Draft lesson.

Ind-completion and Serre quotients

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

A Serre quotient removes objects while preserving exact sequences. Ind-completion adds filtered families of objects. We prove that these operations commute for a small abelian category, construct the right adjoint explicitly, and identify the objects it recovers. The counit proof retains the source restrictions and target quotients in a quotient morphism. Examples distinguish the exact quotient functor from its possibly nonexact right adjoint.

Fix a universe and a small abelian category \(\mathcal C\). A small skeleton of an essentially small abelian category works equally well. Let \(\mathcal S\subseteq\mathcal C\) be a full Serre subcategory, closed under subobjects, quotients, extensions and isomorphisms within \(\mathcal C\). Write \[ Q:\mathcal C\longrightarrow\mathcal B=\mathcal C/\mathcal S, \qquad \mathcal A=\operatorname{Ind}(\mathcal C),\quad \mathcal D=\operatorname{Ind}(\mathcal B). \tag{0.1} \] Choose the quotient with objects labelled by those of \(\mathcal C\). Its Hom sets are small by the quotient formula below. Write \(c_{\mathcal C},c_{\mathcal B}\) for constant embeddings.

Ind-objects through their elements, Sections 1–3, proves the element-category criterion, the Hom formula and extension of a functor through filtered presentations. Limits and colimits of formal objects, Section 1, recognizes Ind presheaves by their finite-colimit tests. Two cokernels, controlled rows and abelian completion, Sections 2–5, proves the abelian structure, exact constant embeddings and exactness of induced Ind functors. Serre quotients and local saturation, Sections 1–2 and 8, constructs the exact quotient and its fractions. Short exact extensions and the Baer group supplies the extension interpretation of \(\operatorname{Ext}^1\). Module examples use Formal colimits and compact presentations, Section 4, and the integer-module proofs in Right derived tensor and torsion-free quotients, Section 1.

1. The exact induced functor and the fraction formula

The induced functor \[ L=\operatorname{Ind}(Q):\mathcal A\longrightarrow\mathcal D \tag{1.1} \] is exact by the Ind exactness theorem, and preserves filtered colimits by construction. Explicitly it takes \(\operatorname{colim}_i c_{\mathcal C}C_i\) to \(\operatorname{colim}_i c_{\mathcal B}QC_i\), with the induced stage arrows.

We need the precise quotient Hom formula. A morphism \(QB\to QA\) can be represented by \[ B'\subseteq B,\quad B/B'\in\mathcal S,\qquad S'\subseteq A,\quad S'\in\mathcal S,\qquad a:B'\longrightarrow A/S'. \tag{1.2} \] Its value is \(Q\pi^{-1}\,Qa\,Qi^{-1}\), for the target quotient \(\pi:A\to A/S'\) and source inclusion \(i:B'\hookrightarrow B\). Representatives are identified after decreasing the source and increasing the negligible target. The complete quotient proof shows that this is a filtered colimit formula for Hom, compatible with composition. In particular \(Qt=0\) for an ordinary morphism \(t\) exactly when \(\operatorname{im}t\in\mathcal S\), since \(Q\) is exact and its object kernel is \(\mathcal S\).

All subsequent completions use this small quotient category; no cofinal-smallness claim about an arbitrary large coefficient category is implicit.

2. Construct the right adjoint as a presheaf

For \(Y\in\mathcal D\), put \[ (RY)(A)=\operatorname{Hom}_{\mathcal D}(c_{\mathcal B}QA,Y), \qquad A\in\mathcal C, \tag{2.1} \] with precomposition along \(Qu\) for \(u:A\to A'\). These are abelian groups, and their maps are additive.

Proposition 2.1. The presheaf (2.1) is an Ind-object of \(\mathcal C\) and defines an additive right adjoint \(R\) of \(L\).

Proof. The composite \(c_{\mathcal B}Q\) preserves finite colimits: \(Q\) is exact and additive, and the constant embedding preserves finite colimits. Hom from a colimit to \(Y\) is the limit of the corresponding Hom sets. Thus (2.1) passes the finite-colimit test. Its category of elements is small, since \(\mathcal C\) is small and every value in (2.1) is a small set. The proved recognition criterion therefore makes it an Ind-object.

For \(X=\operatorname{colim}_i c_{\mathcal C}C_i\), the Hom formula and Yoneda give \[ \begin{aligned} \operatorname{Hom}_{\mathcal A}(X,RY) &=\lim_i (RY)(C_i)\\ &=\lim_i\operatorname{Hom}_{\mathcal D}(c_{\mathcal B}QC_i,Y)\\ &=\operatorname{Hom}_{\mathcal D}(LX,Y). \end{aligned} \tag{2.2} \] Every equality is its natural universal-property bijection. Precomposition and postcomposition commute with all three steps, so this is an adjunction on every arrow, independent of the chosen presentation. Addition agrees in (2.1), hence in the represented Hom maps; \(R\) is additive. The units and counits are those given by identity maps under (2.2), and its naturality proves both triangle identities. \(\square\)

More explicitly, the element category \(E_{RY}\) has objects \((A,f)\), where \(f:c_{\mathcal B}QA\to Y\), and arrows \(t:(A,f)\to(A',f')\) with \(f'c_{\mathcal B}Qt=f\). The recognized presentation is \(RY=\operatorname{colim}_{E_{RY}}c_{\mathcal C}A\). Its counit is the cocone \[ \epsilon_Y:\operatorname{colim}_{(A,f)\in E_{RY}}c_{\mathcal B}QA \longrightarrow Y \tag{2.3} \] with components \(f\).

3. Prove the counit is invertible

Theorem 3.1. Every \(\epsilon_Y\) is an isomorphism. Consequently \(R\) is fully faithful and \(L\) is essentially surjective.

Proof. Test (2.3) by \(c_{\mathcal B}QB\), for every \(B\in\mathcal C\). These are all the representable tests in \(\mathcal D\). They are finitely presented, so maps into its filtered source are represented by \[ g:QB\longrightarrow QA\quad\hbox{at some }(A,f). \] The induced map sends that representative to \(f\,c_{\mathcal B}g\).

It is surjective: a map \(u:c_{\mathcal B}QB\to Y\) itself supplies the object \((B,u)\), and \(1_{QB}\) represents its preimage.

For injectivity, first move two representatives to the single element \((D,f_D)\), where \(D=A\oplus A'\) and \(f_D\) has components \(f,f'\). Let \(v:QB\to QD\) be the difference of their two maps. Equality of their images says \(f_Dc_{\mathcal B}v=0\). Represent \(v\) by (1.2), now as \(a:B'\to D/S'\). Put \(\pi:D\to D/S'\) and \[ f_{D/S'}=f_D(c_{\mathcal B}Q\pi)^{-1}. \tag{3.1} \] Then \(f_{D/S'}c_{\mathcal B}Qa=0\), because \(Qi\) is invertible. Let \(I=\operatorname{im}a\subseteq D/S'\). Exactness of \(Q\) and of the constant embedding makes \(c_{\mathcal B}QB'\to c_{\mathcal B}QI\) epic. Therefore \(f_{D/S'}\) kills \(c_{\mathcal B}QI\), and factors uniquely through \[ E=(D/S')/I. \tag{3.2} \] Denote the factor by \(f_E:c_{\mathcal B}QE\to Y\). The composite \(t:D\to E\) is an arrow \((D,f_D)\to(E,f_E)\) in \(E_{RY}\). Its quotient map kills \(a\), so \(Qt\,v=0\). Thus the two representatives agree at this later filtered stage, proving injectivity.

The image \(I\) in (3.2) need not belong to \(\mathcal S\); this transition is a map in the element category, not a quotient denominator. The argument uses exactly the vanishing after mapping to \(Y\).

We have proved that \(\operatorname{Hom}(c_{\mathcal B}QB,\epsilon_Y)\) is bijective for every \(B\). Ind-objects are full presheaves on \(\mathcal B\), and these bijections are all their pointwise tests, so \(\epsilon_Y\) is invertible. The adjunction now gives \[ \operatorname{Hom}_{\mathcal A}(RY,RZ) \simeq\operatorname{Hom}_{\mathcal D}(LRY,Z) \simeq\operatorname{Hom}_{\mathcal D}(Y,Z), \] with the usual \(R\)-map correspondence. Hence \(R\) is fully faithful. The presentation \(Y\simeq LRY\) proves essential surjectivity of \(L\). \(\square\)

4. The objects killed by completion of the quotient

Regard \(\operatorname{Ind}(\mathcal S)\) as a full subcategory of \(\mathcal A\): its full faithfulness follows from the Ind Hom formula and fullness of \(\mathcal S\subseteq\mathcal C\).

Theorem 4.1. The object kernel of \(L\) is exactly \[ \mathcal T=\operatorname{Ind}(\mathcal S). \tag{4.1} \] It is a localizing Serre subcategory, meaning a Serre subcategory closed under small coproducts.

Proof. A filtered colimit of constants from \(\mathcal S\) is killed by \(L\), since their quotient objects are zero. Conversely let \(X=\operatorname{colim}_i c_{\mathcal C}C_i\) and \(LX=0\). The structural map from \(c_{\mathcal B}QC_i\) to this zero colimit is zero. Finite presentation of that constant test and the filtered equality rule give an arrow \(i\to j\) for which \(Q(C_i\to C_j)=0\). Its ordinary image \(S_{ij}\) belongs to \(\mathcal S\). The structural map \(c_{\mathcal C}C_i\to X\) thus factors through \(c_{\mathcal C}S_{ij}\).

Every map from a constant \(c_{\mathcal C}A\) to \(X\) factors at some \(C_i\), and hence through such a constant from \(\mathcal S\). In the filtered element category of \(X\), take the full subcategory on the objects \((S,s)\) with \(S\in\mathcal S\). It is nonempty and cofinal by this factorization. It is filtered too: a finite diagram of its objects has a cocone in the full element category, and factorization of that cocone's vertex through a constant from \(\mathcal S\) supplies a cocone within the subcategory. It is small, since the category and all its element sets are small. The element presentation therefore realizes \(X\) as a filtered colimit of constants from \(\mathcal S\), proving (4.1).

Exactness of \(L\) makes its object kernel Serre. The adjunction makes \(L\) preserve all small colimits, so its kernel is closed under coproducts as well. These colimits exist: small abelian Ind categories have filtered colimits, finite colimits and coproducts formed from the filtered finite partial sums; arbitrary colimits are the cokernel of the two arrows between coproducts indexed by diagram arrows and objects. This completes the asserted localizing property. \(\square\)

5. The equivalence of abelian quotients

Theorem 5.1. The exact comparison functor is an equivalence \[ \operatorname{Ind}(\mathcal C)/\operatorname{Ind}(\mathcal S) \ \simeq\ \operatorname{Ind}(\mathcal C/\mathcal S). \tag{5.1} \] It is induced by \(L\), with its specified identification on constants. An exact functor from \(\mathcal A\) annihilating \(\mathcal T\) factors through \(L\), uniquely up to the unique compatible natural isomorphism.

Proof. The quotient on the left is locally small. A subobject of an Ind presheaf is a pointwise subpresheaf, since a monomorphism is injective under every representable Hom test. For small \(\mathcal C\), possible such subpresheaves lie in the small product of power sets of its values. Thus each object has only a small set of subobject classes. The proved Serre fraction formula gives the required small Hom sets and the exact quotient \[ \pi:\mathcal A\longrightarrow\mathcal A/\mathcal T. \] Exactness of \(L\) and (4.1) give an induced exact \(\bar L\), with \(\bar L\pi=L\). Define \(H=\pi R\). The counit gives \(\bar LH\simeq1_{\mathcal D}\).

For the other direction, the unit \(\eta_X:X\to RLX\) has kernel and cokernel in \(\mathcal T\). Indeed \(L\eta_X\) is invertible: its inverse is \(\epsilon_{LX}\) by the triangle identity and Theorem 3.1. Exactness of \(L\) proves the kernel and cokernel assertion. Hence \(\pi\eta_X\) is invertible, giving a natural comparison \(1_{\mathcal A/\mathcal T}\simeq H\bar L\). Naturality passes through quotient arrows because it holds on ordinary maps and therefore on their inverted denominators. These two comparisons prove (5.1), with their compatibility inherited from the adjunction triangles.

Finally an exact functor annihilating \(\mathcal T\) factors through \(\pi\) by the exact quotient theorem. Transport that factor through (5.1) to obtain the factor through \(L\). The localization universal property includes natural transformations, so it also gives the stated uniqueness. No preservation of filtered colimits by the test functor is required for this exact quotient property. \(\square\)

6. Local objects and the extension obstruction

Let \(W_{\mathcal T}\) be the maps of \(\mathcal A\) whose kernel and cokernel belong to \(\mathcal T\). Call \(H\in\mathcal A\) local if every \(w:X\to X'\) in this class induces a bijection \(\operatorname{Hom}(X',H)\to\operatorname{Hom}(X,H)\).

Proposition 6.1. The following are equivalent: \(H\) belongs to the essential image of \(R\); \(H\) is local; and \[ \operatorname{Hom}_{\mathcal A}(T,H)=0,\qquad \operatorname{Ext}^1_{\mathcal A}(T,H)=0 \quad\hbox{for every }T\in\mathcal T. \tag{6.1} \] Here \(\operatorname{Ext}^1\) is the Baer group of short exact extensions.

Proof. Objects \(RY\) are local, since adjunction identifies their restriction map with precomposition by \(Lw\), an isomorphism. Conversely, for a local \(H\), its unit \(\eta_H:H\to RLH\) belongs to \(W_{\mathcal T}\). Locality extends \(1_H\) across it to \(v:RLH\to H\), so \(v\eta_H=1_H\). The object \(RLH\) is already local; injectivity of its restriction map along \(\eta_H\) shows \(\eta_Hv=1_{RLH}\), by precomposing both sides with \(\eta_H\). Thus \(\eta_H\) is invertible and \(H\) belongs to the image.

If \(H\) is local, applying its definition to \(T\to0\) gives the Hom vanishing. In any extension \(0\to H\to E\to T\to0\), the inclusion belongs to \(W_{\mathcal T}\). Extending identity gives a retraction, so the extension splits. The extension interpretation gives the second vanishing.

Conversely suppose (6.1). For an epimorphism with kernel in \(\mathcal T\), every map to \(H\) kills that kernel and factors uniquely through its target, by Hom vanishing. For a monomorphism \(i:A\hookrightarrow B\) with cokernel \(T\in\mathcal T\), push a given \(a:A\to H\) out along \(i\). The abelian pushout gives \(0\to H\to E\to T\to0\). It splits by the Ext vanishing; compose its retraction with \(B\to E\) to extend \(a\). Two extensions differ by a map \(T\to H\), so are equal. Every \(w\in W_{\mathcal T}\) factors as its epimorphism onto the image followed by its monomorphism into the target; the two arguments apply and prove the required bijection. \(\square\)

The functor \(R\) is left exact as an additive right adjoint. It need not preserve cokernels; Solution 8.4 computes an explicit failure.

7. Filtered colimits and finite tests survive the comparison

The right adjoint \(R\) preserves filtered colimits, although it need not be exact. Indeed for every \(A\in\mathcal C\), formula (2.1) commutes with a filtered colimit because \(c_{\mathcal B}QA\) is finitely presented. Filtered colimits of Ind-objects are their pointwise presheaf colimits, so these pointwise isomorphisms give the asserted natural comparison for \(R\).

The images of constant objects under (5.1) are precisely \(c_{\mathcal B}QA\), hence are finitely presented. They form a small separating family: every Ind-object has a filtered presentation by them, and Yoneda detects different arrows. Their coproduct is a generator, since a nonzero arrow is detected by a map from one member, which extends to a map from that coproduct by taking zero on the other members. These statements concern finite presentation and filtered colimits in the quotient category, with its actual constant tests.

For completeness, its filtered colimits are exact. Theorem 4.1 of the limits-and-colimits lesson proves that filtered colimits in \(\operatorname{Ind}(\mathcal B)\) commute with finite limits and finite colimits, since \(\mathcal B\) has both. Kernels are finite limits and cokernels are finite colimits; hence the filtered-colimit functor preserves both and is exact. Thus both sides of (5.1) have the same abelian and filtered-colimit interfaces. This does not assert exactness of arbitrary inverse limits or of \(R\); neither follows from finite presentation.

8. Four graded exercises with complete solutions

Exercise 8.1 — Finitely generated groups modulo finite groups (introductory)

Take \(\mathcal C\) to be a small skeleton of finitely generated abelian groups and \(\mathcal S\) its finite groups. Identify both sides of (5.1), the induced \(L,R\), and the unit on \(\mathbb Z\oplus\mathbb Z/n\).

Solution. A finitely generated group \(M\) has a finite torsion subgroup \(T_M\), and \(M/T_M\) is finite free. Indeed the torsion-free finite-generation theorem gives the latter statement; lift its basis to \(M\) to split the quotient map. Projection of a finite generating set of \(M\) then generates \(T_M\). Finitely many torsion generators generate a finite group, by the product of their finite cyclic groups.

Rationalization is exact: a localized element is zero precisely when a nonzero integer kills a representative. A kernel equation can therefore be cleared by an integer and then lifted through the original exact row; surjections lift representatives directly. It kills precisely finite groups in \(\mathcal C\).

The quotient \(\mathcal C/\mathcal S\) is the category of finite-dimensional rational vector spaces. Every such space is the rationalization of \(\mathbb Z^r\). For fullness, remove the finite torsion subgroups and use the lifted free bases just described. A rational matrix between the free rationalizations becomes integral after multiplying by one positive common denominator \(d\). It therefore comes from an ordinary map \(f:M\to N\), with the quotient morphism \(Qf\,(Q(d\,1_M))^{-1}\). Multiplication by \(d\) has finite kernel and cokernel on a finitely generated group, so it is a quotient denominator.

For faithfulness take a fraction represented by (1.2). If its rationalization is zero, its numerator has torsion image. That image is finitely generated and torsion, hence finite; exactness of the quotient kills the numerator. Its fraction is zero. Thus the induced functor is fully faithful and essentially surjective.

Finitely generated integer modules are finitely presented: their relations are submodules of a finite free integer module and are finite free by the earlier PID proof. The module-presentation theorem therefore identifies \(\operatorname{Ind}(\mathcal C)\) with all abelian groups. Similarly the Ind category of finite-dimensional rational spaces is all rational vector spaces. The subcategory \(\operatorname{Ind}(\mathcal S)\) is all torsion groups: every torsion group is the filtered union of its finitely generated subgroups, each of which is finite, and every element of a colimit of finite groups is torsion.

Consequently (5.1) reads \[ \mathsf{Ab}/\mathsf{Tors}\simeq\mathsf{Vect}_{\mathbb Q}. \tag{8.1} \] Here \(L(M)=M\otimes_{\mathbb Z}\mathbb Q\) and \(R\) forgets the rational action. The adjunction is explicit: a group map \(M\to V\) to a rational space extends uniquely by \(m\otimes(a/b)\mapsto(a/b)f(m)\). On the stated group, the unit is \((z,\bar t)\mapsto z\in\mathbb Q\); its kernel is \(\mathbb Z/n\), and its cokernel is \(\mathbb Q/\mathbb Z\), both torsion. The whole group need not be a local object just because its free summand survives rationalization. \(\square\)

Exercise 8.2 — Hom vanishing is not the local-object criterion (intermediate)

In (8.1), characterize local objects, show that \(\mathbb Z\) fails despite having no maps from torsion groups, and compute \(\operatorname{Ext}^1(\mathbb Z/p,\mathbb Z)\) for a prime \(p\).

Solution. Multiplication by each positive integer \(d\) on \(\mathbb Z\) belongs to \(W_{\mathsf{Tors}}\). Applying Hom into a local group \(H\) says that multiplication by \(d\) on \(H\) is bijective. Such a group has a unique rational vector-space structure: define \((a/b)h\) as the unique solution \(x\) of \(bx=ah\). Equality of fractions and the vector-space equations follow after multiplying by a common denominator, which is injective. Conversely any rational space is in the image of \(R\) and hence is local by Proposition 6.1. Thus local means uniquely divisible.

A map from a torsion group to \(\mathbb Z\) is zero, since the image of each torsion element must be torsion. But the exact sequence \[ 0\to\mathbb Z\xrightarrow{p}\mathbb Z\to\mathbb Z/p\to0 \tag{8.2} \] does not split: a retraction of multiplication by \(p\) would give an integer \(r\) with \(pr=1\). This is the missing extension condition.

For the full Ext calculation, in any extension choose a lift \(e\) of the generator of \(\mathbb Z/p\). Its relation is \(pe=i(z)\) for some \(z\in\mathbb Z\). Replacing \(e\) by \(e+i(t)\) changes \(z\) to \(z+pt\). Conversely the group \[ E_z=(\mathbb Z\oplus\mathbb Z)/\langle(-z,p)\rangle \tag{8.3} \] with injection \(a\mapsto[(a,0)]\) and quotient \([(a,b)]\mapsto\bar b\) realizes that parameter. The injection is monic, since a multiple of \((-z,p)\) with second coordinate zero is the zero multiple. The quotient kernel is its image, because \(b=pt\) permits subtraction of \(t(-z,p)\). These formulas give a short exact sequence.

Every extension is isomorphic to (8.3) by sending the two generators to \(i(1),e\); its kernel relation is precisely the one displayed, by looking first at the quotient and then the injected group. Parameters differing by \(pt\) give equivalent extensions by the indicated change of lift, and any equivalence forces such a change. A splitting exists exactly when the parameter is a multiple of \(p\). Finally Baer addition chooses the pair of lifts in the pullback and adds their injected relations in the pushout, so its parameter is \(z+z'\). Therefore \[ \operatorname{Ext}^1_{\mathbb Z}(\mathbb Z/p,\mathbb Z)\simeq\mathbb Z/p, \tag{8.4} \] with (8.2) the parameter \(1\). This proves both the group calculation and the obstruction to locality. \(\square\)

Exercise 8.3 — Invert only one prime (advanced)

Let \(\mathcal S_p\) be the finite \(p\)-primary groups inside the finitely generated abelian groups. Identify the Ind quotient and its local objects. Specify its action on groups with torsion at another prime.

Solution. Put \(D=\mathbb Z[1/p]\). Localization \(M\mapsto M[1/p]\) is exact: an element represented by \(m/p^a\) maps to zero exactly when a further power of \(p\) kills its image; moving to that representative reduces the kernel problem to the original exact row. The same representative argument lifts a quotient element. Its kernel on finitely generated groups is exactly \(\mathcal S_p\), since finitely many generators individually killed by powers of \(p\) are jointly killed by one power.

Every ideal of \(D\) is generated by an integer: multiplication by the denominator, a unit in \(D\), puts each ideal element in its integer contraction, an ideal \(d\mathbb Z\). Hence the ideal is \(dD\). The finite-submodule induction in the PID proof applies to \(D\). Thus its finitely generated modules have finite presentations. Clearing a common power-of-\(p\) denominator in a finite presentation matrix shows every such module is the localization of a finitely presented integer module.

For an arrow \(M[1/p]\to N[1/p]\), choose finite integer generators and relations for \(M\). Lift the generator images to \(N\) after multiplying by one power of \(p\). Each of the finitely many relation images vanishes after localization; another common power kills all of them. This yields an ordinary \(f:M\to N\) and a representation of the arrow as \(f/p^r\). Multiplication by \(p^r\) on \(M\) has kernel and cokernel in \(\mathcal S_p\). These arrows therefore come from quotient morphisms. If a quotient fraction becomes zero after localization, its numerator has a finitely generated \(p\)-power-torsion image, so its image belongs to \(\mathcal S_p\) and the fraction is zero. This proves \[ \mathcal C/\mathcal S_p\simeq\mathsf{Mod}_{\mathrm{fg}}(D). \tag{8.5} \]

The module-presentation theorem and (5.1) now give \[ \mathsf{Ab}/\mathsf{Tors}_p\simeq\mathsf{Mod}(D), \tag{8.6} \] where \(\mathsf{Tors}_p\) comprises groups whose elements are individually killed by powers of \(p\). Such groups are filtered unions of finite \(p\)-primary subgroups, proving that this is exactly \(\operatorname{Ind}(\mathcal S_p)\).

The right adjoint forgets the \(D\)-module action: a group map into a group with invertible multiplication by \(p\) extends uniquely by \(m/p^r\mapsto p^{-r}f(m)\). Consequently local objects are precisely groups on which multiplication by \(p\) is bijective; the inverse of that multiplication specifies the unique \(D\)-action. For a different prime \(\ell\), multiplication by \(p\) on \(\mathbb Z/\ell^a\) is invertible by Bézout, so that group is already local and survives unchanged. Multiplication by \(p\) on a \(p\)-primary group eventually kills every element under localization, so those groups disappear. The unit's kernel and cokernel are \(p\)-power torsion: a kernel element is killed by a power, and the class of \(m/p^r\) modulo the unit image is killed by \(p^r\). \(\square\)

Exercise 8.4 — An explicit nonexact section functor (advanced)

Let \(B=\mathbb Z/4\), \(k=\mathbb F_2\), and \(U=k\) as a \((B,k)\)-bimodule via reduction modulo two. Consider the finite triangular ring \[ \Lambda=\begin{pmatrix}B&U\\0&k\end{pmatrix}. \tag{8.7} \] Let \(\mathcal C\) be a small skeleton of its finitely generated left modules, and let \(\mathcal S\) be those with zero upper \(B\)-component. Identify the quotient and section functor, and show that the latter is not exact.

Solution. A left \(\Lambda\)-module is data \((V,W,t)\), where \(V\) is a left \(B\)-module, \(W\) a left \(k\)-module, and \(t:U\otimes_k W\to V\) is \(B\)-linear. The two diagonal idempotents split its underlying group into \(V\oplus W\); multiplying by an upper off-diagonal element gives \(t\). Conversely let a matrix act by \[ \begin{pmatrix}b&u\\0&a\end{pmatrix}(v,w) =(bv+t(u\otimes w),aw). \tag{8.8} \] The balancing and \(B\)-linearity of \(t\) verify matrix multiplication and the identity. Morphisms are component pairs commuting with \(t\), and kernels and cokernels are componentwise with their induced structure maps.

The ring is finite, so every finitely generated module is finite. Its submodules and quotients are finite and finitely generated; hence \(\mathcal C\) is abelian. Evaluation \(E(V,W,t)=V\) is exact, with kernel \(\mathcal S\), which is therefore Serre. A right adjoint is \[ J(H)=(H,H[2],\mathrm{ev}),\qquad H[2]=\{h\in H:2h=0\},\quad \mathrm{ev}(\bar a\otimes h)=a h. \tag{8.9} \] Its adjunction bijection is explicit. Given \(f:V\to H\), the second component of the corresponding map to \(J(H)\) is \(w\mapsto f(t(\bar1\otimes w))\). Its value lies in \(H[2]\) by \(B\)-linearity, and \(k\)-linearity and the structure square follow from balancing. Conversely that square forces this second component, so projection onto \(f\) is the inverse bijection.

The counit \(EJ(H)\to H\) is identity. The unit has first component \(1_V\) and second component \(w\mapsto t(\bar1\otimes w)\); its kernel and cokernel have zero first component and belong to \(\mathcal S\). Thus \(E\) induces \(\mathcal C/\mathcal S\simeq\mathsf{Mod}_{\mathrm{fg}}(B)\), by the exact quotient and these unit/counit comparisons. All the modules in (8.9) remain finite when \(H\) is finite.

A finitely generated module over either finite ring \(\Lambda\) or \(B\) is finitely presented: a finite free cover has a finite kernel, generated by its finitely many elements. Their Ind categories are therefore all module categories by the module-presentation theorem. Applying (5.1) gives \[ \mathsf{Mod}(\Lambda)/\{(0,W,0)\}\simeq\mathsf{Mod}(B), \tag{8.10} \] with section (8.9) on all \(B\)-modules.

Now take the epimorphism \(B\to k\). Its map on the second component of (8.9) is \[ B[2]=\{0,2\}\longrightarrow k[2]=k, \qquad 2\longmapsto0. \tag{8.11} \] It is zero and not surjective. The induced arrow \(J(B)\to J(k)\) consequently is not epic, even though its first component is the given epimorphism. This disproves exactness of the fully faithful right adjoint while retaining its proved left exactness and filtered-colimit preservation. \(\square\)

9. Further reading

The Stacks Project, Serre subcategories and the exact quotient, provides the abelian quotient and its universal property.

Pierre Schapira, An Introduction to Categories and Homological Algebra, Section 2.7, introduces Ind-objects and their filtered presentations. The programme lessons linked above provide the complete recognition and exactness proofs used in the comparison.