Cofiltered derived limits and morphisms of pro-objects

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

An inverse system has two interpretations. It is a diagram on a specified category, and it presents an object of a Pro-category. These interpretations have different morphisms: a Pro morphism can change the index and need not be a transformation on the original diagrams. We prove that their derived limits agree, including the maps obtained from every Pro morphism. We then compute extensions into constant Pro objects and give four examples that distinguish level maps, formal products and ordinary limits.

Fix universes and choice. Throughout, \(\mathcal A\) is a locally small abelian category with enough injectives and exact small products. Every indexing category is small; a cofiltered category is nonempty. Products and their equalizers give all small ordinary limits. Write \(c:\mathcal A\to\operatorname{Pro}(\mathcal A)\) for the constant embedding, \(\pi\) for its right adjoint, and \(P_I:\mathcal A^I\to\operatorname{Pro}(\mathcal A)\) for formal realization when \(I\) is cofiltered. No essential-smallness assumption on \(\mathcal A\) is made.

We use Finite diagrams and comma objects, Theorems 1.1 and 2.1 and Section 2.8: finite acyclic shapes have common level models with cofinal projections, and the comparison for every finite shape is fully faithful. Two cokernels, controlled rows and abelian completion, Theorems 2.2, 3.2 and 4.1, applied to the opposite category, supplies abelian Pro-categories and level models of exact rows. Formal Pro products and classes that compute derived limits, Sections 1–4 supplies formal products, constant injectives, enough Pro injectives and \(R\pi\) on the full bounded-below derived category.

The other main input is Derived limits on small categories and cofinal restriction, Sections 1–6. We retain its covariant convention, its Roos complex \(C_I^\bullet(F)\), its restriction map \(\rho_u\), and its actual prism homotopies. Universal cohomology and acyclic resolutions, Sections 2, 4 and 5 gives uniqueness of effaceable delta-functors and normalized bounded-below acyclic computation. The extension calculation also uses exact filtered colimits of abelian groups from Hom, tensor and the exactness of module limits, Section 4.

Basic references are Prosmans's Derived Limits in Quasi-Abelian Categories, especially its Pro-category construction, Schapira's An Introduction to Categories and Homological Algebra, and The Stacks Project's Rlim of abelian groups. Our proofs below use the linked course results. All complexes used to construct the comparison are bounded below; there is no assertion that an arbitrary unbounded complex has one common level presentation.

1. Cofinal restriction and finite level models

For a cofiltered \(I\), formal realization is exact. Indeed a transformation has its kernel and cokernel computed objectwise in \(\mathcal A^I\). The common-index kernel and cokernel theorem identifies their formal realizations with the kernel and cokernel of the realized Pro arrow. The ordinary limit universal property gives the normalized equality

Here and below equalities of these functors mean their specified cone identifications. They commute with every transformation of diagrams. Exactness does not imply that \(P_I\) preserves arbitrary products or injective objects.

Call \(u:J\to I\) cofinal for inverse systems when every \((u\downarrow i)\) is nonempty and connected. The arrows in this comma category are arrows \(b:j\to j'\) such that \(a=a'u(b)\), for objects \((j,a:u(j)\to i)\).

Lemma 1.1. If \(J\) is cofiltered and \(u\) is cofinal in this sense, then every \((u\downarrow i)\) is cofiltered. Consequently

is a quasi-isomorphism, and the corresponding restriction is invertible on full bounded-below derived limits.

Proof. Connect two comma objects by a finite zigzag. Cofilteredness of \(J\) supplies an incoming cone to the finite diagram of all the underlying objects and arrows of that zigzag. Composing its cone arrows with the arrows to \(i\) gives equal arrows to \(i\), successively along the zigzag. Thus the cone vertex, with that common composite, is a comma object mapping to both given objects. For parallel comma arrows, equalize their underlying arrows in \(J\); composing the resulting arrow to either source with its arrow to \(i\) makes it a comma morphism. This proves the cofiltered axioms. The acyclic-comma theorem in the small-category lesson applies, because its proof contracts every finite-support integral cycle by an incoming cone. It gives (1.2), and the same theorem gives its bounded-below total comparison. \(\square\)

We also use the following concrete form of finite strictification. Given finitely many Pro objects in an arrow, a composable pair, a parallel pair, or a commuting square, choose their stated presentations. The finite acyclic shape theorem supplies a common cofiltered \(K\), cofinal projections to these presentations and actual natural stage maps satisfying all the shape equations. For a finite shape, full faithfulness also compares any two such presentations by the unique formal transformation that induces a specified transformation of the realized diagrams. One may then strictify that transformation as an arrow in the coefficient category of diagrams. Only finite acyclic shapes are used for essential surjectivity; no cyclic strictification claim is needed.

2. Why a model of the identity gives the identity

This is the point at which equality of formal morphisms must be converted into equality of derived maps. Let \(X:I\to\mathcal A\), let \(K\) be cofiltered, and let \(r,s:K\to I\) be cofinal. Suppose a transformation

represents the identity of \(P_I X\), using the canonical cofinal identifications.

Define \(W\) to have objects \((k,i,\alpha,\beta)\), where \(\alpha:i\to r(k)\), \(\beta:i\to s(k)\), and

A morphism to \((k',i',\alpha',\beta')\) is a pair \((t:k\to k',h:i\to i')\) with \(r(t)\alpha=\alpha'h\) and \(s(t)\beta=\beta'h\). Composition is componentwise. Local smallness and the smallness of \(I,K\) make \(W\) small.

Lemma 2.1 (finite witnesses). Given \(k\), any incoming finite index diagram containing \(r(k),s(k)\) can be refined so that (2.2) holds. More generally the refinement can preserve any finite list of already prescribed index-arrow equations.

Proof. Choose an incoming cone in \(I\), with vertex \(i_0\), on that finite index diagram. The equality saying that (2.1) represents the identity, projected to \(cX(s(k))\), is an equality in

At the chosen cone, its two representatives are \(a_kX(i_0\to r(k))\) and \(X(i_0\to s(k))\). Eventual equality in a filtered colimit gives a further arrow \(i\to i_0\) on which they agree. Precomposing every cone component with this arrow preserves all the installed index equations. Finitely many coefficient equations can be imposed successively, or by one further incoming cone on their witnesses. No equality of coefficient maps is used to infer equality of index arrows. \(\square\)

Lemma 2.2. The category \(W\) is cofiltered and its projection \(t:W\to K\) is cofinal.

Proof. A witness exists over each \(k\) by Lemma 2.1. To cone finitely many objects of \(W\), first choose an incoming cone \(k\to k_h\) in \(K\). In \(I\), take an incoming cone on the finite graph containing \(r(k),s(k)\), all old witness indices \(i_h\), their arrows to \(r(k_h),s(k_h)\), and the images of \(k\to k_h\). This installs exactly the index equations required for morphisms to those witnesses. Lemma 2.1 then installs (2.2) over \(k\) without disturbing them. The result is a witness mapping to all the old objects. For two parallel morphisms in \(W\), first equalize their \(K\) components, and then form an incoming cone on the finite index graph including their \(I\) components. It equalizes those components as well; a further use of Lemma 2.1 preserves all these equations. Hence \(W\) is cofiltered.

For a fixed \(k_0\), the comma \((t\downarrow k_0)\) is nonempty by a witness over \(k_0\). Given finitely many of its objects and arrows, make the preceding construction with the specified arrows \(k_h\to k_0\) included in the finite diagram in \(K\). The resulting cone has a single arrow \(k\to k_0\) and the required commuting equations. Parallel arrows are equalized by the same construction. Thus this comma is cofiltered, proving cofinality. \(\square\)

Let \(v:W\to I\) select the witness index. Its arrows \(\alpha:v\Rightarrow rt\), \(\beta:v\Rightarrow st\) are natural, and (2.2) says \(X\beta=(at)X\alpha\). The prism formula from the small-category lesson gives equalities in \(D^+(\mathcal A)\):

Naturality of restriction in coefficients and its strict composition law rewrite the last equality as

All \(\rho_t,\rho_r,\rho_s\) here are invertible by Lemmas 1.1 and 2.2. Cancellation proves

We did not need to assert cofinality of \(v\). The two prism equalities use it only as a functor.

3. Every Pro morphism acts on the complex

For a presented Pro object \(\mathbf X=P_I X\), put \(L(\mathbf X)=C_I^\bullet(X)\). For a Pro morphism \(f:P_I X\to P_JY\), take a common model \(r:K\to I\), \(s:K\to J\), \(a:Xr\to Ys\) and define

Proposition 3.1. Formula (3.1) is independent of the common model. It defines an additive functor from \(\operatorname{Pro}(\mathcal A)\) to \(D^+(\mathcal A)\), with its canonical comparisons between different presentations and with \(H^0L=\pi\).

Proof. Take two common models of \(f\). They are Pro objects in the category of arrows of \(\mathcal A\). Full faithfulness of the finite-arrow comparison lifts the identity commuting square on \(f\) to the unique isomorphism between these formal arrow objects. Strictify this isomorphism as a single arrow in the coefficient category \(\operatorname{Arr}(\mathcal A)\). Its level maps are commuting squares, and its two vertex maps represent identities of the original source and target Pro objects. Equation (2.6) makes their induced maps identities under the canonical source and target comparisons. Applying \(C\) to the commuting squares therefore identifies the two expressions (3.1). This also proves coherence when the stated presentation itself is changed.

For composable \(f,g\), strictify the finite shape \(0\to1\to2\), including its composite. It supplies actual stage maps \(a,b,ba\) and one cofinal projection at each vertex. In (3.1) the two middle inverse comparisons cancel, leaving \(C(ba)=C(b)C(a)\). Equation (2.6) gives the identity law. For parallel \(f,g\), strictify their parallel-pair shape; at every stage the map representing \(f+g\) is \(a+b\). The product cochain functors and their differentials are additive, so \(C(a+b)=C(a)+C(b)\), proving additivity. In degree zero, \(H^0C_I(X)=\Gamma_I X\), and all restrictions and level maps have their ordinary cone maps. Thus the normalization (1.1) identifies \(H^0L(f)\) with \(\pi(f)\). \(\square\)

The construction is on actual formal morphisms. Choosing separate isomorphisms only for the higher-limit groups of individual presentations would not establish this proposition.

4. Exact rows and their connecting maps

Let \(T^q(\mathbf X)=H^qL(\mathbf X)\) for \(q\geq0\). Every short exact Pro row has a common cofiltered presentation by actual short exact rows in \(\mathcal A\), by the dual exact-row theorem in the prerequisites. Exact products make

termwise exact. Give its cohomology the connecting maps of its exact-sequence triangle, with the course's fixed cone convention.

Proposition 4.1. These maps are independent of the exact-row presentation and natural for every morphism of short exact Pro rows. Together they make \(T^\bullet\) a cohomological delta-functor with \(T^0=\pi\).

Proof. Let \(\mathcal E\) be the full subcategory of \(\mathcal A^{[2]}\) whose objects are short exact rows, writing the two maps and their composite zero as the diagram on \(0\to1\to2\). It is locally small. Choose exact-row presentations of two Pro rows. Full faithfulness of \(\operatorname{Pro}(\mathcal A^{[2]})\to\operatorname{Pro}(\mathcal A)^{[2]}\) lifts a morphism of the Pro rows to a formal morphism of their presented diagrams. Since \(\mathcal E\) is full, the Hom limit–colimit formula shows that \(\operatorname{Pro}(\mathcal E)\to\operatorname{Pro}(\mathcal A^{[2]})\) is fully faithful as well. Thus the lift is a morphism in \(\operatorname{Pro}(\mathcal E)\).

Strictify that one arrow in \(\mathcal E\). Its stage squares are genuine maps of short exact rows, with cofinal projections to the selected exact-row indices. Their cochain maps are maps of (4.1). Naturality of the exact-sequence triangle makes every connecting square commute. Formula (3.1) identifies their vertex maps with the prescribed Pro maps. Apply this also to the identity map between two presentations of one exact row; Proposition 3.1 identifies the vertex comparisons with identities, proving independence of the connecting maps. The long cohomology sequence of (4.1) is exact, and negative cohomology is zero. This proves every delta-functor axiom. \(\square\)

5. Formal products give effacement

For a small family \((A_s)_{s\in S}\), write

Its finite-subset presentation is \(F(T)=\prod_{s\in T}A_s\) on \(D=\operatorname{Fin}(S)^{\mathrm{op}}\), with restriction projections. The empty product in an abelian category is zero.

Lemma 5.1. This particular diagram is a product of evaluation-coinduced diagrams:

Proof. There is one arrow \(T\to\{s\}\) in \(D\) if \(s\in T\), and no such arrow otherwise. Thus \((R_{\{s\}}A_s)(T)\) is \(A_s\) in the first case and zero in the second. Its transition maps are exactly the appropriate identity maps and projections to zero. Taking the product over \(s\) gives \(F\), including all its diagram maps. Lemma 1.2 of the small-category lesson makes this product of coinduced diagrams \(\Gamma_D\)-acyclic. Its ordinary limit is the product of all coefficients by compatible finite tuples. This proves (5.2), also for empty \(S\), where the entire diagram is zero. \(\square\)

Every Pro object \(P_I X\) embeds in \(\mathcal P(X(i))_{i\in\operatorname{Ob}I}\): take its coordinate projections. If two maps into \(P_I X\) have the same composites with this arrow, they have the same projection to every \(cX(i)\); the Pro Hom formula then makes them equal. This proves monicity without requiring the individual projections to be monic. By (5.2), this one embedding effaces every positive \(T^q\).

Theorem 5.2. There is a unique isomorphism of cohomological delta-functors

whose degree-zero map is (1.1). It preserves all Pro morphisms and every exact-row connecting map.

Proof. Proposition 4.1 and the preceding embedding make \(T\) effaceable. The Pro-category has enough injectives by the formal-product lesson, so \(R^\bullet\pi\) exists and is effaceable by injective embeddings. The uniqueness theorem for effaceable delta-functors extends the degree-zero identity and its inverse. The composites extend identities and hence are identities. The same uniqueness includes all naturality and connecting-map squares. \(\square\)

The use of (5.2) concerns the finite-subset presentation of a formal product. It does not assert that arbitrary products in a fixed diagram category are carried by \(P_I\) to formal products.

6. The full bounded-below comparison

Theorem 6.1. For every small cofiltered \(I\), there is a normalized natural isomorphism of exact functors

It agrees on objects in degree zero with Proposition 3.1, and its cohomology comparison is (5.3). For a cofinal \(u:J\to I\), the diagram formed by \(\theta_I,\theta_J\), derived restriction and the canonical cofinal Pro isomorphism commutes. Its ordinary normalization is (1.1).

Proof. If \(Q\) is injective in \(\mathcal A^I\), then \(R^{q>0}\Gamma_I(Q)=0\). Theorems 5.2 and 3.1 therefore give \(R^{q>0}\pi(P_IQ)=0\). Thus \(P_I\) sends injectives to \(\pi\)-acyclic objects. It is exact, so \(RP_I=D^+P_I\). The normalized composition theorem applies to \(P_I\) followed by \(\pi\), with \(\pi P_I=\Gamma_I\); its canonical comparison is (6.1), and is invertible on the full bounded-below category. The theorem's normalization identifies its augmentation, shifts and maps of replacements.

We verify the particular cofinal compatibility, rather than infer a derived equality merely from cohomology. For a diagram \(F\), use its augmented coinduced resolution \(F\to B_I^\bullet(F)\) from the small-category lesson. Its terms are \(\Gamma_I\)-acyclic, and hence their realizations are \(\pi\)-acyclic by (5.3). Applying \(\pi P_I\) to this resolution is exactly \(C_I(F)\). Therefore acyclic computation identifies (6.1) with this actual augmented model. Its resulting cohomology maps form a delta-functor comparison normalized in degree zero; uniqueness identifies them with (5.3).

There is a natural augmented chain map

At \(j\), its target coordinate \(j\to j_0\to\cdots\to j_n\) selects the source coordinate \(u(j)\to u(j_0)\to\cdots\to u(j_n)\). Deleting a vertex, composing the initial arrow, or applying the last coefficient map commutes with this selection. Thus (6.2) commutes with the differential, diagram maps and augmentation. By Lemma 1.1, restriction preserves \(\Gamma_I\)-acyclicity, so its source terms are \(\Gamma_J\)-acyclic; their formal realizations are \(\pi\)-acyclic by (5.3). The target terms have the same property. Hence both augmented complexes compute the required derived objects. Applying the ordinary cone identification to (6.2) gives precisely \(\rho_u\) on the Roos complexes. The comparison of these two acyclic models therefore computes the canonical derived image of the cofinal Pro isomorphism, by naturality of acyclic computation. This proves the claimed commuting square in degree-zero input, including its actual derived arrows.

For a bounded-below complex \(F^\bullet\), apply this construction in each degree and totalize. In total degree \(n\), only the finitely many pairs \((p,q)\) with \(p\geq0\), \(q\) above the common lower bound and \(p+q=n\) occur. The total terms are finite biproducts of the acyclic objects just used. The augmented total is a quasi-isomorphism by the pointwise contractions, and (6.2) totalizes with differential \(d_B+(-1)^p d_F\). The same bounded-below acyclic comparison proves the square on the full stated domain and on its localized morphisms. Restriction selection composes strictly, so the squares also respect successive cofinal functors. \(\square\)

For a Pro arrow represented by \(a:Xr\to Ys\), the two cofinal squares just proved and naturality of \(\theta_K\) in \(a\) identify its \(R\pi\)-image with (3.1). In particular every Pro isomorphism gives an isomorphism of derived limits, with its actual functorial map. This conclusion does not say that derived limit detects Pro isomorphisms; Exercise 3 shows it does not. Nor does it require strictifying infinitely many chain maps in an arbitrary object of \(D^+(\operatorname{Pro}(\mathcal A))\).

7. Extensions into constant objects

For \(A\in\mathcal A\), choose an injective resolution \(A\to J^\bullet\) in \(\mathcal A\). The constant functor is exact, and each \(cJ^n\) is injective in the Pro-category by Theorem 1.1 of the formal-product lesson. Hence \(cA\to cJ^\bullet\) is an actual injective resolution there.

Theorem 7.1. For every \(\mathbf X=P_I X\), every \(A\in\mathcal A\) and every \(n\geq0\), there is a natural isomorphism

The transition maps on the right precompose with the inverse-system arrows. The comparison includes every Pro morphism in the first argument, maps in \(A\), and the exact-sequence connecting maps in the second argument.

Proof. The constant-target Hom formula is a natural isomorphism of complexes of abelian groups:

Filtered colimits in abelian groups are exact. Applying them to the short exact descriptions of cycles, boundaries and their quotient identifies cohomology of (7.2) with the filtered colimit of the cohomologies of its terms on the right. Those are \(\operatorname{Ext}_{\mathcal A}^n(X(i),A)\) by the chosen injective resolution. This proves (7.1), and its degree-zero map is the original Hom formula.

For a Pro arrow, precomposition on the left of (7.2) is defined before taking cohomology. At each degree its right-hand description is the Pro Hom formula, which is independent of representatives and respects composition. Thus it induces exactly the corresponding filtered Ext map; equality of representatives is preserved on passing from chain maps to cohomology classes. For a map \(A\to A'\), injective comparison maps exist and are unique up to homotopy. Applying (7.2) respects these homotopies, so the induced map is independent of the resolution.

For completeness, reduce a connecting-map square for \(0\to A\to B\to C\to0\) to dimension shifting as follows. Embed \(B\) in an injective \(J\). There is a map of this row to \(0\to A\to J\to J/A\to0\), with identity on \(A\) and the induced map \(C\to J/A\). In both Ext theories, naturality of connecting maps expresses the original boundary as the boundary of this latter row after that induced map. Resolve \(A\) starting with \(A\to J\), and then resolve \(J/A\). The auxiliary boundaries are the degree-zero quotient map and the higher dimension-shift maps on the same tail resolution, with the same cone signs. Formula (7.2), followed by exact filtered colimit, commutes with those quotients and tail maps. Thus its cohomology comparison commutes with the auxiliary boundaries, and hence with the original ones. This proves all the stated compatibilities. \(\square\)

When the first object is constant, (7.1) recovers \(\operatorname{Ext}_{\mathcal A}^n(B,A)\). For a general Pro object it is a formula for extensions from that formal object, not for extensions from its ordinary limit.

8. Four exercises with complete solutions

Exercise 1 (introductory: a level map need not be an isomorphism). Let \(Q_n=\mathbb Z/(n+2)!\) for \(n\geq0\), with zero transitions, and put \(X_n=\mathbb Z\oplus Q_n\), with transition \((z,q)\mapsto(z,0)\). Show that the projections \(X_n\to\mathbb Z\) are never isomorphisms, while their Pro morphism is an isomorphism and induces an isomorphism on derived limits. Compute both derived limits.

Solution. Every \(Q_n\) is nonzero, so a level projection has nonzero kernel. The identity of the formal system \(Q\) is zero: at each target stage, the transition from the next stage is already zero, and hence its identity class vanishes in the filtered Hom colimit. An object whose identity is zero is a zero object in an additive category. Thus \(P X=c\mathbb Z\oplus P Q\simeq c\mathbb Z\); the maps given by projection and the inclusion of the first summand are inverse in Pro.

For the zero-transition tower, the two-term limit complex has difference map \(\Delta=1\) on \(\prod_nQ_n\). Its cohomology vanishes. For the constant tower \(\mathbb Z\), \(\Delta(z)_n=z_n-z_{n+1}\) has kernel the constant sequences and is surjective: given \(b\), start with \(z_0=0\) and recursively put \(z_{n+1}=z_n-b_n\). Consequently

The projection induces this specified identification of the first summand. Theorem 6.1 identifies it with the derived image of the Pro isomorphism, including its inverse.

Exercise 2 (intermediate: finite labelled probes). Let \(S\) be any small set and \((A_s)_{s\in S}\) any family in \(\mathcal A\). Let \(D=(\operatorname{FinSet}/S)^{\mathrm{op}}\), using finite subsets of a fixed countable set as finite domains. A map before taking opposites is any label-preserving function, not just an injection. Define \(F(L\xrightarrow{\ell}S)=\prod_{x\in L}A_{\ell(x)}\), with coordinate pullback along those functions. Compute its derived limit. If \(S\) is uncountable, show that finite subsets of \(S\) have no countable cofinal family.

Solution. The category \(\operatorname{FinSet}/S\) is filtered: a disjoint union receives two labelled finite sets, and a pair of parallel label-preserving functions is coequalized by its finite quotient, whose labels are well defined. It is nonempty because it contains the empty set. The use of the chosen finite domains only replaces these finite colimits by isomorphic representatives. Thus \(D\) is cofiltered. Coordinate pullback is functorial because composite coordinate selections compose.

Choose finite representatives of each finite subset \(T\subseteq S\), labelled bijectively by \(T\). They give a functor \(u:\operatorname{Fin}(S)^{\mathrm{op}}\to D\). For a labelled \(L\), an object in \((u\downarrow L)\) is a finite \(T\) containing \(\ell(L)\); its label-preserving map \(L\to T\) is unique. The image subset \(\ell(L)\) is terminal in this comma category, since the arrows are reverse inclusions. Hence \(u\) is cofinal. Its restricted coefficient diagram is exactly the finite-subset model in Lemma 5.1. Lemma 1.1 and (5.2) give

If finite subsets \(T_0,T_1,\ldots\) were cofinal, every singleton \(\{s\}\) would be contained in some \(T_n\). Their countable union would be all of \(S\), contradicting uncountability. Thus (8.2) is a genuinely noncountable presentation in this case. The availability of noninjective functions matters: injections alone would not coequalize parallel maps and would not give the cofiltered category used above.

Exercise 3 (advanced: derived limit is not conservative). Let \(k\) be a nonzero field and \(V=k^{\mathbb N}\). Let \(\mathcal P=\prod_{n\geq0}^{\mathrm{Pro}}ck\), with finite-prefix presentation \(F_N=k^N\), \(N\geq0\). Consider the canonical arrow \(cV\to\mathcal P\). Prove that it is epic with a nonzero kernel \(\mathbf K\), but that \(R\pi(\mathbf K)=0\). Decide whether its kernel tower is Mittag–Leffler.

Solution. The level maps are the surjections \(V\to k^N\). Their kernels are

Common-index kernels and cokernels therefore give a short exact Pro row \(0\to\mathbf K\to cV\to\mathcal P\to0\). The finite prefixes form a cofinal subsystem of finite subsets of \(\mathbb N\): the prefixes containing a given finite subset have a least member and hence a terminal object in the inverse comma convention. Thus they really present \(\mathcal P\).

For any \(N\), all later maps \(K_M\to K_N\) are nonzero, since the vector with its only nonzero coordinate at \(M\) survives. The identity class of \(K_N\) consequently never becomes zero in \(\operatorname{colim}_M\operatorname{Hom}(K_M,K_N)\). The identity of \(\mathbf K\) is not zero, so \(\mathbf K\ne0\).

Both \(cV\) and \(\mathcal P\) are \(\pi\)-acyclic. The first assertion follows from \(R\pi\,c=1\) in the formal-product lesson; the second follows from (5.2) and (5.3). Their ordinary limits are both \(V\), and the induced map is identity in every coordinate. The long derived-limit sequence of their exact row gives \(\pi\mathbf K=0\), \(R^1\pi\mathbf K=0\), and all higher groups zero. Hence \(R\pi\mathbf K=0\) in the derived category, despite \(\mathbf K\ne0\). At fixed \(N\), the images of \(K_M\) in \(K_N\) form a strictly decreasing tail sequence: the vector at coordinate \(M\) belongs to the image of \(K_M\) but not \(K_{M+1}\). It is not Mittag–Leffler. Thus that condition is sufficient, but not necessary, for a tower to be limit-acyclic.

Exercise 4 (advanced: a filtered extension group). Fix a prime \(p\), and let \(\mathbf Z_p\) be the Pro object given by \(\mathbb Z/p^n\), \(n\geq1\), with quotient transitions. Compute \(\operatorname{Ext}^q_{\operatorname{Pro}(\mathsf{Ab})}(\mathbf Z_p,c\mathbb Z)\) for all \(q\geq0\), including the filtered transition maps. Also compute its derived limit.

Solution. The free resolution \(0\to\mathbb Z\xrightarrow{p^n}\mathbb Z\to\mathbb Z/p^n\to0\) gives zero Hom to \(\mathbb Z\), first Ext \(\mathbb Z/p^n\), and zero higher Ext. The quotient \(\mathbb Z/p^{n+1}\to\mathbb Z/p^n\) lifts on these resolutions by identity in degree zero and multiplication by \(p\) in degree minus one. The square commutes because \(p^n p=p^{n+1}\). Applying Hom into \(\mathbb Z\) therefore induces multiplication by \(p\) on first Ext. The sign convention for the Hom differential changes neither its cokernel nor this chain comparison.

The injective maps \(\mathbb Z/p^n\to\mathbb Z[1/p]/\mathbb Z\), \(\bar a\mapsto a/p^n+\mathbb Z\), are compatible with those multiplication-by-\(p\) transitions: \(pa/p^{n+1}=a/p^n\). Their images have union the indicated quotient. Theorem 7.1 now gives

The limit of the quotient tower is the group \(\mathbb Z_p\) of compatible residue classes. Its difference map is surjective: having chosen \(x_n\), choose \(x_{n+1}\) mapping to \(x_n-b_n\), possible by surjectivity of the quotient. The two-term tower model and Theorem 6.1 therefore give \(R\pi(\mathbf Z_p)\simeq\mathbb Z_p[0]\). Formula (8.4) computes extensions from the formal system; it is not obtained by replacing that system by its ordinary limit in an Ext group.

References

• Fabienne Prosmans, Derived Limits in Quasi-Abelian Categories, Bulletin de la Société Royale des Sciences de Liège 68 (1999), 335–401. Freely accessible publisher edition. Section 7 treats the limit of Pro objects and its derivation.
• Pierre Schapira, An Introduction to Categories and Homological Algebra. Freely accessible notes.
• The Stacks Project Authors, The Stacks Project, Rlim of abelian groups. Its fixed-diagram tower model is compared with the Pro-category functor by Theorem 6.1 here.