeMod R ?(cid:31) O O (6) A functor C F Ð→ D is essentially surjective when each object D in D is isomorphic toan object of the form F C , for some object C in C . Whence an equivalence of categoriesis a fully faithful and essentially surjective functor (see [13]). Correspondingly for a field ( k , τ ) with a ring topology, TopVect op ( k ,τ ) Top ( k ,τ ) / / Vect k TopFreeVect op ( k ,τ ) ?(cid:31) O O ❦❦❦❦❦❦❦❦❦❦ (5) he topological dual of the algebraic dual of a free module. Let ( R , τ ) be a topological ring. Let M be a R -module, and let us consider as inSection 3.1, the R -linear map M Λ M ÐÐ→ ( M ∗ , w ∗( R ,τ ) ) ′ ( Λ M ( v ))( ℓ ) = ℓ ( v ) , v ∈ M , ℓ ∈ M ∗ .
43 Lemma
Let M be a projective R -module. Then, Λ M is one-to-one. Thisholds in particular when M is a free R -module. Proof:
Let us consider a dual basis for M , i.e., sets B ⊆ M and { ℓ e ∶ e ∈ B } ⊆ M ∗ , such that for all v ∈ M , ℓ e ( v ) = for all but finitely many ℓ e ∈ B ∗ and v = ∑ e ∈ B ℓ e ( v ) e ([10, p. 23]). Let v ∈ ker Λ M , i.e., ( Λ M ( v ))( ℓ ) = ℓ ( v ) = for each ℓ ∈ M ∗ . Then, in particular, Λ M ( v )( ℓ e ) = ℓ e ( v ) = for all e ∈ B ,and thus v = . ◻ Let ( R , τ ) (resp. ( k , τ ) ) be a rigid ring (resp. field). Let us still de-note by FreeMod opR
Alg ( R ,τ ) ÐÐÐÐ→
TopFreeMod ( R ,τ ) (resp. Vect op k Alg ( k ,τ ) ÐÐÐÐ→
TopFreeVect ( k ,τ ) ) and by TopFreeMod op ( R ,τ ) T op ( R ,τ ) ÐÐÐÐ→
FreeMod R (resp. TopFreeVect op ( k ,τ ) T op ( k ,τ ) ÐÐÐÐ→
Vect k ) the functors provided by Corollaries 39and 42.
44 Proposition
Let us assume that ( R , τ ) is rigid. Λ ∶ = ( Λ M ) M ∶ id ⇒ T op ( R ,τ ) ○ Alg op ( R ,τ ) ∶ FreeMod R → FreeMod R is a natural isomorphism . Proof:
Naturality is clear. Let ( R , τ ) be a topological ring. Let M be afree R -module. For each free basis X of M , the following diagram commutesin Mod R , where M θ X Ð→ R ( X ) is the canonical isomorphism given by θ X ( x ) = δ R x , x ∈ X . Consequently, when ( R , τ ) is rigid, then for each free R -module M , M Λ M ÐÐ→ ( M ∗ , w ∗( R ,τ ) ) ′ is an isomorphism. M Λ M / / ( M ∗ , w ∗( R ,τ ) ) ′ ❲❲❲❲❲❲❲ ( θ ∗ X ) ′ (( R ( X ) ) ∗ , w ∗( R ,τ ) ) ′ R ( X ) θ X λ X / / (( R, τ ) X ) ′ ❣❣❣❣❣❣❣ ρ ′ X (7) A natural isomorphism α ∶ F ⇒ G ∶ C → D is a natural transformation the componentsof which are isomorphisms in D . Two functors F, G ∶ C → D are said to be naturallyisomorphic , which is denoted F ≃ G , if there is a natural isomorphism α ∶ F ⇒ G ∶ C → D .
45 Corollary
Let us assume that ( k , τ ) is a field with a ring topology. Then, Λ = ( Λ M ) M ∶ id ⇒ T op ( k ,τ ) ○ Alg op ( k ,τ ) ∶ Vect k → Vect k is a natural isomor-phism. The algebraic dual of the topological dual of a topologically-freemodule.
Let ( M, σ ) be a topological ( R , τ ) -module. Let us consider the R -linear map M Γ ( M,σ ) ÐÐÐÐ→ ((
M, σ ) ′ ) ∗ by setting ( Γ ( M,σ ) ( v ))( ℓ ) ∶ = ℓ ( v ) .
46 Proposition
Let us assume that ( R , τ ) is a rigid ring. Then, Γ ∶ id ⇒ Alg ( R ,τ ) ○ T op op ( R ,τ ) ∶ TopFreeMod ( R ,τ ) → TopFreeMod ( R ,τ ) is a natural iso-morphism, with Γ ∶ = ( Γ ( M,σ ) ) ( M,σ ) . Proof:
Naturality is clear. Let Θ ∶ ( M, σ ) ≃ ( R, τ ) X be an isomorphism (in TopMod ( R ,τ ) ). Since ( R , τ ) is rigid, λ X ∶ R ( X ) ≃ (( R, τ ) X ) ′ is an isomor-phism. Therefore R ( X ) λ X ÐÐ→ ((
R, τ ) X ) ′ Θ ′ Ð→ (
M, σ ) ′ is an isomorphism tooin Mod R . In particular, ( M, σ ) ′ is a free with basis { Θ ′ ( λ X ( δ R x )) ∶ x ∈ X } .By Lemma 12, the weak- ∗ topology on (( M, σ ) ′ ) ∗ is the same as the initialtopology given by the maps (( M, σ ) ′ ) ∗ Λ ( M,σ )′ ( π x ○ Θ ) ÐÐÐÐÐÐÐÐ→ (
R, τ ) , x ∈ X , because Θ ′ ( λ X ( δ R x )) = Θ ′ ( π x ) = π x ○ Θ . Therefore, Γ ( M,σ ) is continuous if, and onlyif, for each x ∈ X , Λ ( M,σ ) ′ ( π x ○ Θ ) ○ Γ ( M,σ ) = π x ○ Θ is continuous. Continuityof Γ ( M,σ ) thus is proved.That Γ ( M,σ ) is an isomorphism in TopMod ( R,τ ) follows from the com-mutativity of the diagram (in TopMod ( R,τ ) ) below (which may be checkedby hand). ( M, σ ) Θ s s ❢❢❢❢❢❢❢❢❢❢ Γ ( M,σ ) / / ((( M, σ ) ′ ) ∗ , w ∗( R ,τ ) )( R, τ ) X ρ X + + ❲❲❲❲❲❲ (( R ( X ) ) ∗ , w ∗( R ,τ ) ) ( λ − X ) ∗ / / (((( R, τ ) X ) ′ ) ∗ , w ∗( R ,τ ) ) (( Θ ′ ) ∗ ) − O O (8) ◻
47 Corollary
Let us assume that ( k , τ ) is a field with a ring topology. Γ ∶ id ⇒ Alg ( k ,τ ) ○ T op op ( k ,τ ) ∶ TopFreeVect ( k ,τ ) → TopFreeVect ( k ,τ ) with Γ ∶ = ( Γ ( M,σ ) ) ( M,σ ) , is a natural isomorphism. he equivalence and some of its immediate consequences. Collect-ing Proposition 44 and Lemma 46, one immediately gets the following.
48 Theorem
Let us assume that ( R , τ ) is rigid. TopFreeMod op ( R ,τ ) T op ( R ,τ ) ÐÐÐÐ→
FreeMod R is an equivalence of categories , with equivalence inverse thefunctor FreeMod R Alg op ( R ,τ ) ÐÐÐÐ→
TopFreeMod op ( R ,τ ) .
49 Corollary
TopFreeVect op ( k ,τ ) T op ( k ,τ ) ÐÐÐÐ→
Vect k is an equivalence of cat-egories, and Vect k Alg op ( k ,τ ) ÐÐÐÐ→
TopFreeVect op ( k ,τ ) is its equivalence inverse,whenever ( k , τ ) is a field with a ring topology. Finite-dimensional vector spaces.
Let k be a field. Let ( M, σ ) be atopologically-free ( k , d ) -vector space with M finite-dimensional. Then, σ isthe discrete topology on M . It follows that ( M, σ ) ′ = M ∗ , and the equiva-lence established in Corollary 49 coincides with the classical dual equivalence FinDimVect k ≃ FinDimVect op k under the algebraic dual functor, where FinDimVect k is the category of finite-dimensional k -vector spaces. Linearly compact vector spaces.
Let k be a field. A topological ( k , d ) -vector space ( M, σ ) is said to be a linearly compact k -vector space when ( M, σ ) ≃ ( k , d ) X for some set X (see [4, Proposition 24.4, p. 105]). The fullsubcategory LCpVect k of TopVect ( k , d ) spanned by these spaces is equalto TopFreeVect ( k , d ) .
50 Corollary (of Theorem 48) Let R be a ring. For each rigid topologies τ, σ on R , the categories TopFreeMod ( R ,τ ) and TopFreeMod ( R ,σ ) are equiva-lent. Moreover, for each field ( k , τ ) with a ring topology, TopFreeVect ( k ,τ ) is equivalent to LCpVect k . In particular, one recovers the result of J. Dieudonné [8] that
Vect op k ≃ LCpVect k . C ≃ D means that the categories C and D are equivalent , i.e., that there is anequivalence of categories C F Ð→ D . In this situation an equivalent inverse of F is a functor D G Ð→ C such that there are two natural isomorphisms η ∶ id C ⇒ G ○ F ∶ C → C and ǫ ∶ F ○ G ⇒ id D ∶ D → D . he universal property of ( R, τ ) X . For a ring R , the forgetful functor Mod R ∣⋅∣ Ð→ Set (see Remark 5) may be restricted as indicated in the followingcommutative diagram, and the restriction still is denoted
FreeMod R ∣⋅∣ Ð→ Set . Mod R ∣⋅∣ / / SetFreeMod R ?(cid:31) O O ❦❦❦❦❦❦❦❦ (9)Likewise Set F R Ð→ Mod R (see again Remark 5) may be co-restricted asindicated by the commutative diagram below, and the co-restriction is giventhe same name Set F R Ð→ FreeMod R . Set F R / / ) ) ❙❙❙❙❙❙❙❙ Mod R FreeMod R ?(cid:31) O O (10)(Of course, when R is a field k , there is no need to consider the correspondingco-restrictions.)The adjunction F R ⊣ ∣ ⋅ ∣ ∶ Set → Mod R gives rise to a new one F R ⊣∣ ⋅ ∣ ∶ Set → FreeMod R [13, p. 147], and by composition, for each rigid ring ( R , τ ) , there is also the adjunction Alg op ( R ,τ ) ○ F R ⊣ ∣ ⋅ ∣ ○ T op ( R ,τ ) ∶ Set → TopFreeMod op ( R ,τ ) . Since ( R, τ ) X ≃ (( R ( X ) ) ∗ , w ∗( R ,τ ) ) (Lemma 9), this maybe translated into a universal property of ( R, τ ) X , as explained below, whichsomehow legitimates the terminology topologically-free .
51 Proposition
Let us assume that ( R , τ ) is rigid. Let X be a set. Foreach topologically-free module ( M, σ ) and any map X f Ð→ ∣(
M, σ ) ′ ∣ , there isa unique continuous ( R , τ ) -linear map ( M, σ ) f ♯ Ð→ (
R, τ ) X such that ∣( f ♯ ) ′ ∣ ○ ∣ λ X ∣ ○ δ R X = f (recall that δ R X ( x ) = δ R x , x ∈ X ). Proof:
There is a unique R -linear map R ( X ) ˜ f Ð→ (
M, σ ) ′ such that ∣ ˜ f ∣ ○ δ R X = f . Let us define the continuous linear map ( M, σ ) f ♯ Ð→ (
R, τ ) X ∶ = ( M, σ ) Γ ( M,σ ) ÐÐÐÐ→ (((
M, σ ) ′ ) ∗ , w ∗( R ,τ ) ) ( ˜ f ) ∗ ÐÐ→ (( R ( X ) ) ∗ , w ∗( R ,τ ) ) ρ − X ÐÐ→ (
R, τ ) X . One20as ∣( f ♯ ) ′ ∣ ○ ∣ λ X ∣ ○ δ R X = ∣ Γ ′( M,σ ) ∣ ○ ∣(( ˜ f ) ∗ ) ′ ∣ ○ ∣( ρ − X ) ′ ∣ ○ ∣ λ X ∣ ○ δ R X = ∣ Γ ′( M,σ ) ∣ ○ ∣(( ˜ f ) ∗ ) ′ ∣ ○ ∣ Λ R ( X ) ∣ ○ δ R X (because ( ρ − X ) ′ ○ λ X = Λ R ( X ) ) = ∣ Γ ′( M,σ ) ∣ ○ ∣ Λ ( M,σ ) ′ ∣ ○ ∣ ˜ f ∣ ○ δ R X (by naturality of Λ ) = ∣ ˜ f ∣ ○ δ R X (triangular identities for an adjunction [13, p. 85]) = f. (11)It remains to check uniqueness of f ♯ . Let ( M, σ ) g Ð→ (
R, τ ) X be a continuouslinear map such that ∣ g ′ ∣ ○ ∣ λ X ∣ ○ δ R X = f . Then, g ′ ○ λ X = ˜ f . Thus, λ ∗ X ○ ( g ′ ) ∗ = ˜ f ∗ = ρ X ○ f ♯ ○ Γ − ( M,σ ) . So ρ X ○ Γ − ( R,τ ) X ○ ( g ′ ) ∗ = ρ X ○ f ♯ ○ Γ − ( M,σ ) because Γ ( R,τ ) X = ( λ − X ) ∗ ○ ρ X (by direct inspection), and thus Γ − ( R,τ ) X ○ ( g ′ ) ∗ = f ♯ ○ Γ − ( M,σ ) . Then, by naturality of Γ − , g ○ Γ − ( M,σ ) = f ♯ ○ Γ − ( M,σ ) . ◻
52 Corollary
Let ( R , τ ) be a rigid ring. Set P op ( R ,τ ) ÐÐÐ→
TopFreeMod op ( R,τ ) isa left adjoint of TopFreeMod op ( R,τ ) T op ( R ,τ ) ÐÐÐÐ→
FreeMod R ∣−∣ Ð→ Set , and thusis naturally equivalent to
Set
Alg op ( R ,τ ) ○ F R ÐÐÐÐÐÐ→
TopFreeMod op ( R,τ ) . Proof:
A quick calculation shows that P ( R ,τ ) ( f ) = (∣ λ Y ∣ ○ δ R Y ○ f ) ♯ for aset-theoretic map X f Ð→ Y . The relation f ↦ ( λ Y ○ δ R Y ○ f ) ♯ provides a functorfrom Set op to TopFreeMod ( R ,τ ) whose opposite is, by construction, a leftadjoint of ∣ − ∣ ○ T op ( R ,τ ) (this is basically the content of Proposition 51). ◻ In this section most of the ingredients so far introduced and developed fittogether so as to lift the equivalence
FreeMod ≃ TopFreeMod op ( R ,τ ) to aduality between some (suitably generalized) topological algebras and coalge-bras. Let us briefly describe how this goal is achieved:1. A topological tensor product ⍟ ( R ,τ ) for topologically-free modules (overa rigid ring ( R , τ ) ) is provided by transport of the algebraic tensorproduct ⊗ R of R -modules along the dual equivalence from Section 5.21. A topological basis (Definition 28) for ( M, σ ) ⍟ ( R ,τ ) ( N, γ ) is describedin terms of topological bases of ( M, σ ) , ( N, γ ) .3. The aforementioned equivalence is proved to be compatible with ⊗ R and ⍟ ( R ,τ ) (i.e., it is a monoidal equivalence).4. Accordingly, for category-theoretic reasons (see Appendix A), one ob-tains a dual equivalence between some topological algebras and coal-gebras, still under the algebraic and topological dual functors. The results recalled below are well-known (cf. [5] for instance) but they giveus the opportunity to introduce some notations used hereafter.Let R be a ring. For each R -modules M, N , one denotes by M ⊗ R N their (algebraic) tensor product, and by M × N ⊗ Ð→ M ⊗ R N the universal R -bilinear map. As usually the image of ( x, y ) ∈ M × N by ⊗ is denoted x ⊗ y .Furthermore, with the following isomorphisms ( M ⊗ R N ) ⊗ R P α M,N,P
ÐÐÐÐ→ M ⊗ R ( N ⊗ R P ) , α M,N,P (( x ⊗ y ) ⊗ z ) = x ⊗ ( y ⊗ z ) , M ⊗ R R ρ M ÐÐ→ M , ρ M ( x ⊗ R ) = x , R ⊗ R N λ N ÐÐ→ N , λ N ( R ⊗ y ) = y , M ⊗ R N σ M,N
ÐÐÐ→ N ⊗ R M , σ M,N ( x ⊗ y ) = y ⊗ x , Mod R = ( Mod R , ⊗ R , R ) is a symmetric monoidal category ([13, p. 184]).Given two sets X, Y , R ( X ) ⊗ R R ( Y ) ≃ R ( X × Y ) under the unique R -linear map R ( X ) ⊗ R R ( Y ) Φ X,Y
ÐÐÐ→ R ( X × Y ) which maps f ⊗ g to ∑ ( x,y )∈ X × Y f ( x ) g ( y ) δ R ( x,y ) ,with inverse R ( X × Y ) Ψ X,Y
ÐÐÐ→ R ( X ) ⊗ R R ( Y ) the unique R -linear whose valueson the basis elements are given by Ψ X,Y ( δ R ( x,y ) ) = δ R x ⊗ R δ R y .By functoriality, it is clear that given free modules M, N , M ⊗ R N is freetoo. As a consequence, FreeMod R = ( FreeMod R , ⊗ R , R ) is a (symmetric)monoidal subcategory of Mod R . In other words, each R -bilinear map M × N f Ð→ P uniquely factors as M × N f / / ⊗ (cid:15) (cid:15) PM ⊗ R P ˜ f ♠♠♠♠♠♠ (12)where ˜ f is R -linear. By a (symmetric) monoidal subcategory of a (symmetric) monoidal category C =( C , − ⊗ − , I ) we mean a subcategory C ′ of C , closed under tensor products, contain-ing I , and the coherence constraints of C between C ′ -objects. (The last condition isautomatically fulfilled when C ′ is a full subcategory.) The embedding E C ′ of C ′ into C then is a strict monoidal functor E C ′ (see e.g., [15, Def. 2, p. 4876]. .2 Topological tensor product of topologically-free modules We now wish to take advantage of the equivalence of categories
FreeMod R ≃ TopFreeMod op ( R ,τ ) (Theorem 48) for a rigid ring ( R , τ ) , to introduce a topo-logical tensor product of topologically-free modules. From here to the endof Section 6.2, ( R , τ ) denotes a rigid ring.The bifunctor ⍟ ( R ,τ ) . Let ( M, σ ) , ( N, γ ) be two topologically-free ( R , τ ) -modules. One defines their topological tensor product over ( R , τ ) as ( M, σ ) ⍟ ( R ,τ ) ( N, γ ) ∶ = Alg ( R ,τ ) (( M, σ ) ′ ⊗ R ( N, γ ) ′ ) . (13)One immediately observes that ( M, σ ) ⍟ ( R ,τ ) ( N, γ ) still is a topologically-free ( R , τ ) -module as ( M, σ ) ′ and ( N, γ ) ′ are free R -modules (Lemma 40), ( M, σ ) ′ ⊗ R ( N, γ ) ′ also is (Section 6.1), and the algebraic dual of a free moduleis topologically-free (Lemma 36).Actually, this definition is just the object component of a bifunctor TopFreeMod ( R ,τ ) × TopFreeMod ( R ,τ ) −⍟ ( R ,τ ) − ÐÐÐÐÐ→
TopFreeMod ( R ,τ ) namely TopFreeMod ( R ,τ ) × TopFreeMod ( R ,τ ) T op op ( R ,τ ) × T op op ( R ,τ ) ÐÐÐÐÐÐÐÐÐÐ→
Mod opR × Mod opR ⊗ opR ÐÐ→
Mod opR
Alg ( R ,τ ) ÐÐÐÐ→
TopMod ( R ,τ ) .
53 Remark
Let
X, Y be sets. One has ( R, τ ) X ⍟ ( R ,τ ) ( R, τ ) Y = (((( R, τ ) X ) ′ ⊗ R (( R, τ ) Y ) ′ ) ∗ , w ∗( R ,τ ) ) ≃ (( R ( X ) ⊗ R R ( Y ) ) ∗ , w ∗( R ,τ ) ) (under ( λ X ⊗ R λ Y ) ∗ ) ≃ (( R ( X × Y ) ) ∗ , w ∗( R ,τ ) ) (under Ψ ∗ X,Y ; see Section 6.1) ≃ ( R, τ ) X × Y . (under ρ − X × Y ) (14)Given f i ∈ TopFreeMod ( R ,τ ) (( M i , σ i ) , ( N i , γ i )) , i = , , then f ⍟ ( R ,τ ) f ∶ = ( M , σ ) ⍟ ( R ,τ ) ( M , σ ) ( f ′ ⊗ R f ′ ) ∗ ÐÐÐÐÐ→ ( N , γ ) ⍟ ( R ,τ ) ( N , γ ) . By a bifunctor is meant a functor with domain a product of two categories. For every categories C , D , ( C × D ) op = C op × D op .
23n details, let L ∈ (( M , σ ) ′ ⊗ R ( M , σ ) ′ ) ∗ , ℓ ∈ ( N , γ ) ′ and ℓ ∈ ( N , γ ) ′ . Then, (( f ⍟ ( R ,τ ) f )( L ))( ℓ ⊗ ℓ ) = ((( f ′ ⊗ R g ′ ) ∗ )( ℓ ))( ℓ ⊗ ℓ ) = L (( f ′ ⊗ R f ′ )( ℓ ⊗ ℓ )) = L (( ℓ ○ f ) ⊗ ( ℓ ○ f )) . (15) A topological basis of ( M, σ ) ⍟ ( R ,τ ) ( N, γ ) . Our next goal will be toexplicitly describe a topological basis (Definition 28) of ( M, σ ) ⍟ ( R ,τ ) ( N, γ ) in terms of topological bases of ( M, σ ) and ( N, γ ) .
54 Definition
Given a ring S , for every S -modules M, N , one has a natural R -linear map M ∗ ⊗ S N ∗ Θ M,N
ÐÐÐ→ ( M ⊗ S M ) ∗ given by ( Θ M,N ( ℓ ⊗ ℓ ))( u ⊗ v ) = ℓ ( u ) ℓ ( v ) , ℓ ∈ M ∗ , ℓ ∈ N ∗ , u ∈ M and v ∈ N . Let ( M, σ ) , ( N, γ ) be two topologically-free ( R , τ ) -modules. Let u ∈ M and v ∈ N . Let us define u ⍟ v ∶ = Θ ( M,σ ) ′ , ( N,γ ) ′ ( Γ ( M,σ ) ( u ) ⊗ R Γ ( N,γ ) ( v )) ∈ ( M, σ ) ⊗ ( R ,τ ) ( N, γ ) . (16)In details, given ℓ ∈ ( M, σ ) ′ and ℓ ∈ ( N, γ ) ′ , ( u ⍟ v )( ℓ ⊗ ℓ ) = ℓ ( u ) ℓ ( v ) .
55 Lemma
Let ( M, σ ) and ( N, γ ) be both topologically-free ( R , τ ) -modules,with respective topological bases B, D . The map B × D −⍟− ÐÐ→ (
M, σ ) ⍟ ( R ,τ ) ( N, γ ) given by ( b, d ) ↦ b ⍟ d , is one-to-one.
56 Lemma
Let ( M, σ ) and ( N, γ ) be both topologically-free ( R , τ ) -modules.The map M × N −⍟− ÐÐ→ (
M, σ ) ⍟ ( R ,τ ) ( N, γ ) is R -bilinear and separately contin-uous in both variable. Moreover, if τ = d , then ⍟ is even jointly continuous. Proof: R -bilinearity is clear. Since ( M, σ ) ′ ⊗ R ( N, γ ) ′ is free on { x ⊗ y ∶ x ∈ X, y ∈ Y } , where X (resp. Y ) is a basis of ( M, σ ) ′ (resp. ( N, γ ) ′ ), byLemma 12, the topology w ∗( R ,τ ) on ( M, σ ) ⍟ ( R ,τ ) ( N, γ ) is the initial topologyinduced by (( M, σ ) ′ ⊗ R (( N, µ ) ′ ) ∗ Λ ( M,σ )′⊗ R ( N,γ )′ ( x ⊗ y ) ÐÐÐÐÐÐÐÐÐÐÐÐ→ (
R, τ ) , x ∈ X , y ∈ Y .Let x ∈ X , y ∈ Y , u ∈ M and v ∈ N . Then, Λ ( M,σ ) ′ ⊗ R ( N,γ ) ′ ( x ⊗ y )( u ⍟ v ) = ( u ⍟ v )( x ⊗ y ) = x ( u ) y ( v ) = m R ( x ( u ) , x ( v )) , and this automaticallyguarantees separate continuity in each variable of ⍟ .Let us assume that τ = d . According to the above general case, to see that ⍟ is continuous, by [19, Theorem 2.14, p. 17], it suffices to prove continuity atzero of ⍟ . Let A ⊆ X × Y be a finite set, and for each ( x, y ) ∈ A , let U ( x,y ) be24n open neighborhood of zero in ( R, d ) . Let A ∶ = { x ∈ X ∶ ∃ y ∈ Y, ( x, y ) ∈ A } and A ∶ = { y ∈ Y ∶ ∃ x ∈ X, ( x, y ) ∈ A } . A , A are both finite and A ⊆ A × A .Let u ∈ M such that x ( u ) = for all x ∈ A , and v ∈ N such that y ( v ) = forall y ∈ A . Then, ( u ⍟ v )( x ⊗ y ) = ∈ U ( x,y ) for all ( x, y ) ∈ A × A . ◻
57 Remark
Let
X, Y be sets, and let f ∈ R X , g ∈ R Y . Since ⍟ is separatelycontinuous by Lemma 56, f ⍟ g = ( ∑ x ∈ X f ( x ) δ R x ) ⍟ ( ∑ y ∈ Y g ( y ) δ R y ) = ∑ ( x,y )∈ X × Y f ( x ) g ( y ) δ R x ⍟ δ R y . (as a sum of a summable family) (17) For the same reason as above, if B is a topological basis of ( M, σ ) and D isa topological basis of ( N, γ ) , then u ⍟ v = ∑ ( b,d )∈ B × D b ′ ( u ) d ′ ( v ) b ⍟ d , u ∈ M , v ∈ N . In particular, one observes that ( b ⍟ d ) ′ ( u ⍟ v ) = b ′ ( u ) d ′ ( v ) , b ∈ B , u ∈ M , d ∈ D , and v ∈ N .
58 Proposition
Let ( M, σ ) and ( N, γ ) be topologically-free ( R , τ ) -modules,with respective topological bases B, D . Then, ( b ⍟ d ) ( b,d )∈ B × D is a topologicalbasis of ( M, σ ) ⍟ ( R ,τ ) ( N, γ ) . Proof:
By virtue of Lemma 30 and Remark 53, { ( δ R x ⍟ δ R y ) ( x,y )∈ X × Y ∶ ( x, y ) ∈ X × Y } is a topological basis of ( R, τ ) X ⍟ ( R ,τ ) ( R, τ ) Y since one has (( λ − X ⊗ R λ − Y ) ∗ ( Φ ∗ X,Y ( ρ X × Y ( δ R ( x,y ) )))) = δ R x ⍟ δ R y . Let us consider the isomorphism Θ B ∶ ( M, σ ) ≃ ( R, τ ) B , Θ B ( b ) = δ b , b ∈ B (resp. Θ D ∶ ( N, γ ) ≃ ( R, τ ) D ). Byfunctoriality, Θ B ⍟ ( R ,τ ) Θ D ∶ ( M, σ ) ⍟ ( R ,τ ) ( N, γ ) ≃ ( R, τ ) B ⍟ ( R ,τ ) ( R, τ ) D , andsince ( θ B ⍟ ( R ,τ ) θ D )( b ⍟ d ) = δ b ⍟ δ d , ( b, d ) ∈ B × D , { b ⍟ d ∶ ( b, d ) ∈ B × D } isa topological basis of ( M, σ ) ⍟ ( R ,τ ) ( N, γ ) . ◻
59 Corollary
Let ( M, σ ) and ( N, γ ) be topologically-free ( R , τ ) -modules.Then, { u ⍟ v ∶ u ∈ M, v ∈ N } spans a dense subspace in ( M, σ ) ⍟ ( R ,τ ) ( N, γ ) . Proof:
It is clear in view of Lemma 33. ◻ Let ( M i , σ i ) and ( N i , γ i ) , i = , , be topologically-free ( R , τ ) -modules.Let ( M i , σ i ) f i Ð→ ( N i , γ i ) , i = , , be continuous ( R , τ ) -linear maps. Let ( u, v ) ∈ M × M . By Eq. (15) it is clear that ( f ⍟ ( R ,τ ) f )( u ⍟ v ) = f ( u ) ⍟ f ( v ) . (18) The second equality in Eq. (17) follows from the proof of [19, Theorem 10.15, p. 78]which, by inspection, shows that the cited result still is valid more generally after thereplacement of a jointly continuous bilinear map by a separately continuous bilinear map. B and D are topological bases of ( M , σ ) and ( M , σ ) respectively, ( f ⍟ ( R ,τ ) f )( u ⍟ v ) = ∑ ( b,d )∈ B × D b ′ ( u ) d ′ ( v ) f ( b ) ⍟ f ( d ) (in view of Re-mark 57). ⍟ ( R ,τ ) and its direct consequences Since most of the proofs from this section mainly consist in rather tedious,but simple, inspections of commutativity of some diagrams, essentially byworking with given topological or linear bases , and because they did notprovide much understanding, they are not included in the presentation. Thereader is kindly invited to consult Appendix A where are summarized somenotions and notations about monoidal category theory.
60 Proposition
Let ( R , τ ) be a rigid ring. TopFreeMod ( R ,τ ) ∶ = ( TopFreeMod ( R ,τ ) , ⍟ ( R ,τ ) , ( R, τ )) is a symmetric monoidal category .
61 Corollary
For each field ( k , τ ) with a ring topology, TopFreeVect ( k ,τ ) ∶ = ( TopFreeVect ( k ,τ ) , ⍟ ( k ,τ ) , ( k , τ )) is a symmetric monoidal category.
62 Example
Let ( R , τ ) be a rigid ring. Let X be a set. Let us define acommutative monoid M ( R ,τ ) ( X ) ∶ = (( R, τ ) X , µ X , η X ) in TopFreeMod ( R ,τ ) by µ X ( f ⍟ g ) = ∑ x ∈ X f ( x ) g ( x ) δ R x , f, g ∈ R X (under ( R, τ ) X ⍟ ( R ,τ ) ( R, τ ) X ≃ ( R, τ ) X × X from Remark 53) and η X ( R ) = ∑ x ∈ X δ R x . This provides a functor Set op M ( R ,τ ) ÐÐÐ→ c Mon ( TopFreeMod ( R ,τ ) ) . Let ( R , τ ) be a rigid ring. For each free R -modules M, N , let us de-fine Φ M,N ∶ = ( M ∗ , w ∗( R ,τ ) ) ⍟ ( R ,τ ) ( N ∗ , w ∗( R ,τ ) ) = Alg ( R ,τ ) (( M ∗ , w ∗( R ,τ ) ) ′ ⊗ R ( N ∗ , w ∗( R ,τ ) ) ′ ) ( Λ M ⊗ R Λ N ) ∗ ÐÐÐÐÐÐÐ→ (( M ⊗ R N ) ∗ , w ∗( R ,τ ) ) . According to Proposition 44, Φ M,N is an isomorphism in
TopFreeMod ( R ,τ ) . Naturality in M, N is clear,so this provides a natural isomorphism Φ ∶ Alg ( R ,τ ) ( − ) ⍟ ( R ,τ ) Alg ( R ,τ ) ( − ) ⇒ Alg ( R ,τ ) ( − ⊗ R − ) ∶ FreeMod opR × FreeMod opR → TopFreeMod ( R ,τ ) . (19) E.g., associativity of ⍟ ( R ,τ ) is given by the isomorphism ( b ⍟ d ) ⍟ e ↦ b ⍟ ( d ⍟ e ) onbasis elements (Lemma 35). See [13] for the definition of a symmetric monoidal category. ( R, τ ) φ Ð→ ( R ∗ , w ∗( R ,τ ) ) givenby φ ( R ) ∶ = id R , with inverse φ − ( ℓ ) = ℓ ( R ) .Let ( M, σ ) , ( N, γ ) be topologically-free ( R , τ ) -modules. One defines themap Ψ ( M,σ ) , ( N,γ ) ∶ = ( M, σ ) ′ ⊗ R ( N, γ ) ′ Λ ( M,σ )′⊗ R ( N,γ )′ ÐÐÐÐÐÐÐÐ→ (
Alg ( R ,τ ) (( M, σ ) ′ ⊗ R ( N, γ ) ′ )) ′ = (( M, σ ) ⍟ ( R ,τ ) ( N, γ )) ′ . This gives rise to a natural isomorphism Ψ ∶ T op ( R ,τ ) ( − ) ⊗ R T op ( R ,τ ) ( − ) ⇒ T op ( R ,τ ) ( − ⍟ ( R ,τ ) − ) ∶ TopFreeMod op ( R ,τ ) × TopFreeMod op ( R ,τ ) → FreeMod R . (20)Let also R ψ Ð→ (
R, τ ) ′ be given by ψ ( R ) = id R and ψ − ( ℓ ) = ℓ ( R ) .
63 Theorem
Let ( R , τ ) be a rigid ring.1. Alg ( R ,τ ) = ( Alg ( R ,τ ) , Φ , φ ) ∶ FreeMod op R → TopFreeMod ( R ,τ ) is a strongsymmetric monoidal functor.2. Top ( R ,τ ) = ( T op ( R ,τ ) , Ψ , ψ ) is a strong symmetric monoidal functorfrom TopFreeMod op ( R ,τ ) to FreeMod R , so is Top d ( R ,τ ) (Remark 83, Ap-pendix A.2) from TopFreeMod ( R ,τ ) to FreeMod op R .3. Λ op ∶ Top d ( R ,τ ) ○ Alg ( R ,τ ) ⇒ id ∶ FreeMod op R → FreeMod op R is a monoidalisomorphism.4. Γ ∶ id ⇒ Alg ( R ,τ ) ○ Top d ( R ,τ ) ∶ TopFreeMod ( R ,τ ) → TopFreeMod ( R ,τ ) is amonoidal isomorphism.In particular, FreeMod op R and TopFreeMod ( R ,τ ) are monoidally equivalent(Appendix A.2).
64 Corollary
For each field ( k , τ ) with a ring topology, the monoidal cate-gories Vect opk and
TopFreeVect ( k ,τ ) are monoidally equivalent.
65 Corollary
For each rigid ring ( R , τ ) , the induced natural transforma-tions (see Remark 85 in Appendix A.2 and Example 79 in Appendix A.1)• ̃ Λ op ∶ ̃( Top d ( R ,τ ) ) ○ ̃ Alg ( R ,τ ) ⇒ id ǫ Coalg opR ∶ ǫ Coalg opR → ǫ Coalg opR ,• ̃ Λ op ∶ ̃( Top d ( R ,τ ) ) ○ ̃ Alg ( R ,τ ) ⇒ id ǫ,coc Coalg opR ∶ ǫ,coc Coalg opR → ǫ,coc Coalg opR ,• ˜Γ ∶ id Mon ( TopFreeMod ( R ,τ ) ) ⇒ ̃ Alg ( R ,τ ) ○ ̃( Top d ( R ,τ ) ) ∶ Mon ( TopFreeMod ( R ,τ ) ) → Mon ( TopFreeMod ( R ,τ ) ) , ˜Γ ∶ id c Mon ( TopFreeMod ( R ,τ ) ) ⇒ ̃ Alg ( R ,τ ) ○ ̃( Top d ( R ,τ ) ) ∶ c Mon ( TopFreeMod ( R ,τ ) ) → c Mon ( TopFreeMod ( R ,τ ) ) .are all natural isomorphisms.Thus ǫ Coalg opR (resp., ǫ,coc
Coalg opR ) and
Mon ( TopFreeMod ( R ,τ ) ) (resp., c Mon ( TopFreeMod ( R ,τ ) ) ) are equivalent for each rigid topology τ on R .In particular, Mon ( TopFreeMod ( R ,τ ) ) ≃ Mon ( TopFreeMod ( R ,σ ) ) (resp., c Mon ( TopFreeMod ( R ,τ ) ) ≃ c Mon ( TopFreeMod ( R ,σ ) ) ), for each rigid topolo-gies τ, σ on R . Proof:
Follows from Theorem 63 together with Remarks 85 and 86 inAppendix A.2. ◻
66 Example
Let us make explicit the domain and codomain of the naturalisomorphism ˜Γ from Corollary 65.Let (( M, σ ) , m, e ) be an object of Mon ( TopFreeMod ( R ,τ ) ) . Let its topo-logical dual coalgebra be (( M, σ ) ′ , δ, ǫ ) ∶ = ̃( Top d ( R ,τ ) )(( M, σ ) , m, e ) = (( M, σ ) ′ , Λ − ( M,σ ) ′ ⊗ R ( M,σ ) ′ ○ m ′ , ψ − ○ e ′ ) . (21) In details, for ℓ ∈ ( M, σ ) ′ , ǫ ( ℓ ) = ψ − ( e ′ ( ℓ )) = ψ − ( ℓ ○ e ) = ℓ ( e ( R )) , and δ ( ℓ ) = (( Λ − ( M,σ ) ′ ⊗ R ( M,σ ) ′ ○ m ′ )( ℓ )) = ∑ ni = ℓ i ⊗ r i for some ℓ i , r i ∈ ( M, σ ) ′ .Thus, ℓ ( m ( u ⍟ v )) = ∑ ni = ℓ i ( u ) r i ( v ) , u, v ∈ M .Now, ((( M, σ ) ′ ) ∗ , M, E ) ∶ = ̃ Alg ( R ,τ ) (( M, σ ) ′ , δ, ǫ ) is given by M ∶ = δ ∗ ○ ( Λ ( M,σ ) ′ ⊗ R Λ ( M,σ ) ′ ) ∗ and E ∶ = ǫ ∗ ○ φ . Thus E ( R ) = ǫ ∗ ( φ ( R )) = ǫ ∗ ( id R ) = ǫ ,and given L , L ∈ (( M, σ ) ′ ) ∗ , and ℓ ∈ ( M, σ ) ′ , ( M ( L ⍟ L ))( ℓ ) = ( δ ∗ (( Λ ( M,σ ) ′ ⊗ R Λ ( M,σ ) ′ ) ∗ ( L ⍟ L )))( ℓ ) = (( Λ ( M,σ ) ′ ⊗ R Λ ( M,σ ) ′ ) ∗ ( L ⍟ L ))( δ ( ℓ )) = ( L ⍟ L )(( Λ ( M,σ ) ′ ⊗ R Λ ( M,σ ) ′ )( δ ( ℓ ))) = ( L ⍟ L )( ∑ ni = Λ ( M,σ ) ′ ( ℓ i ) ⊗ Λ ( M,σ ) ′ ( r i )) = ∑ ni = Λ ( M,σ ) ′ ( ℓ i )( L ) Λ ( M,σ ) ′ ( r i )( L ) = ∑ ni = L ( ℓ i ) L ( r i ) . (22)
67 Corollary
The equivalence from Corollary 65 restricts to an equivalencebetween the category ǫ FinDimCoalg k (resp. ǫ,coc FinDimCoalg k ) of finite-dimensional (resp. cocommutative) coalgebras and Mon ( FinDimVect k ) (resp. c Mon ( FinDimVect k ) ), where FinDimVect k = ( FinDimVect k , ⊗ k , k ) . Relationship with finite duality
Over a field, there is a standard and well-known notion of duality betweenalgebras and coalgebras, known as the finite duality [1, 7] and we have theintention to understand the relations if any, between the equivalence of cat-egories from Corollary 65 and this finite duality.Let ( − ) ∗ ∶ Mod opR → Mod R be the usual algebraic dual functor. Then, D ∗ ∶ = (( − ) ∗ , Θ , θ ) is a lax symmetric monoidal functor from Mod op R to Mod R (where Θ is as in Definition 54, and θ ∶ R → R ∗ is the isomorphism θ ( R ) = id R (and θ − ( ℓ ) = ℓ ( R ) )). When k is a field, there is the finite dual functor D fin ∶ Mon ( Vect k ) op → ǫ Coalg k (denoted by ( − ) in [1, 7]).The aforemen-tioned finite duality is the adjunction D op fin ⊣ ̃ D ∗ ∶ ǫ Coalg k → Mon ( Vect k ) (see e.g., [7, Theorem 1.5.22, p. 44], where ̃ D ∗ is denoted by ( − ) ∗ ). Let ( R , τ ) be a rigid ring. Let ( M, σ ) , ( N, γ ) be topologically-free ( R , τ ) -modules. According to Lemma 56, M × N −⍟− ÐÐ→ (
M, σ ) ⍟ ( R ,τ ) ( N, γ ) is R -bilinear. Denoting by TopFreeMod ( R ,τ ) ∥−∥ ÐÐ→
Mod R the canonical for-getful functor, this means that there is a unique R -linear map ∥( M, σ )∥ ⊗ R ∥( N, γ )∥ Ξ ( M,σ ) , ( N,γ ) ÐÐÐÐÐÐÐ→ ∥(
M, σ ) ⍟ ( R ,τ ) ( N, γ )∥ such that for each u ∈ M , v ∈ N , Ξ ( M,σ ) , ( N,γ ) ( u ⊗ v ) = u ⍟ v .
68 Lemma A ∶ = (∥ − ∥ , ( Ξ ( M,σ ) , ( N,γ ) ) ( M,σ ) , ( N,γ ) , id R ) is a lax symmetricmonoidal functor from TopFreeMod ( R ,τ ) to Mod R . Let ˜ A ∶ Mon ( TopFreeMod ( R ,τ ) ) → Mon ( Mod R ) be the functor inducedby A as introduced in Appendix A.2. Using the functorial isomorphism O ∶ Mon ( Mod R ) ≃ Alg R (Example 79, Appendix A.1), to any monoid in TopFreeMod ( R ,τ ) is associated an ordinary algebra.
69 Definition
Let us define
U A ∶ = Mon ( TopFreeMod ( R ,τ ) ) O ○ ˜ A ÐÐ→ Alg R .Given a monoid (( M, σ ) , µ, η ) in TopFreeMod ( R ,τ ) , then U A (( M, σ ) , µ, η ) = O ( ˜ A (( M, σ ) , µ, η )) is referred to as the underlying (ordinary) algebra of themonoid (( M, σ ) , µ, η ) . In details, U A (( M, σ ) , µ, η ) = ( M, µ bil , η ( R )) with µ bil ∶ M × M → M given by µ bil ( u, v ) ∶ = µ ( u ⍟ v ) .
70 Remark
Since by Lemma 68, A is symmetric, it induces a functor (seeRemark 83, Appendix A.2) c Mon ( TopFreeMod ( R ,τ ) ) ˜ A Ð→ c Mon ( Mod R ) . Be- ause one has the co-restriction c Mon ( Mod R ) O Ð→ ,c Alg R , one may considerthe underlying algebra functor U A = c Mon ( TopFreeMod ( R ,τ ) ) O ○ ˜ A ÐÐ→ ,c Alg R .
71 Example (Continuation of Example 62)
U A ( M ( R ,τ ) ( X )) = A R X . ̃ D ∗ Let ( R , τ ) be a rigid ring. Let FreeMod R E Ð→ Mod R be the canonical embed-ding functor. Since FreeMod R is a symmetric monoidal subcategory of Mod R it follows that E = ( E, id, id ) is a strict monoidal functor from FreeMod R to Mod R (see Appendix A.2).One claims that D ∗ ○ E op = A ○ Alg ( R ,τ ) . In particular, if k is a field (and τ is a ring topology on k ), then this reduces to D ∗ = A ○ Alg ( k ,τ ) .That ∥ − ∥ ○ Alg ( R ,τ ) = ( − ) ∗ ○ E op is due to the very definition of Alg ( R ,τ ) . Ofcourse, ∥ φ ∥ = θ . That for each free modules M, N , ∥( Λ M ⊗ Λ N ) ∗ ∥ ○ Ξ M ∗ ,N ∗ = Θ M,N is easy to check. So (( − ) ∗ ○ E op , Θ , θ ) = (∥ − ∥ , Ξ , id R ) ○ ( Alg ( R ,τ ) , Φ , φ ) = (∥ − ∥ ○ Alg ( R ,τ ) , (∥ Φ M,N ∥ ○ Ξ M ∗ ,N ∗ ) M,N , ∥ φ ∥) .It follows that the finite dual monoid ̃ D ∗ ( C ) of a k -coalgebra C , is equalto ̃ A ( ̃ Alg ( k ,τ ) ( C )) whatever the ring topology τ on the field k , and thus asordinary algebras, O (̃ D ∗ ( C )) = U A ( ̃
Alg ( k ,τ ) ( C )) .
72 Proposition
Let ( k , τ ) be a field with a ring topology. The functor Mon ( FreeTopVect ( k ,τ ) ) O ○ ˜ A ÐÐ→ Alg k has a left adjoint, namely ̃ Alg ( k ,τ ) ○ D op fin ○ O − . Proof:
One has ̃ D ∗ = ˜ A ○ ̃ Alg ( k ,τ ) , whence ̃ D ∗ ○ ̃( Top d ( k ,τ ) ) = ˜ A ○ ̃ Alg ( k ,τ ) ○ ̃( Top d ( k ,τ ) ) op ≃ ˜ A (natural isomorphism) by Theorem 63. Since ̃ Alg ( k ,τ ) ○ D op fin is a left adjoint of ̃ D ∗ ○ ̃( Top d ( k ,τ ) ) , it follows that it is also the left adjointof ˜ A . ◻ Let R be a ring. Let A = (( A, σ ) , µ, η ) be an object of Mon ( TopFreeMod ( R , d ) ) .One knows that ( A, σ ) is an object of TopMod ( R , d ) and U A ( A ) is an ob- In Vect k . Because if
F, G are two naturally isomorphic functors and L is a left adjoint of F ,then it is also a left adjoint of G . Alg R . Moreover, ( A, σ ) × ( A, σ ) µ bil ÐÐ→ (
A, σ ) is continuous, since it isequal to the composition ( A, σ ) × ( A, σ ) −⍟− ÐÐ→ (
A, σ ) ⍟ ( R , d ) ( A, σ ) µ Ð→ (
A, σ ) of continuous maps (see Lemma 56). Now, let (( A, σ ) , µ, η ) f Ð→ ((
B, γ ) , ν, ζ ) be a morphism in Mon ( TopFreeMod ( R , d ) ) . In particular, ( A, σ ) f Ð→ (
B, γ ) is linear and continuous, and the following diagram commutes. ( A, σ ) × ( A, σ ) f × f (cid:15) (cid:15) −⍟− + + ❱❱❱❱❱❱❱❱ µ bil ) ) ( A, σ ) ⍟ ( R , d ) ( A, σ ) f ⍟ ( R , d ) f (cid:15) (cid:15) µ / / ( A, σ ) f (cid:15) (cid:15) ( B, γ ) ⍟ ( R , d ) ( B, γ ) ν / / ( B, γ )( B, γ ) × ( B, γ ) −⍟− ❤❤❤❤❤❤ ν bil (23)Since by assumption, one also has f ○ η = ζ , it follows that f ( η ( R )) = ζ ( R ) , and thus f is a continuous algebra map from (( A, σ ) , µ bil , η ( R )) to (( B, ν bil , ζ ( R )) .)This provides a topological algebra functor Mon ( TopFreeMod ( R, d ) ) T A
ÐÐ→ TopAlg ( R , d ) and the following diagram commutes (the unnamed arrowsare either the obvious forgetful functors or the evident embedding functor),so that T A is concrete over TopMod ( R , d ) , whence faithful. Mon ( TopFreeMod ( R , d ) ) (cid:15) (cid:15) ˜ A ~ ~ T A / / TopAlg ( R , d ) (cid:15) (cid:15) x x TopFreeMod ( R , d )∥⋅∥ (cid:15) (cid:15) (cid:31) (cid:127) / / TopMod ( R , d ) s s ❣❣❣❣❣❣❣❣❣❣❣❣❣❣ Mon ( Mod R ) / / O Mod R Alg R o o (25) A concrete category C over D is a pair ( C , C U Ð→ D ) with U a faithful functor. Givenconcrete categories ( C i , U i ) , i = , , over D , by a concrete functor ( C , U ) F Ð→ ( C , U ) is meant an ordinary functor C F Ð→ C such that the following diagram commutes. Sucha functor is necessarily faithful. C F / / U ( ( ❘❘❘❘❘❘ C U v v ❧❧❧❧❧❧ D (24) When A is a commutative monoid in TopFreeMod ( R , d ) , then T A ( A ) is a commutative topological algebra.
74 Example
For each set X , T A ( M ( R , d ) ( X )) = A ( R , d ) ( X ) .
75 Proposition
T A is a full embedding (injective on objects and faithful).
Proof:
Let A = (( A, σ ) , µ, η ) and B = (( B, γ ) , ν, ζ ) be two monoids in TopFreeMod ( R , d ) . Let T A ( A ) g Ð→ T A ( B ) be a morphism in TopAlg ( R , d ) .In particular, g ∈ Alg R ( U A ( A ) , U A ( B )) ∩ Top ((∣ A ∣ , σ ) , (∣ B ∣ , γ )) . (Recallfrom Remark 5 that Mod R ∣⋅∣ Ð→ Set is the usual forgetful functor, and
Top is the category of Hausdorff topological spaces.)By assumption, for each u, v ∈ A , one has g ( µ ( u ⍟ v )) = g ( µ bil ( u, v )) = ν bil ( g ( u ) , g ( v )) = ν ( g ( u ) ⍟ g ( v )) . Thus, g ○ µ = ν ○ ( g ⍟ ( R , d ) g ) on { u ⍟ v ∶ u, v ∈ A } . Since this set spans a dense subset of ( A, σ ) ⊗ ( R,τ ) ( A, σ ) (according toCorollary 59), by linearity and continuity, g ○ µ = ν ○ ( g ⍟ ( R , d ) g ) on the wholeof ( A, σ ) ⍟ ( R , d ) ( A, σ ) .Moreover, g ( η ( R )) = ζ ( R ) , then g ○ η = ζ . Therefore, g may be seen asa morphism A f Ð→ B in Mon ( TopFreeMod ( R , d ) ) with T A ( f ) = g , i.e., T A isfull.Let A = (( A, σ ) , µ, η ) , B = (( B, γ ) , ν, ζ ) be monoids in TopFreeMod ( R , d ) such that T A ( A ) = T A ( B ) . In particular, ( A, σ ) = ( B, γ ) , and η = ζ . Byassumption µ bil = ν bil . Whence µ = ν on { u ⍟ v ∶ u ∈ A, v ∈ B } , and bycontinuity they are equal on ( A, σ ) ⍟ ( R , d ) ( B, γ ) . So A = B , i.e., T A isinjective on objects. ◻ As a consequence of Proposition 75,
Mon ( TopFreeMod ( R , d ) ) is isomor-phic to a full subcategory of TopAlg ( R , d ) ([2, Proposition 4.5, p. 49]). Ac-cordingly a monoid in TopFreeMod ( R , d ) is essentially a topological algebra.It is clear that c Mon ( TopFreeMod ( R , d ) ) T A
ÐÐ→ ,c TopAlg ( R , d ) of T A (seeRemark 73) also is a full embedding functor. D fin Let V be any vector space on a field k . Then, V ∗ has a somewhat natu-ral topology called the V -topology ([1]) or the finite topology ([7]), with afundamental system of neighborhoods of zero consisting of spaces W † ∶ = { ℓ ∈ V ∗ ∶ ∀ w ∈ W, ℓ ( w ) = } (26)32here W runs over the finite-dimensional subspaces of V . This is manifestelythe same topology as our w ∗( k , d ) (see Section 3.1). Accordingly this turns V ∗ into a linearly compact k -vector space (p. 19). The closed subspaceof ( V ∗ , w ∗( k , d ) ) are exactly the subspaces of the form W † , where W is anysubspace of V ([4, Proposition 24.4, p. 105]).
76 Lemma
Let W be a subspace of V . codim ( W † ) is finite if, and only if, dim ( W ) is finite. In this case, codim ( W † ) = dim ( W ) . Proof:
One observes that V ∗ / W † ≃ W ∗ because the map V ∗ incl ∗ W ÐÐÐ→ W ∗ is onto, where W incl W ÐÐÐ→ V is the canonical inclusion, and ker incl ∗ W = W † .Since V ∗ / W † ≃ W ∗ , it follows that codim ( W † ) = dim W ∗ . ◻
77 Theorem
Let k be a field. For each monoid A in TopFreeVect ( k , d ) , ̃( Top d ( k , d ) )( A ) is a subcoalgebra of D op fin ( ˜ A ( A )) . Furthermore, the assertionsbelow are equivalent.1. In T A ( A ) every finite-codimensional ideal is closed.2. ˜ A ( A ) is reflexive .3. The coalgebra ̃( Top d ( k , d ) )( A ) is coreflexive .4. ̃( Top d ( k , d ) )( A ) = D op fin ( ˜ A ( A )) . Proof:
Let A = (( A, σ ) , µ, η ) be a monoid in TopFreeMod ( k , d ) . Whenceits underlying topological vector space is a linearly compact vector space(p. 19). Let C ∶ = ̃( Top d ( k , d ) )( A ) . Since ˜ A ○ ̃ Alg ( k , d ) = ̃ D ∗ it follows that ˜ A ≃ ˜ A ○ ̃ Alg ( k , d ) ○ ̃( Top d ( k , d ) ) ≃ ̃ D ∗ ○ ̃( Top d ( k , d ) ) (naturally isomorphic). Inparticular, ˜ A ( A ) ≃ ̃ D ∗ ( C ) . By construction, the underlying topological vectorspace of A , namely ( A, σ ) , is also the underlying topological vector space of T A ( A ) . Also A , T A ( A ) and ˜ A ( A ) share the same underlying vector space A , which is isomorphic to C ∗ , where C is the underlying vector space of thecoalgebra C . Of course, ( A, σ ) ≃ Alg ( k , d ) ( C ) = ( C ∗ , w ∗( k , d ) ) . Therefore, upto such an isomorphism, ( A, σ ) has a fundamental system of neighborhoods A monoid A in Vect k is reflexive when A ≃ ̃ D ∗ ( D op fin ( A )) under the linear map u ↦ ( ℓ ↦ ℓ ( u )) , which is the unit of the adjunction D op fin ⊣ ̃ D ∗ . A coalgebra C is coreflexive , when C ≃ D op fin (̃ D ∗ ( C )) under the natural inclusion u ↦ ( ℓ ↦ ℓ ( u )) , which is the counit of D op fin ⊣ ̃ D ∗ .
33f zero consisting of V † = { ℓ ∈ A ∶ ∀ v ∈ V, ℓ ( v ) = } where V is a finite-dimensional subspace of C (see Eq. (26)).Let ℓ ∈ ( A, σ ) ′ . By continuity of ℓ , there exists a finite-dimensionalsubspace V of C such that V † ⊆ ker ℓ . Let B be a (finite) basis of V , and let D be the (necessarily finite-dimensional, by [11, Thm 1.3.2, p. 21]) subcoalgebraof C it generates. Then, V ⊆ D , which implies that D † ⊆ V † ⊆ ker ℓ . But D † is a finite-codimensional ideal of ˜ A ( A ) (by Lemma 76 and [1, Theorem 2.3.1,p. 78]), whence ℓ ∈ A . It remains to check that the above inclusion incl ( A,σ ) ′ is a coalgebramap from ̃( Top d ( k , d ) )( A ) to D op fin ( ˜ A ( A )) , which would equivalently mean that ( A, σ ) ′ is a subcoalgebra of D op fin ( ˜ A ( A )) . One thus needs to make explicitthe two coalgebra structures so as to make possible a comparison. By con-struction the comultiplication of ̃( Top d ( k , d ) )( A ) is given by the composition Λ − ( A,σ ) ′ ⊗ k ( A,σ ) ′ ○ µ ′ . So for ℓ ∈ ( A, σ ) ′ , ( Λ − ( A,σ ) ′ ⊗ k ( A,σ ) ′ ○ µ ′ )( ℓ ) = ∑ ni = ℓ i ⊗ r i ,for some ℓ i , r i ∈ ( A, σ ) ′ . Therefore, given ℓ ∈ ( A, σ ) ′ , u, v ∈ A , ℓ ( µ ( u ⍟ v )) = ∑ ni = ℓ i ( u ) r i ( v ) . The counit of ̃( Top d ( k , d ) )( A ) is ( A, σ ) ′ η ′ Ð→ ( k , d ) ′ ψ − ÐÐ→ k ,i.e., ℓ ↦ ℓ ( η ( k )) . It follows easily, from the explicit description of D fin ( B ) provided in [7, p. 35], for a monoid B in Vect k , that the above comultipli-cation coincides with that of D fin ( ˜ A ( A )) , and because it is patent that thecounit of ̃( Top d ( k , d ) )( A ) is the restriction of that of D op fin ( ˜ A ( A )) , ( A, σ ) ′ is asubcoalgebra of D op fin ( ˜ A ( A )) .It remains to prove the equivalence of the four assertions given in thestatement. 2 ⇔ ˜ A ( A ) ≃ ̃ D ∗ ( ̃( Top d ( k , d ) )( A )) .The coalgebra C ∶ = ̃( Top d ( k , d ) )( A ) is coreflexive if, and only if, every finite-codimensional ideal of ̃ D ∗ ( C ) ≃ ˜ A ( A ) is closed in the finite topology of C ∗ ([1, Lemma 2.2.15, p. 76]), which coincides with our topology w ∗( k , d ) , andthus it turns out that (̃ D ∗ ( C ) , w ∗( k , d ) ) ≃ T A ( A ) (since C ∗ under the finitetopology is equal to Alg ( k , d ) ( C ) ≃ ( A, σ ) by functoriality). Whence 3 ⇔ T A ( A ) every finite-codimensional ideal is closed.Let ℓ ∈ D fin ( ˜ A ( A )) . By definition ker ℓ contains a finite-codimensional idealsay I of ˜ A ( A ) . Since I is closed, there exists a finite-dimensional subspace A ∶ = { ℓ ∈ A ∗ ∶ ker ℓ contains a finite-codimensional (two-sided) ideal of UA ( A ) } is theunderlying vector space of the finite dual coalgebra D fin ( ˜ A ( A )) . of C such that D † = I (since the closed subspaces are of the form D † fora subspace D of C and by Lemma 76, codim ( I ) = codim ( D † ) = dim ( D ) ),whence I is open, which shows that ( A, σ ) ℓ Ð→ ( k , d ) is continuous so 1 ⇒ C ∶ = ̃( Top d ( k , d ) )( A ) = D op fin ( ˜ A ( A )) , so that ˜ A ( A ) ≃ ̃ D ∗ ( C ) , as above.Whence C = D op fin ( ˜ A ( A )) ≃ D op fin (̃ D ∗ ( C )) . This is not sufficient to ensurecoreflexivity of C , since there is at this stage no guaranty that the aboveisomorphism corresponds to the counit of the adjunction D op fin ⊣ ̃ D ∗ (seeFootnote 39). One knows from the beginning of the proof that ˜ A ( ˜Γ A ) ∶ A ( A ) ≃ ˜ A ( ̃ Alg ( k , d ) ( ̃( Top d ( k , d ) )( A ))) which, in this case where ̃( Top d ( k , d ) )( A ) = D op fin ( ˜ A ( A )) , is the isomorphism ∥ Γ ( A,σ ) ∥ ∶ A ≃ (( A, σ ) ′ ) ∗ = ( A ) ∗ , u ↦ ( ℓ ↦ ℓ ( u )) . So ˜ A ( A ) is reflexive, andthus its finite dual coalgebra C is coreflexive. Thus 4 ⇒ ◻
78 Example
Let R be a ring. Let C R X = ( R ( X ) , d X , e X ) be the group-likecoalgebra on X , i.e., d X ( δ x ) = δ x ⊗ δ x , and e X ( δ x ) = R , x ∈ X . The followingdiagram commutes (this may be checked by hand) for a rigid ring ( R , τ ) . (( R, τ ) X ) ′ λ X (cid:15) (cid:15) µ ′ X / / (( R, τ ) X ⍟ ( R ,τ ) ( R, τ ) X ) ′ R ( X ) d X / / R ( X ) ⊗ R R ( X ) λ − X ⊗ R λ − X / / (( R, τ ) X ) ′ ⊗ R (( R, τ ) X ) ′ Λ (( R,τ ) X )′⊗ R (( R,τ ) X )′ O O (27) Moreover η ′ X ( ℓ ) = ψ ( e X ( λ X ( ℓ ))) for each ℓ ∈ (( R, τ ) X ) ′ . All of this showsthat λ X ∶ ̃( Top d ( R ,τ ) )( M ( R ,τ ) ( X )) ≃ C R X is an isomorphism of coalgebras.Let k be a field. It follows from Theorem 77 and Example 74 that in A ( k , d ) ( X ) every finite-codimensional ideal is closed if, and only if C k X iscoreflexive if, and only if, { π x ∶ x ∈ X } = Alg k ( A k ( X ) , k ) ([17, Corol-lary 3.2, p. 528]). This holds in particular if ∣ X ∣ ≤ ∣ k ∣ (see [17, Corollary 3.6,p. 529]). If k is a finite field, then C k ( X ) is coreflexive if, and only if, X isfinite (see [17, Remark 3.7, p. 530]). References [1] Abe, E. (2004).
Hopf algebras (Vol. 74). Cambridge University Press.[2] Adámek, J., Herrlich, H., & Strecker, G. E. (2004). Abstract and con-crete categories. The joy of cats.353] Aguiar, M., & Mahajan, S. A. (2010).
Monoidal functors, species andHopf algebras (Vol. 29). Providence, RI: American Mathematical Soci-ety.[4] Bergman, G. M., & Hausknecht, A. O. (1996).
Cogroups and co-ringsin categories of associative rings (No. 45). American Mathematical Soc.[5] Bourbaki, N. (1998).
Algebra. I. Chapters 1-3 . Springer-Verlag, Berlin.[6] Bourbaki, N. (2013).
General Topology: Chapters 1-4 (Vol. 18). SpringerScience & Business Media.[7] Dăscălescu, S., Năstăsescu, C., & Raianu, Ş. (2001).
Hopf Algebras .Monographs and Textbooks in Pure and Applied Mathematics, 401.[8] Dieudonné, J. (1951). Linearly compact spaces and double vector spacesover sfields.
American Journal of Mathematics, 73 (1), 13-19.[9] Kelley, J. L. (1955).
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Introduction to the quantumYang-Baxter equation and quantum groups: an algebraic approach (Vol.423). Springer Science & Business Media.[12] Lambek, J., & Scott, P. J. (1988).
Introduction to higher-order categor-ical logic (Vol. 7). Cambridge University Press.[13] MacLane, S. (1998).
Categories for the working mathematician (Vol. 5).Springer Science & Business Media.[14] Poinsot, L. (2015). Rigidity of topological duals of spaces of formal serieswith respect to product topologies.
Topology and its Applications , ,147-175.[15] Poinsot, L., & Porst, H.-E. (2015). Free monoids over semigroups in amonoidal category: Construction and applications. Communications inAlgebra , (11), 4873-4899.[16] Porst, H. E. (2015). The formal theory of hopf algebras Part I: Hopfmonoids in a monoidal category. Quaestiones Mathematicae , (5),631-682. 3617] Radford, D. E. (1973). Coreflexive coalgebras. Journal of Algebra, 26 (3), 512-535.[18] Street, R. (2007).
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Topological rings (Vol. 178). Elsevier.
A Monoidal categories and functors
This appendix contains basic facts about monoidal categories and monoidalfunctors, and a part of it is reprinted from [15]. See [13, Chap. VII] for moredetails.Throughout C = ( C , − ⊗ − , I, α, λ, ρ ) = ( C , − ⊗ − , I ) (resp. C = ( C , − ⊗− , I, α, λ, ρ, σ ) ) denotes a (resp. symmetric) monoidal category with α theassociativity and λ and ρ the left and right unit constraints (resp., and σ the symmetry), referred to as coherence constraints . These constraints haveto make commute some diagrams to ensure coherence of the (resp. symmet-ric) monoidal category (see [13, Chap. VII, p. 165]). If C is a (symmetric)monoidal category, then so is C op ∶ = ( C op , −⊗ op − , I, ( α − ) op , ( ̺ − ) op , ( λ − ) op ) ,called the dual monoidal category of C . A.1 Monoids and comonoids A monoid in C is a triple ( C, C ⊗ C m Ð→ C, I e Ð→ C ) such that the diagrams ( C ⊗ C ) ⊗ C α C,C,C (cid:15) (cid:15) m ⊗ id C / / C ⊗ C m (cid:15) (cid:15) C ⊗ ( C ⊗ C ) id C ⊗ m (cid:15) (cid:15) C ⊗ C m / / C C ⊗ I id C ⊗ e / / ρ C % % ❑❑❑❑❑❑❑❑❑❑ C ⊗ C m (cid:15) (cid:15) I ⊗ C e ⊗ id C o o λ C y y ssssssssss C commute, while a monoid morphism ( C, m, e ) Ð→ ( C ′ , m ′ , e ′ ) is any f ∶ C → C ′ making the diagrams C ⊗ C m / / f ⊗ f (cid:15) (cid:15) C f (cid:15) (cid:15) C ′ ⊗ C ′ m ′ / / C ′ I e / / e ′ (cid:31) (cid:31) ❅❅❅❅❅❅❅ C f (cid:15) (cid:15) C ′ Mon C of monoids in C .The category Comon C of comonoids over C is defined to be ( Mon C op ) op ,the opposite of the category of monoids in C op .A monoid ( C, m, e ) is called commutative if, and only if, m = m ○ σ C,C with σ C,C ∶ C ⊗ C → C ⊗ C the symmetry; dually, a comonoid ( C, µ, ǫ ) is called co-commutative , provided that µ = σ C,C ○ µ . By c Mon C and coc Comon C we de-note the categories of commutative monoids and cocommutative comonoidsrespectively, with all (co)monoid morphisms as morphisms. Of course, onehas coc Comon C = ( c Mon C op ) op .
79 Example Mon ( Set ) is (isomorphic to) the category of monoids,where Set = ( Set , × , ) ( ∶ = { ∅ } ).2. Let Mod R be the monoidal category of R -modules and R -linear maps fora commutative unital ring R with its usual tensor product ⊗ R .(a) Mon ( Mod R ) is isomorphic to the category Alg R of “ordinary”unital R -algebras under the functor O , concrete over Mod R , suchthat O ( A ) ∶ = ( A, m A , A ) , with m A ( x, y ) ∶ = µ ( x ⊗ y ) , x, y ∈ A , and A ∶ = η ( R ) , where A = ( A, µ A , η A ) is a monoid in Mod R .(b) Likewise c Mon ( Mod R ) ≃ ,c Alg R under the (co-)restriction of theabove functor O .(c) Comon ( Mod R ) is the category ǫ Coalg R of counital R -coalgebras([1, 7]), and the category of cocommutative coalgebras ǫ,coc Coalg R is equal to coc Comon ( Mod R ) . A.2 Monoidal functors and their induced functors
We briefly recall the following definitions and facts, too, which are funda-mental for this note. See e.g. [3, 16, 18] for a more detailed treatment andfor the missing proofs.
80 Definition
Let C = ( C , − ⊗ − , I ) and C ′ = ( C ′ , − ⊗ ′ − , I ′ ) be monoidalcategories. A (lax) monoidal functor from C to C ′ is a triple F ∶ = ( F, Φ , φ ) ,where F ∶ C → C ′ is a functor, Φ C ,C ∶ F C ⊗ ′ F C → F ( C ⊗ C ) is a naturaltransformation and φ ∶ I ′ → F I is a C -morphism, subject to certain coherenceconditions. A lax monoidal functor is called strong monoidal (resp. strictmonoidal ), if Φ and φ are isomorphisms (resp. identities). Φ , φ are the coherence constraints of F . C , C ′ be symmetric. A monoidal functor F ∶ C → C ′ is said to be sym-metric when furthermore it satisfies yet another coherence condition relativeto the symmetry constraints.
81 Facts
Let C ∶ = ( C , ⊗ , I ) , C ′ ∶ = ( C ′ , ⊗ ′ , I ′ ) and C ′′ ∶ = ( C ′′ , ⊗ ′′ , I ′′ ) be(symmetric) monoidal categories.1. id C ∶ = ( id C , id −⊗− , id I ) , or simply id , is a monoidal functor from C toitself, and serves as a unit for the composition of monoidal functorsgiven below.2. Given monoidal functors F = ( F, Φ , φ ) ∶ C → D and G = ( G, Ψ , ψ ) ∶ D → E ,one defines a monoidal functor G ○ F ∶ = H = ( H, Θ , θ ) ∶ C → E with(a) H = G ○ F .(b) Given objects C , C of C , Θ C ,C is the composite C ′′ -morphism G ( F ( C )) ⊗ ′′ GF ( C ) Ψ F ( C ) ,F ( C ) ÐÐÐÐÐÐÐ→ G ( F ( C ) ⊗ ′ F ( C )) G ( Φ C ,C ) ÐÐÐÐÐÐ→ G ( F ( C ⊗ C )) .(c) θ ∶ = I ′′ ψ Ð→ G ( I ′ ) G ( φ ) ÐÐÐ→ G ( F ( I )) . H is strong (resp. symmetric) when F, G so are.
82 Proposition
Let F = ( F, Φ , φ ) ∶ C → C ′ be a monoidal functor. ˜ F ( M, m, e ) = ( F M, F M ⊗ F M Φ M,M
ÐÐÐ→ F ( M ⊗ M ) F m
ÐÐ→
F M, I ′ φ Ð→ F I
F e Ð→ F M ) and ˜ F f = F f define an induced functor ˜ F ∶ Mon C → Mon C ′ , such thatthe diagram below commutes (with forgetful functors U m and U ′ m ). Mon C ˜ F / / U m (cid:15) (cid:15) Mon C ′ U ′ m (cid:15) (cid:15) C F / / C ′ (28)
83 Remark
1. When F is symmetric, then ˜ F also provides a functor c Mon C to c Mon C ′ with similar properties as above.2. ̃ id C = id Mon ( C ) and ̃ G ○ F = ˜ G ○ ˜ F . A strong monoidal functor F = ( F, Φ , φ ) ∶ C → C ′ may be considered as astrong monoidal functor F d ∶ = ( F op , ( Φ − ) op , ( φ − ) op ) ∶ C op → ( C ′ ) op , the dualof F ([16, Proposition 17, p. 639]). 39 Let F = ( F, Φ , φ ) and G = ( G, Ψ , ψ ) be monoidal functorsfrom C to C ′ . A natural transformation α ∶ F ⇒ G ∶ C → C ′ is a monoidaltransformation α ∶ F ⇒ G ∶ C → C ′ when the following diagrams commute, forevery C -objects C , C . F C ⊗ ′ F C α C ⊗ ′ α C / / Φ C ,C (cid:15) (cid:15) GC ⊗ ′ GC C ,C (cid:15) (cid:15) I ′ φ / / ψ $ $ ■■■■■■ F I α I (cid:15) (cid:15) F ( C ⊗ C ) α C ⊗ C / / G ( C ⊗ C ) GI (29)
85 Remark
1. Let α ∶ F ⇒ G ∶ C → C ′ be a monoidal transformation. Itinduces ˜ α ∶ ˜ F ⇒ ˜ G ∶ Mon C → Mon C ′ with ˜ α ( C,m,e ) ∶ = α C .2. When the monoidal functors and categories are symmetric, then α alsoinduces ˜ α ∶ ˜ F ⇒ ˜ G ∶ c Mon C → c Mon C ′ [3, Prop. 3.38].3. Let α ∶ F ⇒ G ∶ C → C ′ be a monoidal transformation between strongmonoidal functors, then α op ∶ G op → F op ∶ C op → C ′ op also provides amonoidal transformation α d ∶ G d → F d ∶ C op → C ′ op , called the dual of α . A monoidal isomorphism is a monoidal transformation which is a naturalisomorphism.
86 Remark α − ∶ G ⇒ F ∶ C → C ′ is a monoidal isomorphism when so is α . Accordingly, the induced natural transformation ˜ α ∶ ˜ F ⇒ ˜ G ∶ Mon ( C ) → Mon ( C ′ ) is a natural isomorphism with inverse ( ˜ α ) − = ̃( α − ) . A monoidal equivalence of monoidal categories is given by a monoidalfunctor F ∶ C → C ′ such that there are a monoidal functor G and monoidalisomorphisms η ∶ id ⇒ G ○ F and ǫ ∶ F ○ G ⇒ id . C , C ′ are said monoidallyequivalent .
87 Remark If F is a monoidal equivalence, then ˜ F is an equivalence betweenthe corresponding categories of monoids.
88 Remark
Let F , G ∶ C ′ → C be strong monoidal functors. Let η ∶ id ⇒ G ○ F and ǫ ∶ F ○ G ⇒ id be monoidal isomorphisms. Whence η d ∶ F d ○ G d = ( G ○ F ) d ⇒ id dC = id C op and ǫ d ∶ id C ′ op = ( id C ′ ) d ⇒ G d ○ F d are also monoidal isomorphisms,and this provides a monoidal equivalence between C op and C ′ op ..