Draft lesson.

Finitely presented generators and coherent Ind reconstruction

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

An abelian category can have enough finite pieces to reconstruct every object, while kernels between those pieces still escape the finite part. Reconstruction and coherence answer different questions. We will build the reconstruction directly from finitely presented generators and then identify the extra kernel condition that makes its small coefficient category abelian.

Fix a universe. All sets, families and diagrams called small belong to it. Let $\mathcal A$ be a locally small Grothendieck abelian category: it has small coproducts, exact small filtered colimits and a generator. Suppose a small separating family $(G_i)_{i\in I}$ consists of finitely presented objects. Here separating means that the functors $\operatorname{Hom}(G_i,-)$ jointly distinguish arrows. Finite presentation means preservation of every small filtered colimit by $\operatorname{Hom}(P,-)$. We do not assume the $G_i$ projective, or their coproduct finitely presented.

Generators and small quotient families, Sections 2–4, proves detection, canonical epimorphic presentations and smallness of subobjects. Formal colimits and compact presentations, Theorem 2.1 and Section 4, proves the realization Hom formula and the algebraic module criterion. Retracts and stabilization of formal objects, Sections 1–3, identifies finitely presented Ind objects with retracts of constants. Ind-objects through their elements, Sections 1–2, proves the element presentation and pointwise filtered colimits. We use the exact abelian completion and functor theorem in Two cokernels, controlled rows and abelian completion, Sections 5–6. The quotient comparison in Section 7 comes from Ind-completion and Serre quotients.

1. Finite colimits preserve finite presentation

Write $\mathcal A^{\mathrm{fp}}$ for the full subcategory of finitely presented objects.

Lemma 1.1. This subcategory contains zero and is closed under finite colimits and retracts in $\mathcal A$.

Proof. Mapping out of a finite colimit is the corresponding finite limit of Hom sets. Small filtered colimits of sets commute with finite limits: a tuple has finitely many components, so all its representatives can be moved to one stage; its finitely many compatibility equations hold at a further common stage. Equality of two such tuples can likewise be witnessed by finitely many later-stage equalities. The empty limit is a singleton, also preserved by a nonempty filtered colimit. Applying these facts to the finite diagram of Hom sets proves the finite-colimit claim. This includes finite direct sums and cokernels of arrows between finitely presented objects.

If $P$ is a retract of $Q$, maps $P\to M$ identify naturally with the maps $a:Q\to M$ satisfying $ae=a$, where $e$ is the splitting idempotent on $Q$. This is the equalizer of precomposition by $e$ and by $1_Q$. The same finite-limit argument proves finite presentation of $P$ when $Q$ is finitely presented. $\square$

This lemma does not claim that an arbitrary subobject of a finitely presented object is finitely presented. Kernel closure will be an additional condition.

2. There is a small category of finite pieces

Proposition 2.1. Every $P\in\mathcal A^{\mathrm{fp}}$ is a quotient of a finite direct sum of members of $(G_i)$. Consequently $\mathcal A^{\mathrm{fp}}$ is essentially small.

Proof. The canonical evaluation map $\bigoplus_{(i,u),\ u:G_i\to P}G_i\longrightarrow P \tag{2.1}$ is epic by the separating-family theorem. Its index is small. For each finite subset $F$ of that index, let $P_F\subseteq P$ be the image of its partial sum. These subobjects form a small directed system. Exactness of filtered colimits makes $\operatorname{colim}_F P_F\to P$ monic: take the colimit of the inclusions into the constant diagram $P$. Every component of (2.1) factors through this map, so (2.1) makes it epic as well. It is therefore an isomorphism.

Finite presentation of $P$ now factors $1_P$ through some $P_F\to P$. This inclusion has a right inverse, and a monomorphism with a right inverse is invertible. Thus that finite partial sum already surjects onto $P$. No lifting through an arbitrary epimorphism by a generator was used.

An abelian category is balanced, so the separating family is detecting by the generator theorem. That theorem bounds each object's subobject classes by a small set. Quotients in an abelian category are determined, up to compatible isomorphism, by their kernel subobjects: an epimorphism is the cokernel of its kernel. There are small many finite lists of members of $(G_i)$, and small many quotients of each corresponding sum. Every finitely presented object belongs to these quotient classes. Choose representatives of the finitely presented classes to obtain a small skeleton $\mathcal C$. Local smallness also makes its arrow collection small. $\square$

We use such a skeleton, with its full inclusion $j:\mathcal C\to\mathcal A$. Finite colimits in $\mathcal A^{\mathrm{fp}}$ transfer to it by representative isomorphisms. The skeleton need not contain every literal finite sum as an object; it contains its isomorphism class.

3. Construct the filtered presentation of every object

For $M\in\mathcal A$, define $E_M$ to have objects $(P,u)$, with $P\in\mathcal C$ and $u:jP\to M$. Its arrows $(P,u)\to(Q,v)$ are maps $a:P\to Q$ with $vja=u$.

Theorem 3.1. The category $E_M$ is small and filtered, and the evaluation cocone is an isomorphism $\pi_M:\operatorname{colim}_{(P,u)\in E_M}jP\ \xrightarrow{\sim}\ M. \tag{3.1}$

Proof. Smallness follows from the small skeleton and the small Hom sets into $M$. Zero supplies an object. A direct sum of two finite pieces, replaced by its representative, supplies a common target with the map induced by their two maps to $M$. For parallel arrows $a,b:(P,u)\rightrightarrows(Q,v)$, take the cokernel of $a-b$. Lemma 1.1 makes it a finite piece. Since $v j(a-b)=0$, the map $v$ factors through this cokernel, giving an arrow of $E_M$ equalizing $a,b$. Thus $E_M$ is filtered.

Every map $G_i\to M$ occurs in this cocone after replacing $G_i$ by its skeleton representative. Therefore the canonical evaluation epimorphism from the sum of all these maps factors through $\pi_M$. Cancellation makes $\pi_M$ epic.

To prove monicity, let $x:G_i\to\operatorname{colim}_{E_M}jP$ satisfy $\pi_Mx=0$. Finite presentation of $G_i$ gives a stage $(P,u)$ and $t:G_i\to jP$ representing $x$. We have $ut=0$. The cokernel $jP\to Q'$ of $t$ is finitely presented by Lemma 1.1, because both $G_i$ and $jP$ are. The map $u$ descends to $Q'\to M$. Replace $Q'$ by its skeleton representative. This is a transition in $E_M$ that kills $t$, so $x=0$.

Hence every probe $\operatorname{Hom}(G_i,\pi_M)$ is injective: the difference of two maps with equal images is zero by the preceding argument. The generator theorem makes $\pi_M$ monic. Balancedness gives (3.1). $\square$

The presentation uses all finite pieces mapping to $M$, including nonmonic maps. A category of finite subobjects would not provide the cokernel transitions that killed the maps $t$. For $f:M\to N$, postcomposition gives $E_M\to E_N$, sending $(P,u)$ to $(P,fu)$; the evaluation maps commute with $f$. This is the naturality of (3.1).

4. Realization and restricted Yoneda are inverse

Let $H(M)=\operatorname{Hom}_{\mathcal A}(j(-),M)$, considered as a set-valued presheaf on $\mathcal C$, with its natural abelian group structure.

Theorem 4.1. Realization and restricted Yoneda give inverse equivalences $J:\operatorname{Ind}(\mathcal C)\longrightarrow\mathcal A, \qquad H:\mathcal A\longrightarrow\operatorname{Ind}(\mathcal C). \tag{4.1}$ The first takes a formal filtered presentation $(P_a)$ to $\operatorname{colim}_a jP_a$. Both equivalences preserve small filtered colimits.

Proof. Apply a finite test $jQ$ to (3.1). Finite presentation gives $H(M)(Q)=\operatorname{colim}_{(P,u)\in E_M} \operatorname{Hom}_{\mathcal C}(Q,P). \tag{4.2}$ The comparison is postcomposition with $u$, natural in $Q,M$. Thus $H(M)$ is an Ind presheaf with the indicated small filtered presentation.

The realization theorem already proves full faithfulness of $J$. Its actual map formula is $\begin{aligned} \operatorname{Hom}_{\mathcal A}(\operatorname{colim}_a jP_a, \operatorname{colim}_b jQ_b) &=\lim_a\operatorname{colim}_b \operatorname{Hom}_{\mathcal C}(P_a,Q_b). \end{aligned} \tag{4.3}$ All identifications are natural universal-property maps and preserve composition. Theorem 3.1 supplies its essential-surjectivity hypothesis. More explicitly, (3.1) is the natural isomorphism $JH(M)\to M$; finite presentation of each test gives $HJ(X)(Q)=X(Q)$ for an arbitrary Ind presentation $X$. These comparisons are induced by the same structural maps, so their composites are identities on the presentations and obey the triangle identities. They establish the specified inverse equivalences. Filtered colimits of Ind presheaves are pointwise, and each test $jQ$ preserves them in $\mathcal A$. This proves the filtered-colimit comparisons; alternatively any equivalence preserves existing colimits. $\square$

In particular every transformation $H(M)\to H(N)$ comes from a unique arrow $M\to N$. The restricted tests encode the whole category, including maps, rather than only its object classes.

5. When is the category of pieces abelian?

A Grothendieck category with a small family of finitely presented generators is called locally finitely presented. We call it locally coherent if its finitely presented objects are also closed under kernels in the ambient category.

Proposition 5.1. Under our hypotheses, kernel closure is equivalent to $\mathcal A^{\mathrm{fp}}$ being abelian. In that case its inclusion is exact. Its small skeleton $\mathcal C$ is abelian, and (4.1) is an exact equivalence of abelian categories.

Proof. If kernels are closed, finite sums, kernels and cokernels of arrows in the full subcategory are the ambient ones. Their universal properties restrict there. Images and coimages consequently agree by the ambient abelian axiom, so the subcategory is abelian and its inclusion exact.

Conversely suppose the full subcategory is abstractly abelian. For an arrow $f:P\to Q$ of finite pieces, let $K\to P$ be its kernel there, and let $K'\to P$ be its kernel in $\mathcal A$. There is a comparison $K\to K'$. Testing by any $G_i$, which is a finite piece, gives the same kernel of $\operatorname{Hom}(G_i,P)\to\operatorname{Hom}(G_i,Q)$ on both sides. The comparison is therefore bijective on every probe. Detection makes it an isomorphism. Thus the ambient kernel is finitely presented. Cokernels were already inherited by Lemma 1.1. Finally an equivalence between abelian categories preserves kernels and cokernels by their universal properties and is exact. The abelian Ind-completion theorem supplies the structure on $\operatorname{Ind}(\mathcal C)$. $\square$

Theorem 5.2. Every small abelian category $\mathcal B$ has a locally coherent Grothendieck completion $\operatorname{Ind}(\mathcal B)$, whose finitely presented part is exactly its constant copy of $\mathcal B$, up to isomorphism. Conversely every locally coherent Grothendieck category is an Ind completion of its small abelian category of finite pieces.

Proof. The abelian completion theorem gives the Grothendieck property and a separating family of constants. The element-colimit theorem makes these constants finitely presented. The retract theorem says that every finitely presented Ind object is a retract of a constant. Idempotents split in an abelian category: for $e^2=e$, factor $e$ through its image $p\colon B\twoheadrightarrow\operatorname{im}e$, $i:\operatorname{im}e\hookrightarrow B$; cancellation in $ipip=ip$ gives $pi=1$. A retract of the constant on $B$ therefore represents a constant on that image. Exactness of the constant embedding makes kernels between constants constant as well. This proves local coherence and identifies the finite part. The converse is Proposition 5.1 and Theorem 4.1. $\square$

Coherence here concerns the inherited category of finite pieces. It does not say every subobject of a finite piece is finite, nor that the category is noetherian. Exercise 8.2 separates these conditions explicitly.

6. Reconstruct functors and their transformations

Let $\mathcal A,\mathcal A'$ both be locally coherent, with small abelian finite parts $\mathcal C,\mathcal C'$. Choose the equivalences just proved, denoted $J,H$ and $J',H'$.

Proposition 6.1. If $F:\mathcal A\to\mathcal A'$ preserves filtered colimits and finitely presented objects, it is naturally isomorphic to $J'\operatorname{Ind}(F_{\mathrm{fp}})H, \tag{6.1}$ where $F_{\mathrm{fp}}:\mathcal C\to\mathcal C'$ is its restriction, transferred to the chosen skeletons. Natural transformations between two such functors correspond bijectively to transformations of their restrictions. If $F_{\mathrm{fp}}$ is additive and exact, then $F$ is additive and exact. Conversely an additive exact $F$ has additive exact restriction.

Proof. Applying $F$ to (3.1) yields the natural comparison $\operatorname{colim}_{(P,u)\in E_M}F(jP)\ \xrightarrow{\sim}\ F(M). \tag{6.2}$ The left side is exactly the realization of the induced Ind presentation. This proves (6.1), with naturality following from postcomposition on $E_M$. For a transformation on finite pieces, its components on this diagram give a unique map between the two colimits. Equation (6.2) transports that map to a component on $M$. The equations on all structural maps establish naturality in $M$. Conversely naturality of any transformation on the whole categories determines these components from its values on finite pieces, since a colimit cocone determines a map out uniquely. Restriction and extension are inverse.

The induced Ind functor of an additive exact functor between small abelian categories is additive and exact by the earlier kernel/cokernel theorem. Both realizations are exact equivalences, so (6.1) transfers this conclusion to $F$. For the converse, the finite parts have their inherited kernels, cokernels and sums; restriction of an additive exact functor preserving them retains exactness. $\square$

Thus exactness can be tested on finite pieces for this specified class of functors. A functor that does not preserve filtered colimits is not covered by the argument.

7. Finite pieces in an Ind Serre quotient

Take a small abelian $\mathcal B$, a Serre subcategory $\mathcal S$, and the exact quotient $Q:\mathcal B\to\mathcal B/\mathcal S$. The earlier quotient lesson constructs $L=\operatorname{Ind}(Q):\operatorname{Ind}(\mathcal B) \longrightarrow\operatorname{Ind}(\mathcal B/\mathcal S), \quad L\dashv R, \tag{7.1}$ with invertible counit, kernel $\operatorname{Ind}(\mathcal S)$, and a right adjoint $R$ preserving filtered colimits.

Corollary 7.1. Both categories in (7.1) and the intrinsic category $\operatorname{Ind}(\mathcal S)$ are locally coherent. The finite pieces of the first two are $\mathcal B$ and $\mathcal B/\mathcal S$. The quotient $L$ preserves finite presentation, and the finitely presented objects of its source that it kills are exactly the constants from $\mathcal S$. These are also precisely the finite pieces of $\operatorname{Ind}(\mathcal S)$ under its inclusion. The section $R$ need not preserve finite presentation.

Proof. A Serre subcategory of an abelian category is itself abelian with inherited kernels and cokernels. It and the quotient are small (or can be replaced by small skeletons). Theorem 5.2 applies to all three. The full Ind inclusion of $\mathcal S$ sends its constants to the same constants in the larger completion, and the finite-piece identifications prove the claimed agreement. On those source constants $L$ is the constant of $Q$, so finite presentation and the asserted killed objects follow from $\ker Q=\mathcal S$.

There is also an adjunction proof of the preservation claim. For any filtered diagram $(Y_a)$ and finite source object $P$, the natural adjunction and the filtered-colimit comparison for $R$ give $\begin{aligned} \operatorname{Hom}(LP,\operatorname{colim}_aY_a) &=\operatorname{Hom}(P,\operatorname{colim}_a RY_a)\\ &=\operatorname{colim}_a\operatorname{Hom}(P,RY_a) =\operatorname{colim}_a\operatorname{Hom}(LP,Y_a). \end{aligned} \tag{7.2}$ This makes $LP$ finitely presented. It is a statement about the left adjoint; the failure for the right adjoint is computed in Exercise 8.1. $\square$

8. Four graded exercises with complete solutions

Exercise 8.1 — A section can enlarge a finite piece (introductory)

For rationalization $L:\mathsf{Ab}\to\mathsf{Vect}_{\mathbb Q}$ and its section $R$, determine whether each functor preserves filtered colimits and finitely presented objects. Give an explicit finite object whose section fails finite presentation.

Solution. The full quotient calculation in the preceding lesson identifies $L(M)=M\otimes_{\mathbb Z}\mathbb Q$ and $R$ with forgetting the rational vector-space action. It proves the exact quotient from the small category of finitely generated abelian groups modulo finite groups and the induced adjunction. As a left adjoint, $L$ preserves filtered colimits. Vector-space and group filtered colimits are computed by representatives with addition at a common stage; forgetting scalars leaves precisely that group colimit. Hence $R$ preserves them as well.

The finite integer modules and finite-dimensional rational vector spaces are the finite parts of their categories by the module criterion and the PID argument in that lesson. Rationalizing a finite generating set yields a finite spanning set, so $L$ preserves finite presentation. Equivalently use (7.2).

The one-dimensional space $\mathbb Q$ is finitely presented over $\mathbb Q$, but its underlying group is not even finitely generated. Given finitely many rational numbers, choose a positive common denominator $d$. Their integer span is contained in $d^{-1}\mathbb Z$, which does not contain $1/(2d)$. Thus they cannot generate $\mathbb Q$. The module criterion implies that a finitely presented group is finitely generated, proving the failure for $R$. One can see it in the defining comparison too: $\mathbb Q$ is the directed union of the groups $d^{-1}\mathbb Z$, ordered by divisibility of positive integers, and its identity cannot factor through any such subgroup. Preservation of filtered colimits by a functor does not imply preservation of finite objects by that same functor. $\square$

Exercise 8.2 — Coherence without noetherianity (advanced)

Let $k$ be a field and $R=\prod_{n\geq1}k$. Prove directly that the category of finitely presented $R$-modules is abelian with its inherited structure, while $R$ is not noetherian. Identify the kernel and cokernel splittings needed for this proof.

Solution. A finite matrix $A:R^r\to R^s$ is a sequence of matrices $A_n:k^r\to k^s$. Choose a complement to $\ker A_n$ and a complement to $\operatorname{im}A_n$. Invert $A_n$ from its chosen first complement to its image, and put zero on the second complement. This gives $B_n:k^s\to k^r$ with $A_nB_nA_n=A_n$. Finite matrix sizes allow these coordinate entries to assemble to a matrix $B$ over $R$, and $ABA=A$.

Both $BA$ and $AB$ are idempotent. We have $\ker A=\operatorname{im}(1-BA),\qquad \operatorname{im}A=\operatorname{im}(AB),\qquad \operatorname{coker}A\simeq\operatorname{im}(1-AB). \tag{8.1}$ For the kernel, $A(1-BA)=0$, and a killed vector satisfies $(1-BA)x=x$. The image equality follows from $ABA=A$, and the cokernel formula is the direct-sum splitting by $AB$. Thus these are finitely generated direct summands of finite free modules. Every finitely presented module, being a cokernel of a finite matrix, is such a summand. Conversely such a summand is finitely presented by the retract criterion for modules, already proved in the module-presentation lesson.

For a map between two such summands, write $P=eR^r$, $T=fR^s$, with idempotents $e,f$, and extend its matrix to $A=fAe:R^r\to R^s$. Choose a matrix $B_0$ with $AB_0A=A$ by the coordinate argument. Set $B=eB_0f$; then $ABA=A$, $BA=eBA=BAe$, and $AB=fAB=ABf$. The idempotents $e-BA$ and $f-AB$ give $\ker(P\to T)=\operatorname{im}(e-BA),\qquad \operatorname{coker}(P\to T)\simeq\operatorname{im}(f-AB). \tag{8.2}$ For example $(e-BA)^2=e-BA$, and on $P$ it fixes exactly the killed vectors. On $T$, the complement of $AB$ represents the quotient by the image. Both results are again finite direct summands, so both are finitely presented. This proves the inherited abelian structure and, since $R$ is a finitely presented generator of $\mathsf{Mod}(R)$, local coherence and its Ind reconstruction.

It also proves ring coherence in the usual ideal sense: a finitely generated ideal is the image of a finite row matrix, hence a finite direct summand and finitely presented. But the ideal of finitely supported sequences is not finitely generated. A finite list of its elements has support in one finite subset of $\mathbb N$; multiplying by arbitrary sequences and adding cannot enlarge that support. The coordinate idempotent supported at an index outside it is in the ideal and not in their span. Thus the ideal is not finitely generated and $R$ is not noetherian. $\square$

Exercise 8.3 — An actual kernel escapes the finite part (intermediate)

Let $V$ be an infinite-dimensional vector space over $k$, and define the commutative ring $R=k\oplus V$ by $(a,v)(b,w)=(ab,aw+bv). \tag{8.3}$ For $0\ne v\in V$, compute the kernel and cokernel of multiplication by $(0,v)$ on $R$. Prove that $\mathsf{Mod}(R)$ has Ind reconstruction from its finite pieces but is not locally coherent.

Solution. The formula is associative: the second component of either triple product is $abz+acw+bcv$; its unit is $(1,0)$. The ideal $V$ has square zero. Multiplication by $(0,v)$ sends $(a,w)$ to $(0,av)$. Hence its kernel is $V$, its image is the one-dimensional ideal $kv$, and its cokernel is $R/kv$. Both copies of $R$ are finite free, and this one-relation cokernel is finitely presented.

As an $R$-module, $V$ is acted on only by the scalar $k$ part. A finite list therefore generates only its finite-dimensional $k$-span. Since $V$ is infinite-dimensional it is not finitely generated, and thus not finitely presented. The kernel of an arrow between finite pieces has escaped that subcategory. Proposition 5.1 proves it is not abelian and the module category is not locally coherent.

Nevertheless $R$ is a finitely presented separating generator: $\operatorname{Hom}_R(R,M)$ is the underlying module, and representatives compute its filtered colimits. Module filtered colimits are exact, since a finite kernel equation or representative lift is witnessed at one later stage; small coproducts exist. Thus $\mathsf{Mod}(R)$ meets the hypotheses of Theorem 4.1, or one can use the earlier arbitrary-ring module theorem directly. Its realization from $\operatorname{Ind}(\mathsf{Mod}_{\mathrm{fp}}(R))$ is an equivalence. Here the small category of pieces has finite colimits but fails to be abelian. Reconstruction itself has survived. $\square$

Exercise 8.4 — Watch the missing finite-presentation lift (advanced)

For the ring in Exercise 8.3, set $P=R/V\simeq k$. Give a filtered diagram of finitely presented modules whose colimit is $P$, and show explicitly why the identity of this cyclic module does not lift to any stage.

Solution. Direct the finite-dimensional subspaces $F\subset V$ by inclusion. This is a small directed set; two subspaces are contained in their finite-dimensional sum. Each $F$ is an ideal because multiplication on $V$ is scalar multiplication. A finite basis of $F$ gives a presentation $R^{\dim F}\longrightarrow R\longrightarrow M_F=R/F\longrightarrow0, \tag{8.4}$ so every $M_F$ is finitely presented. For $F\subset F'$ use the quotient map. The colimit is $R/V=P$: a representative in $R$ dies precisely when its difference lies in some finite-dimensional $F$, and the union of these subspaces is $V$. These maps and equalities respect the $R$-action.

An $R$-map $P\to R/F$ is determined by the image $y=(a,\overline w)$ of $1$, with $Vy=0$. For $z\in V$, this condition says $az\in F$. If $a\ne0$, it would imply $V\subseteq F$, contrary to infinite dimension. Thus $a=0$; conversely every $y=(0,\overline w)$ is annihilated by $V$ and defines such a map. Their composites with $R/F\to P$ are all zero. The identity of $P$, a nonzero element of $\operatorname{Hom}_R(P,P)$, is absent from the image of $\operatorname{colim}_F\operatorname{Hom}_R(P,M_F) \longrightarrow\operatorname{Hom}_R(P,P). \tag{8.5}$ In fact the source is zero: the Hom group at a stage is $V/F$, and every representative $w$ vanishes after enlarging $F$ by its span. The target is $k$. Equation (8.5) is exactly the failed surjectivity in the definition of finite presentation. The module $P$ is cyclic, but its annihilator contains infinitely many independent relations. This computes the failure, beyond merely counting generators. $\square$

9. Freely accessible further reading

The Stacks Project, Characterizing finite and finitely presented modules, gives the module filtered-colimit criterion and finite-presentation approximations. Its Coherent rings discusses coherent modules, inherited kernels and cokernels, and the comparison with noetherianity. These are additional readings; the categorical construction and the examples above have their proofs here or in the linked earlier lessons.

Henning Krause's freely accessible Deriving Auslander’s formula, the paragraph after Theorem 4.9 and the functoriality discussion after Corollary 4.11, places finite-piece reconstruction and exact Ind functors in a wider setting. That paper's additional derived-category results are further topics rather than prerequisites for this lesson.