Draft lesson.

Formal Pro products and classes that compute derived limits

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

A formal inverse system retains maps that its ordinary limit can forget. Its formal product of constant objects also has fewer maps to a constant target than an ordinary product can have. We first establish that distinction, then construct the ordinary-limit functor on the entire Pro category. A cogenerating class with three explicit properties supplies its bounded-below derived functor. For a countable tower, a finite-prefix calculation gives a two-term complex computing the result, including its maps under reindexing.

Fix the universe and choice conventions. Let \(\mathcal C\) be a locally \(\mathcal U\)-small abelian category; its objects need not form a small set. All presenting categories and families below are \(\mathcal U\)-small, and presheaves on a large \(\mathcal C\) are housed in the stated larger universe. Write

Ordinary completeness of \(\mathcal C\) is imposed starting in Section 2. It is unnecessary for the formal products in Section 1. Complexes are cochain complexes, and \(D^+\) means bounded-below derived category.

1. Formal products and their finite-support maps

For cofiltered presentations \(\mathbf X=(X_i)_{i\in I}\) and \(\mathbf Y=(Y_j)_{j\in J}\), the dual of the Ind Hom formula is

The inner transitions precompose with the inverse-system maps. The outer transitions postcompose with those of \(\mathbf Y\). Two cokernels, controlled rows and abelian completion, Theorems 2.2 and 3.2, applied to \(\mathcal C^{\mathrm{op}}\), make Pro abelian and \(c\) exact and fully faithful. On a common cofiltered stage model, kernels and cokernels are the formal inverse systems of the stage kernels and cokernels. These assertions require local smallness, not essential smallness.

The same lesson's Theorem 6.1, or the dual of formal colimits, Theorems 3.1 and 4.1, supplies all small formal limits and exact small cofiltered limits in Pro. In particular its small products are exact. Here is the additional step for products: the product of a family is the inverse limit of its finite partial products, indexed by finite subsets with restriction projections. Finite products are biproducts and preserve exact rows. Exactness of this cofiltered limit therefore makes the product of any family of exact Pro rows exact. This argument asserts exactness inside Pro; it asserts nothing about arbitrary ordinary products in \(\mathcal C\).

For objects \((A_s)_{s\in S}\) of \(\mathcal C\), define their formal product by

The empty biproduct is zero. The arrows go from a larger finite subset to a smaller one. Formula (1.1), with finite products commuting with filtered colimits of Hom groups, proves

Indeed a map to each finite partial product is a finite tuple of maps to its coordinates, and compatibility under restrictions is exactly a family of coordinate maps. Thus (1.2) has the actual product universal property for every Pro test object.

For a constant target \(cB\), the other direction gives a different answer:

At a finite stage the Hom group is the finite product of the coordinate Hom groups. Precomposing with a restriction projection extends a tuple by zero on the new coordinates. Its filtered union consists precisely of finite-support tuples; this proves both surjectivity and injectivity of the last isomorphism. In particular (1.4) is not a formula for maps from an ordinary product to \(B\). Exercise 1 exhibits a map missed by that mistaken identification.

We shall use one further exact interface from the Ind lesson, Theorem 4.4. Its dual says: if \(j:\mathbf Y\hookrightarrow\mathbf A\) is monic in Pro and \(p:\mathbf Y\to cB\) is any map, then there are a monomorphism \(k:B\hookrightarrow E\) in \(\mathcal C\) and a map \(a:\mathbf A\to cE\) such that

To check the dualization, in \(\operatorname{Ind}(\mathcal C^{\mathrm{op}})\) the arrow \(j^{\mathrm{op}}:\mathbf A^{\mathrm{op}}\to\mathbf Y^{\mathrm{op}}\) is epic and \(p^{\mathrm{op}}:cB^{\mathrm{op}}\to\mathbf Y^{\mathrm{op}}\) is its represented test map. The epimorphic cover in that theorem reverses to \(k\); its lift reverses to \(a\), giving exactly (1.5). The map \(a\) is not required to be monic.

Theorem 1.1 (constant injectives and their formal products). For any injective object \(I\) of \(\mathcal C\), the constant object \(cI\) is injective in \(\operatorname{Pro}(\mathcal C)\). For every small family of injective objects \((I_s)_{s\in S}\), the formal product \(P(I_s)\) is injective. Neither ordinary completeness of \(\mathcal C\) nor enough injectives in \(\mathcal C\) is required for these assertions.

Proof. Let \(j:\mathbf Y\hookrightarrow\mathbf A\) be any Pro monomorphism and \(g:\mathbf Y\to cI\) any map. Apply (1.5) with \(B=I\). It gives a monomorphism \(k:I\hookrightarrow E\) and a map \(a:\mathbf A\to cE\) with \(aj=(ck)g\). Injectivity of \(I\) extends \(1_I\) along \(k\) to \(r:E\to I\), so \(rk=1_I\). Hence

This is the extension property for the actual constant object. For \(g:\mathbf Y\to P(I_s)\), let \(g_s\) be its composite with the \(s\)-th product projection. Extend each \(g_s\) along \(j\) as just proved, making the small family of choices in the fixed ambient universe. Formula (1.3) assembles these extensions into \(\widetilde g:\mathbf A\to P(I_s)\). Its composite with \(j\) has every coordinate equal to \(g_s\), so it equals \(g\) by the product universal property. The empty family gives the zero object and its unique extensions. This proves the entire formal-product statement. \(\square\)

Theorem 1.2 (enough injectives before ordinary completeness). If \(\mathcal C\) has enough injectives, then every object of \(\operatorname{Pro}(\mathcal C)\) admits a monomorphism into a formal product of constant injectives. Consequently \(\operatorname{Pro}(\mathcal C)\) has enough injectives. This assertion also does not require ordinary completeness of \(\mathcal C\).

Proof. Present \(\mathbf X=(X_i)_{i\in I}\). For the small set of objects of \(I\), choose monomorphisms \(e_i:X_i\hookrightarrow I_i\) with \(I_i\) injective. The structure maps \(p_i:\mathbf X\to cX_i\), followed by \(ce_i\), assemble into

There is no requirement that the chosen embeddings commute with the arrows of \(I\). To prove that \(e\) is monic, let \(u,v:\mathbf T\to\mathbf X\) satisfy \(eu=ev\). Each \(ce_i\) is monic because \(c\) is exact, so \(p_i u=p_i v\) for every \(i\). Formula (1.1) identifies a map into \(\mathbf X\) with exactly its compatible family of these coordinate maps. Thus \(u=v\). The target is injective by Theorem 1.1, proving both claims for every Pro object, not only constant objects or towers. \(\square\)

2. The presentation-independent limit adjunction

Assume henceforth that \(\mathcal C\) has all small limits. For a presentation \(\mathbf X=(X_i)\), put \(\pi\mathbf X=\lim_i X_i\) in \(\mathcal C\). The following bijections identify its universal maps:

For any Pro morphism \(f:\mathbf X\to\mathbf Y\), postcomposition on the left gives a natural transformation of the represented Hom functors on the right. Yoneda gives its unique map \(\pi f:\pi\mathbf X\to\pi\mathbf Y\). Identities and composition follow from uniqueness. Another presentation represents the same Hom functor, so its unique comparison isomorphism respects every such map. Thus this construction is a functor on Pro, with presentation-independent values up to the unique adjunction comparison.

The bijections preserve addition, so the functor is additive, and (2.1) proves \(c\dashv\pi\). Its unit identifies \(T\) with \(\pi cT\). Its counit \(c\pi\mathbf X\to\mathbf X\) is the family of ordinary limit projections. A right adjoint preserves existing limits: a cone from a test object into its image corresponds under the adjunction to a cone before applying it, whose unique factorization gives the required factorization afterwards. In particular \(\pi\) preserves kernels and finite products, and is left exact. Applying (2.1) and (1.3) gives, with their specified projections,

These are ordinary products in \(\mathcal C\). The functor \(\pi\) need not be exact, and (2.2) does not transfer formal-product exactness to ordinary products without additional hypotheses.

3. A class criterion for deriving the limit

Let \(\mathcal J\subset\mathcal C\) be a full additive subcategory, containing zero, closed under finite biproducts and understood up to isomorphism. Impose three hypotheses:

1. Every object of \(\mathcal C\) embeds in an object of \(\mathcal J\).
2. The cokernel of a monomorphism between objects of \(\mathcal J\) belongs to \(\mathcal J\).
3. For every small family of short exact rows \(0\to J'_s\to J_s\to J''_s\to0\) whose terms lie in \(\mathcal J\), the corresponding ordinary product row in \(\mathcal C\) is exact.

Write \(\mathcal J_{\mathrm{pro}}\) for the full subcategory of objects isomorphic to some \(P(J_s)\), with every \(J_s\in\mathcal J\). It contains zero, and the biproduct of two such objects is the formal product indexed by the disjoint union of the two index sets. Thus it is full and additive.

Theorem 3.1. Under these hypotheses, \(\mathcal J_{\mathrm{pro}}\) is a \(\pi\)-injective resolution class. The exact universal functor

exists and is computed by applying \(\pi\) to bounded-below replacements in this class. For every \(P(J_s)\), its positive \(R^q\pi\) vanish. For heart objects, \(R^0\pi=\pi\) with the derived unit normalization.

Proof. We verify the actual hypotheses of the compatible extension criterion, Theorem 1.1.

First present \(\mathbf A=(A_i)_{i\in I}\) and choose \(e_i:A_i\hookrightarrow J_i\), with \(J_i\in\mathcal J\), for the small set of objects of \(I\). The structure projections followed by \(ce_i\) define

This map is monic. If two maps \(\mathbf T\to\mathbf A\) have the same composite with (3.2), monicity of each \(ce_i\) makes their composites to \(cA_i\) equal. Formula (1.1) determines a map into \(\mathbf A\) by exactly these components, so the two original maps agree. No compatibility of the chosen embeddings \(e_i\) with the arrows of \(I\) was needed. This proves cogeneration of \(\mathcal J_{\mathrm{pro}}\).

Now let \(j:P(J_s)\hookrightarrow\mathbf A\) be monic. Apply (1.5) separately to each projection \(p_s:P(J_s)\to cJ_s\). It gives \(k_s:J_s\hookrightarrow E_s\) and \(a_s:\mathbf A\to cE_s\) with \(a_sj=(ck_s)p_s\). By hypothesis 1 embed \(E_s\) into \(K_s\in\mathcal J\), and replace these maps by their composites. Hence we have actual monomorphisms \(J_s\hookrightarrow K_s\), actual maps \(\mathbf A\to cK_s\), and the same commuting equations. Put \(L_s=K_s/J_s\); hypothesis 2 gives \(L_s\in\mathcal J\).

The small formal product of these coordinate rows is exact by Section 1:

The maps \(\mathbf A\to cK_s\) assemble by (1.3) into \(v:\mathbf A\to P(K_s)\), with \(vj=i\). Both the middle object and the actual cokernel of \(i\) are in \(\mathcal J_{\mathrm{pro}}\). We neither need nor assert that \(v\) is monic. Under \(\pi\), row (3.3) is precisely

because (2.2) identifies its coordinate maps. Hypothesis 3 makes this row exact. We have therefore checked both compatible-extension equations and the required \(\pi\)-exact row, rather than assuming that \(\pi\) preserves every exact Pro row.

The cited criterion now proves that every bounded-below Pro complex admits an outgoing class replacement and that applying \(\pi\) to an acyclic class complex remains acyclic. Its full class localization theorem gives (3.1), its exactness, and its normalized, choice-independent computation. For \(P(J_s)\) in degree zero, the identity is already a class replacement; its computed image is \(\prod_s J_s\) in degree zero. This proves positive vanishing. For any heart object the replacement can start in degree zero, and left exactness identifies its degree-zero kernel with \(\pi\) of that object. It also proves negative vanishing on heart objects. \(\square\)

4. The identity after the constant embedding

Theorem 4.1. With the hypotheses of Theorem 3.1, there is a natural normalized isomorphism of exact functors

Proof. The functor \(c\) is exact, so it carries every acyclic complex to an acyclic complex. Cogeneration of \(\mathcal J\) supplies bounded-below class replacements by the arbitrary-class replacement theorem. Therefore \(\mathcal J\) is a computing class for \(c\), whose derived functor is its already descended exact functor \(D^+c\). Also \(c\mathcal J\subset\mathcal J_{\mathrm{pro}}\), using singleton formal products. The composite \(\pi c\) is naturally the identity by (2.1), so its derived functor is naturally the identity as well.

Apply the normalized composition theorem, Theorem 9.1 to \(F=c\), \(F'=\pi\), and these two computing classes. It proves that the specified unit comparison

is an isomorphism. Its natural inverse gives (4.1). On a \(\mathcal J\)-complex \(I\), the computation is the actual identity \(\pi cI=I\); for an arbitrary replacement \(X\to I\), the theorem transports that component through the quasi-isomorphism. Thus this is the comparison normalized by the adjunction and derived units, independent of the choice of replacement or computing class. \(\square\)

5. Every countable tower has a finite-prefix difference row

Let \(X_0\xleftarrow{v_0}X_1\xleftarrow{v_1}X_2\leftarrow\cdots\) be a tower in \(\mathcal C\), and let \(\mathbf X\) be its formal Pro object. Set \(v_{a,b}=v_a\cdots v_{b-1}\) for \(a<b\), and \(v_{a,a}=1\). Write \(P=P(X_n)\). The formal difference map \(d:P\to P\) is specified by its coordinate \(p_nd=p_n-c(v_n)p_{n+1}\). Formula (1.3) makes it an actual Pro morphism.

For \(P_N=\bigoplus_{r=0}^{N}X_r\), represent this map at stage \(N\) by

Coordinate notation here specifies maps using biproduct projections and sums; it does not assume that \(\mathcal C\) is a category of modules. Define \(s_N:P_N\to P_{N+1}\) by its last coordinate zero and recursively

This is a finite recursion of actual morphisms, and substitution proves \(d_Ns_N=1\). The kernel inclusion is

Its last coordinate is the identity. A map from any test object killed by \(d_N\) satisfies its finite recursive coordinate equations, so it uniquely factors through (5.3) by its last coordinate. Thus each row \(0\to X_{N+1}\to P_{N+1}\to P_N\to0\) is short exact. The source and target restriction projections commute with \(d_N\), and the induced map on kernels is \(v_{N+1}\). This gives a common cofiltered stage model of the entire exact row.

Finite initial subsets are cofinal among finite subsets of \(\mathbb N\). Omitting the first stage of an inverse tower is likewise coinitial for its formal presentation. The source and target prefix systems therefore both represent \(P\), and the kernel represents \(\mathbf X\). The stage kernel and cokernel formulas yield the exact Pro row

The splittings (5.2) are finite-stage splittings. They need not commute with restriction projections, so (5.4) is not asserted to split formally or after taking the ordinary limit.

Theorem 5.1. If every \(X_n\) belongs to a class \(\mathcal J\) satisfying Section 3, then \(R^+\pi\mathbf X\) is computed by the complex in degrees zero and one

Consequently

Proof. The augmentation in (5.4) is a quasi-isomorphism from \(\mathbf X\) in degree zero to the two-term class complex \([P\xrightarrow dP]\). Theorem 3.1 computes its derived image by applying \(\pi\). Formula (2.2) identifies the resulting differential with the actual ordinary product map \(\Delta\), proving (5.5) and all cohomology assertions. Its kernel is the ordinary limit because the coordinate equations are exactly the cone compatibility equations.

Equivalently, the exact functor \(R^+\pi\) applied to the exact-row triangle of (5.4) gives its long exact cohomology sequence. The higher groups of both formal products vanish by Theorem 3.1. The remaining segment is \(0\to\pi\mathbf X\to\prod X_n\xrightarrow\Delta\prod X_n\to R^1\pi\mathbf X\to0\), and the later groups vanish. This verifies the exact-sequence interpretation with its connecting map. \(\square\)

6. Maps of towers and changes of indexing

For commuting componentwise maps \(X_n\to Y_n\), the two product maps in (5.5) are the same componentwise product map, and commute with the two differences. We now check arbitrary Pro morphisms between these countable presentations, rather than limiting naturality to that special case.

By (1.1), such a morphism has representatives \(t_n:X_{m_n}\to Y_n\) for a strictly increasing sequence \(m_n\), chosen with \(m_n\geq n\), satisfying

To obtain them, choose a representative at the first target stage. At each successive stage choose any representative and move to a later source stage where its equality with the preceding target component holds. That equality exists by the filtered-colimit equality test. Move further to exceed the preceding source index and the target index. Precomposition preserves every equality already imposed, so induction gives (6.1). This only requires countably many choices.

For \(P_X=\prod_rX_r\), \(P_Y=\prod_nY_n\), define actual ordinary product maps

Every sum is finite; hence every displayed coordinate is an actual map from a finite partial product, and the ordinary product property assembles them. Finite telescoping and (6.1) give

These formulas also define maps of the formal products, because each coordinate has finite support. They give a morphism between the exact rows (5.4), whose map on kernels is the original Pro morphism: its target coordinate is exactly \(t_n\) on source stage \(m_n\). Therefore the normalized class computation makes (6.2) the map computing \(R^+\pi\) and its cohomology maps.

Independence of representatives can be checked directly. Refine \(m_n\) to a strictly increasing \(r_n\geq m_n\), and put \(t'_n=t_nv_{m_n,r_n}\). Let \(T'\) be the corresponding two maps. Define the degree-minus-one map \(H:P_X^1\to P_Y^0\) between the two complexes by

Empty sums are zero. Telescoping gives \(T^0-T'^0=H\Delta_X\). Equation (6.1) and subdivision of finite intervals give \(T^1-T'^1=\Delta_YH\): the intervals on either side express the same difference between the sums over \([m_n,r_n)\) and \([m_{n+1},r_{n+1})\). Thus the two chain maps are homotopic. Two representatives of the same Pro map admit such a common refinement, since their maps to each fixed \(Y_n\) are eventually equal; choose the refinements increasingly. Both are homotopic to the common one, so their maps in the derived category agree.

Composition also agrees with (6.2). For a second representative \(u_n:Y_{l_n}\to Z_n\), the composite uses source indices \(m_{l_n}\) and maps \(u_nt_{l_n}\). In degree zero the formula composes immediately. In degree one, substitute the block sum for each \(Y_q\) coordinate into the block sum over \(q\in[l_n,l_{n+1})\). Compatibility (6.1) moves each transition past \(t_q\); the adjacent blocks \([m_q,m_{q+1})\) partition \([m_{l_n},m_{l_{n+1}})\), yielding the composite formula. Identity indices give identity chain maps. Hence (5.5)–(5.6) are natural for all Pro morphisms between these towers, including reindexing and pro-isomorphisms.

This construction uses an actual class resolution in an abelian Pro category. An arbitrarily chosen completion of a morphism between product triangles in a triangulated category need not supply this same canonical map. The free Stacks treatment of derived limits, Definition 13.34.1 and Remark 13.34.4, makes that distinction explicit.

7. Injectives and arbitrary unital modules

Example 7.1. Suppose \(\mathcal C\) is complete and has enough injectives, and let \(\mathcal J\) be its injective objects. It is cogenerating by definition. If \(I'\hookrightarrow I\) is monic with both objects injective, injectivity of \(I'\) supplies a retraction. The quotient is a direct summand of \(I\), and is injective: extend a map into the whole \(I\), then retract to the summand. Thus hypothesis 2 holds. Every exact row of injectives splits for the same reason. Choose a splitting for each row of a small family. Their ordinary product sections and retractions make the product row split exact, proving hypothesis 3 even when arbitrary ordinary products in \(\mathcal C\) fail to be exact. Finite biproducts of injectives are injective by extending each of their finitely many components. All hypotheses of Theorem 3.1 hold.

In this example, Theorem 1.1 makes each formal product of constant injectives injective in Pro, and Theorem 1.2 supplies enough actual injectives. The cogenerating embeddings (3.2) are the specified levelwise construction. The class construction and the injective-resolution construction compute the same normalized derived functor by its universal property.

Example 7.2. For left modules over any \(\mathcal U\)-small unital ring \(R\), take \(\mathcal J=\mathcal C=\operatorname{Mod}(R)\). The ring need not be commutative. Every object embeds in itself, and every cokernel is another module. Small limits exist as the submodules of products satisfying the relevant equations. Products of short exact module rows are exact: the product map is injective coordinatewise, its kernel consists of the tuples in the coordinate kernels, and a tuple in the quotient product has a lift by choosing one lift in each coordinate. These choices respect the existing module homomorphism and require no linear splitting. Coordinate scalar action makes these the actual module products. Thus hypothesis 3 holds and Theorem 3.1 derives \(\pi\) on the entire Pro category, without requiring the individual modules in the tower to be injective.

For countable module towers, this agrees with the earlier module construction: both derive the same additive left exact \(\pi\) with its unit, so universal uniqueness identifies their functors and computations. The free Stacks comparison, Lemmas 15.88.1 and 15.88.4, gives the corresponding difference complex and reindexing maps for abelian groups. Its category of inverse sequences and the general abelian Pro category are different domains; the present proof supplies the latter interface directly.

8. Four graded exercises with complete solutions

Exercise 1 (10 points). For a nonzero field \(k\), let \(P=P(k)_{n\geq0}\) in \(\operatorname{Pro}(\operatorname{Mod}(k))\). Compute \(\operatorname{Hom}(P,ck)\), and construct an ordinary linear map \(\pi P=k^{\mathbb N}\to k\) which is not the image under \(\pi\) of such a Pro map.

Solution. Formula (1.4) identifies the Pro Hom group with \(k^{(\mathbb N)}\); its tuple \((\lambda_n)\) induces the finite-coordinate functional \(a\mapsto\sum_n\lambda_na_n\). Let \(V=k^{\mathbb N}/k^{(\mathbb N)}\). The class of the all-ones sequence is nonzero, since that sequence has infinite support. Choose a basis of \(V\) containing this nonzero class, and define a linear functional on \(V\) taking that basis element to one and all other basis elements to zero. Compose with the quotient map to obtain \(\ell:k^{\mathbb N}\to k\). It vanishes on all finitely supported sequences and takes the all-ones sequence to one. Any finite-coordinate functional vanishing on each standard basis vector has every coefficient zero, so cannot equal \(\ell\). Thus \(\ell\) is not the image of a Pro map, and \(\pi\) is not full. The construction uses the stated choice convention, not a continuity condition on ordinary linear maps.

Exercise 2 (14 points). In abelian groups, take \(\mathcal J\) to be the groups on which multiplication by two is surjective. Verify all three hypotheses of Section 3. Show that \(c\mathbb Z[1/2]\) is \(\pi\)-acyclic but is not an injective object of Pro.

Solution. The class contains zero and is closed under finite biproducts. Every group embeds into a divisible group by the character-evaluation construction and divisible-injectivity proof. Such a group is in \(\mathcal J\), proving cogeneration. Any quotient of a two-divisible group is two-divisible: lift a quotient element to \(x\), choose \(y\) with \(2y=x\), and take the class of \(y\). This proves hypothesis 2, including all required monic rows. Ordinary small products of abelian-group rows are exact by Example 7.2 with \(R=\mathbb Z\), proving hypothesis 3. Hence \(\mathcal J_{\mathrm{pro}}\) computes \(R^+\pi\).

The group \(\mathbb Z[1/2]\) is two-divisible, since \(a/2^r\) has half \(a/2^{r+1}\). Its constant Pro object is a singleton formal product in the class, so all its positive \(R^q\pi\) vanish. But the monomorphism \(c(3\mathbb Z)\hookrightarrow c\mathbb Z\) and the map \(c(3\mathbb Z)\to c\mathbb Z[1/2]\) sending \(3\) to \(1\) have no extension. Full faithfulness of \(c\) would turn an extension into a group homomorphism \(\mathbb Z\to\mathbb Z[1/2]\), whose value \(a/2^r\) at one would satisfy \(3a=2^r\), impossible. Thus a computing-class object need not be an actual injective object of Pro.

Exercise 3 (12 points). For a class \(\mathcal J\) as in Section 3, compute all the derived limits of a tower of \(\mathcal J\)-objects with zero transitions, and of the constant tower \(J\xleftarrow{1}J\xleftarrow{1}\cdots\) for \(J\in\mathcal J\). Construct a product section in the second case.

Solution. For the zero-transition tower, the ordinary difference map is the identity, so (5.6) gives zero in every degree. Its formal Pro object is itself zero: for every constant target, (1.1) is a filtered colimit whose transition maps are zero, so every representative dies at the next stage. Taking the outer limit over any target presentation then gives zero Hom group to every Pro target. In particular its identity is zero.

For the identity tower, define \(s:\prod_n J\to\prod_n J\) by \((sb)_0=0\) and \((sb)_n=-\sum_{r=0}^{n-1}b_r\) for \(n>0\). Each coordinate uses only a finite sum of product projections, so this is an actual morphism in \(\mathcal C\). Subtraction gives \((\Delta sb)_n=b_n\); hence \(\Delta\) is split epic. Its kernel consists precisely of equal coordinates, and the diagonal identifies it with \(J\). Thus \(R^0\pi=J\), \(R^1\pi=0\), and all higher groups are zero. This is also (4.1) on a heart object, since the identity tower is the constant Pro object.

Exercise 4 (24 points). Let \(p\) be a prime, and consider \(\mathbf X=(\mathbb Z\xleftarrow{p}\mathbb Z\xleftarrow{p}\cdots)\). Define \(\mathbb Z_p=\lim_{N\geq0}\mathbb Z/p^{N+1}\mathbb Z\). Prove that \(R^0\pi\mathbf X=0\), \(R^1\pi\mathbf X\simeq\mathbb Z_p/\mathbb Z\), and all higher groups vanish. Exhibit a nonzero class explicitly.

Solution. Example 7.2 applies with \(R=\mathbb Z\) and the whole module category, so (5.6) applies despite the terms not being injective. A compatible integer sequence satisfies \(a_0=p^na_n\) for every \(n\). Thus \(a_0\) is divisible by every power of \(p\), forcing \(a_0=0\); torsion freeness then forces every \(a_n=0\). This proves the ordinary limit and \(R^0\pi\) vanish. The integer embedding \(\mathbb Z\to\mathbb Z_p\) is injective for the same divisibility reason.

For \(b\in\prod_{n\geq0}\mathbb Z\), set \(S_N(b)=\sum_{r=0}^{N}p^rb_r\), and define

The residues are compatible, since adding the next term changes the sum by a multiple of the preceding modulus. The map is surjective. Given compatible residues, choose \(b_0\in\{0,\ldots,p-1\}\) representing the first one. After constructing \(S_{N-1}\), compatibility says that the desired residue at level \(N\) differs from \(S_{N-1}\) by a multiple of \(p^N\); choose its quotient modulo \(p\) as \(b_N\). This constructs all digits and proves (8.1) has the prescribed residues.

For \((\Delta a)_n=a_n-pa_{n+1}\), finite telescoping gives \(S_N(\Delta a)=a_0-p^{N+1}a_{N+1}\), so \(\Psi(\Delta a)\) is the embedded integer \(a_0\). Conversely, if \(\Psi(b)\) is an integer \(z\), every \(z-S_N(b)\) is divisible by \(p^{N+1}\). Set

These are integers, and subtracting successive expressions proves \(a_n-pa_{n+1}=b_n\), including \(n=0\). Therefore the kernel of the surjective map \(\prod\mathbb Z\xrightarrow\Psi\mathbb Z_p\to\mathbb Z_p/\mathbb Z\) is exactly \(\operatorname{im}\Delta\). It induces the claimed isomorphism of cokernels. Formula (5.6) gives all higher vanishing.

For an explicit nonzero class, take \(b_{2r}=0\) and \(b_{2r+1}=1\). Finite geometric sums give \((p^2-1)S_N(b)\equiv-p\pmod{p^{N+1}}\), so \((p^2-1)\Psi(b)=-p\) in \(\mathbb Z_p\). If \(\Psi(b)\) were an integer \(z\), injectivity of the integer embedding would imply \((p^2-1)z=-p\) in \(\mathbb Z\). This is impossible since \(p^2-1>p>0\). Thus its class in \(\mathbb Z_p/\mathbb Z\) is nonzero. The same argument also works at \(p=2\), where the all-ones digit sequence would instead give an integer and would not prove nonvanishing.

References

The formal abelian structure and the controlled extension used here are proved in Two cokernels, controlled rows and abelian completion, Sections 2–4 and 6. The resolution-class theorem is Compatible injective classes and exact refinements; the normalization of the composite is Universal derived functors and compatible composition. Their proofs retain the full universe and class hypotheses.

For freely accessible comparisons, see the Stacks Project, Rlim of abelian groups, Lemmas 15.88.1 and 15.88.4, and Derived limits, Definition 13.34.1 and Remark 13.34.4. The controlled extension and class criterion establish the general Pro interface used in this lesson.