Draft lesson.

Comodules, rational modules and cofree injective resolutions

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

A coaction records finitely many coefficients for each vector. On the dual algebra, this becomes a continuity condition on each orbit map. We will prove the equivalence, construct the largest continuous part of any algebraic module, and build a cofree injective resolution with every differential and contraction specified. The resulting Ext formula works for an arbitrary coalgebra over a field. Its examples range from graded semisimplicity to nonzero Ext in every degree.

Prerequisites: Finite subcoalgebras and profinite algebra reconstruction, §§1–5; Completed tensors and continuous algebra objects, §§1–5; the locally nilpotent specialization in Locally nilpotent operators and coinduced duals, §§2–4 and Exercise 3; and the resolution comparisons in Injective replacements and right derived functors, §§2–4.

Fix a field \(k\) and a coalgebra \(H\), with coproduct \(\Delta\colon H\to H\otimes H\) and counit \(\epsilon\colon H\to k\). All tensors are ordinary tensors over \(k\). Let \(A=H^*\) have its finite topology and convolution multiplication \((ab)(h)=(a\otimes b)\Delta h\), with unit \(\epsilon\). The first prerequisite proves that \(A\) has a basis of open finite-codimension two-sided ideals. We work with right \(H\)-comodules and left \(A\)-modules, with no extra group action.

1. The abelian comodule category

A right comodule has \(\rho_M\colon M\to M\otimes H\), with \((\rho_M\otimes1)\rho_M=(1\otimes\Delta)\rho_M\) and \((1\otimes\epsilon)\rho_M=1_M\). A morphism satisfies \((f\otimes1)\rho_M=\rho_N f\). These maps form a vector space; addition and scalar multiplication preserve that equation.

Kernels and cokernels are the ordinary vector-space ones. For a kernel inclusion \(i\colon K\to M\), exactness of tensor over the field identifies \(\ker(f\otimes1)=K\otimes H\). The equation for \(f\) therefore gives the unique \(\rho_K\) with \((i\otimes1)\rho_K=\rho_Mi\). Composing with the monic \(i\otimes1\otimes1\) and with \(i\) proves its coassociativity and counit equations. The same monicity proves the kernel factorization property for every comodule map into \(K\).

For a cokernel \(q\colon N\to Q\), the map \((q\otimes1)\rho_N\) kills \(f(M)\) and descends to \(\rho_Q\). The structural identities hold after precomposing with the epic \(q\), hence hold. Every factor through \(q\) respects coactions by that same epic test. Zero and finite biproducts have their component coactions. The underlying coimage-to-image isomorphism is a comodule map, and its inverse respects coactions by multiplying its intertwining equation by the inverse. Thus \(\mathsf{Comod}(H)\) is abelian, and its faithful forgetful functor \(U\) to vector spaces is exact and reflects exactness.

Every small colimit is also the underlying vector colimit. Its coaction is the unique map satisfying \(\rho u_i=(u_i\otimes1)\rho_i\), since tensor with \(H\) preserves colimits. Coassociativity, counit and the universal property are tested on the jointly spanning structure maps \(u_i\). Filtered colimits are exact: for a filtered diagram of exact vector rows, a vector is represented at one stage; a zero or equality is witnessed after a later stage. This proves injectivity and the kernel equality in the colimit, and surjectivity follows by lifting a representative at its own stage. Finitely many witnesses can be put into one common stage, including equalization of parallel arrows. The same colimit and exactness statements follow for comodules through \(U\).

2. Discrete continuous modules and both inverse constructions

Give a left \(A\)-module \(M\) the discrete topology. Its action \(A\times M\to M\) is jointly continuous exactly when

\operatorname{Ann}_A(m)=\{a:am=0\}\text{ is open for every }m. \tag{2.1}

Indeed \(A\times M\) is the disjoint union of its slices \(A\times\{m\}\). On a slice the orbit map is linear; it is continuous exactly when its kernel is open, since all its fibres are kernel cosets. Call these modules rational and write \(\mathsf{Rat}(A)\) for their category with all module maps, which are automatically continuous between discrete spaces.

Theorem 2.1. Right \(H\)-comodules and rational left \(A\)-modules are equivalent, with inverse constructions on all objects and morphisms.

Proof. Write \(\rho(m)=\sum_i m_i\otimes h_i\) and set \(am=\sum_i a(h_i)m_i\). This is independent of the tensor expression. Coassociativity gives

a(bm)=\sum a(h_{(1)})b(h_{(2)})m_{(0)}=(ab)m, \qquad\epsilon m=m, \tag{2.2}

where the iterated-coaction notation denotes the actual finite tensor from coassociativity. Thus it is a left action in the stated convolution order. The annihilator of the finite span of the \(h_i\) lies in \(\operatorname{Ann}(m)\), proving continuity.

Conversely suppose (2.1). Choose an open finite-codimension two-sided ideal \(I\subset\operatorname{Ann}(m)\). It kills the whole cyclic module \(Am\), since \(I\) is two-sided, and \(Am\) is finite-dimensional as an image of \(A/I\). Write \(B=A/I\), and identify \(B^*\) with its finite subcoalgebra in \(H\), by the first prerequisite's annihilator theorem. In a basis \((b_i)\) of \(B\) with dual basis \((h_i)\), define

\rho(m)=\sum_i b_i m\otimes h_i. \tag{2.3}

The action of \(b_i\) is well-defined by the quotient. Evaluation against any \(a\in A\) in the last factor gives exactly \(am\). Such evaluations separate every finite tensor in \(M\otimes H\), so this characterizes (2.3) uniquely, proving independence of basis and ideal. For two vectors choose the intersection of their defining ideals; the same characterization proves linearity of \(\rho\). Evaluating its two proposed iterated coactions against arbitrary \(a,b\in A\) gives \(a(bm)\) and \((ab)m\); the module law makes them equal. Scalar functionals in both last factors separate their finite tensor coefficient spans, so coassociativity holds. Evaluating against \(\epsilon=1_A\) gives the counit identity.

An intertwining comodule map is \(A\)-linear by evaluation. Conversely, for an \(A\)-linear map, evaluation of \((f\otimes1)\rho_M\) and \(\rho_Nf\) against every \(a\) gives \(f(am)=af(m)\), so the tensors agree. Starting with either a coaction or action and applying both constructions recovers it by the same separation identity. Identities and composition are unchanged on the underlying linear maps. This gives the full inverse equivalence. \(\square\)

3. Rationalization and intrinsic limits

For any algebraic left \(A\)-module \(X\), put \(R(X)=\{x:\operatorname{Ann}(x)\text{ is open}\}\). This is a submodule. If open two-sided ideals kill \(x,y\), their intersection kills every linear combination. If \(I\) kills \(x\), then it kills \(ax\), since \(ia\in I\) for \(i\in I\). Thus all those vectors satisfy (2.1). Module maps preserve \(R(X)\); it is the largest rational submodule.

Submodules and quotients of rational modules are rational, by retaining an ideal killing each vector or its representative. Finite biproducts are rational by taking intersections. The full inclusion \(i\colon\mathsf{Rat}(A)\to A\text{-}\mathsf{Mod}\) is therefore exact. All its small colimits are the ambient ones: direct-sum vectors have finite support and hence a common killing ideal, and quotients retain rationality.

There are inverse natural bijections

\operatorname{Hom}_A(iM,X)\simeq\operatorname{Hom}_{\mathsf{Rat}(A)}(M,R(X)). \tag{3.1}

Every map on the left has rational image because a killing ideal for \(m\) kills its image. Restrict the codomain to obtain the forward map; compose with \(R(X)\hookrightarrow X\) for the inverse. They recover every vector and commute with both input maps. Thus \(i\dashv R\), with unit \(M=R(iM)\) and counit the literal submodule inclusion. Restricting that inclusion to its rational part is identity, proving both triangles and \(R^2=R\).

For a small rational diagram, take its ambient module limit and its projections. Every cone from a rational module gives the unique ambient map, which (3.1) factors uniquely through the rational part. Hence the intrinsic limit is \(R(\lim_A iM_j)\), with the restricted projections. The same cone argument makes \(R\) preserve every ambient limit into intrinsic ones. Finite intrinsic limits are the ambient kernels and finite biproducts already proved, so \(R\) is left exact. It need not preserve epimorphisms, as Exercise 7.3 shows.

4. Finite objects, generators and all filtered Hom maps

The finite-subcomodule theorem of the first prerequisite gives every comodule as the directed union of its finite-dimensional subcomodules. Their set is small, and finite sums give common upper bounds. A small skeleton \(\mathcal D\) of the finite-dimensional objects exists: a coaction on \(k^r\) is specified by a finite matrix of elements of the fixed set \(H\), subject to its structural equations.

These finite-dimensional comodules are compact. Given a comodule map \(f\colon D\to\varinjlim_jN_j\), lift a finite vector basis linearly to one common stage, obtaining \(f_j\). The coaction defect \(\rho_jf_j-(f_j\otimes1)\rho_D\) has finitely many basis values in \(N_j\otimes H\), all zero in the colimit. Tensor preserves that filtered colimit, so one common later stage kills them all, making the lift a comodule map. Equality of two lifts is also witnessed after one common stage, by their finite basis differences. This proves both directions of the compact-Hom bijection for general filtered indexing. Conversely a compact object's identity factors through one member of its finite-subcomodule union, forcing its whole underlying space to be finite-dimensional.

Consequently realization gives

\operatorname{Ind}(\mathcal D)\simeq\mathsf{Comod}(H),\qquad \operatorname{Hom}_H(\varinjlim_iD_i,\varinjlim_jE_j) \simeq\varprojlim_i\varinjlim_j\operatorname{Hom}_H(D_i,E_j). \tag{4.1}

The forward map restricts to each finite source piece and lifts it using compactness; stage equality gives compatible classes. The reverse map composes representatives with the target structure maps and glues over the source colimit. They are inverse on every structure map and respect identities, composition and both-variable naturality. Every object is a finite-piece union, proving essential surjectivity.

The finite objects are noetherian because a strictly increasing chain of their subspaces strictly increases dimension. An infinite-dimensional object has a strictly increasing chain of finite-dimensional subcomodules: successively add an outside vector and apply the finite-subcomodule theorem. Thus the noetherian objects are exactly the finite ones. Their small skeleton generates: the direct sum of all finite subcomodules of an object maps onto it, and each nonzero morphism is detected on a finite subcomodule containing a vector where it is nonzero. Taking the coproduct of skeleton representatives gives one small generator by these maps. Together with §1's colimits and filtered exactness, this proves the Grothendieck property and gives a set of noetherian generators.

5. The cofree adjunction and injective closure

For a vector space \(V\), let \(F(V)=V\otimes H\), with coaction \(1_V\otimes\Delta\). A linear map \(t\) acts by \(t\otimes1\). There are inverse natural bijections

\operatorname{Hom}_H(M,F(V))\simeq\operatorname{Hom}_k(UM,V),\quad \alpha\mapsto(1\otimes\epsilon)\alpha,\quad \ell\mapsto(\ell\otimes1)\rho_M. \tag{5.1}

The second map is a comodule map by coassociativity. Applying the counit to it recovers \(\ell\). Starting with \(\alpha\), its intertwining equation and \((\epsilon\otimes1)\Delta=1\) recover \(\alpha\) from its counit composite. These formulas prove both inverse identities and naturality on all maps. Thus \(U\dashv F\); its unit is \(\rho_M\), and its counit is \(1_V\otimes\epsilon\). The first triangle is the comodule counit identity. The second is \((1_V\otimes\epsilon\otimes1)(1_V\otimes\Delta)=1_{F(V)}\), the other coalgebra counit identity.

Every \(F(V)\) is injective. For a monomorphism of comodules, extend the linear map corresponding in (5.1) to a linear map on the larger vector space by extending a basis. The inverse in (5.1) gives the required comodule extension. The unit \(\rho_M\) is monic because its underlying counit composite is identity. Hence every object embeds into a cofree injective, proving enough injectives. An injective object splits its own unit by its extension property, so every injective is a retract of a cofree object. Conversely a retract of an injective is injective by composing with the inclusion, extending, then composing with the retraction.

Arbitrary coproducts of injectives are injective: take those cofree retract presentations and their coproducts, and use \(\bigoplus_sF(V_s)=F(\bigoplus_sV_s)\). Products are injective as well. The actual cofree product is \(F(\prod_sV_s)\), with maps \(F(\pi_s)\), since (5.1) identifies its cone bijection with the vector-product cone bijection. Taking products of the retract maps makes a product of injectives a retract of that cofree injective. These are intrinsic comodule products from §3, rather than a claim that tensor with an infinite-dimensional \(H\) preserves vector-space products.

Filtered colimits of injectives are injective. Let \(I=\varinjlim_jI_j\), each \(I_j\) injective, and consider a map \(f\colon M\to I\) on a subcomodule \(M\subset N\). Order its extensions to intermediate subcomodules by inclusion. A chain has its union extension, so Zorn gives a maximal one on \(P\). If a vector of \(N\) is outside \(P\), choose a finite-dimensional subcomodule \(D\) containing it. The intersection \(D\cap P\) is finite-dimensional. By §4, its restricted map to \(I\) lifts to some \(I_j\). Injectivity extends that lift to \(D\); composing into \(I\) agrees with the existing map on \(D\cap P\). Define the extension on \(P+D\) by adding the two maps. It is well-defined by that intersection equality and is a comodule map by their intertwining equations. This contradicts maximality. Thus \(P=N\), proving the extension property of \(I\).

6. A signed cofree resolution and its Ext complex

For a comodule \(N\), set \(J^{-1}(N)=N\) and \(J^n(N)=N\otimes H^{\otimes(n+1)}\) for \(n\geq0\), with coaction on the last \(H\) factor. Every \(J^n\) is cofree, hence injective. Let \(d^{-1}=\rho_N\). On \(J^n\), define the cofaces \(\delta_0\) by inserting \(\rho_N\), and \(\delta_i\), \(1\leq i\leq n+1\), by applying \(\Delta\) to its \(i\)-th \(H\) factor. Put

d^n=\delta_0+\sum_{i=1}^{n+1}(-1)^i\delta_i. \tag{6.1}

Each coface is a comodule map: insertion before the last factor commutes with its coaction, while splitting that last factor respects its coaction by coassociativity. The identities \(\delta_j\delta_i=\delta_i\delta_{j-1}\) for \(i<j\) follow from disjoint insertions, the comodule equation for the first adjacent insertions, and coalgebra coassociativity for adjacent \(H\)-insertions. In the expansion of \(d^{n+1}d^n\), these pair every term with the equal term of opposite sign. Thus \(d^{n+1}d^n=0\); the augmented equation is \(d^0\rho_N=0\) by coassociativity.

The underlying vector complex has contraction

s^n=(-1)^n\,1_N\otimes1_H^{\otimes n}\otimes\epsilon \colon J^n(N)\to J^{n-1}(N),\qquad n\geq0. \tag{6.2}

For \(n=0\) this is \(1_N\otimes\epsilon\) into \(N\). It satisfies \(s^0\rho_N=1_N\), and

s^{n+1}d^n+d^{n-1}s^n=1_{J^n(N)}. \tag{6.3}

Indeed deleting the last factor commutes with every earlier insertion. Their coefficients in the two composites differ by the signs \((-1)^{n+1}\) and \((-1)^n\), so cancel. The remaining last insertion splits the last factor and then applies its right counit; its two signs multiply to \(1\), leaving identity. This also proves (6.3) at \(n=0\), with \(d^{-1}=\rho_N\). The contraction is asserted on underlying vector spaces; it need not intertwine coactions. It proves exactness there, and §1 reflects exactness in comodules. We have therefore constructed the entire injective resolution \(0\to N\to J^0(N)\to J^1(N)\to\cdots\).

For a source comodule \(M\), (5.1) identifies its Hom into this resolution with

K^n(M,N)=\operatorname{Hom}_k(M,N\otimes H^{\otimes n}),\qquad n\geq0. \tag{6.4}

The induced differential on \(\ell\in K^n\) is

\begin{aligned} \partial\ell={}&(\rho_N\otimes1^{\otimes n})\ell\\ &+\sum_{i=1}^n(-1)^i (1_N\otimes1^{\otimes(i-1)}\otimes\Delta\otimes1^{\otimes(n-i)})\ell\\ &+(-1)^{n+1}(\ell\otimes1_H)\rho_M. \end{aligned} \tag{6.5}

To verify it, lift \(\ell\) to \((\ell\otimes1)\rho_M\), apply (6.1), then apply the final counit for (5.1). The first coface becomes the first term by the comodule counit, the middle splits remain the displayed middle terms, and the last split becomes the final term by the coalgebra right counit. This derives every term and sign. In degree zero, \(\partial\ell=\rho_N\ell-(\ell\otimes1)\rho_M\), so its kernel is exactly \(\operatorname{Hom}_H(M,N)\).

Applying Hom to an injective resolution computes the right derived functors of \(\operatorname{Hom}_H(M,-)\). The resolution comparison in Injective replacements and right derived functors applies to the abelian category and enough injectives proved above. Consequently

\operatorname{Ext}_H^n(M,N)=H^n(K^\bullet(M,N)),\qquad n\geq0, \tag{6.6}

with independence from the chosen injective resolution. A comodule map in \(N\) acts on every resolution term by its tensor with identities, and commutes with all cofaces. A comodule map in \(M\) acts on (6.4) by precomposition; its coaction equation makes (6.5) commute. These are the actual chain maps for both-variable naturality, and their identities and compositions are the corresponding ones on every term. Ext here is computed in the comodule category.

If \(M\) is finite-dimensional, every filtered target comparison is an isomorphism

\varinjlim_j\operatorname{Ext}_H^n(M,N_j) \xrightarrow{\sim}\operatorname{Ext}_H^n(M,\varinjlim_jN_j). \tag{6.7}

Each cofree resolution term commutes with filtered colimits, since its underlying tensor does and colimits are created by \(U\). Section 4's compactness identifies the colimit of the Hom terms with Hom into that same resolution of the colimit, and naturality identifies their differentials. Exact filtered colimits of vector spaces preserve kernels and images, hence the cohomology of this comparison. This proves (6.7) with its actual maps in every degree.

7. Four worked exercises

Exercise 7.1 (intermediate: all group-like comodules). Let \(H=k^{(S)}\) have basis \(e_s\), coproduct \(\Delta e_s=e_s\otimes e_s\), and counit \(\epsilon(e_s)=1\). Classify its comodules and every morphism. Prove that every object is injective and all positive Ext groups vanish.

Solution. Write \(\rho(m)=\sum_s m_s\otimes e_s\), a finite sum. Comparing coefficients of the last factor in coassociativity gives \(\rho(m_s)=m_s\otimes e_s\), and the counit gives \(m=\sum_s m_s\). Let \(M_s=\{m:\rho(m)=m\otimes e_s\}\). The sum of these spaces is direct: applying \(\rho\) to a finite relation and comparing the independent \(e_s\) forces every component zero. Thus \(M=\bigoplus_sM_s\). Conversely this grading gives the coaction on every finite-support vector and satisfies both structural equations. A linear map is a comodule map exactly when it preserves every \(M_s\), proving the classification on all morphisms.

On the dual algebra \(k^S\), the action is \(am=\sum_s a_s m_s\). Each vector's grading support is finite, so its annihilator contains the corresponding open coordinate kernel. For a monomorphism of graded objects, extend a given map to a graded target separately in every degree by a vector basis extension. This gives a comodule extension, so every target is injective. Its degree-zero injective resolution therefore gives zero Ext in every positive degree, with \(\operatorname{Hom}_H(M,N)=\prod_s\operatorname{Hom}_k(M_s,N_s)\).

Exercise 7.2 (advanced: square-zero Ext in every degree). Let \(H=kg\oplus V\), where \(\dim V=r\geq2\), with \(\Delta g=g\otimes g\), \(\Delta v=g\otimes v+v\otimes g\), \(\epsilon(g)=1\), \(\epsilon(V)=0\). For the one-dimensional comodule \(k_g\), compute every \(\operatorname{Ext}_H^n(k_g,k_g)\) using an explicit injective resolution.

Solution. Coassociativity of \(g\) is immediate, and for \(v\) both iterated coproducts are \(g\otimes g\otimes v+g\otimes v\otimes g+v\otimes g\otimes g\). Both counits return the input, so this is a coalgebra. Its dual algebra is \(k\oplus V^*\), with the second summand a square-zero ideal: multiplying two such functionals evaluates zero on both \(g\) and \(V\). Its finite topology is discrete, so all its algebraic modules are rational. A functional \(w\in V^*\) acts on the cofree comodule \(H\) by \(wg=0\), \(wv=w(v)g\).

Let \(q\colon H\to V\) be the projection along \(kg\). Take \(I^n=V^{\otimes n}\otimes H\), with last-factor coaction, augmentation \(1\mapsto g\), and

d^n(z\otimes h)=z\otimes q(h)\otimes g. \tag{7.1}

Each term is cofree and injective. The quotient \(q\) has the trivial coaction \(v\mapsto v\otimes g\): applying \(q\otimes1\) to \(\Delta(ag+v)\) gives \(v\otimes g\). Its insertion \(v\mapsto v\otimes g\) into the next cofree term is a comodule map since \(g\) is group-like. Thus (7.1) is a comodule differential. Its square is zero because \(q(g)=0\); its kernel is \(V^{\otimes n}\otimes kg\), and the preceding image is exactly that space since \(q\) is onto. At degree zero it equals the augmentation image. This proves exactness in every degree.

A map \(k_g\to I^n\) has image in exactly \(V^{\otimes n}\otimes kg\). Indeed, the displayed primitive coaction and scalar evaluation by all \(w\in V^*\) force its \(V\)-part to be zero; every vector with last factor \(g\) works. The Hom differential kills these vectors. Therefore \(\operatorname{Ext}_H^n(k_g,k_g)\simeq V^{\otimes n}\), with dimension \(r^n\) for every \(n\geq0\). No finite bound on positive Ext follows merely from enough injectives.

Exercise 7.3 (advanced: completed-series rationalization and products). Let \(H\) be the divided-power coalgebra with basis \(\epsilon_n\), \(n\geq0\), and coproduct \(\Delta\epsilon_n=\sum_{i+j=n}\epsilon_i\otimes\epsilon_j\). Its dual is \(A=k[[t]]\). Compute rationalization of any algebraic \(A\)-module, test it on the surjection \(A\to A/(t)\), and compute the intrinsic product of copies of the cofree comodule \(H\).

Solution. The ideals \((t^d)\) form the open ideal basis, so (2.1) becomes

R(X)=\bigcup_{d\geq1}\ker(t^d:X\to X). \tag{7.2}

In \(A\), multiplication by \(t^d\) is injective: shifting a nonzero series cannot remove its least nonzero coefficient. Hence \(R(A)=0\), while \(A/(t)=k\) is rational. The surjection becomes \(0\to k\) after rationalization and is no longer onto, proving the failure of exactness claimed in §3.

On \(H\), \(t\epsilon_n=\epsilon_{n-1}\), with \(\epsilon_{-1}=0\). Identify it with the principal-parts module \(k((t))/k[[t]]\) by \(\epsilon_n\mapsto[t^{-n-1}]\); a series acts using its finite truncation on any vector killed by \(t^d\). This is exactly the locally nilpotent specialization of Locally nilpotent operators and coinduced duals, whose §3 proves the uniform-power product criterion and whose Exercise 3 gives the cofree maps. Thus, for an indexing set \(B\), the intrinsic product is

R(H^B)=\bigcup_{d\geq1}(\operatorname{span}(\epsilon_0,\ldots,\epsilon_{d-1}))^B \simeq k^B\otimes H. \tag{7.3}

The displayed inverse maps read the finitely many global coefficient tuples of a uniformly killed tuple, or send a finite sum of coefficient tuples tensored with \(\epsilon_n\) to its coordinate family. They commute with the shift and every coordinate projection. Cofree adjunction identifies those projections with the universal product maps of §5. For \(B=\mathbb N\), the tuple \((\epsilon_n)_n\) is in the ambient module product but has no common killing power, so is excluded. This computes the intrinsic product and all its factorization maps while reusing the previously proved local nilpotence argument.

Exercise 7.4 (advanced: a finite-dimensional algebraic module need not be rational). For \(A=\mathbb F_2^{\mathbb N}\), construct a one-dimensional unital algebraic \(A\)-module whose action is not continuous. Prove that it cannot come from a comodule over the coordinate coalgebra.

Solution. Start with the filter of cofinite subsets of \(\mathbb N\). A union of a chain of proper filters extending it is a proper filter: each finite intersection requirement lies in one chain member, and none contains the empty set. Zorn therefore gives a maximal proper such filter \(\mathcal U\). If a set \(S\) is absent, the filter generated by adding \(S\) must be improper; some \(F\in\mathcal U\) then has \(F\cap S=\varnothing\), giving \(S^c\in\mathcal U\). Both a set and its complement cannot belong to a proper filter. No finite set belongs because its cofinite complement already does. Thus \(\mathcal U\) is nonprincipal.

For \(a\in A\), exactly one of its two value fibres belongs to \(\mathcal U\). Define \(\chi(a)\in\mathbb F_2\) to be that value. If \(\chi(a)=c\), \(\chi(b)=d\), the intersection of their chosen fibres belongs to \(\mathcal U\); on it their sum has value \(c+d\) and product value \(cd\). Upward closure makes those the chosen values, proving additivity and multiplicativity. Constant functions give \(\chi(1)=1\) and scalar linearity. Thus \(am=\chi(a)m\) defines a one-dimensional unital algebraic module.

Its annihilator of \(1\) is \(\ker\chi\). Every finitely supported function has cofinite zero fibre, so belongs to that kernel. These functions are dense in the product topology, while \(1\) is absent. The kernel is therefore dense and proper and cannot be open: an open additive subspace is closed, since its complement is a union of open cosets. Criterion (2.1) fails. Hence this finite-dimensional algebraic module is not rational and Theorem 2.1 rules out an inducing coordinate-coalgebra coaction. Algebraic finite dimension alone does not supply the required topology.