##
**Catégories tannakiennes. (Tannaka categories).**
*(French)*
Zbl 0727.14010

The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. II, Prog. Math. 87, 111-195 (1990).

[For the entire collection see Zbl 0717.00009.]

This article gives P. Deligne’s long awaited version with complete proof of Saavedra Rivano’s main theorem on Tannakian categories [N. Saavedra Rivano, “Catégories tannakiennes”, Lect. Notes Math. 265 (1972; Zbl 0241.14008)]. The inaccuracies in Saavedra’s formulation are remedied, thereby justifying all results of the cited book. From the outset the notion of a groupoid acting transitively on a scheme replaces Saavedra’s more intrinsic way to describe a Tannakian category as a category of representations of a gerb. The equivalence of both methods is demonstrated. The paper consists of nine sections, the first one giving the terminology, the statement of the theorem, the introduction of the concept of (coaction of) a coalgebroid and, grosso modo, the strategy for the proof of the theorem. In sections 2 to 6 (46 pages!) one is gradually led to this proof. The theory of Tannakian categories, groupoids, comonads, the theorem of Barr-Beck and tensor products of categories are explained in some detail. - Section 7 gives a characterization of a Tannakian category over a field of characteristic zero: A tensor category \({\mathcal T}\) over a field k of characteristic zero is a Tannakian category over k (cf. below) iff every object has non-negative integer dimension. The rudiments of algebraic geometry (affine \({\mathcal T}\)-schemes, \({\mathcal T}\)-group schemes, \({\mathcal T}\)-vector bundles,...) are carried over to tensor categories \({\mathcal T}\). In section 8 the fundamental group \(\pi\) (\({\mathcal T})\) of a Tannakian category \({\mathcal T}\) over a field k is introduced: \(\pi\) (\({\mathcal T})\) is a \({\mathcal T}\)-group, and for a fibre functor \(\omega\) of \({\mathcal T}\) on a k-scheme S (cf. below) one has \(\omega(\pi({\mathcal T}))\overset \sim \rightarrow \underline{Aut}_ S^{\otimes}(\omega)\). Furthermore, it is proved (actually something more general) that an object X of \({\mathcal T}\) is a sum of copies of the unit object 1 iff \(\pi({\mathcal T})\) acts trivially on X. The last section gives an application of the theory of Tannakian categories to the theory of Picard-Vessiot. For a differential field \((K,\partial)\) with algebraically closed field of constants \(K_ 0=\{x| \partial x=0\}\) of characteristic zero and an n-th-order ordinary differential equation with coefficients in K there exists an extension \((E,\partial)\) of \((K,\partial)\) with the same field of constants \(K_ 0\) such that the differential equation admits n solutions, linearly independent over \(K_ 0\), in \((E,\partial)\). The whole exposition is virtually self-contained.

Let k be a commutative field. By a tensor category \({\mathcal T}\) over k is meant a rigid abelian k-linear \(\otimes\)-category ACU (i.e. subject to compatible associativity and commutativity constraints and with 1) satisfying \(X\otimes 1\overset \sim \rightarrow X\) and \(1\otimes X\overset \sim \rightarrow X)\) with \(k\overset \sim \rightarrow End(1)\). In particular, 1 is a simple object and \(\otimes: {\mathcal T}\times {\mathcal T}\to {\mathcal T}\) is a k-bilinear bifunctor, exact in each variable. ‘Rigid’ means that \({\mathcal T}\) has internal \(\underline{Hom}\)’s or, more precisely, for every \(X\in {\mathcal O}b({\mathcal T})\) there exists a dual \(X^{\vee}\) and morphisms \(ev: X\otimes X^{\vee}\to 1\) and \(\delta: 1\to X^{\vee}\otimes X\) such that \[ X\to^{X\otimes \delta}X\otimes X^{\vee}\otimes X\to^{ev\otimes X}X \] and \[ X^{\vee}\to^{\delta \otimes X^{\vee}}X^{\vee}\otimes X\otimes X^{\vee}\to^{X^{\vee}\otimes ev}X^{\vee} \] are the identity. For a k-scheme S, an exact k-linear \(\otimes\)-functor \(\omega: {\mathcal T}\to {\mathcal Q}coh_ S\), where \({\mathcal Q}coh_ S\) is the category of quasi- coherent sheaves on S, is called a fibre functor on S. In fact, \(\omega\) takes values in the category of locally free sheaves of finite rank on S, and it commutes with taking duals. If S is non-empty, \(\omega\) is also faithful. A fibre functor \(\omega\) on S extends to a \(\otimes\)- functor, also written \(\omega\), of the category of Ind-objects of \({\mathcal T}\), \(\omega: Ind{\mathcal T}\to {\mathcal Q}coh_ S\). For \(u: T\to S\), \(\omega_ T=u^*\omega\) is a fibre functor on T. For two fibre functors \(\omega_ 1\) and \(\omega_ 2\) on S one writes \(\underline{Isom}_ S^{\otimes}(\omega_ 1,\omega_ 2)\) for the functor that, with every \(u: T\to S\), associates the set of isomorphisms of fibre functors \(u^*\omega_ 1\overset \sim \rightarrow u^*\omega_ 2\). It is representable by an affine scheme over S, also denoted by \(\underline{Isom}_ S^{\otimes}(\omega_ 1,\omega_ 2)\). For fibre functors \(\omega_ i\) on schemes \(S_ i\), \(i=1,2\), one writes \(\underline{Isom}_ k^{\otimes}(\omega_ 2,\omega_ 1)\) for \(\underline{Isom}^{\otimes}_{S_ 1\times S_ 2}(pr^*_ 2\omega_ 2,pr^*_ 1\omega_ 1)\). For a fibre functor \(\omega\) on S, one writes \(\underline{Aut}_ S^{\otimes}(\omega)=\underline{Isom}_ S^{\otimes}(\omega,\omega)\) and \(\underline{Aut}_ k^{\otimes}(\omega)=\underline{Isom}_ k^{\otimes}(\omega,\omega)\). We use the same notation for the affine schemes representing these functors. If the tensor category \({\mathcal T}\) over k admits a fibre functor on a nonempty scheme S, \({\mathcal T}\) is called a Tannakian category over k. A Tannakian category satisfies

(*): its objects have finite length and the Hom’s are finite dimensional k-vector spaces.

A k-groupoid acting on the k-scheme S is a k-scheme G equipped with two morphisms (‘target’ and ‘source’) b,s: \(G\to S\) and a composition law \(\circ: G\times G\to G\) which is a morphism of schemes over \(S\times S\) and such that, for any k-scheme T, these data define a category (‘groupoid’) with objects \(S(T)=Hom(T,S)\) and arrows \(G(T)=Hom(T,G)\) such that every arrow is invertible. G is said to act transitively on S (for the fpqc-topology) if there exists T, faithfully flat and quasi-compact over \(S\times S\), with \(Hom_{S\times S}(T,G)\neq \emptyset\). A representation of G is a quasi-coherent sheaf \({\mathcal V}\) on S with G-action, i.e. a morphism \(\rho\) (g): \({\mathcal V}_{s(g)}\to {\mathcal V}_{b(g)}\), \(g\in G(T)\), between the inverse images of \({\mathcal V}\) under s(g),b(g): \(T\to S\) for any k-scheme T. One assumes the \(\rho\) (g) to be compatible with base change and to satisfy some natural conditions to make them isomorphisms. For a non-empty k-scheme S with transitive G- action one writes Rep(S:G) for the tensor category of locally free sheaves of finite rank on S with G-action. For u: \(T\to S\) one has an induced groupoid \(G_ T\) on T and an equivalence of categories \(Rep(S:G)\overset \sim \rightarrow Rep(T:G_ T)\). For a tensor category \({\mathcal T}\) with fibre functor \(\omega\) on S, the scheme \(\underline{Aut}_ k^{\otimes}(\omega)\) is a k-groupoid acting on S. The main theorem of the article can now be stated:

Let \({\mathcal T}\) be a tensor category over k with fibre functor \(\omega\) on the nonempty k-scheme S. Then:

(i) the groupoid \(\underline{Aut}_ k^{\otimes}(\omega)\) is faithfully flat on \(S\times S;\)

(ii) \(\omega\) induces an equivalence \({\mathcal T}\overset \sim \rightarrow Rep(S:\underline{Aut}_ k^{\otimes}(\omega)).\)

Conversely, let G be a k-groupoid acting on \(S\neq \emptyset\), affine and faithfully flat on \(S\times S\), and let \(\omega\) be the ‘forgetful’ fibre functor of Rep(S:G) on S, then:

(iii) \(G\overset \sim \rightarrow \underline{Aut}_ k^{\otimes}(\omega). (break?) \)

It suffices to consider the case of affine \(S=Spec(B)\) for a k-algebra B. Then \(\omega\) takes values in \(\Pr oj_ B\), the category of projective right B-modules of finite type. Now the proof can be divided into several steps:

A. Preliminaries. 1. Let R be a commutative ring, and let \(B_ 1\) and \(B_ 2\) be two R-algebras. For a (small) category \({\mathcal C}\) with functors \(\omega_ i: {\mathcal C}\to \Pr oj_{B_ i}\), \(i=1,2\), one has the coend \(L_ R(\omega_ 1,\omega_ 2)\). It is a \((B_ 1,B_ 2)\)- bimodule, whose induced R-module structures coincide, and which is equipped with a morphism of bimodules \(\omega_ 1(X)^{\vee}\otimes_ R\omega_ 2(X)\to L_ R(\omega_ 1,\omega_ 2)\), i.e. a morphism of \(B_ 2\)-modules \(\omega_ 2(X)\to \omega_ 1(X)\otimes_{B_ 1}L_ R(\omega_ 1,\omega_ 2)\), functorial in X, for all \(X\in {\mathcal O}b({\mathcal C})\). In particular, for \(B_ 1=B_ 2=B\) and \(\omega_ 1=\omega_ 2=\omega\), \(L_ R(\omega)=L_ R(\omega,\omega)\) becomes a R-coalgebroid acting on B. One has a coaction of \(L_ R(\omega)\) on the \(\omega\) (X), \(\omega (X)\mapsto \omega (X)\otimes_ BL_ R(\omega)\), functorial in X. \(L_ R(\omega)\) is called the coalgebroid of R- endomorphisms of \(\omega\).

2. If \({\mathcal C}\) is a \(\otimes\)-category ACU and \(\omega_ 1,\omega_ 2: {\mathcal C}\to \Pr oj_ R\) are \(\otimes\)-functors, then one has a product \(L_ R(\omega_ 1,\omega_ 2)\otimes_ RL_ R(\omega_ 1,\omega_ 2)\to L_ R(\omega_ 1,\omega_ 2)\) and \(L_ R(\omega_ 1,\omega_ 2)\) becomes a commutative R-algebra. The functor \(\underline{Hom}_ S^{\otimes}(\omega_ 2,\omega_ 1)\), \(S=Spec(R)\), is represented by the affine scheme \(Spec(L_ R(\omega_ 1,\omega_ 2))\). For a \(\otimes\)-category ACU \({\mathcal C}\) over k, commutative k- algebras \(B_ 1\) and \(B_ 2\), and \(\otimes\)-functors \(\omega_ i: {\mathcal C}\to \Pr oj_{B_ i}\), \(i=1,2\), extension of scalars, written \(\omega_ 1\mapsto \omega_ 1\otimes 1 (\omega_ 2\mapsto 1\otimes \omega_ 2)\), turns the \(\omega_ i\) into \(\otimes\)-functors with values in \(\Pr oj_{B_ 1\otimes_ kB_ 2}\). One has: \(L_ k(\omega_ 1,\omega_ 2)=L_{B_ 1\otimes_ kB_ 2}(\omega_ 1\otimes 1,1\otimes \omega_ 2)\) and \(Spec(L_ k(\omega_ 1,\omega_ 2))\) represents the functor \(\underline{Hom}_ k^{\otimes}(\omega_ 2,\omega_ 1)=\underline{Isom}_ k^{\otimes}(\omega_ 2,\omega_ 1)\). Thus \(\underline{Aut}_ k^{\otimes}(\omega)\) is represented by \(Spec(L_ k(\omega))\). In particular, for \({\mathcal C}={\mathcal T}\) a tensor category over k, with fibre functor \(\omega\) on \(S=Spec(B)\), B a k- algebra, the action of \(\underline{Aut}_ k^{\otimes}(\omega)\) on the \(\omega(X)\) is given by the coaction of \(L_ k(\omega)\), \(\omega(X)\mapsto \omega(X) \otimes_ BL_ k(\omega)\), and the composition law of the groupoid \(\underline{Aut}_ k^{\otimes}(\omega)\) is defined by the comultiplication of the coalgebroid \(L_ k(\omega).\)

3. Let \({\mathcal A}\) be a k-linear abelian category satisfying (*) and let \(\omega: {\mathcal A}\to Proj_ B\), B a k-algebra, be a k-linear exact faithful functor. \({\mathcal A}\) is the filtered union of full abelian subcategories \(<X>\), \(X\in {\mathcal O}b({\mathcal A})\), where the objects of \(<X>\) are subquotients of the \(X^ n\), and \(L_ k(\omega)\) is the inductive limit of the \(L_ k(\omega | <X>)\). The following result is due to O. Gabber: For \(X\in {\mathcal O}b({\mathcal A})\), \(<X>\) admits a projective generator, say P. The functor \(Y\mapsto Hom(P,Y)\) gives an equivalence of \(<X>\) with the category of right modules of finite type over the k-algebra \(A=End(P)\). Write \({}_ AM_ B=\omega(P)\). Then \({}_ AM_ B\) is an \((A,B)\)-bimodule, projective of finite type over B and faithfully flat over A. One proves \(L_ k(\omega | <X>)\overset \sim \rightarrow_ BM_ A^{\vee}\otimes_ A{}_ AM_ B\), and the theorem of Barr-Beck says that \(\omega\) induces an equivalence between \(<X>\) and the category of right B-modules of finite type with a coaction of \(L_ k(\omega | <X>)\). A limit argument leads to the result: \(\omega\) induces an equivalence between \({\mathcal A}\) and the category of right B-modules of finite type with coaction of \(L_ k(\omega).\)

4. One introduces the notion of a tensor product \(\otimes {\mathcal A}_ i \) of a finite family of k-linear abelian categories \(\{{\mathcal A}_ i\}_{i\in I}\). By definition, a k-multilinear functor, right exact in each variable, \(\otimes: \prod_{i}{\mathcal A}_ i \to {\mathcal A}\), where \({\mathcal A}\) is a k-linear abelian category, makes \({\mathcal A}\) into a tensor product over k of the \({\mathcal A}_ i\) if, for any k-linear abelian category \({\mathcal C}\), the category of right exact functors from \({\mathcal A}\) to \({\mathcal C}\) is equivalent to the category of multilinear functors, right exact in each variable, from \(\prod_{i}{\mathcal A}_ i \) to \({\mathcal C}\). As a matter of fact, if such a product exists it is unique (up to isomorphism). If the \({\mathcal A}_ i\) satisfy (*), then the tensor product exists and has nice properties. For a finite family \(\{{\mathcal T}_ i\}_{i\in I}\) of Tannakian categories over k the tensor product \(\otimes {\mathcal T}_ i \) over k is a Tannakian category over k. It satisfies (*).

B. A.2. and A.3. above now imply part (ii) of the theorem.

C. For part (i) one proceeds as follows: for any (small) category \({\mathcal C}\), tensor category \({\mathcal T}\) over k and functors \(T_ 1,T_ 2: {\mathcal C}\to {\mathcal T}\), one constructs an Ind-object \(\Lambda_{{\mathcal T}}(T_ 1,T_ 2)\) of \({\mathcal T}\) with a morphism \(T_ 1(X)^{\vee}\otimes T_ 2(X)\to \Lambda_{{\mathcal T}}(T_ 1,T_ 2)\) for all \(X\in {\mathcal O}b({\mathcal C})\), in analogy with the coend construction of A.1. In case \({\mathcal T}={\mathcal T}_ 1\otimes {\mathcal T}_ 2\) is the tensor product (cf. A.4. above) over k of two tensor categories \({\mathcal T}_ 1\) and \({\mathcal T}_ 2\) over k, and \(T_ i: {\mathcal C}\to {\mathcal T}_ i(i=1,2)\) are functors, one writes \(\Lambda_ k(T_ 1,T_ 2)=\Lambda_{{\mathcal T}}(inj_ 1T_ 1,inj_ 2T_ 2)\), where the \(inj_ i\) are the injections of the \({\mathcal T}_ i\) in \({\mathcal T}\), i.e. \(inj_ 1: X\mapsto X\otimes 1\) and \(inj_ 2: X\mapsto 1\otimes X\). If the \({\mathcal T}_ i\) admit fibre functors \(\omega_ i\) on \(Spec(B_ i)\), \(B_ i\) commutative k-algebras, then the functor \((X,Y)\mapsto \omega_ 1(X)\otimes_ k\omega_ 2(Y)\) factors over a fibre functor \(\omega\) of \({\mathcal T}={\mathcal T}_ 1\otimes {\mathcal T}_ 2\) on \(Spec(B_ 1)\times Spec(B_ 2)\) and one has \(\omega \Lambda_ k(T_ 1,T_ 2)=L_ k(\omega_ 1T_ 1,\omega_ 2T_ 2)\). Apply this to the situation with \({\mathcal C}={\mathcal T}_ 1={\mathcal T}_ 2={\mathcal T}\) and \(T_ i\) the identity \(Id_{{\mathcal T}}\). This gives an Ind-object \(\Lambda =\Lambda_ k(Id_{{\mathcal T}},Id_{{\mathcal T}})\) of \({\mathcal T}\otimes {\mathcal T}\). One proves that \(\Lambda\neq 0\). A general result on Tannakian categories says that, if X is a nonzero Ind-object of a Tannakian category over k with fibre functor \(\omega\) on the k-scheme S, \(\omega\) (X) is faithfully flat on S. This implies (i) of the theorem.

D. Finally, for part (iii), let G be a groupoid acting transitively on S and affine on \(S\times S\), say \(G=Spec(L)\), where L is a \(B\otimes_ kB\)-module. The composition law of G makes L into a k-coalgebroid acting on B. It is sufficient to prove the isomorphism \(G\overset \sim \rightarrow \underline{Aut}_ k^{\otimes}(\omega)\) for one fibre: For \(T\to S\), \(T\neq 0\), \(Rep(S:G)\overset \sim \rightarrow Rep(T:G_ T)\) with forgetful functor \(\omega_ T\) on T and induced groupoid \(\underline{Aut}_ k^{\otimes}(\omega_ T)\) on T. Taking for T a point of S one may suppose T to be the spectrum of a field and the result follows from the fact that in this situation one can show that there is an isomorhism \(L_ k(\omega_ T)\overset \sim \rightarrow L\). This concludes the proof of the theorem.

This article gives P. Deligne’s long awaited version with complete proof of Saavedra Rivano’s main theorem on Tannakian categories [N. Saavedra Rivano, “Catégories tannakiennes”, Lect. Notes Math. 265 (1972; Zbl 0241.14008)]. The inaccuracies in Saavedra’s formulation are remedied, thereby justifying all results of the cited book. From the outset the notion of a groupoid acting transitively on a scheme replaces Saavedra’s more intrinsic way to describe a Tannakian category as a category of representations of a gerb. The equivalence of both methods is demonstrated. The paper consists of nine sections, the first one giving the terminology, the statement of the theorem, the introduction of the concept of (coaction of) a coalgebroid and, grosso modo, the strategy for the proof of the theorem. In sections 2 to 6 (46 pages!) one is gradually led to this proof. The theory of Tannakian categories, groupoids, comonads, the theorem of Barr-Beck and tensor products of categories are explained in some detail. - Section 7 gives a characterization of a Tannakian category over a field of characteristic zero: A tensor category \({\mathcal T}\) over a field k of characteristic zero is a Tannakian category over k (cf. below) iff every object has non-negative integer dimension. The rudiments of algebraic geometry (affine \({\mathcal T}\)-schemes, \({\mathcal T}\)-group schemes, \({\mathcal T}\)-vector bundles,...) are carried over to tensor categories \({\mathcal T}\). In section 8 the fundamental group \(\pi\) (\({\mathcal T})\) of a Tannakian category \({\mathcal T}\) over a field k is introduced: \(\pi\) (\({\mathcal T})\) is a \({\mathcal T}\)-group, and for a fibre functor \(\omega\) of \({\mathcal T}\) on a k-scheme S (cf. below) one has \(\omega(\pi({\mathcal T}))\overset \sim \rightarrow \underline{Aut}_ S^{\otimes}(\omega)\). Furthermore, it is proved (actually something more general) that an object X of \({\mathcal T}\) is a sum of copies of the unit object 1 iff \(\pi({\mathcal T})\) acts trivially on X. The last section gives an application of the theory of Tannakian categories to the theory of Picard-Vessiot. For a differential field \((K,\partial)\) with algebraically closed field of constants \(K_ 0=\{x| \partial x=0\}\) of characteristic zero and an n-th-order ordinary differential equation with coefficients in K there exists an extension \((E,\partial)\) of \((K,\partial)\) with the same field of constants \(K_ 0\) such that the differential equation admits n solutions, linearly independent over \(K_ 0\), in \((E,\partial)\). The whole exposition is virtually self-contained.

Let k be a commutative field. By a tensor category \({\mathcal T}\) over k is meant a rigid abelian k-linear \(\otimes\)-category ACU (i.e. subject to compatible associativity and commutativity constraints and with 1) satisfying \(X\otimes 1\overset \sim \rightarrow X\) and \(1\otimes X\overset \sim \rightarrow X)\) with \(k\overset \sim \rightarrow End(1)\). In particular, 1 is a simple object and \(\otimes: {\mathcal T}\times {\mathcal T}\to {\mathcal T}\) is a k-bilinear bifunctor, exact in each variable. ‘Rigid’ means that \({\mathcal T}\) has internal \(\underline{Hom}\)’s or, more precisely, for every \(X\in {\mathcal O}b({\mathcal T})\) there exists a dual \(X^{\vee}\) and morphisms \(ev: X\otimes X^{\vee}\to 1\) and \(\delta: 1\to X^{\vee}\otimes X\) such that \[ X\to^{X\otimes \delta}X\otimes X^{\vee}\otimes X\to^{ev\otimes X}X \] and \[ X^{\vee}\to^{\delta \otimes X^{\vee}}X^{\vee}\otimes X\otimes X^{\vee}\to^{X^{\vee}\otimes ev}X^{\vee} \] are the identity. For a k-scheme S, an exact k-linear \(\otimes\)-functor \(\omega: {\mathcal T}\to {\mathcal Q}coh_ S\), where \({\mathcal Q}coh_ S\) is the category of quasi- coherent sheaves on S, is called a fibre functor on S. In fact, \(\omega\) takes values in the category of locally free sheaves of finite rank on S, and it commutes with taking duals. If S is non-empty, \(\omega\) is also faithful. A fibre functor \(\omega\) on S extends to a \(\otimes\)- functor, also written \(\omega\), of the category of Ind-objects of \({\mathcal T}\), \(\omega: Ind{\mathcal T}\to {\mathcal Q}coh_ S\). For \(u: T\to S\), \(\omega_ T=u^*\omega\) is a fibre functor on T. For two fibre functors \(\omega_ 1\) and \(\omega_ 2\) on S one writes \(\underline{Isom}_ S^{\otimes}(\omega_ 1,\omega_ 2)\) for the functor that, with every \(u: T\to S\), associates the set of isomorphisms of fibre functors \(u^*\omega_ 1\overset \sim \rightarrow u^*\omega_ 2\). It is representable by an affine scheme over S, also denoted by \(\underline{Isom}_ S^{\otimes}(\omega_ 1,\omega_ 2)\). For fibre functors \(\omega_ i\) on schemes \(S_ i\), \(i=1,2\), one writes \(\underline{Isom}_ k^{\otimes}(\omega_ 2,\omega_ 1)\) for \(\underline{Isom}^{\otimes}_{S_ 1\times S_ 2}(pr^*_ 2\omega_ 2,pr^*_ 1\omega_ 1)\). For a fibre functor \(\omega\) on S, one writes \(\underline{Aut}_ S^{\otimes}(\omega)=\underline{Isom}_ S^{\otimes}(\omega,\omega)\) and \(\underline{Aut}_ k^{\otimes}(\omega)=\underline{Isom}_ k^{\otimes}(\omega,\omega)\). We use the same notation for the affine schemes representing these functors. If the tensor category \({\mathcal T}\) over k admits a fibre functor on a nonempty scheme S, \({\mathcal T}\) is called a Tannakian category over k. A Tannakian category satisfies

(*): its objects have finite length and the Hom’s are finite dimensional k-vector spaces.

A k-groupoid acting on the k-scheme S is a k-scheme G equipped with two morphisms (‘target’ and ‘source’) b,s: \(G\to S\) and a composition law \(\circ: G\times G\to G\) which is a morphism of schemes over \(S\times S\) and such that, for any k-scheme T, these data define a category (‘groupoid’) with objects \(S(T)=Hom(T,S)\) and arrows \(G(T)=Hom(T,G)\) such that every arrow is invertible. G is said to act transitively on S (for the fpqc-topology) if there exists T, faithfully flat and quasi-compact over \(S\times S\), with \(Hom_{S\times S}(T,G)\neq \emptyset\). A representation of G is a quasi-coherent sheaf \({\mathcal V}\) on S with G-action, i.e. a morphism \(\rho\) (g): \({\mathcal V}_{s(g)}\to {\mathcal V}_{b(g)}\), \(g\in G(T)\), between the inverse images of \({\mathcal V}\) under s(g),b(g): \(T\to S\) for any k-scheme T. One assumes the \(\rho\) (g) to be compatible with base change and to satisfy some natural conditions to make them isomorphisms. For a non-empty k-scheme S with transitive G- action one writes Rep(S:G) for the tensor category of locally free sheaves of finite rank on S with G-action. For u: \(T\to S\) one has an induced groupoid \(G_ T\) on T and an equivalence of categories \(Rep(S:G)\overset \sim \rightarrow Rep(T:G_ T)\). For a tensor category \({\mathcal T}\) with fibre functor \(\omega\) on S, the scheme \(\underline{Aut}_ k^{\otimes}(\omega)\) is a k-groupoid acting on S. The main theorem of the article can now be stated:

Let \({\mathcal T}\) be a tensor category over k with fibre functor \(\omega\) on the nonempty k-scheme S. Then:

(i) the groupoid \(\underline{Aut}_ k^{\otimes}(\omega)\) is faithfully flat on \(S\times S;\)

(ii) \(\omega\) induces an equivalence \({\mathcal T}\overset \sim \rightarrow Rep(S:\underline{Aut}_ k^{\otimes}(\omega)).\)

Conversely, let G be a k-groupoid acting on \(S\neq \emptyset\), affine and faithfully flat on \(S\times S\), and let \(\omega\) be the ‘forgetful’ fibre functor of Rep(S:G) on S, then:

(iii) \(G\overset \sim \rightarrow \underline{Aut}_ k^{\otimes}(\omega). (break?) \)

It suffices to consider the case of affine \(S=Spec(B)\) for a k-algebra B. Then \(\omega\) takes values in \(\Pr oj_ B\), the category of projective right B-modules of finite type. Now the proof can be divided into several steps:

A. Preliminaries. 1. Let R be a commutative ring, and let \(B_ 1\) and \(B_ 2\) be two R-algebras. For a (small) category \({\mathcal C}\) with functors \(\omega_ i: {\mathcal C}\to \Pr oj_{B_ i}\), \(i=1,2\), one has the coend \(L_ R(\omega_ 1,\omega_ 2)\). It is a \((B_ 1,B_ 2)\)- bimodule, whose induced R-module structures coincide, and which is equipped with a morphism of bimodules \(\omega_ 1(X)^{\vee}\otimes_ R\omega_ 2(X)\to L_ R(\omega_ 1,\omega_ 2)\), i.e. a morphism of \(B_ 2\)-modules \(\omega_ 2(X)\to \omega_ 1(X)\otimes_{B_ 1}L_ R(\omega_ 1,\omega_ 2)\), functorial in X, for all \(X\in {\mathcal O}b({\mathcal C})\). In particular, for \(B_ 1=B_ 2=B\) and \(\omega_ 1=\omega_ 2=\omega\), \(L_ R(\omega)=L_ R(\omega,\omega)\) becomes a R-coalgebroid acting on B. One has a coaction of \(L_ R(\omega)\) on the \(\omega\) (X), \(\omega (X)\mapsto \omega (X)\otimes_ BL_ R(\omega)\), functorial in X. \(L_ R(\omega)\) is called the coalgebroid of R- endomorphisms of \(\omega\).

2. If \({\mathcal C}\) is a \(\otimes\)-category ACU and \(\omega_ 1,\omega_ 2: {\mathcal C}\to \Pr oj_ R\) are \(\otimes\)-functors, then one has a product \(L_ R(\omega_ 1,\omega_ 2)\otimes_ RL_ R(\omega_ 1,\omega_ 2)\to L_ R(\omega_ 1,\omega_ 2)\) and \(L_ R(\omega_ 1,\omega_ 2)\) becomes a commutative R-algebra. The functor \(\underline{Hom}_ S^{\otimes}(\omega_ 2,\omega_ 1)\), \(S=Spec(R)\), is represented by the affine scheme \(Spec(L_ R(\omega_ 1,\omega_ 2))\). For a \(\otimes\)-category ACU \({\mathcal C}\) over k, commutative k- algebras \(B_ 1\) and \(B_ 2\), and \(\otimes\)-functors \(\omega_ i: {\mathcal C}\to \Pr oj_{B_ i}\), \(i=1,2\), extension of scalars, written \(\omega_ 1\mapsto \omega_ 1\otimes 1 (\omega_ 2\mapsto 1\otimes \omega_ 2)\), turns the \(\omega_ i\) into \(\otimes\)-functors with values in \(\Pr oj_{B_ 1\otimes_ kB_ 2}\). One has: \(L_ k(\omega_ 1,\omega_ 2)=L_{B_ 1\otimes_ kB_ 2}(\omega_ 1\otimes 1,1\otimes \omega_ 2)\) and \(Spec(L_ k(\omega_ 1,\omega_ 2))\) represents the functor \(\underline{Hom}_ k^{\otimes}(\omega_ 2,\omega_ 1)=\underline{Isom}_ k^{\otimes}(\omega_ 2,\omega_ 1)\). Thus \(\underline{Aut}_ k^{\otimes}(\omega)\) is represented by \(Spec(L_ k(\omega))\). In particular, for \({\mathcal C}={\mathcal T}\) a tensor category over k, with fibre functor \(\omega\) on \(S=Spec(B)\), B a k- algebra, the action of \(\underline{Aut}_ k^{\otimes}(\omega)\) on the \(\omega(X)\) is given by the coaction of \(L_ k(\omega)\), \(\omega(X)\mapsto \omega(X) \otimes_ BL_ k(\omega)\), and the composition law of the groupoid \(\underline{Aut}_ k^{\otimes}(\omega)\) is defined by the comultiplication of the coalgebroid \(L_ k(\omega).\)

3. Let \({\mathcal A}\) be a k-linear abelian category satisfying (*) and let \(\omega: {\mathcal A}\to Proj_ B\), B a k-algebra, be a k-linear exact faithful functor. \({\mathcal A}\) is the filtered union of full abelian subcategories \(<X>\), \(X\in {\mathcal O}b({\mathcal A})\), where the objects of \(<X>\) are subquotients of the \(X^ n\), and \(L_ k(\omega)\) is the inductive limit of the \(L_ k(\omega | <X>)\). The following result is due to O. Gabber: For \(X\in {\mathcal O}b({\mathcal A})\), \(<X>\) admits a projective generator, say P. The functor \(Y\mapsto Hom(P,Y)\) gives an equivalence of \(<X>\) with the category of right modules of finite type over the k-algebra \(A=End(P)\). Write \({}_ AM_ B=\omega(P)\). Then \({}_ AM_ B\) is an \((A,B)\)-bimodule, projective of finite type over B and faithfully flat over A. One proves \(L_ k(\omega | <X>)\overset \sim \rightarrow_ BM_ A^{\vee}\otimes_ A{}_ AM_ B\), and the theorem of Barr-Beck says that \(\omega\) induces an equivalence between \(<X>\) and the category of right B-modules of finite type with a coaction of \(L_ k(\omega | <X>)\). A limit argument leads to the result: \(\omega\) induces an equivalence between \({\mathcal A}\) and the category of right B-modules of finite type with coaction of \(L_ k(\omega).\)

4. One introduces the notion of a tensor product \(\otimes {\mathcal A}_ i \) of a finite family of k-linear abelian categories \(\{{\mathcal A}_ i\}_{i\in I}\). By definition, a k-multilinear functor, right exact in each variable, \(\otimes: \prod_{i}{\mathcal A}_ i \to {\mathcal A}\), where \({\mathcal A}\) is a k-linear abelian category, makes \({\mathcal A}\) into a tensor product over k of the \({\mathcal A}_ i\) if, for any k-linear abelian category \({\mathcal C}\), the category of right exact functors from \({\mathcal A}\) to \({\mathcal C}\) is equivalent to the category of multilinear functors, right exact in each variable, from \(\prod_{i}{\mathcal A}_ i \) to \({\mathcal C}\). As a matter of fact, if such a product exists it is unique (up to isomorphism). If the \({\mathcal A}_ i\) satisfy (*), then the tensor product exists and has nice properties. For a finite family \(\{{\mathcal T}_ i\}_{i\in I}\) of Tannakian categories over k the tensor product \(\otimes {\mathcal T}_ i \) over k is a Tannakian category over k. It satisfies (*).

B. A.2. and A.3. above now imply part (ii) of the theorem.

C. For part (i) one proceeds as follows: for any (small) category \({\mathcal C}\), tensor category \({\mathcal T}\) over k and functors \(T_ 1,T_ 2: {\mathcal C}\to {\mathcal T}\), one constructs an Ind-object \(\Lambda_{{\mathcal T}}(T_ 1,T_ 2)\) of \({\mathcal T}\) with a morphism \(T_ 1(X)^{\vee}\otimes T_ 2(X)\to \Lambda_{{\mathcal T}}(T_ 1,T_ 2)\) for all \(X\in {\mathcal O}b({\mathcal C})\), in analogy with the coend construction of A.1. In case \({\mathcal T}={\mathcal T}_ 1\otimes {\mathcal T}_ 2\) is the tensor product (cf. A.4. above) over k of two tensor categories \({\mathcal T}_ 1\) and \({\mathcal T}_ 2\) over k, and \(T_ i: {\mathcal C}\to {\mathcal T}_ i(i=1,2)\) are functors, one writes \(\Lambda_ k(T_ 1,T_ 2)=\Lambda_{{\mathcal T}}(inj_ 1T_ 1,inj_ 2T_ 2)\), where the \(inj_ i\) are the injections of the \({\mathcal T}_ i\) in \({\mathcal T}\), i.e. \(inj_ 1: X\mapsto X\otimes 1\) and \(inj_ 2: X\mapsto 1\otimes X\). If the \({\mathcal T}_ i\) admit fibre functors \(\omega_ i\) on \(Spec(B_ i)\), \(B_ i\) commutative k-algebras, then the functor \((X,Y)\mapsto \omega_ 1(X)\otimes_ k\omega_ 2(Y)\) factors over a fibre functor \(\omega\) of \({\mathcal T}={\mathcal T}_ 1\otimes {\mathcal T}_ 2\) on \(Spec(B_ 1)\times Spec(B_ 2)\) and one has \(\omega \Lambda_ k(T_ 1,T_ 2)=L_ k(\omega_ 1T_ 1,\omega_ 2T_ 2)\). Apply this to the situation with \({\mathcal C}={\mathcal T}_ 1={\mathcal T}_ 2={\mathcal T}\) and \(T_ i\) the identity \(Id_{{\mathcal T}}\). This gives an Ind-object \(\Lambda =\Lambda_ k(Id_{{\mathcal T}},Id_{{\mathcal T}})\) of \({\mathcal T}\otimes {\mathcal T}\). One proves that \(\Lambda\neq 0\). A general result on Tannakian categories says that, if X is a nonzero Ind-object of a Tannakian category over k with fibre functor \(\omega\) on the k-scheme S, \(\omega\) (X) is faithfully flat on S. This implies (i) of the theorem.

D. Finally, for part (iii), let G be a groupoid acting transitively on S and affine on \(S\times S\), say \(G=Spec(L)\), where L is a \(B\otimes_ kB\)-module. The composition law of G makes L into a k-coalgebroid acting on B. It is sufficient to prove the isomorphism \(G\overset \sim \rightarrow \underline{Aut}_ k^{\otimes}(\omega)\) for one fibre: For \(T\to S\), \(T\neq 0\), \(Rep(S:G)\overset \sim \rightarrow Rep(T:G_ T)\) with forgetful functor \(\omega_ T\) on T and induced groupoid \(\underline{Aut}_ k^{\otimes}(\omega_ T)\) on T. Taking for T a point of S one may suppose T to be the spectrum of a field and the result follows from the fact that in this situation one can show that there is an isomorhism \(L_ k(\omega_ T)\overset \sim \rightarrow L\). This concludes the proof of the theorem.

Reviewer: W.W.J.Hulsbergen (Breda)

### MSC:

14F99 | (Co)homology theory in algebraic geometry |

14L15 | Group schemes |

18B40 | Groupoids, semigroupoids, semigroups, groups (viewed as categories) |

14E20 | Coverings in algebraic geometry |

20L05 | Groupoids (i.e. small categories in which all morphisms are isomorphisms) |

13N10 | Commutative rings of differential operators and their modules |