Draft lesson.

Finite presentations and Ind reconstruction on finite-cover sites

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

Every module sheaf on a finite-cover site can be assembled from finite presentations using compact generators attached to the site objects. Those generators need not be projective. We construct their actual evaluation maps, a directed diagram of finite relations, and the resulting Ind reconstruction of both objects and morphisms. A site need not be subcanonical or have a terminal object for this construction.

Use Dualizable sheaves, finite covers and global projectivity, §§1 and 3, for module sheafification, the abelian category and pointwise filtered colimits. Finitely presented generators and coherent Ind reconstruction, §§1–5, proves the abstract reconstruction and the separate coherence condition. Formal colimits and compact presentations, §§1–2, gives the full realization Hom formula. Generators and small quotient families, §3, proves the canonical evaluation-presentation interface. All finite-presentation claims below concern categorical Hom, not the dimension of a section space.

1. The site generators and their inverse evaluation maps

Fix a universe and choice. Let \(\mathcal C\) be a small site whose topology is specified by finite covering families, stable under pullback and composition; the required cover intersections exist. Empty covers are allowed where specified. Let \(R\) be a sheaf of commutative unital rings and \(\mathsf M=\mathsf{Mod}(R)\). No terminal object is assumed.

For \(U\in\mathcal C\), form the module presheaf

F_U(V)=\bigoplus_{\alpha:V\to U}R(V)[\alpha], \qquad G_U=a_R F_U, \tag{1.1}

where \(a_R\) is module sheafification. For \(\beta:W\to V\), restrict \(r[\alpha]\) to \(R(\beta)(r)[\alpha\beta]\). The presheaf identity, composition and scalar laws follow directly. The arrow set indexing every sum is small. A map \(U\to U'\) acts on these generators by postcomposition of the labelled arrows. Sheafification is necessary in general; the representable presheaf itself is not assumed a sheaf.

There are inverse natural group isomorphisms

\operatorname{Hom}_{\mathsf M}(G_U,M)\simeq M(U). \tag{1.2}

Indeed sheafification first identifies the left side with module-presheaf maps \(F_U\to M\). Such a map is sent to its value on \(1[1_U]\). The inverse for \(m\in M(U)\) sends \(r[\alpha]\) to \(rM(\alpha)m\). This is linear and commutes with every restriction; the displayed formula follows in the other direction from naturality and linearity. Evaluating it at \(1[1_U]\) returns \(m\). Thus both identities hold, naturally in \(U,M\).

The pointwise filtered-colimit proof in the first prerequisite uses only finite matching groups and their common-stage representative and equality tests. Its colimit assertion needs no terminal object. It applies here unchanged to \(R(U)\)-module values and their scalar restrictions. Equation (1.2) therefore makes every \(G_U\) finitely presented, equivalently compact with respect to filtered colimits.

The family is separating: if two maps \(M\rightrightarrows N\) differ, they differ on some section \(m\in M(U)\), and the corresponding map \(G_U\to M\) detects that difference. Since \(\mathcal C\) is small, this is a small family. Its coproduct is a generator: choose the detecting map on the indicated summand and zero maps on the others.

Filtered colimits are exact. Kernels of module-sheaf maps are pointwise. Pointwise filtered colimits commute with those kernels by the group representative/equality test, and with finite sums and zero. They therefore preserve finite limits. The colimit functor also preserves cokernels, since it is left adjoint to the constant-diagram functor; the cocone universal property proves this adjunction. It is consequently exact. With cocompleteness and the generator just constructed, \(\mathsf M\) is a Grothendieck abelian category with finitely presented generators. None of these arguments assumed those generators projective.

2. A full presentation before taking finite parts

For a module \(M\), take the actual evaluation map

P_0(M)=\bigoplus_{(U,m),\ m\in M(U)}G_U \xrightarrow{\epsilon_M}M. \tag{2.1}

All indices are small. It is epic, either by separation and the generator theorem or directly: the summand labelled by \((U,m)\) contains the sheafification image of \(1[1_U]\), and evaluation sends that section to \(m\). Thus every section is in the image of the evaluation map at its indicated object.

Let \(K=\ker\epsilon_M\). Apply the same evaluation construction to \(K\), then compose with its inclusion into \(P_0(M)\). This gives

P_1(M)\xrightarrow{\delta}P_0(M)\xrightarrow{\epsilon_M}M\longrightarrow0, \tag{2.2}

with both \(P_i\) coproducts of generators and \(\operatorname{im}\delta=K\). Thus \(M\) is the cokernel with the specified map. The summand maps into \(P_0(M)\) have been constructed from actual maps into \(K\); there is no lifting through an arbitrary epimorphism by a generator.

Write \(P_0=\bigoplus_{s\in S_0}G_s\) and \(P_1=\bigoplus_{t\in S_1}G_t\), retaining each summand's site-object label. The coproduct \(P_0\) is the directed colimit of its finite partial sums: compatible maps on those sums are exactly a family of maps on all summands. Compactness of \(G_t\) factors each \(\delta_t:G_t\to P_0\) through one finite partial sum. Choose a finite support \(A_t\subset S_0\) and its factor \(\delta_{t,A_t}\). The inclusion of a partial sum is a split monomorphism, using the projection that is zero on its complement, so its factor is unique once that support is fixed.

3. The actual directed system of finite relations

Let \(D_M\) be the poset of pairs of finite subsets \((S,T)\), with \(S\subset S_0\), \(T\subset S_1\), and \(A_t\subset S\) for every \(t\in T\). Order it by inclusion of both subsets. It is small, nonempty and directed: \((\varnothing,\varnothing)\) belongs, and the union of two admissible pairs is admissible.

Define

Q_{S,T}=\operatorname{coker}\left( \bigoplus_{t\in T}G_t\xrightarrow{\delta_{S,T}} \bigoplus_{s\in S}G_s\right), \tag{3.1}

using the chosen support maps followed by the inclusions. Enlarging \((S,T)\) gives commuting maps on the two presentation terms, hence a unique map on their cokernels. The support equations give the commuting squares on every generator; cokernel uniqueness gives identity and composition for the resulting diagram. Evaluation into \(M\) kills each relation and gives a compatible cocone \(q_{S,T}:Q_{S,T}\to M\).

Each \(Q_{S,T}\) is finitely presented. For any target \(N\), maps from it form the kernel of

\prod_{s\in S}\operatorname{Hom}(G_s,N) \longrightarrow\prod_{t\in T}\operatorname{Hom}(G_t,N). \tag{3.2}

The map is precomposition with \(\delta_{S,T}\). Both products are finite, and filtered colimits of groups commute with their products and kernel by the finite witness tests. Every generator Hom preserves those colimits, so the canonical Hom comparison for (3.1) is invertible. This includes empty finite sets.

Theorem 3.1. The cocone gives an isomorphism

\operatorname{colim}_{(S,T)\in D_M}Q_{S,T}\xrightarrow{\sim}M. \tag{3.3}

Proof. We give the inverse universal-property comparison on all maps. A compatible family \(Q_{S,T}\to N\) first determines maps from every \(G_s\) into \(N\), by using \((\{s\},\varnothing)\). Compatibility on common enlargements makes its composite on every finite presentation term the sum of these fixed maps. They define a map \(a:P_0\to N\).

For a relation \(t\), use the stage \((A_t,\{t\})\). Its cokernel kills \(\delta_{t,A_t}\), so the just-constructed \(a\) satisfies \(a\delta_t=0\). This holds for every summand, hence \(a\delta=0\). The cokernel (2.2) therefore gives a unique \(M\to N\). Conversely such a map composes with every \(q_{S,T}\) to give the compatible family. These operations are inverse on all generator components and hence on the finite cokernels and on \(P_0\). They are natural in \(N\). Thus the cocone of \(M\) itself is a colimit cocone, proving the indicated canonical isomorphism. \(\square\)

The maps \(q_{S,T}\) need not be monomorphisms. This is a presentation by finite objects and relations, not an assertion that every sheaf is a union of finite subobjects. Different support choices yield the same reconstructed object through (3.3).

4. All finite pieces and their size

Theorem 4.1. A module sheaf is finitely presented if and only if it is a retract of a cokernel of a map between two finite sums of the \(G_U\). The finitely presented subcategory is essentially small, closed under finite colimits and retracts. Every module is a filtered colimit of these finite pieces.

Proof. Theorem 3.1 supplies such finite cokernel stages for every \(M\). If \(M\) is finitely presented, its identity factors through one stage: there is \(s:M\to Q_{S,T}\) with \(q_{S,T}s=1_M\). This is an actual retract. Conversely (3.2) proves that every finite cokernel is finitely presented; a retract retains finite presentation. Explicitly, if \(P\) is a retract of \(Q\), its Hom functor identifies with the equalizer of precomposition by the splitting idempotent \(e\) and by \(1_Q\), so filtered finite-limit commutation proves the assertion. The same Hom argument gives closure under finite colimits.

There are small many finite lists of site objects and small many maps between their finite sums, since module-sheaf Hom groups are small. Choose one cokernel for each such map. Every finite object is a retract of one of these choices. Retracts are classified by idempotents on their target; each endomorphism set is small. Idempotents split in an abelian category through their images, with inclusion and corestricted projection composing to the identity of the image. Thus these choices and all their idempotent images form a small set of representatives of all finite objects. Choose a small full skeleton \(\mathcal A\) of that subcategory. Theorem 3.1 proves the last assertion. \(\square\)

This verifies local finite presentation and the hypotheses of the existing abstract Ind reconstruction theorem. Coherence remains a separate condition: kernels between finite pieces must stay finite. The one-point square-zero-ring counterexample in Finitely presented generators and coherent Ind reconstruction, Exercise 8.3 is already a finite-cover site and disproves automatic coherence. Its full calculation is unchanged.

5. Ind realization on every morphism

Let \(j:\mathcal A\to\mathsf M\) be the full inclusion of the finite skeleton. Restricted Yoneda is

H(M)(A)=\operatorname{Hom}_{\mathsf M}(jA,M). \tag{5.1}

Replace the finite stages in (3.3) by their skeleton representatives and transport all arrows through the representative isomorphisms. Finite presentation of each test gives \(H(M)\) as the filtered presheaf colimit of their representables. Thus \(H(M)\) belongs to \(\operatorname{Ind}(\mathcal A)\).

For two filtered presentations by finite pieces, the actual comparison is

\operatorname{Hom}_{\mathsf M} \!\left(\operatorname{colim}_i jA_i,\operatorname{colim}_j jB_j\right) \simeq\lim_i\operatorname{colim}_j \operatorname{Hom}_{\mathcal A}(A_i,B_j). \tag{5.2}

Mapping out of the source colimit first gives a compatible family of maps \(jA_i\to\operatorname{colim}_j jB_j\). Finite presentation supplies a target-stage representative for each source piece. Different representatives become equal at a common later stage; the inverse-limit condition is precisely compatibility with the source transitions after those comparisons. Conversely compose those representatives with the target colimit maps. They form a compatible source cocone and hence give the unique global map. These rules are inverse by the two colimit universal properties and the finite-presentation equality test. They preserve identities and composition since they do so after every source structural map. The target stage may depend on the source piece.

The Ind realization is therefore fully faithful by the proved formal Hom formula. Theorem 3.1 makes it essentially surjective. Its inverse is (5.1): realization of the indicated presheaf presentation recovers \(M\) by (3.3), and every Ind presheaf \(\operatorname{colim}_i h_{A_i}\) is recovered after realization because each test \(jA\) is finitely presented. Both comparisons are induced by the same structural maps, so their inverse identities hold on the presentations and are natural. We obtain

\mathsf{Mod}(R)\simeq\operatorname{Ind}(\mathcal A). \tag{5.3}

If the site is empty or the ring sheaf is zero, the module category is zero; the empty presentations and the zero finite object give the same equivalence. A terminal object, points of the site, local freeness of the generators and projectivity of the generators were never required.

6. Four graded exercises with complete solutions

Exercise 1 (grade 1). On the three-point space of the sheaf compactness lesson, compute \(G_U,G_V,G_W,G_X\) in the span model. Construct the cover presentation of \(G_X\), and prove it does not split although all its terms are finitely presented.

Solution. In span order \(M(U)\to M(W)\leftarrow M(V)\), the generators are

G_U=(k\xrightarrow1k\leftarrow0),\quad G_V=(0\to k\xleftarrow1k),\quad G_W=(0\to k\leftarrow0),\quad G_X=(k\xrightarrow1k\xleftarrow1k)=R. \tag{6.1}

These follow from (1.1) on inclusions and the fibre-product value at \(X\); a map from each is exactly the indicated section map in (1.2). For the cover \(X=U\cup V\), there is the exact row

0\longrightarrow G_W\xrightarrow{(1,-1)}G_U\oplus G_V \xrightarrow{\epsilon}G_X\longrightarrow0. \tag{6.2}

At \(U,V\) the last maps are identities and the kernel is zero. At \(W\), the last map is \((a,b)\mapsto a+b\), whose kernel is \(\{(t,-t)\}\). These maps commute with both restrictions, proving the whole exact row. Global sections of \(G_U\oplus G_V\) are zero: compatibility demands \((a,0)=(0,b)\) in \(k^2\). Global sections of \(G_X\) are \(k\). Consequently no map \(G_X\to G_U\oplus G_V\) can be a section, by the unit-Hom identification of the prerequisite. The generators are finitely presented by §1, but \(G_X\) is not projective. The cover relation, not projective lifting, controls this finite presentation.

Exercise 2 (grade 2). For an arbitrary finite-dimensional span \(A\xrightarrow f C\xleftarrow g B\), give a finite presentation by \(G_U,G_V,G_W\). Deduce the complete finite-presentation classification in this span category and verify its coherence.

Solution. Choose bases, with dimensions \(a,b,c\). Put \(P_0=G_U^a\oplus G_V^b\oplus G_W^c\). Its outer values are \(A,B\), and its middle value is \(A\oplus B\oplus C\). Evaluation into the given span is identity at the outer values and \((x,y,z)\mapsto f(x)+g(y)+z\) at the middle. It commutes with the two restrictions and is componentwise surjective.

Its kernel has zero outer values and middle value \(\{(x,y,-f(x)-g(y))\}\cong A\oplus B\). Hence it is \(G_W^{a+b}\), with the explicit middle inclusion \((x,y)\mapsto(x,y,-f(x)-g(y))\). This proves the exact finite presentation

G_W^{a+b}\longrightarrow G_U^a\oplus G_V^b\oplus G_W^c \longrightarrow M\longrightarrow0. \tag{6.3}

Every finite-dimensional span is therefore finitely presented. Conversely every finite generator value in (6.1) is finite-dimensional, and finite cokernels and retracts preserve that property. Theorem 4.1 makes every finitely presented span a retract of such a finite cokernel, so all three of its values are finite-dimensional. This proves the classification. Kernels of maps between these spans are pointwise vector-space kernels and stay finite-dimensional, as do cokernels. Their full category is therefore abelian with the ambient exact structure: the module-sheaf category here is locally coherent. This coexistence with the nonprojective generator \(G_X\) in Exercise 1 is consistent, since coherence asks about kernels of finite pieces rather than projectivity of each generator.

Exercise 3 (grade 3). Let the site have one object \(*\), with endomorphism monoid \(\mathbb N\) under addition and the trivial topology, and let \(R\) be the constant field presheaf \(k\). Compute its module sheaves and its generator \(G_*\). Show that a finitely presented generator can have an infinite-dimensional value over \(R(*)\), and compute a finite presentation of the tensor unit with a nonsplit epimorphism onto it.

Solution. Every presheaf is a sheaf for the trivial topology. A module is a vector space \(M\) with an endomorphism \(T=M(1)\); the action of \(n\) is \(T^n\). Its maps are the linear maps commuting with \(T\). This gives inverse constructions with \(k[t]\)-modules by letting \(t\) act as \(T\). Composition in the monoid and polynomial multiplication give all functor and module laws.

The generator has basis \([n]\), \(n\ge0\), and restriction along 1 sends \([n]\) to \([n+1]\). Thus \(G_*\cong k[t]\) as a \(k[t]\)-module. A map from it is determined by the image of 1, with inverse \(t^n\mapsto T^n m\); this verifies (1.2). It is compact projective because this Hom is the underlying vector space and the module category has pointwise exactness and filtered colimits. Yet its value at \(*\) is the infinite-dimensional space \(k[t]\) over \(R(*)=k\). Every cover in this topology includes the identity, so it cannot become a finite-free retract on a covering: evaluating any such retract at the identity slice object would make \(k[t]\) a retract of finite-dimensional \(k^n\). In particular it is not dualizable.

The tensor unit is the constant module \(k\) with \(T=1\), hence \(k[t]/(t-1)\). Its actual finite presentation is

G_*\xrightarrow{t-1}G_*\longrightarrow R\longrightarrow0. \tag{6.4}

It is finitely presented and dualizable as the tensor unit, but is not projective. A section of \(k[t]\twoheadrightarrow k[t]/(t-1)\) would send 1 to a polynomial annihilated by \(t-1\). That polynomial must be zero, since the leading coefficient of \((t-1)f\) is nonzero for every nonzero \(f\). It cannot map to 1. Global unit-Hom is \(\ker(T-1)\), not evaluation at \(*\): this category has no terminal object because \(*\) has many endomorphisms. The generator comparison (1.2) remains valid without one.

Exercise 4 (grade 4). At a point with coefficient field \(k\), let \(M=N=\bigoplus_{n\ge1}k e_n\), presented by their finite partial sums. Describe every morphism using (5.2). Determine the image of \(\operatorname{colim}_G\operatorname{Hom}(M,N_G)\to\operatorname{Hom}(M,N)\), and show why it cannot replace the order of limit and colimit in (5.2).

Solution. A map \(M\to N\) is determined by one finite-support vector for the image of each \(e_n\). Thus it is precisely an infinite matrix with each column finitely supported; a row may have infinitely many nonzero entries. Conversely such a matrix acts on every input, since an input is a finite linear combination of basis vectors, and defines a linear map.

For a finite source set \(F\), the images of its finitely many basis vectors have a common finite target support \(G\). A matrix from \(M_F\) therefore appears in \(\operatorname{colim}_G\operatorname{Hom}(M_F,N_G)\). Compatibility over source inclusions means that these matrices agree on their overlapping columns. Pairing all such compatible columns gives the full column-finite matrix, and restricting a full matrix gives them back. This proves both inverse descriptions of \(\lim_F\operatorname{colim}_G\operatorname{Hom}(M_F,N_G)\).

A map in \(\operatorname{Hom}(M,N_G)\) has every column supported in one fixed finite set of rows. The colimit over \(G\) consequently consists exactly of maps whose image is finite-dimensional: a finite set of rows gives that conclusion, and a finite basis of a finite-dimensional image has a finite union of coordinate supports. The identity matrix is column-finite but has no such uniform bound, so it is absent. Even the single map \(e_n\mapsto e_1\) has an infinite row but is in this finite-image part; column finiteness and row finiteness should not be interchanged. The correct formula allows the target stage to depend on the finite source stage, with equality tested later. Interchanging the order would discard legitimate morphisms.