On a tannakian theorem due to Nori

A. Bruguières

January 28, 2004

Résumé

M. V. Nori a construit en 1997 une catégorie de motifs en caractéristique nulle. Sa construction s’appuie sur un nouveau théorème de reconstruction tannakienne valide sur un anneau noethérien K. Étant donné un carquois Q et une représentation T de Q sur K, Nori construit une catégorie abélienne K-linéaire C(T), qui satisfait une certaine propriété universelle (Th. 1). Dans le présent article, nous donnons une démonstration de ce résultat, à ce jour non publié, pour un anneau K cohérent. En fait, nous établissons un résultat un peu plus fort sur C(T) (Th. 2) dont le théorème de Nori est un corollaire. Enfin, nous donnons une version monoïdale de la construction de Nori (Th. 3, Cor. 3).

Abstract

M. V. Nori constructed in 1997 a category of motives in characteristic 0. This construction relies on a new theorem of tannakian reconstruction over a noetherian ring K. Given a quiver Q and a representation T of Q over K, Nori constructs an abelian K-linear category C(T), which satisfies a certain universal property (Th. 1). In this paper, we give a proof of this as yet unpublished result for K coherent ring. In fact, we prove a somewhat stronger result on C(T) (Th. 2) of which Nori’s theorem is a corollary. Lastly, we give a monoidal version of Nori’s construction (Th. 3, Cor. 3).

2

Introduction

In 1997, M. V. Nori proposed in a yet unpublished text a very promising candidate for a category of motives in characteristic 0. In a nutshell, Nori’s approach goes as follows. He introduces a certain quiver (or diagram) Q_{a} and a representation T_{a} of this quiver with values in the category of ℤ-modules of finite type. The quiver is defined in terms of algebraic varieties over ℂ (or, more generally, a subfield of ℂ), and the representation is defined by singular (co-)homology. This part of the construction is surprisingly straightforward. Now comes a remarkable result : given a noetherian commutative ring K and a representation T of a quiver Q with values in the category K- mod of Kmodules of finite type, one may construct an abelian K-linear category C(T) over K- mod such that T lifts to C(T), and this construction is universal. As a corollary, if Q is a (small) abelian category over K, and T an abelian K-linear faithful exact functor, then Q → C(T) is an equivalence. This is a striking result, in that, until then, this type of tannakian reconstruction was known only over a field ([S], [D], [B]). Applied to the quiver Q_{a} and the representation T_{a}, Nori’s theorem yields an abelian category C(T_{a}) which is his candidate for a category of effective (co)-homological motives.

In Nori’s text, the universal property is stated, but the proof is not written down. The aim of the present paper is to provide a written proof, together with minor improvements :

1) we work over a coherent commutative ring;

2) we deduce Nori’s theorem from a somewhat stronger statement;

3) we give a monoidal version of Nori’s theorem, which requires a rather strong condition on K (essentially, the global dimension of K is at most 2).

I wish to express my deepest gratitude to Madhav Nori for allowing me (and indeed, warmly encouraging me) to publish this text which owes so much to his work.

Plan

In the first section, we recall the definitions of quivers and representations of quivers, give Nori’s construction of the category C(T), and state Nori’s theorem (Theorem 1) together with another theorem (Theorem 2) of which theorem 1 will be a consequence. In section 2, we rehearse a number of constructions related with categories of modules, namely tensor products and external Hom’s.

In section 3, we introduce a notion of external Hom of quiver representations, and explore its properties. In section 4, we prove theorem 2 as a direct application of the notion of external Hom of quiver representations, and deduce theorem 1. In section 5, we recall the notion of pro-objects and re-interpret the category C(T) as the category of left modules on the ‘pro-’ version the algebra of endomorphisms of T. We then give a monoidal version of Nori’s reconstruction (Theorem 3 and corollary 3).

Conventions and notations

Let A be a ring. We denote A- Mod the category of left A-modules, and \overline{A} the ring A seen as a left A-module. We denote A- mod ⊂ A- Mod the full subcategory of finitely presented Amodules. The ring A is coherent (on the left) if the category A- mod is abelian and the inclusion functor A- mod ↪ A- Mod is exact. From now on, K is a commutative coherent ring.

1 Quivers and Nori’s theorem

1.1 Quivers

Quivers (or diagrams in Nori’s terminology) are presheaves on the category \{0\overrightarrow{⟶}1\}. As such they form a category, and even a topos. More concretely, a quiver D consists in the following data : a set D_{0} (the vertices, or objects of the quiver), a set D_{1} (the edges, or arrows), and two maps s, t : D_{1} ⟶ D_{0} respectively called source and target. Example. If C is a category, Fl(C)\overrightarrow{⟶} Ob(C) is a quiver. Any functor is a morphism of quivers. We will denote in the same way a category and the underlying quiver. Let D be a quiver and C a category. A representation of D with values in C is a morphism of quivers T : D → C. Representations of a quiver D with values in a category C form a category. Given two such representations T, T′, a morphism ϕ : T → T′ is a family (ϕ_{i})_{i∈D_{0}}, ϕ_{i} ∈ Hom_{C}(T_{i}, T′_{i}), such that the following squares are commutative:

\xymatrix{T_{j} \ar[r]^{T_{a}} \ar[d]_{ϕ_{i}} & T_{k} \ar[d]^{ϕ_{k}} \\ T′_{j} \ar[r]_{T′_{a}} & T′_{k}}

for any (j →^{a} k) ∈ D_{1} (j = s(a), k = t(a)).

This is a generalization of the notion of natural morphism.

1.2 Nori’s theorem

Let K be a coherent commutative ring. Let D be a quiver, and T : D → K- mod a representation of D. If D is finite, E = End T is a K-algebra in K- mod. For i ∈ D_{0}, T_{i} has a natural structure of left E-module, denoted \overline{T}_{i}, and for (j →^{a} k) ∈ D_{1}, T_{a} : T_{j} → T_{k} is an E-linear map. Thus T lifts to a representation \overline{T} : D → End T - mod. For D arbitrary, let F be the set of finite subsheaves of D, ordered by inclusion. For D′ ∈ F, let E_{D′} = End(T_{|D′}). If D′′ ∈ F, D′′ ⊂ D, we have a canonical morphism E_{D′} → E_{D′′}, hence a functor E_{D′′}- mod → E_{D′}- mod which is K-linear, faithful, exact. We let

C(T) = \underrightarrow{lim}_{D′∈F} E_{D′}- mod .

This 2-limit being filtered, C(T) is an abelian K-linear category, and the forgetful functor Ω : C(T) → K- mod is K-linear, faithful, exact. ^{1} Moreover T lifts to a representation \overline{T} : D → C(T). Remark. In C(T), Hom’s are finitely presented K-modules if D is finite or K noetherian. The representation \overline{T}, together with the forgetful functor Ω, satisfies a universal property.

Theorem 1 (Nori) Let D be a quiver, and T : D → K- mod a representation of D. On the other hand, let C be an abelian K-linear category, F : C → K- mod a K-linear faithful exact functor, and S : D → C a representation of D in C such that F S = T. Then there exists a functor S′ : C(T) → C, unique (up to unique isomorphism) such that the following diagram is commutative (up to isomorphism) :

\xymatrix{ & C(T) \ar[d]^{S′} \ar[ddr]^{Ω} \\  & C \ar[dr]^{F} \\ D \ar[uur]^{\overline{T}} \ar[ur]_{S} \ar[rr]_{T} &  & K- mod .}

1See section 5 for a more conceptual approach

Moreover the functor S′ is K-linear faithful exact.

Remark. Let us clarify the statement of the theorem. The claim is that there exist a functor S′, an isomorphism α : S′\overline{T} ⟶^{∼} S, and an isomorphism β : F S′ ⟶^{∼} Ω, with the compatibility condition: F (α) = β\overline{T}. Uniqueness means that if (S′′, α′, β′) is another solution to the problem, there exists a unique isomorphism γ : S′′ ⟶^{∼} S′ such that α′ = (γ\overline{T})α and β′ = βF (γ).

Corollary 1 If D is a (small) abelian K-linear category and T : D → K- mod is a K-linear, faithful exact functor, then the functor \overline{T} : D → C(T) is an equivalence.

Remark. This corollary generalizes over a coherent ring the fact that, over a field K, the following sets of data are equivalent :

(1) a coalgebra L;

(2) an abelian K-linear category C together with a K-linear faithful exact functor F : C → vect K. This is the fundamental fact which allows one to construct the tannakian dictionary (in the neutral case). Proof. Observe that the data (1_{D} : D → D, T) satisfies the same universal property as the data (\overline{T} , Ω). By the uniqueness assertion, the functor \overline{T} : D → C(T) is an equivalence. □ We will deduce theorem 1 from the following theorem.

Theorem 2 Let D be a finite quiver, C an abelian K-linear category, T (resp. S) a representation of D with values in K- mod (resp. C). There exists a K-linear right exact functor H(T, S) : C(T) → C, equipped with a morphism ev : H(T, S) ∘ \overline{T} → S, and universal (final) for this property.

[figure: C(T); H(T,S); ev; D; \overline{T}; S; C]

Remark. In the statement, universal means that, given any K-linear right exact functor H : C(T) → C, the canonical map

Hom(H, H(T, S)) → Hom(H ∘ \overline{T} , S)

defined by f ↦ ev ∘(f\overline{T}), is a bijection.

2 Categories of modules, tensor products and external Hom

The results presented here are more or less classical, see [P] for instance. In this section, C is an abelian K-linear category. Recall that if A is a Kalgebra, a left (resp. right) A-module in C is an object M of C equipped with a K-algebra morphism A → End_{C}(M) (resp. A^{o} → End_{C}(M)). We will denote A-C (resp. C-A) the category of left (resp. right) A-modules in C.

Proposition 1 Let A be a K-algebra in K- mod, C an abelian K-linear category, and M an object of C-A. 1)There exists a K-linear right exact functor F : A- mod → C such that F (\overline{A}) ≃ M in C-A.

2) Given such a functor F, there exits functorial isomorphism (the ‘adjonction’ isomorphism) ^{2}

Hom_{C}(F (V), X) ≃ Hom_{A}(V, Hom_{C}(M, X)) .

Notation. The functor F, unique up to unique isomorphism thanks to assertion 2), will be denoted M⊗_{A}?.

Corollary 2 The category C-A is equivalent to the category of K-linear right exact functors from A- mod to C.

Proof. If M is a right A-module, there is a canonical map Hom_{A}(\overline{A}^{m}, \overline{A}^{n}) → Hom_{C}(M^{m}, M^{n}), and the aim is to extend this to a functor A- mod → C. Let A = A- mod. Define a new category A′ as follows :

• the objects of A′ are the exact sequences \overline{A}^{m} ⟶^{R} \overline{A}^{n} ⟶^{s} V → 0 in A;

• for V_{•} = (\overline{A}^{m} →^{R} \overline{A}^{n} → V → 0), V′_{•} = (\overline{A}^{m′} →^{R′} \overline{A}^{n′} → V′ → 0) objects

of A′, Hom_{A′}(V_{•}, V′_{•}) = Hom_{A}(V, V′).

The forgetful functor U : A′ → A, (\overline{A}^{m} → \overline{A}^{n} → V → 0) ↦ V is an equivalence, a quasi-inverse thereof is given by arbitrary choice of a finite presentation for each object in A- mod. For a start, we construct a functor F′ : A′ → C as follows. If V_{•} = (\overline{A}^{m} →^{R} \overline{A}^{n} → V → 0) is an object of A′, then F′(V_{•}) = coker(M^{m} →^{R} M^{n}).

2If in C, Hom’s are finitely presented K-modules, Hom_{C}(?, X) is right adjoint to F.

If ϕ is a morphism from V_{•} to another object V′_{•} = (\overline{A}^{m′} →^{R′} \overline{A}^{n′} → V′ → 0) in A′, that is, an element of Hom_{A}(V, V′), pick morphisms ϕ_{1}, ϕ_{2} so that the following diagram commutes in A- mod :

\xymatrix{\overline{A}^{m} \ar[d]_{ϕ_{2}} \ar[r]^{R} & \overline{A}^{n} \ar[d]_{ϕ_{1}} \ar[r] & V \ar[d]_{ϕ} \ar[r] & 0 \\ _{\overline{A}_{m′}} \ar[r]^{R′} & _{\overline{A}_{n′}} \ar[r] & V′ \ar[r]^{R} & 0 ,}

and define F′(ϕ) as being the morphism in C which makes the following diagram commutative :

\xymatrix{M^{m} \ar[d]_{ϕ_{2}} \ar[r]^{R} & M^{n} \ar[d]_{ϕ_{1}} \ar[r] & F′(V_{•}) \ar[d]_{F′(ϕ)} \ar[r] & 0 \\ M^{m′} \ar[r]^{R′} & M^{n′} \ar[r] & F (V′_{•}) \ar[r]^{R} & 0 .}

This construction, being clearly independent of choices, defines a functor F′. Let F be as in the statement of the proposition. For V_{•} = (\overline{A}^{m} →^{R} \overline{A}^{n} →^{s} V → 0) object of A′, we have via F a short exact sequence M^{m} →^{R} M^{n} → F (V) → 0 in C, hence a canonical isomorphism F′(V_{•}) ≃ F (V). Thus F U ≃ F′, and F ≃ F′Q. Left exactness of F = F′Q will immediately follow from assertion 2), which we now prove. For V_{•} = (\overline{A}^{m} → \overline{A}^{n} → V → 0) object of A′ and X object of C, the lines in the following commutative diagram

\xymatrix{0 \ar[r] & Hom(F (V),X) \ar[r] & Hom(M^{n},X) \ar[r] \ar[d] & Hom(M^{m},X) \ar[d] \\ 0 \ar[r] & Hom_{A}(V,Hom_{C}(M,X)) \ar[r] & Hom_{A}(\overline{A}^{n},Hom_{C}(M,X)) \ar[r] & Hom_{A}(\overline{A}^{n},Hom_{C}(M,X))}

are exact, and both vertical arrows are isomorphisms; hence an isomorphism

Hom(F (V), X) ⟶^{∼} Hom_{A}(V, Hom_{C}(M, X)) ,

which is functorial in X and in V_{•}, so in V, too. □ Definition. Let C be an abelian K-linear category, and X an object of C. Define the functor \underline{Hom}_{C}(?, X) : (K- mod)^{o} → C to be the opposite functor to X⊗_{K} ? : K- mod → C^{o}. Thus, for X, Y objects of C and V object of K- mod, one has a functorial isomorphism

Hom_{C}(X, \underline{Hom}_{C}(V, Y)) ≃ Hom_{K}(V, Hom_{C}(X, Y)) .

This defines a K-linear left exact functor

\underline{Hom}_{C}(?,?) : (K- mod)^{o} × C → C .

called external Hom. Remarks.

1) The formation of M⊗_{A}? commutes with K-linear right exact functors; in particular, the construction of \underline{Hom} (V,?) commutes with K-linear left exact functors.

2) Let A be a K-algebra in K- mod, and C = A- mod. Given a finitely presented K-module V, and a finitely presented A-module M, \underline{Hom}_{C}(V, X) is nothing but Hom(V, M), seen as a left A-module.

3 External Hom for quiver representations

Let C be an abelian K-linear category, D a finite quiver, S a representation of D with values in C, and T a representation D → K- mod. To this data we associate a morphism Δ_{T,S} of C :

Δ_{T,S} = ∂^{0} − ∂^{1} : ∏_{i∈D_{0}} \underline{Hom}_{C}(T_{i}, S_{i}) ⟶ ∏_{(j\underrightarrow{a}k)∈D_{1}} \underline{Hom}_{C}(T_{j}, S_{k}) .

Interpreting the ∏’s as ⨁’s, ∂^{0}, ∂^{1} are defined blockwise as follows. For j →^{a} k, consider

S_{a}_{∗} : \underline{Hom}_{C}(T_{j}, S_{j}) → \underline{Hom}_{C}(T_{j}, S_{k}),
T_{a}^{∗} : \underline{Hom}_{C}(T_{k}, S_{k}) → \underline{Hom}_{C}(T_{j}, S_{k}) ;

then ∂_{i,a}^{0} = δ_{i,j}S_{a}_{∗} and ∂_{i,a}^{1} = δ_{i,k}T_{a}^{∗}. Notice that these are morphisms of right End T -modules in C. Notation. Set \underline{Hom}_{C}(T, S) = ker(Δ_{T,S}), so that we have the following exact sequence in C- End T :

(E_{T,S}) 0 → \underline{Hom}_{C}(T, S) → ∏_{i∈D_{0}} \underline{Hom}_{C}(T_{i}, S_{i}) → ∏_{(j\underrightarrow{a}k)∈D_{1}} \underline{Hom}_{C}(T_{j}, S_{k}) .

The object \underline{Hom}_{C}(T, S) deserves the title of external Hom; indeed we have the following lemma.

Lemma 1 For any object Σ of C- End T, there is a canonical isomorphism

Hom_{C- End T} (Σ, \underline{Hom}_{C}(T, S)) ⟶^{∼} Hom(Σ ⊗_{End T} \overline{T} , S) .

Proof. Let E be a K-algebra in K- mod, M an object of E- mod, Σ an object of C-E, and X an object of C. By adjunction, we have Hom_{K}(M, Hom(Σ, X)) ≃ Hom_{C}(Σ, \underline{Hom}_{C}(M, X) hence (by functoriality) Hom_{E}(M, Hom(Σ, X)) ≃ Hom_{C-E}(Σ, \underline{Hom}_{C}(M, X)). By adjunction, Hom_{C}(Σ ⊗_{E} M, X) ≃ Hom_{E}(M, Hom(Σ, X)) hence a canonical isomorphism :

Hom_{C}(Σ ⊗_{E} M, X) ≃ Hom_{C-E}(Σ, \underline{Hom}_{C}(M, X)) .

Applied to E = End T, this yields a commutative diagram :

\xymatrix{0 \ar[r] & Hom(Σ⊗_{E} \overline{T} ,S) \ar[r] & ∏_{i} Hom(Σ⊗_{E} \overline{T}_{i},S_{i}) \ar[r] \ar[d] & ∏_{a} Hom(Σ⊗_{E} \overline{T}_{j},S_{k}) \ar[d] \\ 0 \ar[r] & Hom_{C-E}(Σ,\underline{Hom}_{C}(T,S)) \ar[r] & ∏_{i} Hom_{C-E}(Σ,\underline{Hom}_{C}(T_{i},S_{i})) \ar[r] & ∏_{a} Hom_{C-E}(Σ,\underline{Hom}_{C}(T_{j},S_{k}))}

where lines are exact and vertical arrows are the canonical isomorphisms, hence the isomorphism we were looking for. □ In particular, for Σ = \underline{Hom} (T, S), the identity of \underline{Hom}_{C}(T, S) corresponds with a canonical representation morphism

ev_{T,S} = ev : \underline{Hom}_{C}(T, S) ⊗_{End T} \overline{T} → S .

Lemma 2 The formation of \underline{Hom} (T, S) enjoys the following properties.

1) (Functoriality) if C′ is an abelian K-linear category and F : C → C′ a K-linear exact left functor, then \underline{Hom}_{C′}(T, F S) = F (\underline{Hom}_{C}(T, S)) and ev_{T,F S} = F (ev_{T,S}).

2) (Trivial case) for C = K- mod and S = T, \underline{Hom} (T, T) = End T and ev is the identity.

Proof. The functor F commutes with the formation \underline{Hom}_{C}, and preserves left short exact sequences, hence F (E_{T,S}) = (E_{T,F S}), and in particular \underline{Hom}_{C′}(T, F S) = F (\underline{Hom}_{C}(T, S)). Now one checks that F commutes with the formation of the bijection of lemma 1, and so, of ev_{T,S}. The trivial case is straightforward. □

4 Proof of theorems 1 and 2

Let us prove theorem 2. Setting E = End T, we have C(T) = E- mod. Let H(T, S) = \underline{Hom}_{C}(T, S)⊗_{E}? : C(T) ⟶ C, and let ev be the evaluation morphism \underline{Hom}_{C}(T, S) ⊗_{E} \overline{T} → S. Any right exact K-linear functor S′ :

C(T) → C is of the form Σ⊗_{E}? for some right E-module Σ in C. We have Hom(S′ ∘ \overline{T} , S) ≃ Hom(Σ ⊗_{E} \overline{T} , S) ≃ Hom_{E}(Σ, \underline{Hom}_{C}(T, S)) (by remark 3) ≃ Hom(S′, H(T, S)) (corollary to proposition 1). Hence the theorem.

Now let us prove theorem 1 (Nori’s theorem). Notice that if the functor S′ exists, it is K-linear faithful exact. Indeed, the categories C(T), C and K- mod are abelian, K-linear, and both F and F ∘ S′ ≃ Ω are K-linear faithful exact. Notice also that, since C(T) is the filtered direct limit of End(T_{|D′})’s for D′ ⊂ D finite, we may assume D finite, and so do we. Theorem 2 provides a functor \overline{S} = H(T, S), and we will show that S′ = \overline{S} answers our problem. In order to do so, we will check that α := ev_{T,S} : \overline{S} \overline{T} → S is an isomorphism, and construct an isomorphism β : F \overline{S} ⟶^{∼} Ω such that β\overline{T} = F α. Let E = End T. Recall that, in the short exact sequence in K- mod :

(E_{T,T})  0 → E → ∏_{i∈D_{0}} Hom(T_{i}, T_{i}) ⟶ ∏_{(j\underrightarrow{a}k)∈D_{1}} Hom(T_{j}, T_{k}) ,

each object is naturally an (E, E)-bimodule, and the maps are (E, E)-linear, so (E_{T,T}) lifts to a short exact sequence of right E-modules in E- mod :

(\overline{E})  0 → \overline{E} → ∏_{i∈D_{0}} \underline{Hom} (T_{i}, \overline{T}_{i}) ⟶ ∏_{(j\underrightarrow{a}k)∈D_{1}} \underline{Hom} (T_{j}, \overline{T}_{k})

which is none other than (E_{T,\overline{T}}). Now let α = ev_{T,S} : \overline{S} \overline{T} → S. Since F is exact, F (α) = ev_{T,F S} = ev_{T,T} = 1_{T} (lemma 2); F being also faithful, and therefore conservative ^{3}, we see that α is an isomorphism. On the other hand, F \overline{S} is right exact, so by proposition 1, it is of the form Σ⊗_{E}?, where Σ = F \overline{S}(\overline{E}) = F (\underline{Hom}_{C}(T, S)) = Hom(T, F S) = E (by exactness of F, F S = T, and lemma 2) = Ω(\overline{E}). Hence a canonical isomorphism β : F \overline{S} ⟶^{∼} Ω by proposition 1 again. One checks F (α) = β\overline{T}, hence existence. Now for uniqueness. Assume S′ is a functor as required; it comes equipped with compatible morphisms α′ : S′\overline{T} ⟶^{∼} S, β′ : F S′ ⟶^{∼} Ω. By the preliminary remark, S′ is K-linear faithful exact. In particular, S′ ≃ Σ⊗_{E}?, with Σ = S′(\overline{E}) (by right exactness of F). Now S′(\overline{E}) = S′(\underline{Hom}_{C}(T, \overline{T})) = \underline{Hom}_{C}(T, S′\overline{T}) (by left exactness of S′) ≃ \underline{Hom}_{C}(T, S) via α′. Hence an isomorphism γ : S′ ⟶^{∼} \overline{S}. By construction, α′ = (γ\overline{T})α, and one checks : β′ = F (γ)β. □ We will also need the following result.

3A functor F is conservative if any morphism whose image by F is an isomorphism, is an isomorphism.

Proposition 2 Let T : D → K −mod be a representation of a quiver D. Let A be a full subcategory of C(T) stable under direct sum, kernels and cokernels, and containing \overline{T}_{i} for each i ∈ D_{0}. Then A = C(T).

Proof. This can be deduced from Th. 1 by abstract nonsense. In a more pedestrian way, one checks that A contains \underline{Hom} (N, \overline{T}_{i}) for any N in K- mod, then A contains \overline{End(T_{|D_{0}})} for any finite subquiver D_{0} of D, and finally, all of C(T). □

5 Monoidal version

Throughout this section, we assume a certain familiarity with monoidal categories. See for instance [M], [B]. If K is a field, tannakian theory states that given a monoidal category C and a monoidal functor F : C → vect K, the associated coalgebra L(F) actually is a bialgebra. If C has duals, it is a Hopf bialgebra. Here, we will adapt these results to Nori’s setting. Since we are dealing with quivers, instead of categories, we will define monoidal quivers and monoidal representations. Since K is no longer a field, we will have to make rather strong assertions : we will consider representations taking values in projective modules (P1), together with a mild hypothesis to compensate for the absence of identities in a quiver (P2), and we will make a homological assumption (⋆) on K, which essentially says that the global dimension of K is at most 2.

Instead of the coalgebra of coendomorphisms, we are dealing with the algebra of endomorphisms, which in this setting is a Pro-object ; so we will now rehearse a few general facts on Pro-objects.

5.1 Pro-objects.

Let C be a category. There is a universal way of constructing a completion of C with respect to (small) filtered inverse limits, and this is the category of Pro-objects of C, denoted Pro C. Here follows a sketch of this construction. A Pro-object of C is a small filtered inverse system (X_{i})_{i∈I} in C, which we will denote “\underleftarrow{lim}_{i∈I} X_{i}” for convenience. The Pro-objects of C form a catgory Pro C, with morphisms defined as follows :

Hom_{Pro C}(“\underleftarrow{lim}_{i∈I} X_{i}”, “\underleftarrow{lim}_{j∈J} Y_{j}”) =\underleftarrow{lim}_{j}\underrightarrow{lim}_{i}Hom_{C}(X_{i}, Y_{j}) .

The canonical functor C → Pro C, X ↦ “\underleftarrow{lim}X” being fully faithful, we will identify C to a full subcategory of Pro C. Small filtered inverse limits exist in Pro C, and Pro C enjoys the following universal property. Let E be a category where small filtered inverse limits exist. Any functor F : C → E extends uniquely (up to unique isomorphsm) to a functor F̃ : Pro C → E which commutes to small filtered inverse limits; F̃ is defined by “\underleftarrow{lim}_{i} X_{i}” ↦\underleftarrow{lim}_{i} F (X_{i}). The category Pro K- mod is abelian, K-linear, monoidal (indeed, if C is abelian or monoidal, so is Pro C). If A is an algebra in Pro K- mod, we denote A- mod the category of A-modules whose underlying Pro-object belongs to K- mod. If D is quiver and T : D → K- mod a représentation of T, denote

End(T) = “\underleftarrow{lim}_{D∈F} End(T |_{D_{0}})” .

Then End(T) is an algebra in Pro K- mod, and C(T) = End(T)- mod. Remark. Dually, one may construct a completion of a category C with respect to direct filtered limits : this is the category Ind C of Ind-objects in C. In particular, for a coherent ring K, Ind K- mod is just the category K- Mod of all K-modules. Unfortunately there is no such simple description for Pro K- mod : it seems that this is the price to pay for Nori’s theorem.

5.2 The monoidal setting

A (strict) monoidal category is a data (M, ⊗, I), where

— C is a category;

— ⊗ is a associative functor M × M → M (tensor product);

— I is a unit object, i. e. ? ⊗ I = 1_{M} = I⊗?. In other words, it is just a monoid in the category of categories. (Thanks to MacLane’s coherence theorem we may assume all monoidal categories to be strict without loss of generality, see [M].) By analogy, we define a monoidal quiver. Definition. A monoidal quiver is a quiver D = (D_{1} \overrightarrow{⟶}D_{0}) with monoid structures on D_{0} and D_{1} such that s, t : D_{1} → D_{0} are morphisms of monoids. In other words, it is a quiver in monoids, that is, a data (D, μ, η), where D is a quiver, μ : D × D → D is associative, and η : 1 → D is a unit for μ.

Let (M, ⊗, I) be a monoidal category. Define a new category T /M as follows. Objects of T /M are representations of quivers with values in M. Morphisms from T : D → M to T′ : D′ → M are couples (f, α), where f is a quiver morphism D → D′, and α an isomorphism of representations : α : T ⟶^{∼} T′ ∘ f, with obvious composition. Define on T /M a tensor product ⊠ : given two objects T : D → M, T′ : D → M, T ⊠ T′ is the representation D × D′ → M, (i, j) ↦ T_{i} ⊗ T′_{j}. Thus, T /M becomes a monoidal category, whose unit object is the representation 𝕀 : 1 → M which sends the only vertex of the initial quiver 1 to I and the only edge to 1_{I}. Definition. Let D be a monoidal quiver, and (M, ⊗, I) a monoidal category. A monoidal representation of D with values in M is a representation T : D → M equipped with morphisms (μ, Φ) : T ⊠ T → T and (η, Φ_{0}) : 𝕀 → T in the category T /M, such that :

• Φ_{ij,k}(Φ_{i,j} ⊗ 1_{T_{k}}) = Φ_{i,jk}(1_{T_{i}} ⊗Φ_{j,k});

• Φ_{i,e} = 1_{T_{i}} ⊗Φ_{0} and Φ_{e,i} = Φ_{0} ⊗ Φ_{e,i} (where e denotes the unit of D_{0}).

The representation is usually referred to as (T, Φ, Φ_{0}), or just T.

Lemma 3 Monoidal representations of monoidal quivers with values in M are exactly monoids in T /M.

Proof. A monoid in T /M is an data (T, μ, η), where T is an object of T /M, and μ, η are morphisms, μ : T ⊠ T → T, η : 𝕀 → T, with the usual axioms of associativity : μ(μ⊗1_{T}) = μ(1_{T} ⊗μ) and unity : μ(η ⊗1_{T}) = 1_{T} = μ(1_{T} ⊗η). Worked out explicitly, this boils down to a monoidal representation of a monoidal quiver. □ Definition. Let (T, Φ, Φ_{0}), (T′, Φ′, Φ′_{0}) be two monoidal representations of a monoidal quiver D in a monoidal category C. A morphism of representations α : T → T′ is monoidal if :

— Φ′_{i,j}(α_{i} ⊗ α_{j}) = α_{ij}Φ_{i,j} for i, j ∈ D_{0};

— Φ′_{0} = α_{e}Φ_{0}.

5.3 The theorem in the monoidal setting

Let T = T /K- mod. We have a canonical functor T ↦ End(T) from T to the category of algebras in Pro K- mod. Given any two objects T, T′ in T_{K}, we have a canonical morphism Φ_{T,T′} : End(T) ⊗ End(T′) → End(T ⊗ T′). This is not an isomorphism in general.

Consider the following example. Let V a finitely presented K-module, f ∈ End(V), and denote T_{V,f} the representation of the initial quiver which sends the unique vertex to V and the unique edge to f. Then End(T_{V,f}) = End(T_{V,f}) is the centralizer Z(f) = {g ∈ End(V) | gf = fg}. We have End(T_{V,f}) ⊗ End(T_{K,0}) = Z(f), whereas End(T_{V,f} ⊗ T_{K,0}) = End(T_{V,0}) = End(V); if f is not central, the canonical morphism is not an isomorphism. However we shall see that, under certain assumptions, the canonical morphism is an isomorphism. Definition. We say that the ring K satisfies condition (⋆) if for any morphism of K-modules f : P → P′, with P and P′ projective of finite type, ker(f) is projective. Notice that, since K is coherent, ker(f) is finitely presented, and so, ‘projective’ just means ‘flat’ here. Condition (⋆) means that the homological dimension of the category mod K is at most 2. Recall that, if K is noetherian, the homological dimension of mod K is the global dimension of K. In particular (⋆) holds for K = ℤ. Now, consider the following conditions on a representation T : D → K- mod : P1) ∀i ∈ D_{0}, T_{i} is a projective K-module of finite type;

P2) ∀i ∈ D_{0}, ∃a ∈ D_{1}, i →^{a} i, such that T_{a} = 1_{T_{i}}.

Remark. Condition P2 is automatic if D is a category and T a functor.

Definition. By monoidal right exact abelian category over K, we mean an abelian K-linear category C equipped with a monoidal structure (⊗, I) where ⊗ is K-linear right exact in each variable, and End(I) = K.

Theorem 3 Let T : D → K- mod be a monoidal representation of a monoidal quiver D. Assume that T satisfies P1 and P2, and (⋆) holds for K. Then :

1) the algebra End(T) has a natural structure of bialgebra in Pro K- mod, so that C(T) is a monoidal right exact abelian category over K, \overline{T} is a monoidal representation, and the forgetful functor Ω : C(T) → K- mod is monoidal strict;

2) the above construction satisfies the following universal property. Let C be a monoidal right exact abelian category over K, F : C → K- mod a K-linear faithful exact monoidal functor, and S : D → C a monoidal representation of D in C such that F S = T as monoidal representations. Then there exists a monoidal functor S′ : C(T) → C, unique (up to unique monoidal isomorphism) such that the following diagram is commutative (up to monoidal isomorphism) :

\xymatrix{ & C(T) \ar[d]^{S′} \ar[ddr]^{Ω} \\  & C \ar[dr]^{F} \\ D \ar[uur]^{\overline{T}} \ar[ur]_{S} \ar[rr]_{T} &  & K- mod .}

The functor S′ is (monoidal) K-linear faithful exact.

Remark. The statement of the universal property is to be understood as follows : there exist a monoidal functor S′, a monoidal isomorphism α : S′\overline{T} ⟶^{∼} S, and a monoidal isomorphism β : F S′ ⟶^{∼} Ω, with the compatibility condition: F (α) = β\overline{T}. Uniqueness means that if (S′′, α′, β′) is another solution to the problem, there exists a unique monoidal isomorphism γ : S′′ ⟶^{∼} S′ such that α′ = (γ\overline{T})α and β′ = βF (γ). Proof. Let T_{0} ⊂ T be the full subcategory whose objects are representations T : D → K- mod satisfying P1 and P2. Notice that T_{0} is a monoidal subcategory of T .

Lemma 4 The correspondence T ↦ End(T) is a monoidal functor from T_{0}^{o} to the category of algebras in Pro K- mod.

Proof. Let T : D → mod K and T′ : D → mod K be objects of T , and let T′′ = T ⊗ T′. The whole point is to show that, if T and T′ satisfy P1 and P2, the natural morphism End(T) ⊗ End(T′) → End(T′′) is an isomorphism; in order to prove this, we may and do assume D and D′ finite. Recall that End(T) is given by the short exact sequence

(E_{T,T})  0 → End(T) ⟶ L(T) ⟶^{Δ_{T}} M(T) ,

where L(T) = ∏_{i∈D_{0}} Hom(T_{i}, T_{i}), M(T) = ∏_{(j\underrightarrow{a}k)∈D_{1}} Hom(T_{j}, T_{k}), Δ_{T} = Δ_{T,T}.

By P1, the K-modules L(T) and M(T) are projective of finite type, and so is End(T) by (⋆). We view End(T) as a submodule of L(T). The same holds for T′ and T′′. All modules involved being flat, we have

End(T) ⊗ End(T′) = (End(T) ⊗ L′) ∪ (L ⊗ End(T′)) .

Let ϕ = (ϕ_{i,i′}) ∈ L(T′′) ≃ L(T) ⊗ L(T′). Then :

• ϕ ∈ End(T′′) if an only if the following condition holds :

∀(a, a′) ∈ D_{1} × D′_{1}, (T_{a} ⊗ T_{a′}) ϕ_{s(a),s(a′)} = ϕ_{t(a),t(a′)}(T_{a} ⊗ T_{a′}) ;

• ϕ ∈ End(T) ⊗ End(T′) if and only if the following two conditions hold :

∀a ∈ D_{1}, ∀i′ ∈ D′_{0}, (T_{a} ⊗ 1_{T_{i′}}) ϕ_{s(a),i′} = ϕ_{t(a),i′}(T_{a} ⊗ 1_{T_{i′}}) , ∀a′ ∈ D′_{1}, ∀i ∈ D_{0}, (1_{T_{i}} ⊗ T_{a′}) ϕ_{i,s(a′)} = ϕ_{i,t(a′)}(1_{T_{i}} ⊗ T_{a′}) .

Now clearly, End(T) ⊗ End(T′) ⊂ End(T′′), and equality holds because T and T′ satisfy P2. □ Now we prove assertion 1) of the theorem. The assumptions made on T mean that it is a monoid in T_{0}, hence a co-monoid in T_{0}^{o}; therefore its image by the monoidal functor End(?) is a co-monoid in the category of algebras, that is, a bialgebra. The coproduct Δ and counit ε are obtained by applying End(?) respectively to the multiplication T ⊠ T → T and the unit η : 𝕀 → T. Therefore C(T) = End(C) − mod is a monoidal category, the tensor product being defined as usual for a category of left modules on a bialgebra. The functor Ω and the representation \overline{T} are clearly monoidal.

As for assertion 2) (universal property) : Th. 1 yields S′ as a ‘naked’ functor S′, which is K-linear faithful exact. The whole point is to check that S′ can be endowed with an appropriate monoidal structure. For simplicity, we assume that F is monoidal strict. The difficult point is to construct for X, Y in C(T) a functorial isomorphism Φ_{X,Y} : S′(X) ⊗ S′(Y) ⟶^{∼} S′(X ⊗ Y). Since Ω is monoidal and F S′ ⟶^{∼} Ω, we dispose of a functorial isomorphism Ψ_{X,Y} : F (S′(X) ⊗ S′(Y)) ⟶^{∼} F (S′(X ⊗ Y)). Consider the property : L(X, Y) : there exists Φ_{X,Y} : S′(X) ⊗ S′(Y) ⟶^{∼} S′(X ⊗ Y) such that F (Φ_{X,Y}) = Ψ_{X,Y}. Notice that if L(X, Y) holds, Φ_{X,Y} is uniquely determined because F is faithful. Now L(X, Y) holds for X = \overline{T}_{i}, Y = \overline{T}_{j}. Indeed we have S′\overline{T} ⟶^{∼} S, and S, \overline{T} are monoidal, hence our isomorphism

S′(\overline{T}_{i}) ⊗ S′(\overline{T}_{i}) ⟶^{∼} S_{i} ⊗ S_{j} ⟶^{∼} S_{i,j} ⟶^{∼} S′(\overline{T}_{ij}) ⟶^{∼} S′(\overline{T}_{i} ⊗ \overline{T}_{j}) .

In order to prove L(X, Y) in general, we need the following lemmas.

Lemma 5 In C(T), any object is a quotient of an object which is projective as a K-module.

Proof. For any X in C(T), there exists a finite subquiver D_{0} ⊂ D such that X is an E_{0}-module for E_{0} = End(T_{|D_{0}}). Now, thanks to conditions (⋆) and P1, E_{0} is a projective K-module, and X is a quotient of \overline{E_{0}}^{n} for some integer n. □

Lemma 6 Let A, B, C be abelian categories, G, G′ : A → B be additive functors, and F : B → C be a faithful exact functor. Let α be a morphism of functors : F G → F G′. We say that X ∈ Ob A has the lifting propery if there exists a (necessarily unique) morphism β_{X} : G(X) → G′(X) such that F (β_{X}) = α_{X}. We denote A_{0} ⊂ A the full subcategory of objects having the lifting property. Then :

1) β is a morphism of functors G_{|A_{0}} → G′_{|A_{0}};

2) A_{0} is stable under direct sums and direct summands;

3) if G and G′ are right- (resp left-) exact, A_{0} is stable under cokernels (resp. kernels).

Proof. 1) Holds because F is faithful. 2) Let X, Y be in A_{0}. Then F (β_{X} ⊕ β_{Y}) = α_{X} ⊕ α_{Y} = α_{X⊕Y} so X ⊕ Y is in A_{0}. The assertion on summands is equally trivial. 3) Assume for instance that G is right exact. Let f : X → Y be a morphism in A_{0}, and let Q = coker(f). We have the following commutative diagram, where lines are exact :

\xymatrix{G(X) \ar[r] \ar[d]_{β_{X}} & G(Y) \ar[d]^{β_{Y}} \ar[r] & G(Q) \ar[r] & 0 \\ G′(X) \ar[r] & G′(Y) \ar[r] & G′(Q) \ar[r] & 0} ,

hence a morphism β_{Q} : G(Q) → G′(Q). Applying the exact functor F, we see that F (β_{Q}) = α_{Q}, so coker f is in A_{0}. □ Let Y be an object of C(T). We will apply lemma 6 three times to G = S′(?) ⊗ S′(Y), G′_{Y} = S′(? ⊗ Y) and α = Ψ_{?,Y}. Notice that G, G′ are right exact, and they are actually exact if Y is projective as a K-module. Consider first the case Y = \overline{T_{j}}. Here G and G′ are exact; \overline{T}_{i}’s have the lifting property, and so by proposition 2 and the lemma, we deduce L(X, \overline{T}_{j}) for any X. We also have L(\overline{T}_{j}, X) by symmetry. Now assume Y is projective as a K-module. Here again G and G′ are exact, and \overline{T}_{i}’s have the lifting property, so by the same argument we have L(X, Y) provided that X or Y is projective as a K-module. Now consider arbitrary Y, so that G and G′ are just right exact. If X is projective as a K-module, X has the lifting property. Now by lemma 5, any X in C(T) fits in a short exact sequence P′ → P → X → 0, with P, P′ projective as K-modules. By lemma 6, X has the lifting property, hence L(X, Y) for X, Y arbitrary. The rest is then a fastidious but straightforward verification which we shall omit. □ For End(T) to be a Hopf bialgebra, that is, a bialgebra with bijective antipode, we need duals. Recall that a duality in a monoidal category C is a data (X, Y, e, h), where X, Y are objects, e : X ⊗ Y → I, h : I → Y ⊗ X are morphisms satisfying :

(e ⊗ 1_{X})(1_{X} ⊗h) = 1_{X} and (1_{Y} ⊗e)(h ⊗ 1_{Y}) = 1_{Y} .

If (X, Y, e, h) is a duality, we say that X is a left dual of Y, and Y is a right dual of X. We say that X has duals if X has both a left- and a right dual.

Definition. Let C be a monoidal right exact abelian category over K. We say that C has enough duals if any object of C is a quotient of an object having duals.

Proposition 3 Let C be a monoidal right exact abelian category over K and let ω : C → K- mod be a K-linear faithful monoidal functor.

1) Let X be an object of C. If ω(X) is projective, both ? ⊗ X and X⊗? are exact functors. If X has a left dual in C, ω(X) is projective.

2) Consider a morphism f : X → Y, and let Z = coker(f). Assume that X and Y have left duals ^{∨}X, ^{∨}Y, and that ω(Z) is projective. Then Z has a left dual.

3) If C has enough duals, any object X such that ω(X) is projective has both a left- and a right dual.

Proof. If for some object T in C, ω(T) is a projective K-module we say for short that T is projective over K.

1) As a monoidal functor, ω preserves duals, and an object of K- mod has a dual if and only if it is projective, hence the second assertion. The first results from the fact that ω is faithful exact and projective modules are flat.

2) We have ω(^{∨}X) ≃ ω(X)^{∗}, ω(^{∨}Y) ≃ ω(Y)^{∗}, and ω(^{∨}f) = ω(f)^{∗} via these isomorphisms. Since Z is projective over K, ω(Y) ↠ ω(Z) has a section. Therefore ω(Z′) ↪ ω(^{∨}Y) is a direct summand. In particular Z′ is projective over K. Moreover, ω(Z′) is canonically isomorphic to ω(Z)^{∗}. All that remains to see is that the evaluation and coevaluation morphisms ω(Z) ⊗ ω(Z′) → K and K → ω(Z′) ⊗ ω(Z) lift to an evaluation and a coevaluation Z ⊗ Z′ → I, I → Z′ ⊗ Z. This is an easy diagram-chase, using assertion 1) and the fact that all objects involved are projective over K.

3) Let Z be projective over Z. If C has enough duals, there exists a short exact sequence X → Y → Z → 0, where X, Y have duals. By (2), Z has duals. □

Corollary 3 Assume that K satisfies (⋆). Let C be a monoidal right exact abelian category over K having enough duals, and let ω : C → K- mod be a K-linear faithful monoidal functor. Then End(ω) is a Hopf algebra in Pro - mod, and \overline{ω} : C → C(ω) is a K-linear monoidal equivalence.

Proof. Let C_{0} be the full subcategory of C of objects having duals, and ω_{0} = ω_{|C_{0}}. By assumption, any object of C is a cokernel of a morphism of C_{0}. The canonical bialgebra morphism

End(ω) → End(ω_{0})

is therefore an isomorphsm. Now C_{0} is a monoidal category, and ω_{0} is a monoidal functor satisfying P1 and P2 as a quiver representation, so End(ω_{0}) is a bialgebra by Th. 3. Moreover duals exist in C_{0}, so by standard tannakian theory, End(ω_{|C_{0}}) admits a bijective antipode : it is a Hopf bialgebra. Consider the commutative square :

[figure: C_{0}; ^{\overline{ω}^{0}}; incl.; C(ω_{0}); can.; C; \overline{ω}; C(ω)]

Since End(ω) ⟶^{∼} End(ω_{0}), the canonical functor C(ω_{0}) → C(ω) is an equivalence. By Th. 3, C(ω_{0}) is a monoidal right exact abelian category over K, so C(ω) inherits such a structure. By corollary 1, \overline{ω} is a K-linear equivalence. We only have to check that \overline{ω} is monoidal, in other words we need an isomorphism Φ_{X,Y} : \overline{ω}(X) ⊗ \overline{ω}(Y) ≃ \overline{ω}(X ⊗ Y). By construction we have such an isomorphism for X, Y in C_{0}; for arbitrary X, Y we have ω(X) ⊗ ω(Y) ≃ ω(X ⊗ Y). Using lemma 6 we can lift this to define Φ_{X,Y}. □

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