Draft lesson.

Projective probes and recollements of many-object modules

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

A small family of compact projective modules separates a module category into two module categories, linked by six adjoint functors. The probe category gives one side; a quotient by its trace ideal gives the other. We construct all six functors and prove that every recollement with module categories on both sides has this form. The two outer embeddings need not be exact.

Use Modules and tensor functors over small preadditive categories, §§2–5, for actual presentations, balanced tensor, the tensor–Hom adjunction and reconstruction of every colimit-preserving functor. Cauchy completion and Morita equivalence of many-object modules, §2, identifies compact projectives with finite representable retracts. Ideals, quotients and torsion triples for many-object modules proves the quotient/annihilator adjoint triple and the full idempotent-ideal classification. A freely accessible further reference is Parra–Saorín–Virili, Torsion pairs in categories of modules over a preadditive category, §§4.3–4.4.

1. The small probe category and exact restriction

Fix a universe, a small preadditive category \(\mathcal C\), and a small indexed family \((P_e)_{e\in E}\) of compact projective right \(\mathcal C\)-modules. Compact means that \(\operatorname{Hom}(P_e,-)\) preserves small filtered colimits. Let \(\mathsf M=\mathsf{Mod}\text{-}\mathcal C\). Define a small preadditive category \(\mathcal B\) by

\operatorname{Ob}\mathcal B=E,\qquad \mathcal B(e,f)=\operatorname{Hom}_{\mathsf M}(P_e,P_f). \tag{1.1}

The Hom groups are small, composition is bilinear, and the identity at \(e\) is \(1_{P_e}\). Repeated or zero probes are allowed. Define

H(M)(e)=\operatorname{Hom}_{\mathsf M}(P_e,M). \tag{1.2}

For \(u\colon P_e\to P_f\), its right action is precomposition, \(s\mapsto su\). Postcomposition gives \(H\) on module maps. These actions commute, are additive, and obey both functor laws.

The functor \(H\) preserves all small limits, since compatible maps from a fixed probe into a limit are exactly its compatible component maps. It is exact: Hom preserves kernels, and projectivity supplies surjectivity on epimorphisms. It also preserves all small colimits. Indeed each \(P_e\) is a retract of a finite sum of representables. For that sum, Hom is a finite product of evaluations, hence a finite direct sum of evaluations and preserves coproducts. The retract argument gives coproduct preservation for \(P_e\): apply the finite-sum coproduct comparison and its natural splitting on both sides. Exactness then gives preservation of cokernels. Every colimit is a cokernel of the difference of two maps between coproducts of diagram objects, proving colimit preservation with the canonical comparison maps.

2. The trace ideal and its multiplication radical

Define

I(c,d)=\sum_{e,\ a:P_e\to h_d}\operatorname{im}a_c, \qquad h_d=\mathcal C(-,d). \tag{2.1}

This is a two-sided ideal. Its precomposition stability comes from the module maps \(a\); postcomposing an arrow \(d\to d'\) replaces \(a\) by its composite with \(h_d\to h_{d'}\). All sums are small sums of subgroups.

For each probe choose a finite representable retract. Its splitting gives a finite dual-basis identity

1_{P_e}=\sum_{j=1}^{n_e}\tau_{x_j}\sigma_j, \quad x_j\in P_e(c_j),\quad \sigma_j:P_e\to h_{c_j}, \quad (\tau_x)_c(v)=P_e(v)x. \tag{2.2}

To check it, write \(P_e\xrightarrow{s}\bigoplus_jh_{c_j}\xrightarrow{r}P_e\) with \(rs=1\), take \(\sigma_j\) to be the projection of \(s\), and let \(x_j\) be the value at \(1_{c_j}\) of the restriction of \(r\) to that summand. Additive Yoneda identifies that restriction with \(\tau_{x_j}\), so summing gives \(rs\). The zero probe permits an empty sum.

For \(a:P_e\to h_d\) and \(p\in P_e(c)\), equation (2.2) gives

a_c(p)=\sum_j a_{c_j}(x_j)\,\sigma_{j,c}(p). \tag{2.3}

Here \(\sigma_{j,c}(p)\in I(c,c_j)\) and \(a_{c_j}(x_j)\in I(c_j,d)\); the product is composition in that order. Every generator of (2.1) therefore lies in \(I^2\). The reverse inclusion holds for every ideal, so \(I^2=I\).

The trace of the probes in any module agrees with ideal multiplication:

\operatorname{tr}_{P}(M):= \sum_{e,\ g:P_e\to M}\operatorname{im}g=MI. \tag{2.4}

For one inclusion, (2.2) writes \(g_c(p)=\sum_j M(\sigma_{j,c}(p))g_{c_j}(x_j)\), an element of \((MI)(c)\). Conversely write an ideal arrow as a finite sum of \(a_c(p)\)'s. For \(m\in M(d)\), the value \(M(a_c(p))m\) is \((\tau_m a)_c(p)\), which lies in the trace. Additivity handles the full finite sums in ideal multiplication. Both constructions are natural submodules; thus (2.4) also identifies their restricted maps.

It follows that

\ker H=\{M:MI=0\}=\mathsf T_I, \qquad \operatorname{Gen}(P)=\{M:MI=M\}=\mathsf C_I. \tag{2.5}

The first equality holds because the trace is zero exactly when every map from every probe is zero. For the second, the trace is the image of the small evaluation coproduct of all probe maps. If it is all of \(M\), that coproduct is epic; conversely a generating epimorphism makes its probe images sum to all of \(M\).

3. The balanced tensor left embedding

The family \(P\) is a bimodule \(P\colon\mathcal C^{\mathrm{op}}\times\mathcal B\to\mathsf{Ab}\): at \((c,e)\) use \(P_e(c)\), and act on \(u:e\to f\) by the module map \(u_c\). The actions commute by naturality of \(u\). Define

T(N)(c)=N\otimes_{\mathcal B}P(-)(c), \qquad N\in\mathsf{Mod}\text{-}\mathcal B. \tag{3.1}

Explicitly, start with \(\bigoplus_e N(e)\otimes_{\mathbb Z}P_e(c)\) and quotient by

[N(u)n\otimes p]=[n\otimes u_c(p)], \quad u:e\to f,\ n\in N(f),\ p\in P_e(c). \tag{3.2}

An arrow \(v:c'\to c\) acts by \([n\otimes p]\mapsto[n\otimes P_e(v)p]\). Naturality of \(u\) makes it respect (3.2); additive functor laws follow on pure tensors, which generate the group. A module map in \(N\) acts on its first factor. The tensor–Hom theorem in the prerequisite gives \(T\dashv H\). Its inverse Hom maps have the concrete formulas

F\colon T(N)\to M\ \longmapsto \bigl(n\mapsto(p\mapsto F_c[n\otimes p])\bigr), \qquad \theta\colon N\to HM\ \longmapsto \bigl([n\otimes p]\mapsto\theta_e(n)_c(p)\bigr). \tag{3.3}

The balance equation is exactly the \(\mathcal B\)-naturality of \(\theta\); the \(\mathcal C\)-naturality is exactly that its values are module maps. These formulas recover each other on all pure tensors and all values. They commute with both precomposition and postcomposition, so include all morphisms and naturality. The unit \(\kappa_N:N\to HTN\) sends \(n\in N(e)\) to \(p\mapsto[n\otimes p]\); the counit \(\epsilon_M:THM\to M\) evaluates \([g\otimes p]\mapsto g_c(p)\). Their triangles are the two evaluation identities in (3.3).

Theorem 3.1. The unit \(\kappa\) is invertible, so \(T\) is fully faithful.

Proof. Co-Yoneda identifies \(T(h_e^{\mathcal B})\) with \(P_e\). The identification \(H(P_e)(f)=\operatorname{Hom}(P_f,P_e)=\mathcal B(f,e)\) is \(h_e^{\mathcal B}\). Under these maps \(\kappa_{h_e}\) sends an arrow to itself and is invertible. Both \(HT\) and the identity preserve coproducts and cokernels. Consequently the unit is invertible on coproducts of representables, and hence on every module through its actual two-term representable presentation. More explicitly, apply the inverse comparison on the second coproduct term and then its cokernel map. Naturality makes this composite kill the first image, so it factors through the cokernel. Its two composites with the unit are identities after the corresponding epic cokernel maps; cancellation proves both inverse identities. This gives a natural inverse on every module. Adjunction (3.3) then identifies \(\operatorname{Hom}(TN,TN')\) with \(\operatorname{Hom}(N,HTN')\cong\operatorname{Hom}(N,N')\), proving full faithfulness. \(\square\)

4. The reconstructed Hom right embedding

Since \(H\) preserves all colimits, the reconstruction theorem gives the bimodule

Q(e,c)=H(h_c)(e)=\operatorname{Hom}_{\mathsf M}(P_e,h_c), \qquad Q_c=Q(-,c), \tag{4.1}

with \(\mathcal B\)-action by precomposition and \(\mathcal C\)-action by postcomposition. Its natural comparison with \(H\) is

\alpha_M:M\otimes_{\mathcal C}Q\xrightarrow{\sim}HM, \qquad [m\otimes a]\longmapsto\tau_m a. \tag{4.2}

All balancing and inverse comparison arguments are the full reconstruction proof in the prerequisite; here \(H\) satisfies its colimit hypothesis by §1. Define

K(N)(c)=\operatorname{Hom}_{\mathsf{Mod}\text{-}\mathcal B}(Q_c,N). \tag{4.3}

For \(v:c'\to c\), precompose with \(Q(v):Q_{c'}\to Q_c\). These are the right \(\mathcal C\)-actions; postcomposition supplies \(K\) on maps in \(N\). The two actions commute and are additive. Tensor–Hom, transported through (4.2), proves \(H\dashv K\). A map \(\theta:HM\to N\) transposes to

\widetilde\theta_c(m)_e(a)=\theta_e(\tau_m a). \tag{4.4}

The inverse first evaluates \([m\otimes a]\mapsto\widetilde\theta_c(m)_e(a)\) and then precomposes with \(\alpha_M^{-1}\). The prerequisite proves its well-definedness, naturality and both inverse identities. The unit \(\eta_M:M\to KHM\) is (4.4) with \(\theta=1\). The counit \(\delta_N:HKN\to N\), after (4.2), evaluates \([s\otimes a]\mapsto s_e(a)\). These are actual module maps, and the tensor–Hom evaluation identities prove both triangles.

Theorem 4.1. The counit \(\delta\) is invertible, so \(K\) is fully faithful.

Proof. At an object \(e\), the adjunction and Yoneda give inverse natural bijections

HKN(e)=\operatorname{Hom}_{\mathsf M}(P_e,KN) \simeq\operatorname{Hom}_{\mathcal B}(HP_e,N) \simeq N(e). \tag{4.5}

The first takes \(g\) to \(\delta_N H(g)\); the second evaluates at \(1_{P_e}\), using \(HP_e=h_e\). Since \(H(g)_e(1_{P_e})=g\), their composite is \(g\mapsto\delta_{N,e}(g)\). Thus every component of the actual counit is invertible. Its inverse is natural: invert the counit's naturality equations, including those for every arrow of \(\mathcal B\) and every map in \(N\). Finally \(\operatorname{Hom}(KN,KN')\cong\operatorname{Hom}(HKN,N')\) \(\cong\operatorname{Hom}(N,N')\). \(\square\)

5. All six functors and the canonical comparisons

An abelian recollement of \(\mathsf M\) means two adjoint triples \(i^*\dashv i_*\dashv i^!\) and \(j_!\dashv j^*\dashv j_*\), with \(i_*,j_!,j_*\) fully faithful and \(\operatorname{im}i_*=\ker j^*\). Images here are essential images. Take the functors

Functor	Source and target	Formula
\(i_*\)	\(\mathsf{Mod}\text{-}(\mathcal C/I)\to\mathsf M\)	restriction \(R\)
\(i^*\)	\(\mathsf M\to\mathsf{Mod}\text{-}(\mathcal C/I)\)	\(M/MI\)
\(i^!\)	\(\mathsf M\to\mathsf{Mod}\text{-}(\mathcal C/I)\)	\(\operatorname{Ann}_I M\)
\(j_!\)	\(\mathsf{Mod}\text{-}\mathcal B\to\mathsf M\)	\(T\) in (3.1)
\(j^*\)	\(\mathsf M\to\mathsf{Mod}\text{-}\mathcal B\)	\(H\) in (1.2)
\(j_*\)	\(\mathsf{Mod}\text{-}\mathcal B\to\mathsf M\)	\(K\) in (4.3)

All adjunctions, units and counits of the first triple are the explicit quotient and annihilator maps in the ideal prerequisite. Sections 3–4 prove the second triple and full faithfulness of its embeddings. Equation (2.5) identifies its kernel with the essential image of restriction. Hence these are all six functors of a recollement. The functors \(i_*\) and \(j^*\) are exact; the outer adjoints have only their respective right or left exactness in general.

The evaluation counit has \(\operatorname{im}\epsilon_M=MI\), since its pure-tensor values are exactly the values of all probe maps. The other unit has \(\ker\eta_M=\operatorname{Ann}_I M\). Indeed \(\eta_{M,c}(m)=0\) exactly when \(\tau_m a=0\) for every \(a:P_e\to h_c\). At every \(b\) this says \(M(a_b(p))m=0\) for all \(p\in P_e(b)\). These arrows generate \(I(b,c)\), so this is precisely the annihilator condition. Thus there are natural exact sequences

THM\xrightarrow{\epsilon_M}M\longrightarrow RLM\longrightarrow0, \qquad 0\longrightarrow RAM\longrightarrow M\xrightarrow{\eta_M}KHM. \tag{5.1}

They do not assert injectivity of \(\epsilon_M\) or surjectivity of \(\eta_M\). The associated TTF triple is exactly \((\mathsf C_I,\mathsf T_I,\mathsf F_I)\) of the prerequisite, with the image and kernel just computed.

6. The converse for every module-category recollement

Theorem 6.1. Every abelian recollement with middle category \(\mathsf{Mod}\text{-}\mathcal C\) and both side categories module categories over small preadditive categories is equivalent, by equivalences of the side categories fixing the middle, to the construction above. Its kernel ideal is the trace of compact projective probes.

Proof. Write its right category as \(\mathsf{Mod}\text{-}\mathcal D\), and set \(P_d=j_!(h_d^{\mathcal D})\). Because \(j^*\) has both adjoints, it preserves all limits and colimits and is exact. Adjunction gives

\operatorname{Hom}_{\mathsf M}(P_d,M) \simeq\operatorname{Hom}_{\mathcal D}(h_d,j^*M) \simeq(j^*M)(d). \tag{6.1}

The right side preserves epimorphisms and filtered colimits; the first property proves projectivity by the lifting definition, and the second is compactness. Thus the probes meet all hypotheses. Full faithfulness of \(j_!\) identifies \(\operatorname{Hom}(P_d,P_{d'})\) with \(\operatorname{Hom}(h_d,h_{d'})=\mathcal D(d,d')\), including identities, sums and composition. Its probe category is therefore isomorphic to \(\mathcal D\), with object labels retained. Equation (6.1) is natural in all arrows and all \(M\), so identifies the original \(j^*\) with \(H\), not just their object values.

Adjoints to the same functor are uniquely naturally isomorphic with their adjunction maps. To verify the uniqueness used here, represent \(\operatorname{Hom}(X,-)\) using the two left-adjoint bijections; take the images of the two identity maps to obtain comparison maps in both directions. Naturality makes their composites correspond to identities, hence gives inverse maps commuting with units and counits. The dual argument treats right adjoints. Applying this to \(j^*\cong H\) identifies the original \(j_!,j_*\) with \(T,K\), including all maps.

The original \(i_*\) is fully faithful with essential image \(\ker j^*=\ker H=\mathsf T_I\). Restriction \(R\) gives an equivalence of \(\mathsf{Mod}\text{-}(\mathcal C/I)\) with that same full subcategory. Factoring \(i_*\) through this equivalence gives an equivalence from its original left category to \(\mathsf{Mod}\text{-}(\mathcal C/I)\): it is fully faithful by the two full embeddings and essentially surjective by equality of their essential images. The just-proved uniqueness of adjoints identifies the original \(i^*,i^!\) with \(L,A\) after this equivalence. All six functors and their comparisons have therefore been recovered.

Finally the kernel determines its ideal uniquely in the TTF classification: alternatively an arrow killed by all modules in \(\mathsf T_I\) must belong to \(I\), by applying it to \([1_d]\) in \(h_d/I(-,d)\). Thus the trace ideal just constructed is the original embedded kernel invariant. \(\square\)

Equivalently, an idempotent ideal is the kernel ideal of a module-category recollement exactly when it is the trace of a small family of compact projectives: §5 supplies the forward construction from such a trace, and Theorem 6.1 proves the necessity for any recollement. This does not claim that distinct indexing families give distinct recollements. Repeated probes, finite sums or retracts can give equivalent right categories; the trace ideal remains the embedded invariant.

If \(E\) is empty, \(\mathcal B\) is empty, \(H,T,K\) have zero-category side, \(I=0\), and the left side is all of \(\mathsf M\). Zero probes have zero identities and zero module values, so can be retained or removed. If all representables are taken, \(I=\mathcal C(-,-)\) and the left side is the zero category. Empty \(\mathcal C\) causes all categories here to be zero. All assertions concern the compactness of each probe; their infinite coproduct need not be compact.

7. Four graded exercises with complete solutions

Exercise 1 (grade 1). In the free preadditive path category \(0\to1\) over \(\mathbb Z\), a right module is \(M_1\xrightarrow fM_0\). Take the probe \(P=h_0=(0\to\mathbb Z)\). Compute all six functors, the two \(j\)-adjunction inverse maps, and (5.1).

Solution. Here \(\mathcal B\) is the one-object category \(\mathbb Z\), and \(H(M)=M_0\). The ideal is generated by \(1_0\), so the left category is \(\mathsf{Ab}\) at vertex 1. The first triple is \(L(M)=M_1\), \(R(V)=(V\to0)\), \(A(M)=\ker f\), as proved in the ideal lesson. The other embeddings are \(T(N)=(0\to N)\) and \(K(N)=(N\xrightarrow1N)\).

For \(T\dashv H\), a module map \((0\to N)\to(M_1\to M_0)\) is exactly its arbitrary homomorphism \(N\to M_0\). Restricting and inserting that homomorphism are inverse and natural. For \(H\dashv K\), a map \(M\to(N\xrightarrow1N)\) has components \((a_1,a_0)\) with \(a_1=a_0f\). Hence it corresponds to \(a_0:M_0\to N\); the inverse is \((a_0f,a_0)\). These equations check every module map and both inverse identities.

The counit \(THM=(0\to M_0)\to M\) is inclusion on vertex 0, with image \(MI\); its cokernel is \((M_1\to0)\). The unit \(M\to KHM=(M_0\xrightarrow1M_0)\) has components \((f,1_{M_0})\), whose kernel is \((\ker f\to0)=RAM\). Its vertex 1 component need not be surjective, so the second row of (5.1) cannot be terminated by 0 in general. Both side comparisons \(HTN\cong N\) and \(HKN\cong N\) are the identity on the group \(N\).

Exercise 2 (grade 2). In the same path category take \(P=h_1=(\mathbb Z\xrightarrow1\mathbb Z)\). Compute all six functors and the comparison rows, and exhibit a counit \(THM\to M\) that is not monic.

Solution. Evaluation gives \(H(M)=M_1\). The ideal is generated by \(1_1\); its left quotient is vertex 0. Thus \(L(M)=\operatorname{coker}f\), \(R(V)=(0\to V)\), and \(A(M)=M_0\). The other embeddings are \(T(N)=(N\xrightarrow1N)\) and \(K(N)=(N\to0)\). A map \(TN\to M\) has arbitrary component \(a_1:N\to M_1\) and forced component \(a_0=fa_1\); a map \(M\to KN\) has arbitrary \(a_1:M_1\to N\) and zero component at 0. Taking or inserting the indicated arbitrary component gives inverse natural Hom bijections for both adjunctions.

The counit \(THM=(M_1\xrightarrow1M_1)\to M\) is \((1_{M_1},f)\); its image is \((M_1\to\operatorname{im}f)=MI\), with cokernel \((0\to\operatorname{coker}f)\). For \(M=(\mathbb Z\to0)\), its component at 0 is \(\mathbb Z\to0\), so it is not monic. The other unit is \(M\to(M_1\to0)\), with components \((1_{M_1},0)\), whose kernel is \((0\to M_0)=RAM\). Its two vertex maps are surjective; this is a property of this example, rather than of every recollement.

Exercise 3 (grade 3). Let \(d\ge2\) and \(R=\begin{pmatrix}\mathbb Z&\mathbb Z/d\\0&\mathbb Z\end{pmatrix}\), with the usual triangular multiplication and natural actions on \(\mathbb Z/d\). For either diagonal idempotent \(e\), take \(P=eR\). Derive the formulas \(H(M)=Me\), \(T(N)=N\otimes_{eRe}eR\), \(K(N)=\operatorname{Hom}_{eRe}(Re,N)\), including their actions. Show separately that \(T\) need not preserve monomorphisms and \(K\) need not preserve epimorphisms, although \(P\) is compact projective.

Solution. The summand \(eR\subseteq R_R\) has projection left multiplication by \(e\), making it a finite representable retract. A right-linear map \(eR\to M\) is uniquely determined by \(m=\varphi(e)\in Me\), and its inverse formula is \(er\mapsto mr\). An endomorphism of \(eR\) is left multiplication by its value \(a\in eRe\); composition is multiplication \(ab\), in the same order. Thus the right \(eRe\)-action on Hom is \(m\cdot a=ma\), giving \(H(M)=Me\).

The tensor formula is (3.1), with left \(eRe\)-action on \(eR\) by multiplication and right \(R\)-action on tensors by \((n\otimes p)r=n\otimes pr\). Further, \(Q_R=H(R_R)=Re\), a right \(eRe\)-module with commuting left \(R\)-action. Formula (4.3) gives the indicated Hom group, with right action \((\varphi r)(x)=\varphi(rx)\). One checks \(((\varphi r)s)(x)=\varphi(rsx)\) and \((\varphi1)(x)=\varphi(x)\); additivity and compatibility with the \(eRe\)-linearity follow from multiplication. These formulas recover both tensor–Hom adjunctions of the theorem.

For \(e=e_0=\operatorname{diag}(1,0)\), \(eRe=\mathbb Z\) and \(eR\) has diagonal components \(\mathbb Z,\mathbb Z/d\). Consequently \(T(N)e_0=N\) and \(T(N)e_1=N/dN\). The injective map \(\mathbb Z\xrightarrow{d}\mathbb Z\) induces at the second component the zero endomorphism of the nonzero group \(\mathbb Z/d\). Hence \(T\) does not preserve this monomorphism.

For \(e=e_1=\operatorname{diag}(0,1)\), \(Re\), as a right \(eRe=\mathbb Z\)-module, has components \(\mathbb Z/d,\mathbb Z\). The components of \(K(N)\) are \(K(N)e_0\cong\operatorname{Hom}_{\mathbb Z}(\mathbb Z/d,N)\) and \(K(N)e_1\cong N\). These follow by restricting a homomorphism \(Re\to N\) to \(e_iRe\), since right multiplication by \(e_i\) on \(K\) precomposes with left multiplication by \(e_i\) on \(Re\). For the epimorphism \(\mathbb Z\twoheadrightarrow\mathbb Z/d\), the first component becomes \(0\to\mathbb Z/d\), because an element of \(\mathbb Z\) annihilated by \(d\) is zero while every element of \(\mathbb Z/d\) is annihilated. Thus \(K\) does not preserve this epimorphism. Compact projectivity is a condition on the right \(R\)-probe, not flatness or projectivity of both auxiliary bimodule actions.

Exercise 4 (grade 4). Replace the compact probe \(\mathbb Z\) in \(\mathsf{Ab}\) by the projective group \(G=\bigoplus_{n\ge0}\mathbb Z\). Explain precisely why the preceding six-functor argument fails. Contrast it with a small infinite family of individually compact representables.

Solution. The group \(G\) is projective: lift a map through an epimorphism by lifting the image of each basis vector and assembling those chosen lifts. But \(\operatorname{Hom}(G,-)\) does not preserve coproducts. For the family \(A_n=\mathbb Z\), the identity map \(G\to\bigoplus_n A_n=G\) sends the \(n\)-th basis vector to the \(n\)-th target summand. It is not in the image of \(\bigoplus_n\operatorname{Hom}(G,A_n)\), because a member of that direct sum has only finitely many target-coordinate maps nonzero, and therefore all its values are supported in one fixed finite set of summands. The identity has no such finite bound.

The comparison is a homomorphism of modules over \(\operatorname{End}(G)\), acting by precomposition. Its failure to be surjective even on underlying groups proves failure in that module category. If Hom from \(G\) also had a right adjoint, it would be a left adjoint and preserve this coproduct, a contradiction. Thus the missing \(H\dashv K\) cannot be supplied by the proof: projectivity alone gives exactness but not the necessary colimit preservation or a six-functor recollement of this form. The directed system of finite partial sums of \(G\) also witnesses noncompactness, since its identity fails to factor through any finite partial sum.

By contrast, for the discrete preadditive category with objects \(\mathbb N\), endomorphism rings \(\mathbb Z\) and zero Hom groups between distinct objects, take all representables as an indexed family. Each is compact projective. An additive module on this category is exactly a family of groups: additivity forces an integer endomorphism to act by that integer times the identity, and zero arrows act by zero. Module maps are families of group homomorphisms. Coordinate evaluation and this construction are inverse on objects and maps. Thus the probe category is the original discrete category and \(H\) gives an equivalence \(\mathsf M\cong\mathsf{Ab}^{\mathbb N}\). The theorem applies, with full trace ideal and zero left quotient side. The coproduct of all probes is not compact: the identity cannot factor through a finite partial sum, by inspecting a coordinate outside that sum. The theorem requires compactness of each member of the small family, not compactness of their total coproduct.