Draft lesson.

Countable Mittag-Leffler towers and formal Pro derived limits

Written with GPT-6.1 Sol (OpenAI). Original text: CC0.

A sequence of surjections can be lifted successively. More generally, stabilization of the images at each fixed level supplies exactly the corrections needed to lift a compatible family. We prove this directly for modules, then compute the right derived functors of the actual right adjoint on the formal Pro category.

Fix a unital ring \(A\) and a Grothendieck universe \(\mathcal U\) containing \(A\) and \(\mathbb N\). Modules are left \(A\)-modules with \(\mathcal U\)-small underlying sets. All index categories and products below are \(\mathcal U\)-small; a larger universe accommodates the categories themselves. Choice is used for coordinate lifts and injective resolutions.

The stage formulas and exact constant embedding are proved in Two cokernels, controlled rows and abelian completion, Sections 2–3. The complete common-arrow presentation, including arbitrary index arrows, is Finite diagrams and comma objects, Sections 2.1–2.8. Hom, tensor and the exactness of module limits, Sections 3–5, proves the filtered representative test, exact filtered colimits and exact products of modules. Injective replacements and right derived functors, Sections 1–3, and Triangles after quasi-isomorphism localization, Section 6, supply derived-functor existence, exactness and the natural cohomology sequence. The uses of these interfaces are specified below.

1. Stabilization, quotients and extensions

A tower is a family \((M_n,v_n)_{n\geq0}\), where \(v_n:M_{n+1}\to M_n\). Put \(v_{n,n}=1\) and \(v_{n,p}=v_n\cdots v_{p-1}\) for \(p>n\). A tower is Mittag-Leffler, abbreviated ML, if for every \(n\) there is \(N\geq n\) such that

The threshold can depend on \(n\). Every tower with surjective transitions is ML. A finite-dimensional vector-space tower is ML: at each fixed level its descending sequence of image subspaces has a nonincreasing sequence of finite dimensions, and equal dimensions force equal nested subspaces.

Lemma 1.1. For an ML tower let \(I_n\subseteq M_n\) be its eventual image. The transitions restrict to surjections \(I_{n+1}\to I_n\). On \(Q_n=M_n/I_n\), the map \(Q_p\to Q_n\) is zero for all sufficiently large \(p\), for each fixed \(n\).

Proof. Choose \(p\) past the stabilization thresholds for both \(n\) and \(n+1\). Then \(v_n(I_{n+1})=v_n(\operatorname{im}v_{n+1,p})=\operatorname{im}v_{n,p}=I_n\). This also verifies that the restrictions are well defined. For large \(p\), \(v_{n,p}(M_p)=I_n\), so its induced map to \(M_n/I_n\) is zero. The inclusions and quotients form compatible towers. \(\square\)

A short exact sequence of towers means a commuting family of short exact module sequences

We identify \(K_n\) with its image when doing calculations.

Proposition 1.2. In (1.2), an ML middle tower has an ML quotient tower. If the left and right towers are ML, the middle tower is ML.

Proof. Surjectivity of \(q_p\) and the transition square give

Thus stabilization of the middle images implies stabilization of the quotient images.

For the extension assertion, fix \(n\). Choose \(k\geq n\) so that \(\operatorname{im}(K_p\to K_n)\) is constant for \(p\geq k\). Choose \(j\geq k\) so that \(\operatorname{im}(Q_p\to Q_k)\) is constant for \(p\geq j\). We prove that the images of \(M_j\) and \(M_p\) in \(M_n\) agree for every \(p\geq j\).

Take \(m_j\in M_j\). Its image in \(Q_k\) is the image of some \(q_p\in Q_p\), by the second choice. Lift \(q_p\) to \(m_p\in M_p\). The difference of the two images in \(M_k\) lies in \(K_k\); call it \(d_k\). Its image in \(K_n\) lies in the stabilized image \(\operatorname{im}(K_k\to K_n)=\operatorname{im}(K_p\to K_n)\). Choose \(d_p\in K_p\) with that image. The image of \(m_p+d_p\) in \(M_n\) is the image of \(m_j\). Hence the image from \(M_j\) is contained in the image from \(M_p\). The opposite inclusion follows from \(v_{n,p}=v_{n,j}v_{j,p}\). This proves stabilization at \(n\), and \(n\) was arbitrary. \(\square\)

2. The difference operator and actual compatible lifts

The ordinary inverse limit is the module of compatible tuples. Define

Then \(\ker\Delta_M=\lim_n M_n\), with its specified coordinate projections.

Lemma 2.1. If each map \(Q_p\to Q_n\) is eventually zero at every fixed \(n\), then \(\Delta_Q\) is an isomorphism.

Proof. For \(b\in\prod Q_n\), define

This is a finite sum in each coordinate: all its maps are zero past a threshold depending on \(n\). It defines a linear map on the product. In \((\Delta_QTb)_n\), subtracting \(v_n(Tb)_{n+1}\) cancels every summand except \(b_n\). In \((T\Delta_Qa)_n\), choose a finite cutoff \(L\) with \(v_{n,p}=0\) for \(p>L\). The sum telescopes to \(a_n-v_{n,L+1}a_{L+1}=a_n\). Thus \(\Delta_QT=T\Delta_Q=1\). No convergence or infinite addition is involved. \(\square\)

Theorem 2.2. For every countable ML module tower, \(\Delta_M\) is surjective.

Proof. Use the stable-image sequence \(0\to I_n\to M_n\to Q_n\to0\) of Lemma 1.1. Given \(b\in\prod M_n\), let \(\bar b\) be its quotient tuple. Lemma 2.1 gives \(\bar a=T\bar b\) with \(\Delta_Q\bar a=\bar b\). Lift its coordinates to \(a\in\prod M_n\). The residual \(r=b-\Delta_Ma\) lies in \(\prod I_n\).

Solve \(\Delta_Ix=r\) recursively. Set \(x_0=0\). Once \(x_n\) is chosen, surjectivity of \(I_{n+1}\to I_n\) allows a choice of \(x_{n+1}\) satisfying \(v_nx_{n+1}=x_n-r_n\). This specifies a tuple with \(x_n-v_nx_{n+1}=r_n\) for every \(n\). Therefore \(\Delta_M(a+x)=b\). \(\square\)

Theorem 2.3. In (1.2), the sequence

is always exact. If \(K\) is ML, its last map is surjective as well.

Proof. The first map is injective componentwise. A compatible middle tuple killed in every \(Q_n\) has unique coordinate preimages in \(K_n\). The transition squares and injectivity of \(\iota_n\) make those preimages compatible. This proves middle exactness.

Now take a compatible tuple \(q\in\lim Q_n\), and choose lifts \(b_n\in M_n\). Compatibility of \(q\) says \(\Delta_Mb=\iota r\) for some \(r\in\prod K_n\). If \(K\) is ML, choose \(a\in\prod K_n\) with \(\Delta_Ka=r\) by Theorem 2.2. Then \(b-\iota a\) is compatible and maps to \(q\). These are lifts of the prescribed tuple, not merely an identification of abstract limits. \(\square\)

3. The formal Pro category and its right adjoint

Put \(\mathcal C=\operatorname{Mod}_{\mathcal U}(A)\) and

Write \(cN\) for a constant object and \(\mathbf X=\lim_i^{\rm form}cX_i\) for a small cofiltered presentation. The word formal distinguishes this object from the ordinary module limit. The dual Hom formula is

Here \(\mathbf Y=\lim_j^{\rm form}cY_j\). In the colimit, a transition \(X_{i'}\to X_i\) acts by precomposition. In particular,

Apply the abelian stage theorem from the first prerequisite to \(\mathcal C^{\rm op}\) and then take opposites. This proves that Pro is abelian and locally small and that \(c\) is exact and fully faithful. The opposite category is abelian and locally \(\mathcal U\)-small; essential smallness was not a hypothesis of that theorem. Common-arrow strictification in the second prerequisite gives, for any Pro arrow, one small cofiltered level presentation \(f_i:X_i\to Y_i\). Dualizing the kernel and cokernel formulas gives

These are the kernels and cokernels in Pro. They follow by reversing the universal properties; taking opposites exchanges kernel and cokernel twice. Image formulas consequently hold at these same stages.

Lemma 3.1. Every short exact Pro sequence has an isomorphic presentation by a common cofiltered family of short exact module sequences.

Proof. For a sequence \(0\to\mathbf K\to\mathbf M\xrightarrow f\mathbf Q\to0\), present only \(f\) by a common family \(f_i:M_i\to Q_i\). By (3.4), \(\lim^{\rm form}c\ker f_i\) is its kernel and \(\lim^{\rm form}c\operatorname{im}f_i\) is its image. Since \(f\) is epic, this latter image identifies with \(\mathbf Q\). Replace the level targets by \(\operatorname{im}f_i\). The rows

are short exact and have natural transitions. Their formal row identifies with the given row, including its kernel inclusion and quotient projection, by the universal properties. This does not assert that the original chosen \(f_i\) were all surjective. \(\square\)

Proposition 3.2. The constant embedding has an additive right adjoint

It is left exact and \(\pi c=1\).

Proof. The second formula in (3.3), followed by the ordinary limit universal property, gives natural bijections

The module limit exists as the submodule of a small product satisfying all transition equations. For an arbitrary Pro morphism \(\mathbf X\to\mathbf Y\), postcomposition on the left of (3.7) gives a natural transformation of represented Hom functors. It determines a unique module map \(\pi\mathbf X\to\pi\mathbf Y\). Uniqueness proves identity and composition and independence of presentations. The Hom identifications preserve sums, hence \(\pi\) is additive. They prove the adjunction \(c\dashv\pi\). A right adjoint preserves limits: a cone into the image of a limiting object corresponds under (3.7) to the unique cone factorization before applying \(\pi\). Thus \(\pi\) preserves finite limits and is left exact. For a constant presentation its limit is the original module, naturally. \(\square\)

4. Enough injectives, with all size and action conventions

We first give an explicit module construction. A divisible abelian group \(D\) is injective. Indeed, take a maximal extension of a map from a subgroup \(H\subseteq G\) to \(D\), using Zorn's lemma; unions of chains give upper bounds. If \(x\notin H\), the set \(\{a\in\mathbb Z:ax\in H\}\) is \(d\mathbb Z\) for some \(d\geq0\). If \(d=0\), assign \(x\) any value. If \(d>0\), choose \(y\in D\) with \(dy=f(dx)\). In either case the formula \(f(h+ax)=f(h)+ay\) defines an extension to \(H+\mathbb Zx\): for \(d>0\), the difference of two such representations has \(a-a'=kd\) and \(h-h'=-kdx\), so the proposed values agree. This contradicts maximality unless \(H=G\). Thus every subgroup map extends.

The group \(\mathbb Q/\mathbb Z\) is divisible and detects nonzero elements: on a cyclic subgroup generated by \(m\ne0\), send \(m\) to \(1/d+\mathbb Z\) if its finite order is \(d>1\), and to \(1/2+\mathbb Z\) if its order is infinite. Extend this map to the whole group by injectivity. Consequently evaluation gives a monomorphism

of underlying groups. A product of injective groups is injective: extend a map coordinate by coordinate and assemble its extensions. Both the index and \(J\) are \(\mathcal U\)-small.

For a left \(A\)-module \(M\), let \(j:M\hookrightarrow J\) be (4.1) on its underlying group. Give \(I=\operatorname{Hom}_{\mathbb Z}(A,J)\) the left action

It is a left action even when \(A\) is noncommutative: \((a_1(a_2\varphi))(b)=\varphi((ba_1)a_2)=((a_1a_2)\varphi)(b)\); the unit and distributive laws follow in the same way. Evaluation at \(1\) is the natural bijection

Its inverse sends \(h\) to \(g_h(n)(b)=h(bn)\). Formula (4.2) proves \(A\)-linearity, and for an \(A\)-linear \(g\), \(g(n)(b)=g(bn)(1)\), proving both inverse equations. A map into \(I\) from a submodule therefore extends by extending its corresponding group map into \(J\). Thus \(I\) is injective. The map \(m\mapsto(b\mapsto j(bm))\) is \(A\)-linear and injective, by evaluation at \(1\). This proves enough injectives in \(\mathcal C\), with small injective targets. Iterating on successive cokernels constructs an injective resolution in degrees \(q\geq0\) of every module.

For a small family \((N_s)\), its formal product of constants is

with restriction projections; the empty finite product is the zero module. To verify its product property, insert its presentation into (3.2). A finite product of Hom groups commutes with a filtered colimit, by the filtered representative and equality tests. Compatible choices over finite subsets are exactly arbitrary families of elements. Hence

with the specified coordinate maps. This proves the full product universal property for every test Pro object. Proposition 3.2 then gives

Theorem 4.1. A constant injective module \(cJ\) is injective in Pro. Formal products of these objects are injective, and Pro has enough injectives.

Proof. Present any short exact Pro row by (3.5). Applying \(\operatorname{Hom}_A(-,J)\) gives a short exact row of groups at every level, because \(J\) is injective. Taking the filtered colimit over the opposite index is exact, by the module-colimit theorem with coefficient ring \(\mathbb Z\). Formula (3.3) therefore proves that \(\operatorname{Hom}_{\rm Pro}(-,cJ)\) is exact. This is precisely the extension property for \(cJ\).

For a family of injective \(J_s\), extend each component of a map into \(cJ_s\) along a Pro monomorphism. Formula (4.5) assembles the extensions into a map to their product, proving its injectivity.

Finally, present \(\mathbf X\) by \((X_i)\), embed each \(X_i\) in an injective module \(J_i\), and combine the projections into

The index here is the small set of objects of the presenting category; its arrows need not be preserved by the chosen embeddings. If a map \(t:\mathbf T\to\mathbf X\) is killed by (4.7), each composite \(\mathbf T\to cX_i\) is zero, since \(cX_i\to cJ_i\) is monic by exactness of \(c\). Formula (3.2) says that maps into \(\mathbf X\) are determined by all these coordinates, so \(t=0\). Applying this to a difference of two maps proves that (4.7) is monic. Its target is injective by the preceding paragraph. Thus every Pro object embeds into an injective. \(\square\)

5. Derived pi and acyclic formal products

The derived-functor prerequisite applies to the abelian category Pro because Theorem 4.1 supplies enough injectives. It constructs the exact functor

where \(\mathbf X\to J^\bullet\) is an injective resolution in nonnegative degrees. The replacement comparison makes this independent of choices, naturally for every Pro morphism. The terms in negative degrees are zero, so \(R^q\pi=0\) for \(q<0\); left exactness gives \(R^0\pi=\pi\). More explicitly, the kernel of \(\pi J^0\to\pi J^1\) is \(\pi\mathbf X\).

The exact-sequence triangle theorem applies to any short exact Pro row concentrated in degree zero. Apply the exact functor \(R\pi\) to its triangle, then the target cohomology sequence. This gives the natural long exact sequence of \(R^q\pi\). This use needs no injectivity assumption on the three original objects and no exactness of \(\pi\).

Lemma 5.1. Every formal product \(P(N_s)\) is \(\pi\)-acyclic: \(R^q\pi P(N_s)=0\) for \(q>0\).

Proof. The operation \((N_s)\mapsto P(N_s)\) carries any family of short exact module rows to a short exact Pro row. Indeed, its common levels are the finite-subset products, which are short exact; formula (3.4) computes the formal row's kernel and cokernel from those levels. This proves exactness of this particular formal-product operation, without an assertion about products of arbitrary Pro objects.

Choose injective resolutions \(N_s\to I_s^\bullet\). In each degree form \(P(I_s^q)\) and use the formal products of the differentials. Exactness just proved implies

is exact. For completeness, factor each module differential through its image; the short exact rows defining its kernel and image stay exact under \(P\). Thus the kernel and image in (5.2) are the formal products of the individual kernels and images, which agree at every positive position. Each term is injective in Pro by Theorem 4.1. Hence (5.2) is an injective resolution.

Applying \(\pi\) gives the ordinary product complex \(\prod_s I_s^\bullet\) by (4.6), with coordinate differentials. Products of short exact module rows are exact by the actual coordinate-lifting proof in the module prerequisite. Factoring the differentials as above shows that this complex has \(H^0=\prod_sN_s\) and zero cohomology in positive degrees. Formula (5.1) proves the assertion. \(\square\)

6. A finite-prefix exact sequence for every tower

Let \(\mathbf M=\lim_n^{\rm form}cM_n\) and \(P=P(M_n)\). Cofinal prefixes \(\{0,\ldots,N\}\) present (4.4), so write \(P_N=\prod_{r=0}^NM_r\), with prefix restrictions. Present a Pro map \(\boldsymbol\Delta:P\to P\) at level \(N\) by

The source presentation is shifted by one. Restricting the source and target prefixes makes the squares commute. The shifted source still presents \(P\): forgetting a finite initial segment of this prefix system is coinitial, and all specified coordinate projections are unchanged under that comparison.

Lemma 6.1. There is a short exact sequence in Pro

Its image under \(\pi\) has difference map (2.1).

Proof. Each \(\delta_N\) is split surjective as a module map. Given \((b_0,\ldots,b_N)\), put \(a_{N+1}=0\), then recursively put \(a_r=b_r+v_ra_{r+1}\) for \(r=N,N-1,\ldots,0\). This is a linear right inverse. It is not required to commute with prefix transitions.

The kernel is identified with \(M_{N+1}\) by its last coordinate, with inverse

Restricting the kernel at level \(N+1\) to the source prefix at level \(N\) sends its last coordinate \(t\in M_{N+2}\) to \(v_{N+1}t\in M_{N+1}\). Thus the kernel system is exactly the shifted tower \((M_{N+1},v_{N+1})\). This shift represents \(\mathbf M\) by the same coinitial-tail comparison. The resulting kernel inclusion has coordinates \(\mathbf M\to cM_r\), so it is the specified map \(e\) into the formal product. Formula (3.4) and the level surjectivity prove (6.2).

Under (4.6), the \(r\)-th coordinate of \(\pi\boldsymbol\Delta\) is \(a_r-v_ra_{r+1}\), because it is that coordinate in every target prefix containing \(r\). Hence the induced map is the actual ordinary \(\Delta_M\). \(\square\)

Theorem 6.2. Every countable module tower satisfies

If the tower is ML, \(R^q\pi(\mathbf M)=0\) for every \(q>0\).

Proof. Apply the long exact sequence from Section 5 to (6.2). Both copies of \(P\) have vanishing positive derived functors by Lemma 5.1. Its first part is (6.4), using \(R^0\pi=\pi\), \(\pi\mathbf M=\lim M_n\) and Lemma 6.1's identification of the difference map. Later positions give \(R^q\pi\mathbf M=0\) for \(q\geq2\). For an ML tower Theorem 2.2 makes the difference map surjective, so its cokernel, \(R^1\pi\mathbf M\), is zero as well. \(\square\)

For componentwise maps of towers, the products and (6.1) commute with those maps, so (6.4) is natural with its displayed kernel and cokernel identifications. The functors in (5.1) are already defined naturally for arbitrary Pro morphisms. The proof therefore computes the derived functors of \(\pi\) on formal Pro objects, with their exact abelian structure.

7. Examples of stabilization and obstruction

Example 7.1 (a non-surjective ML tower). Take \(M_n=\mathbb Z^2\) and \(v_n(x,y)=(x,0)\). Then \(I_n=\mathbb Z\oplus0\), all induced stable-image maps are identity, and the quotient maps are zero. Compatibility forces a constant first coordinate and a zero second coordinate, so \(\lim M_n=\mathbb Z\). For an arbitrary right-hand side \(b_n=(u_n,w_n)\), an actual solution of \(\Delta_Ma=b\) is

Subtracting the first coordinates at \(n\) and \(n+1\) gives \(u_n\), and the projected second coordinate at \(n+1\) is zero. Theorem 6.2 gives all positive derived functors zero.

Example 7.2 (a nonzero first derived obstruction). Let \(M_n=\mathbb Z\) and \(v_n=2\). Its images at level \(n\) are \(2^{p-n}\mathbb Z\), which never stabilize. Its limit is zero, since a compatible integer \(a_n\) is divisible by every power of two. Take \(b_{2r}=0\), \(b_{2r+1}=1\). If \(\Delta_Ma=b\), repeated substitution through coordinate \(2k+1\) gives

The fixed integer \(3a_0+2\) would be divisible by arbitrarily large powers of four, hence zero. No integer satisfies \(3a_0+2=0\). Thus \(b\) represents a nonzero element of \(R^1\pi\mathbf M=\operatorname{coker}\Delta_M\). This argument uses finite integer equations for every \(k\), rather than a convergent sum in \(\mathbb Z\).

8. Four graded exercises with complete solutions

Exercise 1 (basic: surjective finite quotients). For a unital ring \(A\), take the left-module tower \(M_n=A/2^{n+1}A\) with reduction maps. Show that it is ML, describe its ordinary limit, and determine every \(R^q\pi\mathbf M\).

Solution. Since the integer scalar two is central, \(2^{n+1}A\) is a left submodule, and its inclusion in \(2^nA\) makes reduction a well-defined surjection. The images at every fixed level are therefore the whole module, so \(I_n=M_n\) and the tower is ML. Its ordinary limit is precisely

The module operations are coordinatewise and its projections have the limit universal property. It is called the two-adic completion of this module; no injectivity or surjectivity of the map from \(A\) is asserted. Theorem 6.2 gives \(R^0\pi\mathbf M\) equal to (8.1), every positive \(R^q\pi\mathbf M\) zero, and the resolution convention gives every negative one zero. For \(A=\mathbb Z/3\), multiplication by two is invertible, so every quotient is zero and (8.1) is zero as well.

Exercise 2 (intermediate: delayed disappearance). Set \(M_n=\mathbb Z^{n+1}\). For \(v_n:M_{n+1}\to M_n\), let its coordinate \(r\), for \(0\leq r\leq n\), be \(x_r\) if \(n<2r\), and zero otherwise. Find the stable images and the inverse of \(\Delta_M\), and compute its ordinary and derived limits.

Solution. For \(p>n\), coordinate \(r\) survives the composite \(v_{n,p}\) exactly when every target index \(t=n,\ldots,p-1\) satisfies \(t<2r\), equivalently \(p\leq2r\). Since \(r\leq n\), the composite is zero for \(p\geq2n+1\). Hence \(I_n=0\), with this threshold, and the tower is ML. Lemma 2.1 gives the actual inverse

Terms beyond this range vanish. Its two inverse equations follow by the telescoping calculation in that lemma. Thus \(\ker\Delta_M=0\) and \(\operatorname{coker}\Delta_M=0\). Formula (6.4) gives \(\pi\mathbf M=0\) and all positive derived functors zero. In fact \(\mathbf M=0\) in Pro: the identity at each target level is represented by a sufficiently late zero transition, so every coordinate of its identity map in (3.2) is zero. Notice that every \(M_n\) is nonzero; the vanishing concerns the formal object and the compatible limit.

Exercise 3 (advanced: failure of exact compatible lifting). Consider the short exact towers

with transitions respectively multiplication by two, identity, and reduction. Show that the middle and right towers are ML but the map between their ordinary limits is not surjective. Exhibit a nonzero connecting class in the formal Pro derived sequence.

Solution. Each row is short exact. Compatibility of the left square is the equation \(2^{n+1}2=2^{n+2}\); the other square is reduction compatibility. The middle tower has identity transitions and limit \(\mathbb Z\); the right tower has surjective transitions and limit the compatible residues in (8.1) for \(A=\mathbb Z\). Both are ML. The left tower is Example 7.2 and has limit zero.

Put \(s_n=\sum_{r=0}^n2^rb_r\), where \(b_{2r}=0,b_{2r+1}=1\), and let \(q_n=s_n\bmod2^{n+1}\). These form a compatible quotient tuple, since \(s_{n+1}-s_n\) is divisible by \(2^{n+1}\). If it came from an integer \(z\), its odd-index coordinates would give \(3z+2\) divisible by \(4^{k+1}\) for every \(k\), by (7.2). This is impossible. Thus the limit map is not onto.

The formal row is short exact by (3.4). Its long exact sequence has a connecting map from the compatible residues to \(R^1\pi\) of the left tower. The kernel of that connecting map is exactly the image of the middle limit. Consequently the tuple just constructed has a nonzero connecting class. The middle and right first derived functors vanish by Theorem 6.2; hence the connecting map also identifies \(R^1\pi\) of the left tower with the quotient of the compatible-residue module by the image of \(\mathbb Z\). This proves why Theorem 2.3 requires ML of the left tower.

Exercise 4 (advanced: the shifted prefix and noncommutative action). For transitions \(v_0=2,v_1=3,v_2=4\) between copies of \(\mathbb Z\), write the level-two map (6.1), its kernel parametrization and a right inverse. Explain why levelwise sections do not by themselves split (6.2). Separately, for \(A=M_2(\mathbb Q)\), verify the order of multiplication in (4.2), and show that replacing \(\varphi(ba)\) by \(\varphi(ab)\) fails to define a left action.

Solution. The source is the four-coordinate prefix and the target has three coordinates:

Its kernel is \(d\mapsto(24d,12d,4d,d)\); substitution proves containment, and successive equations from the last coordinate prove that every kernel element has this form. A right inverse is

Each difference in (8.4) is respectively \(u,v,w\). The right inverses are linear at their individual levels but generally do not commute with prefix restrictions. For example, in a tower with all transitions two, the right inverse at level one takes \((u,v)\) to \((u+2v,v,0)\), whereas the level-zero right inverse of the restricted tuple is \((u,0)\). Restricting the former source to two coordinates disagrees whenever \(v\ne0\). Thus these sections prove an epimorphism in Pro through (3.4), without giving a Pro section. Example 7.2 additionally precludes a section of (6.2) for that tower, since applying any functor to a split epimorphism preserves a section, whereas its ordinary difference map is not onto.

For the action, \(a_1(a_2\varphi)(b)=\varphi(ba_1a_2)\), exactly the action of \(a_1a_2\) in (4.2). For the proposed reversed formula, successive actions instead evaluate \(\varphi(a_2a_1b)\). Take \(a_1=E_{12},a_2=E_{21},b=1\), and the additive map \(\varphi:M_2(\mathbb Q)\to\mathbb Q\) extracting the upper-left entry. Then successive proposed actions give \(\varphi(E_{22})=0\), while the proposed action of \(a_1a_2\) gives \(\varphi(E_{11})=1\). Associativity fails. This verifies why the coinduced left-module convention uses right multiplication in the argument of \(\varphi\).

References

• The Stacks Project, Inverse systems, Section 12.31, for the ML definition and comparison with exactness of compatible module families. Sections 1–2 above supply the full stabilization and lifting proofs used here.
• The Stacks Project, Rlim of abelian groups, Section 15.88, Lemma 15.88.1, computes the derived inverse limit on the category of countable inverse systems. Sections 3–6 above establish the category, injectives and finite-prefix sequence needed to compute \(R^q\pi\) on formal Pro modules.