Finite support and the limits that a category permits

Written by GPT-6.1 Sol (OpenAI), reasoning effort Ultra, October 2026. Self-checked by GPT-6.1 Sol, the writing AI; no independent review. Public domain (CC0).

A limit answers a universal question about maps from the objects of a particular category. If those objects change, the answer can change too. We will see increasing nonzero modules whose colimit becomes zero in a full abelian subcategory, and decreasing quotients whose limit does the same. Formal ind-objects give a second phenomenon: finite support can prevent inverse limits from preserving surjections, even when every individual short exact sequence splits.

The last section connects finite support to a finiteness condition on an object. An object of finite type can detect when two maps eventually agree. An object of finite presentation must also allow every map into a filtered colimit to appear at a stage. These are different requirements, even for an object of length one.

We assume ordinary abelian categories and filtered diagrams. The prerequisites are Full subcategories and exact closure, Theorem 2.2; Composition factors and uniform chain bounds, Theorem 2.1; Ind-objects through their elements, for compact constants and pointwise filtered colimits; Limits and colimits of formal objects, Corollary 2.3 and Theorem 3.1; and Ind-abelian categories and controlled exact sequences, Theorems 3.2 and 6.1. Their constructions and proofs are used throughout. Basic open references are [Stacks, Finite modules], [Stacks, Inverse systems] and [Schapira, Homological Algebra], listed below.

Fix a universe of small indexing categories, and use larger ambient universes for presheaves and iterated completions. Categories are locally small relative to the indexing universe. An essentially small category is not required. Every limit, colimit and countable family below is small in that fixed indexing universe. For an abelian category \(\mathcal C\), write \(\iota X\) for the constant object in \(\operatorname{Ind}(\mathcal C)\). We distinguish formal coproducts in this category from any coproduct in \(\mathcal C\).

1. Zero limits inside uniformly nilpotent modules

Let \(k\) be any field and \(A=k[t]\). Let \(\mathcal N\) be the full subcategory of all \(A\)-modules \(M\) for which \(t^rM=0\) for some integer \(r\geq0\) depending on \(M\). There is no restriction on dimension or on the number of generators. The case \(r=0\) gives only the zero module.

This is an abelian category with exact inclusion in \(\mathsf{Mod}(A)\). Indeed, zero and finite direct sums belong to it. A kernel is a submodule of the source and inherits its bound; a cokernel is a quotient of the target and inherits its bound. Apply the full-subcategory criterion in the prerequisite. It is also extension closed: if \(t^aL=0\) and \(t^bN=0\) in a short exact sequence \(0\to L\to M\to N\to0\), then \(t^bM\subseteq L\) and \(t^{a+b}M=0\). Its abelian structure therefore agrees with the ambient kernels, cokernels and short exact sequences.

For \(n\geq0\), put

The maps \(X_n\to X_{n+1}\) are inclusions, and the maps \(Y_{n+1}\to Y_n\) are reductions. Both diagrams lie in \(\mathcal N\), since \(t^nX_n=0\) and \(t^{n+1}Y_n=0\). In particular \(X_0=0\). The inclusions are monomorphisms and the reductions are epimorphisms, both internally and in the ambient module category.

Proposition 1.1. Their universal objects are as follows.

Proof. The ambient colimit of the increasing submodules \(X_n\) is their union

To verify the colimit property directly, maps defined compatibly on the submodules define a unique map on their union. Every Laurent polynomial modulo \(A\) has only finitely many negative powers, so belongs to some \(X_n\). Multiplication by \(t\) on \(P\) is surjective: divide a Laurent representative by \(t\).

Consider any cocone from the \(X_n\) to \(N\in\mathcal N\). It induces an ambient map \(f:P\to N\). Choose \(r\) with \(t^rN=0\). For every \(p\in P\), surjectivity of multiplication by \(t^r\) supplies \(z\in P\) with \(p=t^rz\). Then

Thus every such cocone is zero. It factors uniquely through the zero cocone with vertex \(0\). This proves the colimit assertion inside \(\mathcal N\), including the required universal property for arbitrarily large modules in that category.

For the second diagram, write a class in \(Y_n\) uniquely as a polynomial of degree at most \(n\). Compatibility of its reductions says that each coefficient is independent of the truncation once that coefficient is present. Hence compatible tuples correspond bijectively to formal power series. This correspondence respects addition and the \(A\)-action, and its coordinate maps are truncations. Conversely a power series supplies all its truncations. Maps into this module are exactly compatible families of maps into the \(Y_n\), so the ambient limit is \(k[[t]]\).

Multiplication by \(t^r\) on \(k[[t]]\) is injective: it shifts the coefficients by \(r\) places. A cone from \(M\in\mathcal N\) induces an ambient map \(g:M\to k[[t]]\). If \(t^rM=0\), then \(t^rg(m)=g(t^rm)=0\), so injectivity gives \(g(m)=0\) for every \(m\). Every cone is therefore zero and factors uniquely through \(0\). This proves the internal limit assertion. \(\square\)

Both ambient answers are nonzero. The class of \(t^{-1}\) is nonzero in \(P\); for every \(r\), the class of \(t^{-(r+1)}\) survives multiplication by \(t^r\). In \(k[[t]]\), multiplication by \(t^r\) sends \(1\) to the nonzero series \(t^r\). Neither ambient answer belongs to \(\mathcal N\). Exactness of the inclusion on finite exact sequences does not make it preserve these particular infinite limits or colimits.

2. Formal tails and failure of exact inverse limits

Let \(\mathcal C\) be any abelian category with all small limits. Its ind-category is abelian and has all small limits and colimits by the prerequisite theorems. Constants are compact for formal filtered colimits, and \(\iota\) preserves the small limits already present in \(\mathcal C\).

For arbitrary objects \((X_m)_{m\geq0}\) of \(\mathcal C\), define formal coproducts

Each finite prefix \(F_n\) is \(\iota(\bigoplus_{m<n}X_m)\), since constants preserve finite biproducts. The formal coproduct construction identifies \(S\) with the filtered colimit of these finite prefixes. There is a finite biproduct decomposition \(S=F_n\oplus T_n\). The inclusions \(T_{n+1}\to T_n\) define an inverse system.

Theorem 2.1. For every such family, including families containing zero objects,

Proof. The limit exists by the formal limit theorem. Test a cone on it from a constant \(\iota Y\). Its maps \(a_n:\iota Y\to T_n\), followed by the inclusions into \(S\), are one common map \(a:\iota Y\to S\), by compatibility.

Compactness of the constant gives a factorization \(a:\iota Y\to F_r\to S\) for some \(r\). For \(n\geq r\), the same map also factors through \(T_n\). Project \(S=F_n\oplus T_n\) onto \(F_n\). It kills that second factorization, whereas its restriction to \(F_r\) is the split inclusion \(F_r\to F_n\). This inclusion is monic, so the map \(\iota Y\to F_r\) is zero. Thus \(a=0\). Every inclusion \(T_n\to S\) is split monic, so all \(a_n=0\).

The limit therefore has zero Hom group from every constant. Constants detect objects: in the presheaf description, \(\operatorname{Hom}(\iota Y,L)=L(Y)\). A presheaf with only its zero element at every \(Y\) is the zero ind-object. This proves (2.2). \(\square\)

An infinite intersection of tails has vanished, but the transition maps need not ever become zero. The assertion concerns the actual inverse limit in the ind-category; it does not assert that the diagram is a zero formal pro-object.

Theorem 2.2. If \(\mathcal C\) contains a nonzero object, cofiltered limits in \(\operatorname{Ind}(\mathcal C)\) are not exact. A countable inverse system of split short exact sequences already demonstrates the failure.

Proof. Choose \(0\ne X\in\mathcal C\) and set \(X_m=X\) for every \(m\). At each \(n\) we have the split short exact sequence

On the first terms use the tail inclusions, on the middle terms use identities, and on the last terms use the prefix projections. These maps make (2.3) a short exact sequence of inverse systems. Its inverse limit has the form

The first limit is zero by Theorem 2.1. The middle diagram is constant, so has limit \(S\). The limit of the finite prefixes in \(\mathcal C\) is \(P\): compatible maps into finite prefixes are precisely a map to each coordinate. Preservation of limits by \(\iota\) identifies the last limit with \(\iota P\). The map \(\alpha\) is induced by embedding every finite prefix in \(P\), with zero in the other coordinates.

We prove that \(\alpha\) is not epic by constructing a nonzero map that kills it. This is necessary: a failure of surjectivity after applying \(\operatorname{Hom}(\iota Y,-)\) alone would not establish failure of epimorphy unless \(\iota Y\) were projective.

Let \(D_n\subseteq P\) be the first \(n\) coordinates, inserted with zero elsewhere; \(D_0=0\). This is a split monomorphism from the finite biproduct. The inclusions \(D_n\subseteq D_{n+1}\) give quotient maps, hence an ind-object

The compatible quotient maps define \(q:\iota P\to Q\), represented at stage \(n\) by \(P\to P/D_n\). Compactness of \(\iota P\) identifies its Hom group into \(Q\) with the filtered colimit of the corresponding Hom groups. Therefore \(q=0\) would mean that one of these quotient maps becomes zero at a later stage.

None is zero. The projection of \(P\) to coordinate \(n\) kills \(D_n\), so factors through \(P/D_n\). It is nonzero because inserting \(X\) in that coordinate is its section and \(1_X\ne0\). Thus \(q\ne0\).

On any finite prefix of \(S\), the composite \(q\alpha\) becomes zero after quotienting by a sufficiently long \(D_n\). It is consequently zero on every summand, and hence zero on \(S\) by the coproduct property. The distinct maps \(q,0:\iota P\to Q\) agree after precomposition by \(\alpha\). This proves that \(\alpha\) is not epic, so the inverse limit of (2.3) is not right exact. \(\square\)

The finite-prefix sections \(F_n\to S\) split each individual row but fail compatibility. For example, inserting the last coordinate of \(F_{n+1}\) into \(S\) is nonzero, whereas projecting that coordinate to \(F_n\) and then taking its section gives zero. A compatible section of the inverse systems would give a right inverse to \(\alpha\), contrary to the theorem.

The quotient construction extends the concrete formal quotient in Finite-rank tests and finite generation, Proposition 2.1. The categorical argument here uses coordinate projections and sections in any abelian \(\mathcal C\), rather than elements of vector spaces. For comparison, [Stacks, Inverse systems] treats Mittag–Leffler hypotheses for ordinary module systems. No general assertion about arbitrary abelian categories is obtained by replacing formal objects with their ordinary realizations.

Corollary 2.3. For nonzero \(\mathcal C\), the abelian categories

are not equivalent.

Proof. Apply the dual of the ind-abelian theorem to \(\mathcal C\). It makes \(\operatorname{Pro}(\mathcal C)\) abelian with all small limits and exact small cofiltered limits. Its constant embedding is fully faithful, so the image of a nonzero object of \(\mathcal C\) is nonzero. Theorem 2.2, with this pro-category as its base, shows that \(\operatorname{Ind}(\operatorname{Pro}(\mathcal C))\) does not have exact cofiltered limits.

On the other hand, the same dual theorem applied to the abelian category \(\operatorname{Ind}(\mathcal C)\) gives exact cofiltered limits in \(\operatorname{Pro}(\operatorname{Ind}(\mathcal C))\). An equivalence preserves all existing limits and colimits by transporting their universal properties. In particular it preserves zero objects, kernels and cokernels, and therefore short exact sequences. It would transport exactness of the inverse-limit functor from one category to the other, a contradiction. All iterations use the fixed small indexing universe and sufficiently large ambient universes as specified above. \(\square\)

3. Noetherian objects detect eventual equality

Let \(\mathcal A\) be an abelian category with all small colimits and exact small filtered colimits. For a small filtered diagram \(Y:I\to\mathcal A\), there is a canonical comparison

Here finite type means that (3.1) is injective for every such diagram. Finite presentation means it is bijective. The arbitrary-ring module criterion, Theorem 3.1 of Finite-rank tests and finite generation, is retained: finite type in a module category is exactly finite generation. Its commutative instance is [Stacks, Finite modules]. We now prove the category-level implication using kernels instead of module elements.

Theorem 3.1. Every Noetherian object of \(\mathcal A\) is of finite type. Consequently every object of finite length is of finite type.

Proof. Suppose two representatives in the left side of (3.1) have the same image. Filteredness moves them to a common stage \(j\). Their difference is a map \(g:X\to Y_j\) whose composite into \(L=\operatorname{colim}_iY_i\) is zero. It is enough to show that \(Y(u)g=0\) for a single arrow \(u:j\to i\).

Use the category \(J=j/I\) whose objects are arrows \(u:j\to i\), and whose maps \(u\to v\) are arrows \(h:i\to i'\) with \(hu=v\). It is small, nonempty and filtered. To combine two objects, first choose a common target of their endpoints and then equalize the resulting two arrows from \(j\). To equalize parallel arrows in \(J\), use filteredness of \(I\) on their underlying arrows. This treats parallel morphisms as well as objects; the index is not assumed to be a poset.

The projection \(J\to I\) preserves the colimit of \(Y\). Here is the required cocone check. Given a cocone on the restricted diagram, define its map from \(Y_i\) by choosing arrows \(i\to k\) and \(j\to k\) and using the cocone map at the latter object of \(J\). Two choices can be moved to a common endpoint, then the two resulting arrows from \(i\), and separately those from \(j\), can be equalized after a further arrow. The cocone identities therefore make the definition independent of choices. The same operation checks compatibility with every arrow from \(i\), and it recovers the original restricted cocone. Conversely any cocone on \(I\) restricts to this one. The two operations are inverse, establishing the colimit claim.

For \(u\in J\), define the subobject

A map \(u\to v\) gives an inclusion \(K_u\subseteq K_v\). These form the kernel diagram of the natural transformation from the constant diagram \(X\) on \(J\) to the restricted \(Y\). A constant diagram on a nonempty filtered category has colimit its constant value: compatibility forces all cocone maps to agree, since any two indices have a common target. Exactness of filtered colimits preserves kernels, by the exact-functor criterion in the prerequisite. As the induced map \(X\to L\) is zero, we obtain

with its canonical maps into \(X\).

The \(K_u\) constitute a small nonempty directed family of subobjects. A Noetherian object makes such a family have a greatest member, by the chain criterion in the composition-factors prerequisite. Choose it as \(K_v\). Every map \(K_u\to X\) factors uniquely through \(K_v\); these factorizations are compatible because its inclusion is monic. The universal property of (3.3) gives \(b:X\to K_v\) with

That inclusion is both monic and epic, so is an isomorphism in an abelian category. Hence \(K_v=X\), which means \(Y(v)g=0\). The two Hom representatives become equal after one arrow and have the same filtered-colimit class. This proves injectivity of (3.1).

An object of finite length is Noetherian by the complete chain-bound theorem in the prerequisite. Apply the implication just proved. Artinianity is not needed for the stronger Noetherian statement. \(\square\)

Nothing in this argument constructs a representative for an arbitrary map \(X\to L\). Exercise 1 shows that surjectivity can fail even for a simple object. Exactness of filtered colimits lets the increasing kernels fill \(X\); Noetherianity makes one kernel already fill it. Those are the two separate steps.

4. Exercises with full solutions

Exercise 1 (intermediate: length one with infinitely many relations). Let \(V\) be an infinite-dimensional vector space over a field \(k\). Give \(R=k\oplus V\) the multiplication

Let \(K=R/V\). Prove that \(K\) has length one and is of finite type, but is not finitely presented. For a concrete failure of surjectivity, use the filtered diagram \(R/W\) indexed by finite-dimensional subspaces \(W\subseteq V\). Compute both sides of (3.1).

Solution. Formula (4.1) makes \(R\) a commutative unital ring with unit \((1,0)\): multiplying three factors gives scalar part \(abc\) and vector part \(abz+acw+bcv\), in either association. The ideal \(V\) has square zero. The quotient \(K\) is \(k\), with \(R\) acting through its scalar part. Its submodules are \(k\)-subspaces of the one-dimensional space \(k\), so are just \(0,K\). Thus it is simple and has length one. It is cyclic, so the retained module criterion gives finite type; Theorem 3.1 gives the same conclusion.

Every subspace \(W\subseteq V\) is an ideal, because multiplication on \(V\) uses only the scalar part. The finite-dimensional subspaces form a small filtered poset under inclusion: \(W+W'\) is a common upper bound. Their union is \(V\), and the quotient maps have colimit \(R/V=K\). Indeed, compatible maps out of \(R/W\) correspond to a map out of \(R\) killing every \(W\), hence all of \(V\).

A homomorphism \(K\to R/W\) is determined by the image of \(1\), which must be annihilated by \(V\). For a class represented by \((a,v)\), multiplication by \((0,u)\) gives \((0,au)\) modulo \(W\). To vanish for every \(u\in V\) requires \(aV\subseteq W\). As \(W\) is finite-dimensional and \(V\) is not, this forces \(a=0\). Conversely every class in \(V/W\) is annihilated by \(V\). Therefore

The middle equality holds because the class of any \(v\) is killed after enlarging \(W\) to \(W+kv\). The Hom comparison is consequently \(0\to k\). It is injective, but its image misses \(1_K\ne0\). The categorical finite-presentation criterion in Formal colimits and compact presentations, Theorem 4.1, proves that \(K\) is not finitely presented. This conclusion does not depend on a choice of basis of \(V\).

Exercise 2 (advanced: a power-series endomorphism ring). For \(A=k[t]\) and \(P=A[t^{-1}]/A\), compute \(\operatorname{End}_A(P)\) as a ring. Classify all submodules of \(P\), decide whether it is Noetherian or Artinian, and determine the kernel and image of every nonzero endomorphism. Finally decide whether \(P\) is of finite type in \(\mathsf{Mod}(A)\).

Solution. Write \(e_n\) for the class of \(t^{-n}\), with \(n\geq1\). The submodule \(X_n=Ae_n\) is isomorphic to \(A/(t^n)\), and

Moreover \(\ker(t^n:P\to P)=X_n\). Indeed, the classes of the negative monomials form a \(k\)-basis, and a class killed by \(t^n\) has no powers below \(t^{-n}\).

An endomorphism \(f\) preserves these annihilator submodules. Thus its restriction to \(X_n\) is multiplication by a unique \(a_n\in A/(t^n)\), determined by \(f(e_n)=a_ne_n\). Compatibility with \(te_{n+1}=e_n\) is exactly

The compatible coefficients give a unique power series \(a\in k[[t]]\). Conversely any such series acts on \(P\): on an element killed by \(t^n\), use its polynomial truncation modulo \(t^n\). Different sufficiently long truncations agree, and (4.4) proves compatibility on all \(X_n\). Addition and composition are addition and multiplication of the truncated coefficients at every \(n\). Hence this correspondence is a ring isomorphism

To classify submodules, an element of exact annihilator exponent \(n\) can be written \(b e_n\) with \(b\in A\) having nonzero constant coefficient modulo \(t^n\). Such a \(b\) is a unit in \(A/(t^n)\): after dividing by its constant coefficient, its remaining positive-degree part is nilpotent, and a finite geometric sum gives an inverse. This element therefore generates \(X_n\). A submodule with elements of unbounded annihilator exponent contains every \(X_n\) and equals \(P\). If its nonzero exponents are bounded, their maximum \(n\) exists; it contains \(X_n\), and all its elements lie in \(X_n\), so it equals \(X_n\). The complete list is

The strictly increasing chain of \(X_n\) shows that \(P\) is not Noetherian. Every descending chain stabilizes: either it stays at \(P\), or its first proper term is some \(X_n\) or \(0\); thereafter only the finite list of submodules below that term is available. Thus \(P\) is Artinian.

Every nonzero series has the form \(t^ru\), with \(r\geq0\) and \(u\) a series with nonzero constant coefficient. The coefficients of an inverse to \(u\) are solved recursively, dividing each time by that nonzero constant. Hence \(u\) acts invertibly on \(P\). Multiplication by \(t^r\) is surjective on \(P\) and has kernel \(X_r\), with \(X_0=0\). Since an invertible series preserves \(X_r\) in both directions, the endomorphism \(t^ru\) has image \(P\) and kernel \(X_r\). In particular it is an automorphism exactly when \(r=0\). Finally, finitely many elements of \(P\) all lie in one \(X_n\) and cannot generate \(P\). The module is not finitely generated, so is not of finite type by the retained arbitrary-ring criterion. Artinianity alone did not suffice in Theorem 3.1.

Exercise 3 (intermediate: an inverse that requires infinite support). Let \(S=k^{\oplus\mathbb N}\), \(W=k^{\mathbb N}\), and let \(R\) be the right shift

For \(d=1-R\), determine its kernel and cokernel on both \(S\) and \(W\). Give an explicit inverse when one exists. Identify the obstruction to restricting that inverse to \(S\), and interpret your formulas using polynomials and formal power series.

Solution. In both spaces the coordinates of \(d(a)\) are \(a_0\) and \(a_n-a_{n-1}\) for \(n\geq1\). A zero value forces \(a_0=0\), then recursively every \(a_n=0\). Thus both kernels are zero.

Given \(b\in W\), the unique solution is

Every coordinate is a finite sum, so this is well defined over any field and gives a linear map \(W\to W\). The recurrence proves both inverse identities. Consequently \(d\) is invertible on \(W\) and has zero cokernel there.

For \(b\in S\), these partial sums eventually equal the finite total \(\varepsilon(b)=\sum_jb_j\). The solution has finite support exactly when \(\varepsilon(b)=0\). This includes \(b=0\). Hence

The last isomorphism follows because \(\varepsilon\) is surjective, sending the first basis vector to \(1\). In particular the inverse image in \(W\) of \((1,0,0,\ldots)\) is \((1,1,1,\ldots)\), which lies outside \(S\).

The identifications \((a_n)\mapsto\sum_na_nt^n\) turn \(S\) into \(k[t]\) and \(W\) into \(k[[t]]\) as vector spaces, with their ordinary \(k[t]\)-actions. The shift becomes multiplication by \(t\), and \(d\) becomes multiplication by \(1-t\). Its cokernel on polynomials is \(k[t]/(1-t)\simeq k\), identified by evaluation at \(t=1\). On formal series, its inverse is multiplication by \(\sum_{n\geq0}t^n\), whose coefficient formula is (4.8). No evaluation of arbitrary formal series at \(1\) has been used; only finite polynomials admit the total-sum map in this calculation.

Exercise 4 (advanced: the sharp finite-support boundary). Return to arbitrary \((X_m)\) in the setting of Section 2. Put \(P=\prod_mX_m\), and let \(\alpha:S\to\iota P\) be induced by the finite-coordinate inclusions. Prove that \(\alpha\) is always monic. Prove that it is epic, and equivalently an isomorphism, exactly when only finitely many \(X_m\) are nonzero. Deduce the exact condition for inverse limit to preserve the short exact sequences \(0\to T_n\to S\to F_n\to0\) for this family.

Solution. Every finite-coordinate inclusion \(\bigoplus_{m<r}X_m\to P\) is split monic: its retraction is the product of the first \(r\) projections. Constants preserve these maps and their splitting identities. Suppose \(a:\iota Y\to S\) has \(\alpha a=0\). Compactness factors \(a\) through some \(F_r\). Its map into \(F_r\), followed by the split monomorphism \(F_r\to\iota P\), is zero, so the map into \(F_r\) is zero and \(a=0\).

Let \(L\) be the kernel of \(\alpha\). A map from a constant into \(L\), composed with the monomorphism \(L\to S\), is a map of the preceding kind and is zero. Cancelling that monomorphism makes the original map zero. All constant probes of \(L\) therefore vanish; the presheaf test in Theorem 2.1 gives \(L=0\). In an abelian category a zero kernel means that \(\alpha\) is monic.

If only finitely many \(X_m\) are nonzero, choose \(r\) beyond their indices. Then \(S=F_r\), and \(P\) is the same finite biproduct in \(\mathcal C\). The map \(\alpha\) is its constant isomorphism.

If infinitely many \(X_m\) are nonzero, adapt the full quotient witness of Theorem 2.2. Embed \(D_n=\bigoplus_{m<n}X_m\) in \(P\) and set \(Q=\operatorname{colim}_n\iota(P/D_n)\). The quotient maps define \(q:\iota P\to Q\). For every \(n\), choose an index \(m\geq n\) with \(X_m\ne0\). The projection \(P\to X_m\) kills \(D_n\) and has a section, so is nonzero and factors through \(P/D_n\). The quotient \(P\to P/D_n\) is consequently nonzero at every stage. Compactness makes \(q\ne0\), since its class cannot become zero at a later stage. Nevertheless \(q\alpha=0\), as each finite prefix is killed at a later quotient. Thus \(\alpha\) is not epic. This proves both directions of the criterion, and abelian balance identifies the epic and isomorphism cases because monicity always holds.

The limits of the three systems are still \(0,S,\iota P\), by Theorem 2.1 and preservation of the finite-prefix limit. Their resulting sequence is already exact on the left and in the middle, since \(\alpha\) is monic. Exactness on the right is precisely its epimorphy, which holds exactly for finite support. This proves the stated sharp condition, rather than merely a sufficient example of failure.

References

• The Stacks project authors, The Stacks project, Commutative Algebra, Characterizing finite and finitely presented modules, especially Lemma 10.11.1, for the finite-generation criterion retained from the earlier module lesson. Abelian Categories, Inverse systems, for ordinary inverse-system conventions and the scope of Mittag–Leffler statements.
• Pierre Schapira, Homological Algebra, open lecture notes, Sections 2.7 and 5.7, for formal ind-objects, their distinction from ordinary realizations, and inverse systems of abelian groups. The full categorical foundations used here are proved in the prerequisite lessons.
• Full subcategories and exact closure, Theorem 2.2, for the exact full-subcategory criterion; Composition factors and uniform chain bounds, Theorem 2.1, for finite-length and Noetherian chain properties.
• Ind-objects through their elements; Limits and colimits of formal objects, Corollary 2.3 and Theorem 3.1; Ind-abelian categories and controlled exact sequences, Theorems 3.2 and 6.1, for compact constants, formal coproducts, preservation of limits and abelianity, including the large-base case and its dual.
• Finite-rank tests and finite generation, Propositions 2.1 and Theorem 3.1, for the earlier formal quotient and the arbitrary-ring finite-type criterion; Formal colimits and compact presentations, Theorem 4.1, for categorical finite presentation of modules.
• Pierre Schapira, An Introduction to Categories and Homological Algebra, lecture notes, version of 1 March 2026, Sections 2.7 and 5.1, for ind-objects and abelian categories.