Draft lesson.

Ind representations and the locally finite coreflector

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

Filtered colimits of finite-dimensional representations give exactly the locally finite representations. This remains true for an infinitely generated group, but compactness must then be checked in the locally finite category rather than assumed in the category of all representations. A concrete right adjoint constructs the largest locally finite subrepresentation and the intrinsic limits. Extension closure and exactness of that right adjoint are separate questions.

Use Modules and tensor functors over small preadditive categories, §§1–2, for module-category exactness, colimits and actual presentations. Formal colimits and compact presentations, §§1–2, proves the realization functor and its full Hom formula. Finitely presented generators and coherent Ind reconstruction, §§1–5, proves the abstract Ind and coherence criteria. We verify their representation hypotheses below; no finite-generation assumption on the group is imposed unless stated.

1. Locally finite objects and their finite pieces

Fix a field \(k\), a universe and choice. Let \(G\) be a small abstract group and \(R=k[G]\) its group algebra. Write \(\mathsf{Rep}(G)\) for left \(R\)-modules: vector spaces with an action of \(G\) by invertible linear maps, and equivariant linear maps as morphisms. The module constructions of the prerequisite give an abelian category with underlying vector-space limits, colimits and exactness, retaining the induced group actions.

A representation \(M\) is locally finite if every vector belongs to a finite-dimensional invariant subspace, equivalently if \(k[G]m\) is finite-dimensional for every \(m\in M\). The forward implication uses that subspace; the reverse uses the invariant span itself. Write \(\mathsf{Rep}_{\mathrm{lf}}(G)\) for this full subcategory.

Its finite-dimensional subrepresentations form a small nonempty directed poset: zero belongs, and the sum of two such subspaces is again finite-dimensional and invariant. Their union is all of \(M\), with the actual inclusions giving

M=\operatorname{colim}_{U\subset M,\ \dim_kU<\infty}U. \tag{1.1}

Compatible maps on those subspaces give a unique linear equivariant map on their union, proving this colimit assertion on all targets.

The finite-dimensional category is essentially small. After a basis choice, its dimension-\(n\) objects have actions \(G\to\operatorname{GL}_n(k)\), a subset of the small set of all such functions. The representation equations cut out a subset. Taking the union over \(n\ge0\), then representatives up to isomorphism, gives a small full skeleton \(\mathcal A\). Its Hom sets are small matrix spaces cut out by equivariance. Kernels and cokernels between finite-dimensional objects are finite-dimensional, so this is a small abelian category, with structure transported through its representative isomorphisms.

2. The exact abelian and colimit constructions

Subrepresentations and quotients of a locally finite object are locally finite: a vector in a subrepresentation has its finite invariant span inside it; a quotient vector is the image of a finite invariant subspace containing a lift. Finite sums are locally finite for the same reason. Thus the ambient kernels and cokernels of arrows between locally finite objects remain in the full subcategory. Their universal properties restrict there, and the ambient coimage–image comparison remains an isomorphism. This proves abelianity and exactness of the full inclusion

I:\mathsf{Rep}_{\mathrm{lf}}(G)\hookrightarrow\mathsf{Rep}(G). \tag{2.1}

Every small direct sum of locally finite objects is locally finite: a vector has finitely many nonzero coordinates, each in a finite invariant subspace, and their finite sum contains its orbit. Every small colimit is an ambient quotient of such a direct sum, by the standard difference-map presentation of a colimit. It therefore remains locally finite and has the same universal property by fullness. Hence the subcategory is cocomplete and all its colimits are the ambient colimits. In particular filtered colimits are exact, by the underlying vector-space finite-witness calculation.

There is a small generating family, namely the representatives in \(\mathcal A\). If two maps from \(M\) differ, choose a vector on which they differ and a finite-dimensional invariant subspace containing it. Its inclusion, transported from its representative, detects that difference. The coproduct of this small family is consequently a generator. Thus \(\mathsf{Rep}_{\mathrm{lf}}(G)\) is Grothendieck abelian. This argument has not asserted ambient extension closure.

3. Kill a finite basis of all equivariance defects

Theorem 3.1. The categorically compact objects of \(\mathsf{Rep}_{\mathrm{lf}}(G)\) are exactly its finite-dimensional objects. Here compact means that Hom preserves small filtered colimits.

Proof. Let \(V\) be finite-dimensional and \(N=\operatorname{colim}_iN_i\) a filtered colimit of locally finite representations. For an equivariant map \(f:V\to N\), choose representatives of the images of a finite basis of \(V\) at one common stage. Extending linearly gives \(f_0:V\to N_i\), with composite \(f\). This map need not yet be equivariant.

The finitely many image vectors lie in one finite-dimensional invariant subspace \(U\subset N_i\): take the sum of their finite invariant spans. For every \(g\in G\), the defect \(g f_0-f_0g\) has values in \(U\). Let \(W\subset U\) be the span of all these defect values for all \(g\) and all inputs. It is finite-dimensional, even when \(G\) is infinitely generated. Every vector of \(W\) maps to zero in \(N\), because \(f\) is equivariant. Choose a finite basis of \(W\). The filtered equality test moves to one later stage that kills every basis vector, hence all of \(W\). Composing \(f_0\) with that stage map now kills every equivariance defect, so gives an actual representation map lifting \(f\).

If two representation maps from \(V\) have the same colimit image, first move them to a common stage. Their differences on a finite basis vanish later, so a common later stage makes the whole maps equal. This proves injectivity as well as surjectivity of the canonical comparison

\operatorname{colim}_i\operatorname{Hom}_G(V,N_i) \xrightarrow{\sim}\operatorname{Hom}_G(V,N). \tag{3.1}

The common-stage tests for a general filtered category include the finite coequalization of parallel arrows; a directed-poset indexing assumption was not used. The construction proves the canonical comparison, including its naturalities.

Conversely, apply compactness of a locally finite \(M\) to (1.1). Its identity factors through a finite-dimensional subrepresentation \(U\hookrightarrow M\). That inclusion is then onto as well, because its image contains the identity image of \(M\). Thus \(M=U\) and is finite-dimensional. \(\square\)

One must not replace the finite-dimensional defect subspace \(U\) by an arbitrary stage representation and silently impose finitely many group equations. Local finiteness of the stage supplied the finite-dimensional witness used above.

4. The full Ind equivalence and its image among all representations

Realization and restricted Yoneda give inverse exact equivalences

\operatorname{Ind}(\mathcal A)\simeq\mathsf{Rep}_{\mathrm{lf}}(G), \qquad H(M)(V)=\operatorname{Hom}_G(V,M). \tag{4.1}

Here are all their hypotheses and maps. A filtered colimit of finite-dimensional representations is locally finite: any vector comes from a stage, whose image is a finite-dimensional invariant subspace. Conversely (1.1), transported to skeleton representatives with conjugated transition maps, presents every locally finite object. Theorem 3.1 makes every test in \(\mathcal A\) compact. For two such presentations the actual morphism comparison is therefore

\operatorname{Hom}_G(\operatorname{colim}_iV_i,\operatorname{colim}_jW_j) \simeq\lim_i\operatorname{colim}_j\operatorname{Hom}_G(V_i,W_j). \tag{4.2}

It first restricts a map to each source stage, then uses (3.1) to choose a target-stage representative, with equality tested later. Conversely compose those representatives with the target cocone; inverse-limit compatibility gives a source cocone and its unique global map. The two operations are inverse and preserve composition, by the proved realization theorem. Applying each finite test to (1.1) gives \(H(M)\) as the corresponding colimit of representables. Realization recovers \(M\); conversely a realized Ind presentation recovers its presheaf because each test is compact. These structural comparisons are natural and their composites are identities on every stage, proving the specified inverse equivalences on objects and morphisms.

The finite-dimensional objects are closed under kernels, so the Grothendieck category is locally coherent. The abstract completion theorem supplies the abelian structure of \(\operatorname{Ind}(\mathcal A)\); equivalence preserves its kernels and cokernels by their universal properties, proving exactness of (4.1).

The same realization into \(\mathsf{Rep}(G)\) is fully faithful, since the inclusion is full and preserves these colimits. Its essential image is precisely the locally finite representations. It is onto all representations if and only if \(G\) is finite. For finite \(G\), every cyclic orbit span has dimension at most \(|G|\). For infinite \(G\), the orbit of the basis vector 1 in the regular representation \(k[G]\) spans its infinite linearly independent group basis, so that representation is not locally finite.

5. The coreflector and every intrinsic limit

For any representation \(M\), put

r(M)=\sum_{U\subset M,\ \dim_kU<\infty,\ U\text{ invariant}}U. \tag{5.1}

This is a subrepresentation and is locally finite: each vector in the sum lies in a finite sum of the indicated subspaces. It contains every locally finite subrepresentation by its definition. A representation map sends a finite invariant subspace to a finite invariant image, so restricts to \(r(M)\to r(N)\). Restrictions preserve identities, composition and addition, defining an additive functor.

There are inverse natural bijections

\operatorname{Hom}_G(I L,M) \simeq\operatorname{Hom}_{\mathrm{lf}}(L,r(M)). \tag{5.2}

Indeed every vector of \(L\) lies in a finite invariant subspace; its image under a representation map lies in a finite invariant image, hence in \(r(M)\). Restricting the codomain gives the unique factor. The inverse composes with the inclusion \(r(M)\hookrightarrow M\). These operations are inverse on all vectors and commute with both input maps. Thus \(I\dashv r\), with unit the identity \(L=r(IL)\) and counit the actual subrepresentation inclusion. Restricting that inclusion to its locally finite part is identity, proving both triangles and \(r(r(M))=r(M)\).

For a small diagram \((L_j)\) of locally finite representations, take its ambient limit \(X\) and restrict its projections to \(r(X)\). Any cone from a locally finite \(L\) gives its unique ambient map \(L\to X\); (5.2) factors it uniquely through \(r(X)\). Thus

\lim_{\mathrm{lf}}L_j=r\!\left(\lim_{\mathsf{Rep}(G)}IL_j\right) \tag{5.3}

with the actual restricted projections. The adjunction also makes \(r\) preserve arbitrary ambient limits into intrinsic ones: a cone \(L\to r(M_j)\) is the same as a cone \(IL\to M_j\) by (5.2), so its universal object is \(r(\lim M_j)\). The finite intrinsic limits agree with the ambient kernels and finite biproducts proved in §2. Hence \(r\) preserves kernels and finite biproducts and is left exact. It need not be exact; an epimorphism may fail to lift finite orbit vectors, as the exercises show.

6. Ambient compactness and the separate extension condition

Theorem 6.1. The following are equivalent: \(G\) is finitely generated; every finite-dimensional representation is compact in \(\mathsf{Rep}(G)\); the trivial representation \(k\) is compact there.

Proof. Suppose \(S\) is a finite set of group generators, including their inverses. For a finite-dimensional \(V\), the evaluation \(R\otimes_kV\to V\) has kernel generated by

d_s(v)=s\otimes v-1\otimes sv\quad(s\in S), \tag{6.1}

where it suffices to use a basis of \(V\). To verify the kernel assertion, define \(d_g\) by the same formula for every \(g\). The identity

d_{ab}(v)=a\,d_b(v)+d_a(bv) \tag{6.2}

and \(d_1=0\) show that every \(d_g(v)\) lies in the submodule generated by (6.1), by induction on a word in \(S\). In the quotient, \(v\mapsto[1\otimes v]\) is therefore equivariant. Evaluation and this map are inverse: on a generator \([g\otimes v]\), its reverse composite is \([1\otimes gv]=[g\otimes v]\). Thus (6.1) gives a finite presentation by finite free \(R\)-modules. Mapping that presentation into an arbitrary target gives a kernel between two finite powers of that target. Filtered colimits commute with these powers and kernels, so its Hom preserves every ambient filtered colimit. This proves the first implication, and the next is immediate.

For the converse, suppose \(G\) is not finitely generated. For finite \(F\subset G\), let \(H_F\) be the subgroup it generates. These form a small directed system under finite-set inclusion. Every \(H_F\) has infinite index: a finitely generated subgroup of finite index, together with finitely many coset representatives, would generate \(G\). Use the permutation representations \(k[G/H_F]\), with transitions taking each coset basis vector to its larger-subgroup coset. Their colimit is \(k\), with the augmentation map. Indeed every finite list of coset representatives belongs to some larger \(H\), where those basis vectors become the identity coset. An augmentation-zero combination then becomes zero. The identity coset persists with augmentation 1, proving exactly the claimed colimit.

An invariant vector in \(k[G/H_F]\) must have equal coefficients on its transitive coset set. That set is infinite, while a vector has finite support, so the invariant vector is zero. Thus \(\operatorname{Hom}_G(k,k[G/H_F])=0\) at every stage, while the identity belongs to \(\operatorname{Hom}_G(k,k)\) at the colimit. Compactness fails, proving the converse. \(\square\)

When \(G\) is finitely generated, the locally finite subcategory is also closed under ambient extensions. Here is the full construction. In an exact row \(0\to A\to E\to B\to0\) with \(A,B\) locally finite, take \(e\in E\). If its quotient is zero, its finite invariant span lies in \(A\). Otherwise choose a finite-dimensional invariant \(B_0\subset B\) containing its quotient and a linear lift \(\sigma:B_0\to E\) that lifts that vector to \(e\). For the finitely many generators and their inverses and a finite basis of \(B_0\), the defects \(s\sigma(b)-\sigma(sb)\) form finitely many vectors of \(A\). Their finite invariant spans have a finite-dimensional invariant sum \(A_0\). The space \(\sigma(B_0)+A_0\) contains \(e\), is finite-dimensional and is stable under every generator and inverse by those defect equations. It is therefore a finite invariant subspace of \(E\). This proves local finiteness for every vector, hence extension closure.

Together with subobject and quotient closure, this makes the image a Serre subcategory under this hypothesis. It does not make \(r\) exact on arbitrary ambient epimorphisms. For an infinitely generated group, ambient extension closure may fail as well, despite the exact full inclusion and the abelian Ind equivalence.

7. Four graded exercises with complete solutions

Exercise 1 (grade 1). For a finite group, prove that (4.1) realizes every representation, in any characteristic. For \(G=C_p\) and \(\operatorname{char}k=p\), prove that its compact trivial representation is not projective.

Solution. For a vector \(v\), the at most \(|G|\) vectors \(gv\) span a finite-dimensional invariant subspace. Hence every representation is locally finite and (4.1) gives the whole category, on all maps. The finite-defect proof or the finite group-generator presentation proves compactness of every finite-dimensional representation.

The augmentation \(k[C_p]\to k\) is onto. In the regular basis indexed cyclically by the group, an invariant vector has all coefficients equal, since a generator cyclically shifts them. Thus its invariant subspace is the line spanned by \(N=\sum_{g\in C_p}g\). A section from the trivial representation would send 1 to an invariant vector with augmentation 1. Every such vector instead has augmentation a scalar multiple of \(p=0\) in \(k\). No section exists. This proves nonprojectivity without changing the Ind or compactness assertions.

Exercise 2 (grade 2). For \(G=\mathbb Z\), identify the locally finite representations and the coreflector explicitly. Give a failure of its exactness even though the group is finitely generated and the image is extension closed.

Solution. A representation is a vector space with an invertible operator \(T\), hence a module over \(R=k[t,t^{-1}]\). A vector in a finite invariant subspace has a finite list of powers that are linearly dependent, giving a nonzero Laurent polynomial annihilator. Conversely, if \(p(T)v=0\), multiply by a power of \(t\) to make its smallest exponent zero and its constant and leading coefficients nonzero. If the resulting degree is zero then \(v=0\). Otherwise the relation expresses the highest power in terms of lower powers, and its inverse-operator shift expresses \(T^{-1}v\) in those same powers. Consequently the span of \(v,Tv,\ldots,T^{d-1}v\) is stable under both \(T,T^{-1}\) and is finite-dimensional. This proves both implications: local finiteness is exactly Laurent torsion, and \(r(M)\) is the torsion submodule.

The ring \(R\) is a domain: clearing negative exponents embeds each nonzero product calculation in ordinary polynomials, whose leading coefficients have nonzero product. Thus \(r(R)=0\). But \(R/(t-1)\cong k\) is torsion and is its own locally finite part. The ambient epimorphism \(R\to R/(t-1)\) becomes \(0\to k\) under \(r\), which is not onto. Theorem 6.1 nevertheless applies to ambient compactness of finite-dimensional objects, and its extension argument applies to two locally finite end terms. The kernel of this epimorphism is not locally finite, so there is no conflict with that extension assertion.

Exercise 3 (grade 3). Work over \(k=\mathbb Q\), let \(\lambda_n=n+1\), and let \(V_n\) be the one-dimensional \(\mathbb Z\)-representation where \(t\) acts by \(\lambda_n\), for \(n\ge1\). Compute their product in the locally finite category, with all universal maps. Compare it with the ambient product.

Solution. The ambient product is \(X=\prod_n\mathbb Q\), with coordinatewise action. By Exercise 2, a vector \((x_n)\) belongs to \(r(X)\) exactly when some nonzero Laurent polynomial \(p\) satisfies \(p(\lambda_n)x_n=0\) for every \(n\). A nonzero polynomial has only finitely many roots: division by \(t-a\) removes each distinct root and lowers its degree, proving this by induction; multiplying by a power of \(t\) gives the same statement for nonzero Laurent polynomials at nonzero inputs. The \(\lambda_n\) are distinct nonzero rationals. Therefore an annihilated tuple has finite support. Conversely a finite-support tuple is annihilated by the product of \((t-\lambda_n)\) over its support. Thus

\prod_{\mathsf{Rep}_{\mathrm{lf}}(\mathbb Z)}V_n =r(X)=\bigoplus_nV_n. \tag{7.1}

Its projections are the ordinary coordinate projections. For any locally finite \(L\) and family of representation maps \(L\to V_n\), the ambient product map sends each vector to an annihilated, hence finite-support tuple. It factors uniquely through that direct sum; composing its projections recovers every original map. Conversely a map to the direct sum supplies those maps. These are the complete inverse product comparisons, not merely a dimension calculation. The ambient tuple \((1,1,\ldots)\) has no polynomial annihilator, so the two products differ.

Exercise 4 (grade 4). Let \(G=\bigoplus_{i\ge1}\mathbb Z g_i\). Give an exact row with locally finite ends and a middle term that is not locally finite. Compute the coreflector on that row. Then exhibit an ambient filtered-colimit test distinguishing compactness of the trivial representation in the two categories.

Solution. Let \(E=k e_0\oplus\bigoplus_{i\ge1}k e_i\), where \(g_i e_0=e_0+e_i\) and \(g_i e_j=e_j\) for \(j\ge1\). The transvections commute because their off-diagonal operators have pairwise zero products; their inverses subtract \(e_i\). A finite-support integer tuple therefore acts by \(e_0\mapsto e_0+\sum_i a_ie_i\), fixing the other vectors. This defines the whole group action in every characteristic.

The invariant subrepresentation \(K=\bigoplus_{i\ge1}k e_i\) has trivial action, as does the quotient \(E/K=k\), so both are locally finite. A vector \(a e_0+\sum b_i e_i\) with \(a\ne0\) has orbit differences \(a e_i\) for every \(i\); these are infinitely linearly independent. It cannot have finite invariant span. Thus \(E\) is not locally finite and \(r(E)=K\). The exact row \(0\to K\to E\to k\to0\) becomes \(0\to K\xrightarrow1K\to k\), with its last map zero. It witnesses both failed ambient extension closure and failed right exactness of \(r\).

For an explicit ambient compactness test, identify \(k[G]\) with the Laurent ring in the variables \(t_i\), using its finite-support exponent group basis. For finite \(F\subset\mathbb N\), put \(M_F=R/(t_i-1:i\in F)\). Substitution \(t_i=1\) on \(F\), and inclusion of the remaining variables, give inverse maps with the Laurent ring in the variables outside \(F\): all monomials involving \(F\) reduce by its relations. The directed colimit of these quotients is \(k\), since each finite list of variables is eventually set to 1, with augmentation retaining its scalar value.

Choose \(j\notin F\). Multiplication by \(t_j\) cannot fix a nonzero Laurent polynomial of finite support: its largest \(j\)-exponent would increase by one. Hence \(M_F\) has no nonzero invariant vector and \(\operatorname{Hom}_G(k,M_F)=0\), while the colimit has the identity map from \(k\). The representation \(k\) is therefore not ambient compact. It is compact in \(\mathsf{Rep}_{\mathrm{lf}}(G)\) by Theorem 3.1. Each test stage \(M_F\) lies outside that subcategory: the orbit of 1 contains the linearly independent powers of \(t_j\). Thus the two filtered-colimit tests genuinely have different allowed target diagrams.