Draft lesson.

Pro representations and continuous duality

Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).

A compatible family of finite-dimensional quotients carries a topology: knowing a vector means knowing each of its finite images. Duality changes those quotients into a union of finite-dimensional subrepresentations. This lets us identify every continuous map, prove that its image is closed, and determine which inverse limits preserve exact sequences. The group action imposes a real condition: the finite quotients must be invariant, and continuous operators alone do not guarantee that condition.

Prerequisites: Ind representations and the locally finite coreflector, especially §§2–5; Formal colimits and compact presentations, §§1–2; and the power-series and principal-parts arguments in Finite support and the limits that a category permits, §1 and Exercise 2.

1. The topology and the two categories

Fix a field \(k\) with its discrete topology and a small group \(G\), also discrete. Representations are left representations. Write \(\mathsf{LF}(G)\) for the category of representations in which every vector lies in a finite-dimensional invariant subspace. Its morphisms are all equivariant linear maps. The first prerequisite proves that this is an abelian category, that its filtered colimits are exact, and that its compact objects are precisely its finite-dimensional objects. Its finite-dimensional subcategory has a small skeleton \(\mathcal A\).

A linear topology has a neighbourhood basis of zero consisting of vector subspaces. In this lesson, a linearly compact space means a Hausdorff linear topological space \(V\) with a basis of open subspaces of finite codimension, for which the canonical map

is a topological isomorphism. Here the limit runs over those subspaces, quotients are discrete, and the limit has its subspace topology in their product. Finite intersections make the index cofiltered. Thus completeness means that every compatible family of finite images comes from exactly one vector. In §2 we prove that these spaces are exactly the products of copies of discrete \(k\), up to a topological linear isomorphism. Linear compactness does not imply topological compactness when \(k\) is infinite: the one-dimensional discrete space \(k\) already demonstrates this.

Let \(\mathsf L(G)\) have as objects linearly compact spaces with a linear \(G\)-action and a neighbourhood basis consisting of invariant open subspaces of finite codimension. Its morphisms are continuous equivariant linear maps. Each group operator is a homeomorphism: an invariant basic subspace is preserved by it and its inverse. Equivalently, the action of the discrete group is continuous. The extra invariant-basis requirement is stronger than this continuity statement.

For a discrete vector space \(M\), give its full algebraic dual \(D(M)=\operatorname{Hom}_k(M,k)\) the finite topology, whose basic subspaces are

Restriction to \(E\) is surjective: extend a basis of \(E\) to a basis of \(M\), and extend any functional by zero on the added basis vectors. Consequently its kernel is (1.2), which has codimension \(\dim E\). Distinct functionals differ on a vector, so the topology is Hausdorff. Restrictions give a topological isomorphism \(D(M)\simeq\varprojlim_E E^*\): its inverse assigns to a compatible family the functional obtained by evaluating a vector in any finite subspace containing it. Compatibility gives independence and linearity. The kernels of these projections are exactly (1.2), proving the topology assertion and completeness. Every finite-codimension open subspace contains one of these kernels, so this also gives (1.1) over all such subspaces.

Choosing a basis \((m_i)_{i\in I}\) identifies \(D(M)\) with \(k^I\), by its values on the basis. Every finite subspace of \(M\) lies in the span of finitely many basis vectors, so (1.2) gives exactly the product topology. This description is independent of the chosen basis because (1.2) was defined without one.

For a representation, put \((g\lambda)(m)=\lambda(g^{-1}m)\). Precomposition makes every group operator continuous: the inverse image of \(\operatorname{ann}(E)\) is \(\operatorname{ann}(g^{-1}E)\). If \(M\) is locally finite, every finite subspace is contained in a finite-dimensional invariant subspace, by summing the finite orbit spans of a finite basis. Its annihilator is invariant. Thus \(D(M)\) belongs to \(\mathsf L(G)\).

An equivariant \(f\colon M\to N\) gives \(D(f)\colon D(N)\to D(M)\), \(\lambda\mapsto\lambda f\). This map is continuous because the inverse image of \(\operatorname{ann}(E)\) is \(\operatorname{ann}(fE)\). The formula commutes with group actions and reverses composition, giving a contravariant functor on all morphisms.

2. Continuous duals and both evaluation inverses

For \(V\in\mathsf L(G)\), put \(C(V)=\operatorname{Hom}_{k,\mathrm{cont}}(V,k)\), with its discrete vector-space structure and the same contragredient group action. A linear functional is continuous exactly when its kernel is open. That kernel contains an invariant basic subspace \(U\), so the functional factors through the finite-dimensional representation \(V/U\). Hence

The union is directed: replacing two subspaces by their intersection includes both duals. Each summand is invariant and finite-dimensional, proving local finiteness. A continuous equivariant map \(F\colon V\to W\) gives \(C(F)(\lambda)=\lambda F\); continuity of the composite and the group formula show that this is a morphism in \(\mathsf{LF}(G)\).

Theorem 2.1. Evaluation gives natural isomorphisms

where the second is a topological isomorphism. They make \(D\) and \(C\) inverse contravariant equivalences between \(\mathsf{LF}(G)\) and \(\mathsf L(G)\).

Proof. For \(m\in M\), evaluation at \(m\) is continuous by (1.2). Functionals separate vectors, so \(e_M\) is injective. Any continuous \(\phi\colon D(M)\to k\) kills \(\operatorname{ann}(E)\) for some finite-dimensional \(E\). Since restriction \(D(M)\to E^*\) is surjective, \(\phi\) is a functional on \(E^*\). Finite-dimensional biduality identifies it with evaluation at a unique \(e\in E\). Thus \(\phi=e_M(e)\), proving surjectivity and giving the inverse explicitly. Finite biduality here follows immediately from a basis and its dual basis.

For \(V\), if \(v\ne0\), Hausdorffness gives a basic \(U\) not containing \(v\). A functional on the finite quotient \(V/U\) detects its nonzero image, so \(e_V\) is injective. Given an arbitrary algebraic functional \(a\colon C(V)\to k\), restrict it to each \((V/U)^*\). Finite biduality gives a vector \(v_U\in V/U\). If \(U'\subset U\), evaluation against every functional on \(V/U\) shows that \(v_{U'}\) maps to \(v_U\). Completeness gives a unique \(v\in V\) with these images. Every continuous functional belongs to some summand of (2.1), so \(a(\lambda)=\lambda(v)\) for every \(\lambda\). This constructs the inverse to \(e_V\).

For a finite subspace \(E\subset C(V)\), the inverse image under \(e_V\) of \(\operatorname{ann}(E)\) is the intersection of the kernels of a finite basis of \(E\), and is open. Conversely \(U\) is precisely the common kernel of \((V/U)^*\), by finite-dimensional separation. Thus every basic open subspace is the inverse image of a basic finite-topology subspace, proving that the inverse is continuous as well.

Both evaluations are equivariant: for example \((g e_V(v))(\lambda)=e_V(v)(g^{-1}\lambda)=\lambda(gv)\). For a map \(f\) or \(F\), evaluation of either naturality composite at \(\lambda\) is respectively \(\lambda(fm)\) or \(\lambda(Fv)\). These are equal, giving naturality on all maps. The two triangular composites

are also identities: the first sends \(\lambda\) to the functional whose value at \(m\) is \(e_M(m)(\lambda)=\lambda(m)\); the second sends \(\lambda\) to the functional whose value at \(v\) is \(e_V(v)(\lambda)=\lambda(v)\). This proves the equivalence with its actual inverse maps. \(\square\)

The same proof with the group omitted shows that every space in (1.1) is the full dual of its continuous dual, hence a product of copies of discrete \(k\). Notice that (2.2) uses the continuous dual of \(D(M)\); it does not identify an infinite-dimensional space with its algebraic double dual.

3. Every Pro presentation and every continuous morphism

Define \(\operatorname{Pro}(\mathcal A)=\operatorname{Ind}(\mathcal A^{\mathrm{op}})^{\mathrm{op}}\). Its objects can be represented by small cofiltered diagrams \((X_i)\) of finite-dimensional representations, and its morphisms have the formula

The inner index is filtered because precomposition reverses the cofiltered arrows. Finite-dimensional duality \(X\mapsto X^*\), with the contragredient action, is an exact equivalence \(\mathcal A^{\mathrm{op}}\simeq\mathcal A\), with finite evaluation as its inverse. Exactness follows by extending a functional from a subspace; kernel and image are then the corresponding annihilators.

For a Pro diagram, set

There is a canonical topological equivariant isomorphism \(D(M_X)\simeq\mathcal R(X)\). A functional on the colimit restricts to compatible functionals on all \(X_i^*\), which finite evaluation identifies with vectors \(x_i\in X_i\). Conversely a compatible family defines a functional on a stage representative by \([\lambda_i]\mapsto\lambda_i(x_i)\). The colimit relations give well-definedness, and these operations are inverse.

To check topology, let \(p_i\colon\mathcal R(X)\to X_i\) be a projection. Its kernel corresponds to the annihilator of the image of \(X_i^*\) in \(M_X\). That image is finite-dimensional. Conversely every finite-dimensional subspace of \(M_X\) lies in the image of one stage: lift a finite basis to finitely many stages and take a common filtered successor. The kernels of the \(p_i\) are therefore cofinal among basic finite-topology subspaces. Finite intersections of projection kernels contain the kernel of one common cofiltered predecessor, so they also form a basis for the limit topology. This proves the assertion without assuming that any transition or projection is surjective. In particular \(\mathcal R(X)\in\mathsf L(G)\), and Theorem 2.1 gives \(C(\mathcal R(X))\simeq M_X\).

Theorem 3.1. Realization is an equivalence \(\operatorname{Pro}(\mathcal A)\simeq\mathsf L(G)\). Its inverse represents \(V\) by its invariant finite quotients \((V/U)\). On all morphisms, realization identifies (3.1) with \(\operatorname{Hom}_{\mathsf L(G)}(\mathcal R(X),\mathcal R(Y))\).

Proof. The finite quotients have surjective transitions and their actual canonical limit is \(V\), by (1.1). A small neighbourhood basis closed under finite intersections suffices: all its subspaces form a set, and compatibility over this basis determines compatibility over every basic quotient. Thus every object is realized.

For full faithfulness, Theorem 2.1 and the compactness theorem for \(\mathsf{LF}(G)\) give inverse bijections

Here is the map and inverse behind the formula. A continuous equivariant \(F\colon\mathcal R(X)\to\mathcal R(Y)\), followed by \(p_j\), dualizes to a map \(Y_j^*\to M_X\). Since \(Y_j^*\) is compact in the locally finite category, this map lifts equivariantly to some \(X_i^*\). Finite duality gives \(f_{ij}\colon X_i\to Y_j\), and \(p_jF=f_{ij}p_i\), as can be tested against all functionals on \(Y_j\). Two lifts agree after passage to a later filtered stage, so their Pro classes agree. Compactness includes this equality test even for general filtered categories with parallel arrows. The compatibility of the \(p_jF\) gives the outer-limit compatibility.

Conversely each class represented by \(f_{ij}\) gives a continuous equivariant map \(f_{ij}p_i\colon\mathcal R(X)\to Y_j\). A later-stage representative gives the same map; the Pro equality relations therefore make it well-defined. Compatibility in \(j\) gives a cone and hence a unique map into \(\mathcal R(Y)\). It is continuous because every target coordinate is continuous, and equivariant because every coordinate is equivariant. The two constructions are inverse by their equality on every target projection, or equivalently by the inverse Hom maps in (3.3). Identity and composite maps have the same projections as the realized identity and composite classes; thus realization is a functor and the bijections respect composition. The finite-quotient inverse, together with these Hom inverses, gives the stated equivalence. \(\square\)

The lifting argument takes place in \(\mathsf{LF}(G)\), where finite-dimensional objects are compact for every small group. It does not require their compactness in the category of all representations. Nor does it assert that a map on the image of \(p_i\) extends equivariantly to \(X_i\).

For an explicit reason why the chosen stage may matter, let \(G=C_p\) and \(\operatorname{char}k=p\). Set \(X_0=k[C_p]\), \(X_n=k\) with trivial action for \(n\geq1\), send \(1\in X_1\) to \(N=\sum_{g\in C_p}g\in X_0\), and use identity transitions thereafter. Its limit is \(k\), with projection to stage zero equal to this nonzero norm inclusion. Every equivariant \(k[C_p]\to k\) is a multiple of augmentation, since its value on every group basis vector equals its value on \(1\). Augmentation kills \(N\), because \(p=0\) in \(k\). The identity map from the limit to \(k\) therefore cannot factor through stage zero, although it factors through stage one. This is fully consistent with (3.3).

4. Closed images, quotient topology and exactness

For an algebraic subspace \(K\subset N\), restriction \(D(N)\to D(K)\) is a surjection with kernel \(K^\perp\). It is an open map for the finite topologies. Indeed, for every finite-dimensional \(E\subset N\),

One inclusion is immediate. For the other, extend a functional on \(K\) vanishing on \(E\cap K\) to \(K+E\) by \(\lambda(k+e)=\lambda(k)\). This is well-defined because the two expressions differ by a vector in \(E\cap K\). Extend further to \(N\) by a basis. It vanishes on \(E\), proving (4.1). Translates of these basic open subspaces prove openness. Thus the ordinary quotient topology on \(D(N)/K^\perp\) is exactly the finite topology on \(D(K)\).

Similarly, the full dual of the algebraic quotient \(N/K\) embeds as \(K^\perp\subset D(N)\), with its subspace topology. It is closed because it is an intersection of kernels of continuous evaluations at vectors of \(K\). For a finite \(E\subset N\), its intersection with \(\operatorname{ann}(E)\) is the annihilator of the image of \(E\) in \(N/K\). Every finite subspace of \(N/K\) has a finite-dimensional lift in \(N\), so these are exactly the basic subspace neighbourhoods. If \(N\) is locally finite and \(K\) invariant, its subspace and quotient are locally finite by the first prerequisite; all these dual spaces therefore belong to \(\mathsf L(G)\).

Let \(F\colon V\to W\) be a morphism in \(\mathsf L(G)\), and put \(M=C(V)\), \(N=C(W)\), \(h=C(F)\colon N\to M\). Under evaluation, \(F=D(h)\). Its actual image is

The forward inclusion follows by evaluating at an element of \(\ker h\). Conversely if \(w\) annihilates \(\ker h\), define a functional on \(\operatorname{im}h\) by \(h(\lambda)\mapsto\lambda(w)\). It is well-defined by the hypothesis. Extend it linearly to \(M\), obtaining \(v\in D(M)=V\). Then \(\lambda(Fv)=h(\lambda)(v)=\lambda(w)\) for every \(\lambda\in N\); separation proves \(Fv=w\). This is a scalar-functional extension, and assumes no equivariant splitting of \(h\).

The image is consequently closed. The kernel is \((\operatorname{im}h)^\perp\subset D(M)\). The preceding topology calculations identify the kernel and cokernel as

The first has the subspace topology, and the second has the ordinary quotient topology, already Hausdorff and complete. The factorization through the image is dual to \(N\twoheadrightarrow\operatorname{im}h\hookrightarrow M\): it factors \(F\) as the open surjection \(D(M)\twoheadrightarrow D(\operatorname{im}h)\) followed by the closed embedding \(D(\operatorname{im}h)\hookrightarrow D(N)\). Hence the canonical coimage-to-image map is a topological isomorphism. Every morphism is strict in this sense.

These are also the categorical kernel and cokernel. Equivalently, the additive equivalence \(\mathsf L(G)\simeq\mathsf{LF}(G)^{\mathrm{op}}\) transports their universal properties and abelianity. Thus \(\mathsf L(G)\) is abelian; its monomorphisms are closed embeddings and its epimorphisms are open surjections. Exact sequences are precisely the algebraically exact sequences whose subspace and quotient topologies are the ones just described. Full duality over a field is algebraically exact because every functional on a subspace extends, which proves the surjectivity required in dual short exact sequences.

5. Products, coproducts and exact cofiltered limits

For a family \(V_s=D(M_s)\), with \(M_s\in\mathsf{LF}(G)\), duality gives

Here \(r\) is the locally finite coreflector from the first prerequisite. The first expression is the ordinary topological product: a functional on the algebraic direct sum is a family of functionals on its summands, and a finite subspace of that sum has finite support. This identifies both topology and projections. Every family of continuous maps to the factors gives its coordinatewise continuous map to the product. Invariant finite-codimension neighbourhoods are obtained by imposing such conditions in finitely many factors, so the product belongs to \(\mathsf L(G)\).

For the second expression the structural injections are the duals of the intrinsic product projections \(r(\prod M_s)\to M_s\). Given continuous maps \(V_s\to W=D(N)\), their dual maps \(N\to M_s\) form an ambient product map. It factors uniquely through \(r(\prod M_s)\) since \(N\) is locally finite. Dualizing gives the unique map from the proposed coproduct to \(W\). The evaluation triangles show that restriction to every summand recovers the original maps. Thus (5.1) includes the actual universal maps, and the coproduct need not be the algebraic direct sum.

All small limits in \(\mathsf L(G)\) are the ordinary topological linear limits. Indeed products were just proved to agree. The equalizer of two maps is the closed kernel of their difference, with its ordinary subspace topology by §4. For a small diagram its limit is the equalizer of the two maps from the product over objects to the product over arrows, one using source maps and the other target projections. Their ordinary equalizer imposes exactly the diagram's compatibility relations. Thus its topology and projections are the ordinary ones. Duality and the existence of all intrinsic limits in \(\mathsf{LF}(G)\) also prove that \(\mathsf L(G)\) has all small colimits.

Theorem 5.1. Every small cofiltered limit is exact in \(\mathsf L(G)\).

Proof. Dualizing a cofiltered diagram of short exact sequences gives a filtered diagram of short exact sequences in \(\mathsf{LF}(G)\), with the arrows and each sequence reversed. Its filtered colimit is exact, by the first prerequisite. Dualizing back gives a short exact sequence in \(\mathsf L(G)\). This is the original diagram's limit sequence: a cone into its proposed limit is, after duality, precisely a cocone from the dual diagram, and the colimit's universal property gives the required unique map. The evaluation triangles identify its projections with the original ones. Section 4 makes its last map an actual open surjection and its first map a closed embedding with the required kernel. This proves exactness, including the underlying vector-space surjectivity, without topological compactness or an infinite-field restriction. \(\square\)

6. Four worked exercises

Exercise 6.1 (intermediate: continuity without invariant finite quotients). Give \(k^{\mathbb Z}\) its product topology and let the generator \(t\) of \(\mathbb Z\) act by \((tx)_n=x_{n-1}\). Prove that this is a continuous action on a linearly compact space, but the representation is not in \(\mathsf L(\mathbb Z)\).

Solution. It is the full dual of the direct sum with basis indexed by \(\mathbb Z\), hence linearly compact by §1. Shifting coordinates and shifting back are homeomorphisms, so the discrete group acts continuously. A continuous functional must kill the subspace on which some finite list of coordinates vanishes, and is therefore a finite linear combination of coordinate evaluations \(\epsilon_n\). Conversely each such combination is continuous. Thus the continuous dual is \(k^{(\mathbb Z)}\). The contragredient action is \(t\epsilon_n=\epsilon_{n+1}\), so it is the regular Laurent module \(k[t,t^{-1}]\).

No nonzero Laurent polynomial has a finite-dimensional orbit span. Otherwise a nontrivial finite relation between its translates would give \(a(t)p(t)=0\) with both Laurent polynomials nonzero. This cannot happen: clearing powers of \(t\) reduces to ordinary polynomials, where the leading coefficient of a product is the nonzero product of leading coefficients. If \(U\) were a proper invariant open subspace of finite codimension, then \((V/U)^*\) would be a nonzero finite-dimensional invariant subrepresentation of the continuous dual, a contradiction. A Hausdorff invariant finite-codimension basis would include proper such subspaces, so it cannot exist. By Theorem 3.1 this continuous representation is outside the Pro essential image.

Exercise 6.2 (advanced: all matrices and a coproduct). Over \(k=\mathbb Q\), let \(\lambda_n=n+1\) for \(n\geq1\). Let \(V=\prod_{n\geq1}\mathbb Q\), with \(t\) acting on coordinate \(n\) by \(\lambda_n^{-1}\). Describe its Pro presentation, every continuous equivariant endomorphism, and the coproduct in \(\mathsf L(\mathbb Z)\) of these one-dimensional coordinate representations.

Solution. Kernels of projections onto finite sets \(F\) of coordinates form an invariant open finite-codimension basis. The product is complete, so \(V\in\mathsf L(\mathbb Z)\), represented by \(V_F=\prod_{n\in F}\mathbb Q\) and coordinate-forgetting transitions. Every vector is exactly its compatible family of finite coordinate tuples. Its continuous dual is \(M=\bigoplus_n \mathbb Q_{\lambda_n}\), where \(t\) acts by \(\lambda_n\), because coordinate functionals transform contragrediently.

Every continuous linear map \(V\to V\) is given by a row-finite matrix \((a_{ij})\): its \(i\)-th coordinate is a continuous functional, hence involves finitely many input coordinates. Conversely any row-finite matrix defines a linear map on every tuple, and the inverse image of a basic condition on finitely many output coordinates depends on the finite union of their input supports, proving continuity. Equivariance says

Distinctness of the \(\lambda_n\) forces all off-diagonal entries to vanish. Every diagonal sequence works; composition multiplies its entries. Therefore \(\operatorname{End}_{\mathrm{cont},G}(V)=\prod_n\mathbb Q\) as a ring, with the correspondence given on every tuple by \((x_n)\mapsto(a_nx_n)\).

For the coproduct, first take the intrinsic product of the dual one-dimensional representations. In the ambient product \(\prod_n\mathbb Q_{\lambda_n}\), a vector with finite support has a finite-dimensional orbit. Conversely a locally finite vector is annihilated by a nonzero polynomial in \(t\), obtained from a finite linear dependence of its iterates. At coordinate \(n\), this says \(p(\lambda_n)x_n=0\). A nonzero polynomial has only finitely many roots: division by a factor \(t-a\) for each distinct root and induction on degree proves this. Thus only finitely many coordinates can be nonzero. The intrinsic product is consequently \(\bigoplus_n\mathbb Q_{\lambda_n}\). Formula (5.1) dualizes it to \(V\). Its coproduct injections are the coordinate inclusions, dual to the direct sum's coordinate projections. Any family of maps from the one-dimensional spaces to \(W\) dualizes to maps \(C(W)\to\mathbb Q_{\lambda_n}\); their tuple lies in the locally finite part, so their dual gives the unique continuous map \(V\to W\). Thus the product \(V\) is also this coproduct, with the asserted universal maps.

Exercise 6.3 (advanced: an adic action and its continuous dual). Give \(A=k[[u]]\) its \(u\)-adic topology and let the generator \(g\) of \(\mathbb Z\) act by multiplication by \((1-u)^{-1}\). Identify the continuous dual as a representation, its finite quotient presentations, and \(\operatorname{End}_{\mathrm{cont},G}(A)\).

Solution. The subspaces \(u^nA\) are invariant and have finite codimension; the inverse operator multiplies by \(1-u\). Compatible polynomial truncations determine exactly one formal series coefficient by coefficient, so \(A\simeq\varprojlim_n k[u]/(u^n)\) with its stated topology. This proves completeness and gives its Pro presentation, with the quotient action of \((1-u)^{-1}\), a finite geometric sum modulo \(u^n\).

The continuous dual has basis \(\epsilon_n\), extracting the coefficient of \(u^n\), for \(n\geq0\). Every continuous functional factors through a truncation, so it is a finite linear combination of these, and each such combination is continuous. Contragredience gives

Identify this module with \(E=k[u,u^{-1}]/k[u]\) by \(\epsilon_n\mapsto[u^{-n-1}]\). Multiplication by \(u\) is the backward shift in (6.2), and \(g\) acts as \(1-u\). It is invertible on \(E\): for a vector killed by \(u^N\), its inverse is \(1+u+\cdots+u^{N-1}\). Each finite initial span of the \(\epsilon_n\) is invariant, proving local finiteness. The dual of \(A/u^N A\) is precisely that span for \(0\leq n<N\); reduction of quotients dualizes to its actual inclusion in the next span. Evaluation pairs a series with the corresponding finite combination of its coefficients, giving both inverse duality maps of §2 here.

A linear map on \(E\) commutes with \(g\) exactly when it commutes with \(u=1-g\). Commuting with \(u\) also commutes with the locally finite geometric formula for \(g^{-1}\). Thus \(\operatorname{End}_G(E)=\operatorname{End}_{k[u]}(E)\). Exercise 2 of Finite support and the limits that a category permits proves, by compatible actions on the submodules killed by \(u^N\), that this ring is \(k[[u]]\). Applying §2 reverses composition, so \(\operatorname{End}_{\mathrm{cont},G}(A)\simeq k[[u]]^{\mathrm{op}}=k[[u]]\).

The actual endomorphism for \(a(u)\in k[[u]]\) is multiplication by \(a(u)\). Indeed a continuous map \(F\) commuting with \(g\) commutes with \(g^{-1}\) and hence with multiplication by \(u=1-g^{-1}\) on \(A\). On polynomials it therefore satisfies \(F(p)=pF(1)\). Polynomial truncations converge to every series; continuity, Hausdorffness and continuity of multiplication extend this equality to \(F(x)=xF(1)\). Multiplication preserves every \(u^nA\), so it is continuous and equivariant. This proves the correspondence on all maps, not just the abstract ring identification.

Exercise 6.4 (advanced: precisely where inverse-limit surjectivity fails). With trivial group actions, take the short exact rows

whose first transitions are inclusions, middle transitions identities, and last transitions reductions. Compute the inverse-limit sequence in ordinary vector spaces. Explain why Theorem 5.1 does not apply. Then replace the polynomial terms by \(A=k[[u]]\), with their adic topologies, and compute the limit in \(\mathsf L(1)\).

Solution. A compatible family in the first system is one polynomial belonging to every \(u^n k[u]\), hence zero: a nonzero polynomial has a least nonzero degree. The middle limit is \(k[u]\), and the last is \(k[[u]]\), since reductions specify exactly the compatible coefficients. The resulting map is the inclusion of polynomials into series. It is not surjective; \(\sum_{j\geq0}u^j\) has infinitely many nonzero coefficients and cannot be a polynomial. Thus the inverse limit of (6.3) is

and fails to be right exact. Infinite-dimensional discrete \(k[u]\) is not linearly compact in the sense of §1: the open neighbourhood \(\{0\}\) would have to contain a finite-codimension basic subspace, which would necessarily be zero and have infinite codimension. Theorem 5.1 therefore makes no assertion about these discrete rows. Giving the polynomial space its adic topology would instead fail completeness: the same compatible truncations of \(\sum_j u^j\) have no polynomial limit.

For the completed rows, \(u^nA\) is closed in \(A\) and is topologically isomorphic to \(A\) by multiplication by \(u^n\); division by \(u^n\) on that subspace gives the continuous inverse. The quotient \(A/u^nA\) is the same finite-dimensional truncation with its discrete topology. Each row is therefore a short exact row in \(\mathsf L(1)\). The first limit is still zero by the least nonzero coefficient argument; the middle is \(A\); and the last is \(A\), with the adic topology. The middle-to-last map now sends each series to all its truncations, and is exactly the topological isomorphism establishing completeness. The limit sequence is \(0\to0\to A\xrightarrow{\mathrm{id}}A\to0\), which is exact as Theorem 5.1 requires.

Further reading

Eric Marberg, Linear compactness and combinatorial bialgebras, §3, discusses the finite-coordinate topology, continuous matrices and continuous duals. The invariant group condition and the Pro morphism comparison above explain how that vector-space topology interacts with arbitrary discrete group representations.