Tannakian duality over Dedekind rings and applications
aa r X i v : . [ m a t h . AG ] M a y TANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS
NGUYEN DAI DUONG AND PH`UNG H ˆO HAI
Dedicated to H´el`ene Esnault, with admiration and affection A BSTRACT . We establish a duality between flat affine group schemes and rigid tensor categoriesequipped with a neutral fiber functor (called Tannakian lattice), both defined over a Dedekind ring.We use this duality and the known Tannakian duality due to Saavedra to study morphisms betweenflat affine group schemes. Next, we apply our new duality to the category of stratified sheaves ona smooth scheme over a Dedekind ring R to define the relative differential fundamental groupscheme of the given scheme and compare the fibers of this group scheme with the fundamentalgroup scheme of the fibers. When R is a complete DVR of equal characteristic we show that thiscategory is Tannakian in the sense of Saavedra. I NTRODUCTION
Tannakian duality for group scheme over a field was studied by Saavedra [21]. The dualityin the neutral case, as shown by Saavedra, is a dictionary between k -linear abelian rigid tensorcategories equipped with a fiber functor to the category of k -vector spaces and affine groupschemes over k . The duality consists of two parts:- The reconstruction theorem which recovers a group scheme from a neutral Tannakiancategory ( T , ω : T −→ Vect ( k )) , as the group of automorphisms of ω preserving thetensor product, the Tannakian group of ( T , ω ) .- The presentation (or description) theorem which claims the equivalence between theoriginal category T and the representation category of the Tannakian group of T .Saavedra also extended this result to the non-neutral case - when the fiber functor goes to amore general category of coherent sheaves over a k -scheme. A complete proof of this theoremwas given by Deligne in [6].An important application of Tannakian duality is to define various fundamental group schemes.Let X be a scheme over a field k . There are certain abelian tensor categories associated to X .For example, if k is perfect, X is reduced and connected, M. Nori introduced the category ofessentially finite bundles; if X is smooth and k has characteristic zero, one has the category offlat connections on X ; if k has positive characteristic, one has the category of stratified bundles(i.e. O X -coherent modules equipped with the action of the sheaf D X/k of algebras of differ-ential operators on X ). Given a k -rational point x of X , the functor taking fibers at x makesthe above categories Tannakian category and Tannakian duality yields the corresponding affine Mathematics Subject Classification. group scheme, which is usually called the fundamental group scheme of X . Tannakian duality isalso used as an alternative approach to the Picard-Vessiot theory of linear differential equations.Let now f : X −→ S := Spec R be a smooth morphism, where R is a Dedekind ring. We areinterested in the category of modules over D ( X/S ) , which are coherent as O X -modules, where D ( X/S ) denotes the sheaf of algebras of differential operators on X/S . Such a sheaf will becalled stratified sheaf over
X/S . The category str ( X/S ) of stratified sheaves on X/S is an abeliantensor category, in which an object is rigid if it is locally free as an O X -module. Assume that f admits a section ξ : S −→ X . Then the functor ξ ∗ provides us a fiber functor for str ( X/S ) . It isthen natural to ask, if there exists a generalization of Tannakian duality to this case.In fact, Tannakian duality over Dedekind rings has already been considered by Saavedra in[21, II.2]. Saavedra gave a condition for an abelian category equipped with an exact faithfulfunctor to the module category (over a given Noetherian ring) to be equivalent to the comodulecategory over the coalgebra reconstructed from this functor. This duality is then developed to aTannakian duality for flat affine group schemes over a Dedekind ring.In many examples, the involving category may not satisfy all the properties in Saavedra’sdefinition. One therefore wants to reconstruct an affine group scheme from a smaller part of itsrepresentation category, for instance, from rigid representations (i.e. representations in finiteprojective modules). There are at least two approaches to this problem, one by Wedhorn [29],and the other by Brugui´eres [3] following an idea of M. Nori, the latter one has been used bydos Santos [23] to define the Galois differential groups of a relative stratified bundle. Wedhorn’sapproach is similar to Saavedra’s approach. He introduced the notion of Tannakian lattice andreconstructed a group scheme from such a lattice. However Wedhorn has not established a fullduality; he did not show that a Tannakian lattice is equivalent to the category of finite projectiverepresentations of a flat affine group scheme.In Section 1 we first recall the definition of the representation category of a flat affine groupscheme over a Dedekind ring. A special property of such a category, as noticed by Serre [25],is that any object can be represented as a quotient of a projective one (over the base ring).Saavedra uses this feature to characterize these representation categories (Theorem 1.2.2). Aself-contained proof of this theorem will be given in the Appendix.In Section 2 we propose a new definition of neutral Tannakian lattices. Saavedra’s method isused to establish a full duality between neutral Tannakian lattices and flat affine group schemesover a Dedekind ring (Theorem 2.3.2). As a corollary we show that a Tannakian lattice is con-tained in a unique (up to equivalence) abelian envelope (Proposition 2.4.4). We also constructa torsor of isomorphisms between two fiber functors.The Tannakian duality is used in Section 3 to study properties of homomorphisms of flat coal-gebras over a Dedekind ring. We first introduce the notions of special coalgebra homomorphismsand special subcoalgebras (Definition 3.1.1). For coalgebras over a field, one has the local finite-ness property: each coalgebra is the union of its finite dimensional subcoalgebras. The situationbecomes more complicated for flat coalgebras over a Dedekind ring. A flat coalgebra over aDedekind ring is still the union of its finite subcoalgebras, but one cannot choose the subcoal-gebras to be at the same time finite and saturated as submodules. We introduced the notion ofspecial locally finite coalgebras over a Dedekind ring, which are those representable as unionof finite saturated subcoalgebras. We show that the coordinate ring of a flat group scheme, the ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 3 generic fiber of which is reduced and connected, is specially locally finite (Proposition 3.1.7).In the second part of this section we provide conditions for a coalgebra homomorphism to be(specially) injective or surjective (Propostions 3.2.1, 3.2.5).In Section 4 the results on homomorphisms of flat coalgebras are used to study homomor-phisms of flat group schemes: to characterize closed immersions and the faithful flatness. In thelast part of Section 4, we give a criterion for the exactness of sequences of homomorphisms offlat affine group schemes over Dedekind rings in terms of Tannakian duality. Such a criterion forgroup schemes over a field has proved to be a useful tool, see [9, 11, 10, 15]. Some applicationsof the results in this section will be presented in a subsequent paper [8].In Section 5 we apply the Tannakian duality to the category str ( X/R ) of stratified sheaves overa smooth scheme X/R , where R is a Dedekind ring. We show that the subcategory str o ( X/R ) ofrelative stratified bundles is a Tannakian lattice (assuming the existence of an R -point of X ). Wethen compare the fibers of the relative fundamental group and the fundamental group of thefibers. When R is a complete local discrete valuation ring of equal characteristics, we show thatthe category of all stratified sheaves str ( X/R ) is a Tannakian category over R .In the Appendix, for the sake of the reader, we recall Saavedra’s proof of Tannakian dualityfor flat affine group schemes over a Dedekind ring.1. P RELIMINARIES
In this article R will denote a Dedekind ring . The fraction field of R is denoted by K , a residuefield will be denoted by k . The category of R -modules is denoted by Mod ( R ) , its full subcategoryof finite modules is denoted by Mod f ( R ) and the full subcategory of finite projective modulesis denoted by Mod o ( R ) . We shall intensively use the facts that over a Dedekind ring, torsionfree modules are flat and finitely generated flat modules are projective. The tensor product of R -modules, when not explicitly indicated, is understood as the tensor product over R .1.1. Flat affine group schemes.
Let G be a flat affine group scheme over R . The coordinatering of G , R [ G ] , is usually denoted by L for short. Thus L is an R -flat (commutative) Hopfalgebra.1.1.1. By definition, a G - module (or a G - representation ) is the same as a right L -comodule, i.e.,an R -module M equipped with a coaction of L : ρ : M −→ M ⊗ R L ; ( ρ ⊗ id ) ρ = ( id ⊗ ∆ ) ρ , ( id ⊗ ε ) ρ = id ,where ∆ : L −→ L ⊗ L denotes the coproduct and ε : L −→ R denotes the counit of L .The flatness of L implies that the category Comod ( L ) of (right) R -modules equipped with acoaction of L is an R - linear abelian category. We call an L -comodule M finite if it is finite asan R -module and finite projective if it is also projective over R . The full subcategory of finite L -comodules is denoted by Comod f ( L ) and the full subcategory of finite projective L -comodules isdenoted by Comod o ( L ) . Each L -comodule is the union of its finite subcomodules (see Appendix).1.1.2. Comod ( L ) is a tensor category with respect to the tensor product over R . The unit objectis R equipped with the trivial coaction of L : R −→ R ⊗ L = L , 1 ⊗
1. The subcategory
Comod o ( L ) is rigid, i.e., each objects possesses a dual. NGUYEN DAI DUONG AND PH`UNG HˆO HAI J is said to be trivial if the coaction maps any element m to m ⊗ ∈ J ⊗ L . A finite trivial comodule is thus a quotient of the (trivial) comodule R n . Bytaking duality we see that a rigid trivial comodule is a subcomodule of R n . For a comodule V ,the maximal trivial subcomodule V triv of consists of elements v such that ρ ( v ) = v ⊗ V be a finite projective L -comodule. The coaction ρ : V −→ V ⊗ L induces a map Cf : V ∨ ⊗ V −→ L , ϕ ⊗ m X ϕ ( m i ) m ′ i , ϕ ∈ V ∨ , m ∈ V , ∆ ( m ) = X i m i ⊗ m ′ i .This map can be considered either as a homomorphism of L -comodules, where L coacts on itselfby the coproduct and coacts on V ∨ ⊗ V by the action on the second tensor component, or as ahomomorphism of coalgebras. The image of this map, denoted by Cf ( V ) , is called the coefficientspace of V . Since L is flat, it is a subcoalgebra of L , i.e., ∆ ( Cf ( V )) ⊂ Cf ( V ) ⊗ Cf ( V ) (cf. Lemma3.1.4 (i)). The coaction of L on V factors through the coaction of Cf ( V ) on V .We note that our requirement here for a subcoalgebra is weaker than in some other literatures,e.g. [14], where a subcoalgebra is a special subcoalgebra in our sense (see. 3.1.1 for definition).1.1.5. Following dos Santos, we call a subcomodule U of V special if U is saturated in V , i.e., V/U is R -flat. For instance, the coaction V −→ V ⊗ L can be considered as a comodule map(where L coacts on the target by the coaction on itself) . It splits as an R -module map by meansof the counit ε : L −→ R of L . Hence, if V is R -flat, V is a special submodule of V ⊗ L .If V is finite projective, then Cf ( V ) contains both Cf ( U ) and Cf ( V/U ) , [23, Lem. 9]. In deed,this inclusion property is a local property, so we can assume that R is a DVR, then V , U and V/U are all free over R . Hence we can choose a basis ( e , e , . . . , e n ) of V such that the first m elements form a basis of U and the cosets of the other elements form a basis of V/U . Thecoaction of L on V with respect to this basis is given in terms of a multiplicative matrix ( a ji ) : ρ ( e i ) = n X j = e j ⊗ a ji , i =
1, . . . , n .Notice that the elements a ji are uniquely determined by the basis elemnents e i ’s. Now theassumption that U is a subcomodule implies that a ji = i =
1, . . . , m and j = m +
1, . . . , n .It follows that the coaction of L on V/U with respect to the basis ( e m + , . . . , e n ) is given by ρ ( e k ) = n X l = m + e l ⊗ a lk , k = m +
1, . . . , n .For example, for an R -flat comodule V , the maximal trivial subcomodule V L of V is special.Indeed, if we have 0 = v = au , a ∈ R , u ∈ V then from the equality ρ ( au ) = au ⊗ V ⊗ L , we conclude that ρ ( u ) = u ⊗
1. Thus
V/V L is torsion free, hence flat. Thisimplies that for any two finite projective comodules V , W , the inclusionHom L ( V , W ) −→ Hom R ( V , W ) is saturated. Indeed, we haveHom L ( V , W ) ∼ = Hom L ( R , W ⊗ V ∨ ) = ( W ⊗ V ∨ ) L ⊂ W ⊗ V ∨ . ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 5 V , we denote by h V i the full subcategory generated by V , i.e.,consisting of subquotients of finite direct sums of V . For a finite projective comodule V , denoteby h V i s the full subcategory specially generated by V , i.e. consisting of special subquotients(quotients of special subobjects or special subobjects of quotients objects) of direct sums ofcopies of V . Lemma 1.1.7.
Let V be a finite projective comodule over a flat R -coalgebra L . Then the category h V i s is equivalent with Comod o ( Cf ( V )) (by means of the obvious functor which is the identityfunctor on the underlying R -modules).Proof. Consider the restriction functor
Comod o ( Cf ( V )) −→ Comod o ( L ) . The condition for a map ϕ : M −→ N to be an L -comodule map reads as follows: the map ρ N ϕ − ( ϕ ⊗ id ) ρ M : M −→ N ⊗ L is the zero map (i.e. the outer square in the diagram below is commutative). M ρ M / / ϕ (cid:15) (cid:15) M ⊗ Cf ( V ) / / ϕ ⊗ id (cid:15) (cid:15) M ⊗ L ϕ ⊗ id (cid:15) (cid:15) N ρ N / / N ⊗ Cf ( V ) / / N ⊗ L .If N is flat over R , the horizontal map N ⊗ Cf ( V ) −→ N ⊗ L in the above diagram is injective.Hence the condition for ϕ to be an L -comodule map is the same as the condition for ϕ to bea Cf ( V ) -comodule map (which amounts to the left square to commute). Thus the restrictionfunctor Comod o ( Cf ( V )) −→ Comod o ( L ) is fully faithful.On the other hand, according to 1.1.5, if W ∈ h V i s then Cf ( W ) ⊂ Cf ( V ) , hence W is asubcomodule of Cf ( V ) . Thus, it remains to show that any finite projective comodule over Cf ( V ) is a special subquotient of a finite direct sum of copies of V .This claim holds for Cf ( V ) itself, as, by definition, Cf ( V ) is a quotient of V ∨ ⊗ V , where Cf ( V ) coacts on V ∨ ⊗ V by the coaction on the second tensor component. Further, if M is a finiteprojective Cf ( V ) -comodule then M is a special subcomodule of M ⊗ Cf ( V ) (cf. 1.1.5), i.e. it is aspecial subquotient of a direct sum of copies of V . The proof is complete. (cid:3) Warning.
The functor
Comod f ( Cf ( V )) −→ Comod ( L ) is faithful and exact but generallynot full, see Section 3 below. It is not clear to us how to specify the image of this functor.1.1.9. For a comodule M , its ( R -) torsion part M tor is also a subcomodule. The quotient M/M tor is R -torsion free, hence flat, hence R -projective if it is finite over R .1.1.10. L -comodules are locally finite, i.e. they are union of their subcomodules of finite type.In fact, for each finite set S of a comodule M , there exists a subcomodule N , finitely generatedover R , which contains S . It follows that L itself is the union of its subcoalgebras, which arefinite over R , c.f. [25, Cor. 2]. See also Appendix A.1.4.1.1.11. Let K be the fractions field of R . Denote L K := L ⊗ R K . Then L K is a Hopf algebra overthe field K . If V is a comodule over L , then V K := V ⊗ R K is a comodule over L K . For any twofinite projective comodules V , W , the natural mapHom L ( V , W ) ⊗ R K −→ Hom L K ( V K , W K ) NGUYEN DAI DUONG AND PH`UNG HˆO HAI is an isomorphism. Indeed, if f ∈ Hom L K ( V K , W K ) , then there exists 0 = a ∈ R such that af : V −→ W . But then we have f = af ⊗ a − .Conversely, let X be a finite dimensional comodule over L K . Since X ⊗ K L K ∼ = X ⊗ R L , X isan L -comodule. Let ( e , . . . , e n ) be a basis of X . Then there exists a finite L -comodule V ⊂ X ,which contains ( e i , . . . , e n ) . Now V is finite hence projective over R . As V contains a basis of X , we have V K ∼ = X . Note that V is not unique, but all such V has the same rank, which is thedimension of X over K .1.2. Tannakian duality for abelian tensor categories.Definition 1.2.1 (Subcategory of definition, cf. [21, II.2.2]) . Let C be an R -linear abelian cat-egory, and ω : C −→ Mod f ( R ) be an R -linear exact faithful functor. Suppose that C o is a fullsubcategory of C such that:(i) for any object X ∈ C o , ω ( X ) is a finitely generated projective R -module;(ii) every object of C is a quotient of an object of C o .Then C o is called a subcategory of definition of C with respect to ω .This definition is motivated by the following fact, due to Serre (see [25, Prop.3]). For anyfinite L -comodule E there exists a short exact sequence of L -comodules0 −→ F ′ −→ F −→ E −→ F ′ and F are R -finite projective. Thus, the subcategory Comod o ( L ) of R -finite projective L -comodules is a subcategory of definition in Comod f ( L ) . Theorem 1.2.2 (cf. [21, Thm. II.2.3.5, II.2.6.1]) . Let R be a Noetherian ring and let C be an R -linear abelian category. Assume that there exist an R - linear exact faithful functor ω : C −→ Mod f ( R ) and a subcategory of definition C o with respect to ω . Then ω factors through an equiva-lence C ≃ Comod f ( L ) and the forgetful functor, for some flat R -coalgebra L . Although this theorem is formulated for any Noetherian ring, We don’t know any examplesof comodule category satisfying the conditions of Definition 1.2.1 when R is a ring of dimensionlarger than 1. A self-contained proof of this theorem will be given in the Appendix.1.2.3. The coalgebra L in the theorem above can be determined from the fiber functor ω asfollows. We claim that there is a natural isomorphism(1) Nat ( ω , ω ⊗ M ) ≃ Hom R ( L , M ) ,for any R -module M , see Appendix A.1.10. If C = Comod f ( L ) and ω is the forgetful functorfrom C to Mod ( R ) , then the above isomorphism implies that Coend ( ω ) ≃ L . In particular, a flatcoalgebra over R can be reconstructed from the category of its comodules. The isomorphism in(1) implies that, for any R -algebra A , Nat A ( ω ⊗ A , ω ⊗ A ) ≃ Hom R ( L , A ) .If C is a tensor category and ω is a (strict) tensor functor, then L is a bialgebra and we have anisomorphism Nat ⊗ A ( ω ⊗ A , ω ⊗ A ) ≃ Hom R − Alg ( L , A ) , ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 7 where
Nat ⊗ denotes the set of natural transformations that are compatible with the tensorproduct.1.2.4. Assume that C is an R -linear abelian tensor category. The reader is referred to [5, Sect. 1]for the notion of dual objects. An object is called rigid if it possesses a dual. Notice that the imageof a rigid object under a tensor functor to Mod f ( R ) is a finite projective module. Denote by C o the full subcategory of C consisting of rigid objects. We say that C is dominated by C o if eachobject of C is a quotient of a rigid object. Definition 1.2.5.
A (neutral) Tannakian category over a Dedekind ring R is an R -linear abeliantensor category C , dominated by C o , together with an exact faithful tensor functor ω : C −→ Mod ( R ) . In this case, C o is a subcategory of definition in C .Let ω o denote the restriction of ω to C o . Then we have, for any R -algebra A , Nat ⊗ A ( ω ⊗ A , ω ⊗ A ) ≃ Nat ⊗ A ( ω o ⊗ A , ω o ⊗ A ) ≃ Aut ⊗ A ( ω o ⊗ A , ω o ⊗ A ) .For the first isomorphism see the proof of Lemma A.1.6, for the second isomorphism see [5,Prop. 1.13]. Theorem 1.2.6 ([21, Thm. II.4.1.1]) . Let ( C , ω ) be a neutral Tannakian category over a Dedekindring R . Then the group functor A Aut ⊗ A ( ω ⊗ A ) is representable by a flat group scheme G and ω factors through an equivalence between C and Rep f ( G ) . Scalar extension.
In [29], Wedhorn proposes to establish a duality for rigid tensor cate-gory over a Dedekind ring. In fact, in many examples, it is easy to specify a rigid tensor category,but it is very difficult to determine a Tannakian category containing the given monoidal cate-gory as a subcategory of definition. The natural problem is to characterize intrinsically thesubcategory of rigid objects in a Tannakian category (over a Dedekind ring). For this, Wedhornintroduces the notion of scalar extension of category to define his Tannakian lattice. Wedhorn’sTannakian lattice is not necessarily equivalent to the full subcategory of rigid objects in a Tan-nakian category. In the next section we shall provide a characterization of such categories. Inthis subsection we will recall the notion of scalar extension of categories and some basis prop-erties.1.3.1. Let ϕ : R −→ S be a ring homomorphism. Let C be an R -linear category. The category C S obtained from C by scalar extension ϕ is defined as follows. The objects of C S are the sameas those of C and for two objects X and Y in C S their hom-set isHom C S ( X , Y ) := Hom C ( X , Y ) ⊗ R S .We have an S -linear category together with an R -linear functor ϕ ∗ : C −→ C S .If the map ϕ : R −→ S is flat, the functor ϕ ∗ preserves monomorphisms and epimorphisms[29, 3.6]. Let D be another R -linear category and ω : C −→ D be an R -linear functor. Then wehave a functor ω S : C S −→ D S .If ϕ is flat and ω is faithful then so is ω S [29, 3.7].Assume that M is an R -linear tensor category. The R -bilinear functor ⊗ : M × M −→ M extends to an S -bilinear functor ⊗ : M S × M S −→ M S . In this way M S is an S -linear tensorcategory. It is rigid if M is rigid. NGUYEN DAI DUONG AND PH`UNG HˆO HAI L be a flat R -coalgebra. Then L K := L ⊗ R K is a coalgebra over K . There is a naturalfunctor from Comod o ( L ) to Comod f ( L K ) , M M ⊗ R K . This induces a functor φ : Comod o ( L ) K −→ Comod o ( L K ) .This functor φ is an equivalence of abelian categories [29, Subsection 6.4]. Consequently, if G is a flat affine group scheme over R , we have an equivalence of abelian tensor categoriesRep o ( G ) K ≡ Rep f ( G K ) .1.3.3. Warning.
Monomorphisms in
Comod o ( L ) are injective comodules maps, as the kernel ofa map in Comod o ( L ) is again in Comod o ( L ) ( L being flat). However an epimorphism in this cat-egory is not necessarily a surjective map, for instance a map [ a ] : V −→ V given by multiplyingwith a non-unit a in R ( a =
0) is an epimorphism in
Comod o ( L ) but is not surjective. Put inanother way, the forgetful functor from Comod o ( L ) to Mod ( R ) does not preserve epimorphisms.Later we shall impose the condition that our fiber functor preserves images (of morphisms) as areplacement for the exactness.1.3.4. For an R -linear abelian tensor category C , we define the special fiber C s at a closed point s of S := Spec ( R ) to be the full subcategory of objects which satisfy m s X =
0, where m s is themaximum ideal of R , which determines s .For a flat affine group scheme G , we can identify Rep f ( G s ) with Rep f ( G ) s , cf. [17, Chapt. 10].Indeed, let L = R [ G ] and L s := L ⊗ R R/m s . Then any L s comodule is an L -comodule in a naturalways, as we have V ⊗ R L ∼ = V ⊗ k s L s for any k s -vector space V (on which R acts through themap R −→ k s .If C is Tannakian then the fiber functor ω yields an equivalence between C s and Rep f ( G s ) where G is the Tannakian group of C .One can show that C s is equivalent to the scalar extension C k s where k s = R/m s is the residuefield of R at s . In fact, the R -linearity yields the functor X X ⊗ R k s from C to C s and hence afunctor from C k s −→ C s which is an equivalence, as the base change R −→ k s does not affect C s . 2. D UALITY FOR T ANNAKIAN LATTICES
The kernel and image of a morphism.
For the definition of a neutral Tannakian lattice,we shall need the notion of the kernel and the image of a morphism. The notion of kernels isstandard in the category theory, we recall it here for the sake of the reader.2.1.1.
Kernel.
Let C be an additive category, i.e. the hom-sets are equipped with abelian groupstructures and the composition of morphisms is bi-additive. For a morphism f : X −→ Y , the ker-nel of f is the equalizer of f and the zero map, i.e. the final object in the category of morphisms h : Z −→ X satisfying f ◦ h = Z h ! ! ❈❈❈❈❈❈❈❈ ∃ ! ϕ (cid:15) (cid:15) ker f i / / X f / / Y . ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 9
The morphism i : ker f −→ X is then a monomorphism.2.1.2. Image-factorization.
The image-factorization of a morphism f : X −→ Y is the initialobject in the category of factorizations of the form f = g ◦ h with g being a monomorphism.That is, if f = g ′ h ′ is another factorization with g ′ being a monomorphis X h ′ ! ! ❉❉❉❉❉❉❉❉❉ h / / f im ( f ) g / / m (cid:15) (cid:15) ✤✤✤ YZ g ′ = = ④④④④④④④④④ then there exists a unique morphism m such that g = g ′ m . Since g ′ is a monomorphism, wewill also have m ◦ h = h ′ . If the image-factorization exists for any morphism we say that C is a category with images .Let C , D be categories with images. A functor ω : C −→ D is said to preserve images if ω preserves the image-factorization for any morphisms in C . Example 2.1.3. (i) Any abelian category obviously has kernels and images. An exact functorbetween abelian categories preserves kernels and images.(ii) The category Mod o ( R ) ( R being a Dedekind ring) has kernels and images, which aredetermined set theoretically; the forgetful functor to Mod f ( R ) preserves kernels and images.(iii) For a flat affine group scheme G over a Dedekind ring R , the category Rep o ( G ) has kernelsand images, and the forgetful functor from Rep o ( G ) to Mod f ( R ) preserves kernels and images.2.2. Tannakian lattice. I is a commu-tative ring. Given a commutative ring R , an additive rigid tensor category T over R , is an R -linearadditive rigid tensor category in which the natural map R −→ End ( I ) in an isomorphism, where I denotes the unit object.An object J in T is called trivial if there is a monomorphism J −→ I n . The full subcategory oftrivial objects of T is denoted by T triv . Definition 2.2.2.
Let R be a Dedekind ring. A neutral Tannakian lattice over R is an additive rigidtensor category T over R , in which kernels and images exist, the category T K given by scalarextension is abelian, and is equipped with an R -linear additive tensor functor ω : T −→ Mod ( R ) ,satisfying the following conditions:T1) ω is faithful and preserves kernels and images.T2) ω restricted to T triv is fully faithful.2.2.3. Since T is rigid, the image of ω is in Mod o ( R ) . Since ω is faithful, the hom-sets in T arefinite flat modules over R , consequenlty the functor ι ∗ : T −→ T K is faithful. Therefore, we shallidentify T with its image in T K . In particular, morphisms in T are considered as morphisms in T K . An object X of T , when considered as object of T K , will be denoted by X K , and the image ofa morphism f under ι ∗ will be denoted by the same symbol f . For an element a ∈ R and objects X , Y in T , we shall use the symbol [ a ] to denote the morphism X −→ Y induced by a . Further we shall write [ a ] ◦ f = f ◦ [ a ] simply as af . If f : X K −→ Y K is amorphism in T K , then there exists a ∈ R (not uniquely determined) such that af is a morphismin T . The proof of the following lemma is obvious. Lemma 2.2.4.
Let g : X −→ Y be such that g : X K −→ Y K is an isomorphism. Then there exist a ∈ R and h : Y −→ X , such that h ◦ g = g ◦ h = [ a ] . More general, given objects X , Y , Z in T andmorphisms f : X −→ Z , g : Y −→ Z . Assume that there exists h : X K −→ Y K such that g ◦ h = f .Then there exists a ∈ R and h ′ : X −→ Y , such that ah = h ′ and g ◦ h ′ = af . X a / / h ′ (cid:15) (cid:15) X f (cid:15) (cid:15) h (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ Y g / / Z Lemma 2.2.5.
Let f : X −→ Y be a morphism in T . If there exists a map ϕ : ω ( X ) −→ ω ( Y ) suchthat aϕ = ω ( f ) for some a ∈ R , then there exists g : X −→ Y with ag = f and ω ( g ) = ϕ . In otherwords, the submodule ω ( Hom T ( X , Y )) is saturated in Hom R ( ω ( X ) , ω ( Y )) .Proof. First, assume that Y = I – the unit object. Consider the image-factorization of f = hg : X −→ J −→ I . Applying ω we have the following commutative diagram ω ( X ) ϕ (cid:15) (cid:15) ω ( g ) / / / / ω ( f ) ❍❍❍ ❍❍❍❍ ω ( J ) ω ( h ) (cid:15) (cid:15) { { R [ a ] / / R Thus ω ( h ) is injective and its image lies in aR , hence it factors as ω ( h ) : ω ( J ) ψ −−→ R [ a ] −−−→ R .Since ω is fully faithful on trivial objects, we have ψ = ω ( h ′ ) , that is, ϕ = ω ( h ′ ◦ g ) . Hence f = a ( h ′ ◦ g ) .The general case follows from this case by means of the isomorphismHom T ( X , Y ) ∼ = Hom T ( X ⊗ Y ∨ , I ) ,which is compatible with the fiber functor ω . (cid:3) Lemma 2.2.6.
The functor ω K : T K −→ Vect ( K ) is exact. Thus ( T K , ω K ) is a neutral Tannakiancategory over K .Proof. Recall that the functor i ∗ : T −→ T K preserves mono- and epimorphisms.Let Y K f −→ Z K be an epimorphism in T K . Multiplying f with an element from R if needed, wecan assume that f is in T . Consider the image-factorization of f in T : f = Y h −−→ im ( f ) g −−→ Z .Thus, in T K , g is both epi- and monomorphism, hence an isomorphism. Now ω ( h ) is surjectivehence so is ω ( h ) K , thus ω ( f ) K is also surjective.Let g : X K −→ Y K be the kernel of f in T K . We can similarly assume that g is in T . Considerthe image-factorization of the map X −→ ker ( f ) induced by g in T : X p −−→ im ( g ) q −−→ ker ( f ) . We ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 11 have, as above, q is epi- and monomorphism in T K . Applying ω to the factorization, we obtain ω ( X ) ω ( p ) −−−→ ω ( im ( g )) ω ( q ) −−−→ ω ( ker ( f )) ,with ω ( q ) being invertible in Vect ( K ) and ω ( p ) being surjective. Consequently, ω ( X ) is thekernel of ω ( f ) . (cid:3) G be a flat affine group scheme over R , then the category Rep o ( G ) of represen-tations of G in finite projective R -modules equipped with the forgetful functor to Mod ( R ) , is aTannakian lattice over R .2.3. Tannakian duality. T , ω ) be a Tannakian lattice over R . The discussion in 1.2.3 yields a coalgebra L and a factorization T ω / / ω L $ $ ■■■■■■■■■■ Mod ( R ) Comod o ( L ) ν qqqqqqqqqq where ν is the forgetful functor. Recall that L is a bialgebra, ω L is a tensor functor and we havean isomorphism End ⊗ S ( ω ⊗ S ) ≃ Hom R − Alg ( L , S ) ,for any R -algebra S . According to [5, Prop. 1.13], we haveAut ⊗ S ( ω ⊗ R S ) ∼ = End ⊗ S ( ω ⊗ R S ) .Hence the functor Aut ⊗ R ( ω ) : S Aut ⊗ S ( ω ⊗ R S ) is representable by L . That is, L is a (commu-tative) Hopf algebra and Aut ⊗ R ( ω )( S ) ∼ = Hom alg ( L , S ) .Thus Aut ⊗ R ( ω ) is an affine group scheme over R . Theorem 2.3.2.
Let ( T , ω ) be a Tannakian lattice over a Dedekind ring R . Then the group scheme G = Aut ⊗ R ( ω ) is faithfully flat over R and ω induces an equivalence between T and Rep o ( G ) . L is flat. We shall use an indirect construction toprove that the Hopf algebra L satisfies the claims of Theorem 2.3.2. Lemma 2.2.6 shows that ω K : T K −→ Vect ( K ) is an exact, faithful functor, i.e. a fiber functor for the abelian rigid tensorcategory T K . Thus ( T K , ω K ) is a Tannakian category. The classical Tannakian duality yields aHopf algebra L over K and an equivalence ω K : T K ∼ = Comod f ( L ) .For each X ∈ T , we have ω ( X ) ⊗ R K = ω K ( X K ) .Thus ω ( X ) ⊗ R L = ω K ( X K ) ⊗ K L .Therefore we can consider ω ( X ) as a comodule over the R -coalgebra L . Denote by L R the unionin L of all the coefficient spaces Cf ( ω ( X )) as X runs in the objects of T . Then L R is a Hopf R -subalgebra of L , which coacts on all ω ( X ) , (moreover we have L R ⊗ K = L ). Thus the functor ω factors through the functor (followed by the forgetful functor):(2) ω : T −→ Comod o ( L R ) .2.3.4. Proof of Theorem 2.3.2.
We proceed by showing that the functor ω in (2) is an equiva-lence of categories. ω is fully faithful : By assumption, ω is faithful. On the other hand, the Tannakian duality for T K says that ω ( Hom T ( X , Y )) ⊗ R K = Hom L ( ω K ( X K ) , ω K ( Y K )) .Let ϕ ∈ Hom L R ( ω ( X ) , ω ( Y )) . Then the above equality ensures the existence of a morphism f ∈ Hom T ( X , Y ) such that ω ( f ) = aϕ for some a ∈ R . Now Lemma 2.2.5 implies that thereexists g ∈ Hom T ( X , Y )) such that ω ( g ) = ϕ . That is, ω : T −→ Comod o ( L R ) is full. ω is essentially surjective : Each finite projective L R -comodule M is a special subquotient of L Rr (direct sum of r copies of L R ) by means of the coaction: δ : M −→ M ⊗ R L R , see 1.1.5.Consequently for any subcomodule N in L Rr containing M , the quotient N/M is also flat. Inparticular, we can choose N to be the subcomodule Cf ( ω ( X )) r for some X in T . Thus we concludethat M is a special subquotient of an object in the image of ω .Next, we show that each N in Comod o ( L R ) is isomorphic in this category to some ω ( X ) . Firstassume that N is a quotient of some ω ( Y ) :0 −→ M −→ ω ( Y ) −→ N −→ Z in T such that N ⊗ K ∼ = ω K ( Z K ) in Comod ( L K ) . Suchan isomorphism yields an injective map N −→ ω ( Z ) in Comod ( L ) . Consider the composition ω ( Y ) −→ N −→ ω ( Z ) . Since ω is fully faithful, this map is the image of a morphism f : Y −→ Z .By assumption, f has image in T and ω preserves it, hence we claim ω ( im ( f )) = N . Dualizingthe above sequence we have an exact sequence0 −→ N ∨ −→ ω ( Y ∨ ) −→ M ∨ −→ M ∨ ∼ = ω ( im g ) for some morphism g in T , thus M ∼ = ω ( T ) with T := ( im f ) ∨ . Thus, all special subobjects of ω ( X ) are also in im ( ω ) . Consequently all specialsubquotients of ω are in im ( ω ) . This completes the proof of the fact that ω is an equivalence ofcategories between T and Comod o ( L R ) . L R is the coend of ω : this follows from the equivalence just established. In deed, by the equiv-alence above, ω : T −→ Mod ( R ) can be identified with the forgetful functor Comod o ( L R ) −→ Mod ( R ) . But the coend of this last functor is just L R , cf. 2.3.1. This finishes the proof of Theorem2.3.2. ✷ Torsors.
Let ( T , ω ) be a Tannakian lattice. Let S be a faithfully flat R -algebra. A fiberfunctor η : T −→ Mod o ( S ) is defined to be an R -linear tensor functor satisfying the followingconditions:(T1) η is faithful and preserves kernels and images. ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 13
In this section we construct out of this data a torsor
Iso ⊗ R ( ω , η ) over Aut ⊗ R ( ω ) . In particularwe will show that for different neutral fiber functors ω and ω ′ , the representation categories of Aut ⊗ R ( ω ) and Aut ⊗ R ( ω ′ ) are equivalent, thus determining a unique abelian envelope of T .2.4.1. The abelian envelope of T . Let ( T , ω ) be a neutral Tannakian lattice and let η : T −→ Mod ( R ) be another fiber functor. Let L ( ω ) be the coend of ω and L ( η ) be the coend of η . ThenTannakian duality for ( T , ω ) and ( T , η ) yields a functor ϕ : Comod o ( L ( ω )) −→ Comod o ( L ( η )) : T ω x x qqqqqqqqqqq η & & ▼▼▼▼▼▼▼▼▼▼▼ Comod o ( L ( ω )) ϕ / / Comod o ( L ( η )) .The subcategory of trivial objects in Comod o ( L ( ω )) is exactly the category Mod o ( R ) of finiteprojective R -modules. The restriction of ϕ to Mod o ( R ) is thus a faithful functor, preservingkernels and images and sending R to itself. It is therefore a fully faithful functor. Thus, applyingTheorem 2.3.2 to ϕ we conclude that this restriction of ϕ is an equivalence of categories. Wewill show that ϕ extends to an equivalence between Comod ( L ( ω )) and Comod ( L ( η )) .2.4.2. Let V be in Comod o ( L ( ω )) . Denote C := Cf ( V ) ⊂ L ( ω ) . Then Comod o ( C ) is equivalentto h V i s , cf. 1.1.7. Let W := η ( V ) ∈ Comod o ( L ( η )) and D := Cf ( W ) ⊂ L ( η ) . Then ϕ induces anequivalence between Comod o ( C ) and Comod o ( D ) .Let A := Hom ( C , R ) , then A is a (non-commutative) algebra, finite projective as an R -moduleand we have an equivalence Comod ( C ) ∼ = Mod ( A ) inducing and equivalence Comod o ( C ) ∼ = Mod o ( A ) .Here, we denote by Mod ( A ) the category of finite left A -modules and by Mod o ( A ) full subcate-gory modules which are finite projective over R . Thus, denoting B := Hom ( D , R ) , we have andequivalence, also denoted by ϕ : Mod o ( A ) ϕ −→ Mod o ( B ) . Lemma 2.4.3.
The functor ϕ extends to an equivalence between Mod f ( A ) and Mod f ( B ) .Proof. Since ϕ preserves kernels and images, it preserves exact sequences in Mod o ( A ) (i.e se-quences in Mod o ( A ) , exact in Mod ( A ) .Denote P := ϕ ( A ) . Then P is a B − A -bimodule (the action of A on P is induced from the rightaction of A on itself, which commutes with morphisms in Mod ( A ) ). For an object M ∈ Mod o ( A ) ,consider a finite resolution A m −→ A n −→ M −→
0. We have an exact sequence ϕ ( A m ) −→ ϕ ( A n ) −→ ϕ ( M ) −→ Mod ( B ) . Since ϕ is additive, we have ϕ ( A n ) = P n , hence we have canonical isomorphism ϕ ( M ) ∼ = P ⊗ A M .Thus ϕ coincides with P ⊗ A − on Mod o ( A ) .We show that P is flat over A . For a finite A -module M , consider a resolution0 −→ N −→ A m −→ M −→ Since N is R -projective being a submodule of A m , and since ϕ (−) = P ⊗ − : Mod o ( A ) −→ Mod ( B ) preserves monomorphisms, the long exact sequence0 −→ Tor A ( P , M ) −→ P ⊗ N −→ P m −→ P ⊗ A M −→ A ( P , M ) =
0. Thus we have an exact functor P ⊗ − : Mod f ( A ) −→ Mod f ( B ) ,which reduces to an equivalence Mod o ( A ) −→ Mod o ( B ) . Hence it is itself an equivalence from Mod f ( A ) to Mod f ( B ) . (cid:3) Proposition 2.4.4.
The functor ϕ : Comod o ( L ( ω )) −→ Comod o ( L ( η )) extends to an equivalencebetween Comod f ( L ( ω )) and Comod f ( L ( η )) .Proof. The equivalence is obtained by extending h V i s larger and larger. (cid:3) The torsor
Iso ⊗ R ( ω , η ) . Consider now the more general situation: η is a fiber functor T −→ Mod ( S ) , where S is an R -algebra. Recall that Iso ⊗ R ( ω , η ) is the functor which associatesto each S -algebra S ′ the set Iso ⊗ ( ω , η )( S ′ ) of natural isomorphisms compatible with the tensorstructure between the two given functors. According to [5, Prop. 1.13], this set is equal to theset Nat ⊗ S ′ ( S ′ ⊗ R ω , S ′ ⊗ S η ) . The algebra L ( ω , η ) represents functor Iso ⊗ R ( ω , η ) if it satisfiesNat ⊗ S ′ ( ω ⊗ R S ′ , η ⊗ S S ′ ) ∼ = Hom S -alg ( L ( ω , η ) , S ′ ) ,for any S -algebra S ′ . As in 1.2.3, we notice that L ( ω , η ) can be defined as the S -module repre-senting the functor S ′ Nat S ′ ( ω ⊗ R S ′ , η ⊗ S S ′ ) ∼ = Nat R ( ω , η ⊗ S S ′ ) .That is, we determine L ( ω , η ) by the functorial isomorphism(3) Nat R ( ω , η ⊗ S N ) ∼ = Hom S ( L ( ω , η ) , N ) ,for any S -module N .2.4.6. Identifying T with Comod o ( L ) by means of ω , we can consider ω as the identity functorand η as a tensor functor Comod o ( L ) −→ Mod ( S ) , where L := L ( ω ) .Let C ⊂ L be an R -finite subcoalgebra, and set A = Hom R ( C , R ) . The discussion in the proofof Lemma 2.4.3 shows that the restriction of η to Comod o ( C ) = Mod o ( A ) extends to an exactfunctor P ⊗ A − : Mod f ( A ) −→ Mod f ( S ) , where P = η ( A ) . Lemma 2.4.7.
Consider ω and η as functors on the category Mod o ( A ) , we have natural isomor-phism Nat R ( ω , η ⊗ S N ) ∼ = P ⊗ S N ∼ = Hom S ( η ( C ) , N ) , for any S -module N .Proof. This is standard. By the functoriality, a natural transformation λ : ω −→ η ⊗ S N isuniquely determined by its value at A , i.e. an R -linear map λ A : A −→ η ( A ) ⊗ S N = P ⊗ S N , ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 15 which is A -linear. Indeed, this is a consequence of the following commutative diagram, for any f in Mod o ( A ) : A λ A / / f (cid:15) (cid:15) P ⊗ S N P ⊗ f ⊗ N (cid:15) (cid:15) M λ M / / ( P ⊗ A M ) ⊗ S N ,forcing λ M = λ A ⊗ A M . (The A -linearity follows from the choice M = A ). Conversely, any such A -linear map determines a natural equivalence.As to the last isomorphism, notice that C is R -finite projective, hence A is the dual of C in Comod o ( L ) , therefore P = η ( A ) is dual to η ( C ) as S -modules. Thus we have P ⊗ S N = Hom S ( η ( C ) , S ) ⊗ S N ∼ = Hom S ( η ( C ) , N ) .This finishes the proof. (cid:3) Proposition 2.4.8.
The scheme
Iso ⊗ R ( ω , η ) is a torsor under the group scheme Aut ⊗ R ( ω ) .Proof. This is a remedy of [5, Prop. 3.2]. According to the isomorphism in (3) and the lemmaabove, the S -module L ( ω , η ) = lim −→ C ⊂ L ( ω ) η ( C ) represents the functor Iso ⊗ R ( ω , η ) . Note that the transition maps in the directed system areinclusions of subcoalgebras C ֒ → C ′ of L , which give rise to injective maps η ( C ) −→ η ( C ′ ) . Thetorsor action is obvious, loc.cit. It remains to check that Iso ⊗ R ( ω , η ) is faithfully flat over S , i.e. tocheck that L ( ω , η ) is faithfully flat over S . As L ( ω , η ) is the direct limit of S -projective modules,it is flat over S . We show the faithfulness.If a finite subcoalgebra C contains the unit element of L ( ω ) , then the inclusion R −→ C splitsin Mod ( R ) (by means of the counit). We obtain an exact sequence in Comod o ( C ) :0 −→ R −→ C −→ C/R −→ Comod o ( S ) (as ω preserves kernels and images)0 −→ S −→ η ( C ) −→ η ( C/R ) −→ S −→ L ( ω , η ) , which is pure, as L ( ω , η ) /R ∼ = lim ←− R ⊂ C ⊂ L η ( C/R ) is flat. This shows that L ( ω , η ) is faithfully flat over S . (cid:3)
3. C
OALGEBRA HOMOMORPHISMS
Specially locally finite coalgebras.
An important property of coalgebras over a field is thelocal finiteness: a coalgebra is the union of its finite dimensional subcoalgebras. This propertyformally generalizes to flat coalgebras over a Dedekind ring. Indeed, let M ⊂ L be a finitesubcomodule, then M is projective over R and is contained in the subcoalgebra Cf ( M ) of L (seeproof of Lemma 3.1.2). However this is not fully reflected in the Tannakian duality. For a subcoalgebra C of coalgebra L over a field k , the category Comod f ( C ) can be identified with a full, exact subcategory of Comod f ( L ) , which is closed under taking subobjects. This is no more the case for flat coalgebrasover a Dedekind ring as the example 3.1.3 below shows. The reason is that the quotient module L/C may be non-flat over the base ring. If we try to enlarge C so that the quotient becomes flat,we cannot guarantee that it remains finite over R . In view of Tannakian duality, this reflects thefact that the full abelian subcategory generated by a single object in a comodule category maybecome quite large.This phenomenon is one of the main obstructions to the study of flat coalgebras and flatgroup schemes over a Dedekind ring. Below we will show that the coordinate ring of a reduced,connected group scheme is specially locally finite as a coalgebra (Proposition 3.1.7).Recall that a subcomodule M of an L -comodule N is said to be special if N/M is flat over R ; a special subquotient M of an L -comodule N is a special submodule of a quotient of N , or,equivalently, a quotient of a special submodule of N . Definition 3.1.1.
Let L be an R -flat coalgebra.(i) A comodule N is said to be specially locally finite if any finite subcomodule of N iscontained in an R -finite special subcomodule;(ii) A subcoagebra C of L is said to be special if L/C is flat. A homomorphism of flatcoalgebras f : L ′ −→ L is said to be special if f ( L ′ ) is a special subcoalgebra of L ;(iii) L is said to be specially locally finite if for any finite subcomodle C , there exists a finitespecial subcoalgebra containing C .Let N be an L -comodule. Then N tor , the R -torsion submodule of N is an L -subcomodule.Hence for any L -subcomodule M , the preimage of ( N/M ) tor in N , denoted M sat , is an L -comodule. Since R is a Dedekind ring, the quotient N/M sat is flat, being torsion free. Thus M sat is the smallest special subcomodule of N , containing M . It is called the saturation of M in N . Lemma 3.1.2. An R -flat coalgebra L is specially locally finite if and only if it L is locally finite as acomodule on itself.Proof. Assume that L is specially locally finite as a right comodule on itself. Let C be a finitesubcoalgebra of L and C sat the saturation. It is to show that C sat is a subcoalgebra.We have the filtration C sat ⊗ C sat ⊂ C sat ⊗ L ⊂ L ⊗ L , the successive quotients of which areflat, hence L ⊗ L/C sat ⊗ C sat is also flat. Thus ( C ⊗ C ) sat ⊂ C sat ⊗ C sat .Hence, by the definition of C sat , we have ∆ ( C sat ) ⊂ ( C ⊗ C ) sat ⊂ C sat ⊗ C sat .For the converse statement we use the well-known fact that each finite subcomodule of L iscontained in a finite subcoalgebra of L . Indeed, given M ⊂ L finite, then it is R -projective and Specially locally finite coalgebras are called IFP coalgebras in [14, I.3.11].
ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 17 the coaction M −→ M ⊗ L induces a coalgebra map (cf. 1.1.4) M ∨ ⊗ M −→ L , ϕ ⊗ m X ϕ ( m i ) m ′ i , where ∆ ( m ) = X i m i ⊗ m ′ i .Its image is the coefficient space Cf ( M ) of M , which contains M . In fact, let ε M be the restrictionof ε to M , then ε M ⊗ m P i ε ( m i ) m ′ i = m . (cid:3) The next example shows that there exist coalgebras which are not specially locally finite.
Example 3.1.3 ([7], Remarque 11.10.1) . Let 0 = x ∈ R be a non-unit element. Let G be theaffine group scheme over R determined by the Hopf subalgebra of K [ G a ] = K [ T ] : R [ G ] := { P ∈ K [ T ] | P ( ) ∈ R } ,that is, R [ G ] consists of polynomials in K [ T ] with the constant coefficients belonging to R . Let C i be the subcoalgebra spanned (over R ) by 1 and x − i T . Then we have C i ( C i + , xC i + ( C i .Thus the saturation of C is not finite.In what follows we will need some standard facts on tensor products and flat modules, cf.[4]. Lemma 3.1.4.
Let A ⊂ B be flat R -modules and M , M , M ⊂ N be arbitrary R -modules. Then (i) M ⊗ A ∩ M ⊗ A = ( M ∩ M ) ⊗ A , M ⊗ A + M ⊗ A = ( M + M ) ⊗ A , as subsetsof N ⊗ A ; (ii) if B/A is also flat, we have N ⊗ A ∩ M ⊗ B = M ⊗ A as submodules in N ⊗ B . The following result is proved in more generality in [14, I.3.11], we recall it here for com-pleteness. The reader is referred to [26, Tag 0599] for the notion of Mittag-Leffler system.
Proposition 3.1.5.
Let R be a Dedekind ring and L be an R -flat coalgebra. (i) If L is specially locally finite then L is Mittag-Leffler as an R -module. Hence, if L is moreovercountably generated over R , it is R -projetive. (ii) If L is R -projective, then it is specially locally finite as an R -coalgebra.Proof. (i) Let { C α } be the directed system of finite special subcoalgebras. Then for any finite R -module N , the system Hom R ( C α , N ) is Mittag-Leffler. In fact, each inclusion C α −→ C β splits,as C β /C α is R -torsion free and finite, hence projective over R . Consequently the map Hom R ( C β , N ) −→ Hom R ( C α , N ) is surjective. Thus by definition, L is Mittag-Leffler as an R -module. It is well-known that a flat,countably generated, Mittag-Leffler module is projective.(ii) We show that any projective comodule is specially locally finite. If N ⊂ M is a subcomod-ule then N sat is the preimage of ( M/N ) tor under the quotient map M −→ M/N . Thus we haveto show that N sat is finite provided that M is projective and N is finite as R -modules. This is apure question of R -modules. Embed M is a free R -module F as a direct summand. Replacing M by F will only enlarge N sat , thus we can assume that M is free over R . Then, as N is finite, we can find a free direct summand of F which contains N . Now F = F sat0 implying N sat ⊂ F ,hence it is finite. (cid:3) Questions 3.1.6.
It is not known if any specially locally finite R -flat coalgebra is R -projective.Another interesting question is: which affine flat R -group scheme of finite type is specially locallyfinite. Proposition 3.1.7.
Let G be a flat group scheme of finite type over a Dedekind ring R . Assume thatthe generic fiber G K is reduced and connected. Then R [ G ] is specially locally finite as an R -coalgebra.Proof. Let I be the augmented ideal of R [ G ] , that is I = ker ( ε ) . Since the R [ G ] is flat over R , themap R [ G ] −→ R [ G ] ⊗ R K = K [ G K ] is injective and as K is flat over R , the augmentation idealof K [ G K ] is I K = I ⊗ R K . With the assumption that G K is reduced and connected, K [ G K ] is anintegral domain. By Krull’s intersection theorem T m ( I K ) m = M ⊂ R [ G ] be a finite submodule. Then there exists m such that M ⊗ K ∩ ( I K ) m = ( I K ) m = I m ⊗ K . Hence ( M ∩ I m ) ⊗ K = M ∩ I m =
0. It follows that M sat ∩ I m =
0. Indeed, if 0 = a ∈ M sat ∩ I m then there exists 0 = r ∈ R such that ra ∈ M ∩ I m ,it forces ra =
0, consequently a = R [ G ] is torsion free. Thus, the map M sat −→ R [ G ] /I m isinjective. But the module R [ G ] /I m is finite, hence so is M sat . (cid:3) On the other hand, the situation for group schemes with finite fibers turns out to be morecomplicated, as in the following example, communicated to us by dos Santos.
Example 3.1.8.
Let R be a DVR of equal positive characteristic p , with uniformizer π . Let G bethe group scheme determined by the Hopf algebra R [ G ] := R [ T ] / ( πT p − T ) , ∆ ( T ) = ⊗ T + T ⊗ G are ´etale group scheme but R [ G ] is not specially locally finite. Indeed, thesaturation of the finite subcomodule spanned by 1 and T contains T p k , k >
1, and is not finite.3.2.
Tannakian description of homomorphisms of coalgebras.
In the second part of thisparagraph we will use the Tannakian duality to characterize (special) injective and surjectivehomomorphisms of flat coalgebras.Let f : L ′ −→ L be homomorphism of flat coalgebras over a Dedekind ring R . We denote by ω f : Comod f ( L ′ ) −→ Comod f ( L ) the “restriction” functor (which considers each L ′ -comodule asan L -comodule by means of f , thus ω f is the identity functor on the underlying R -modules).Then ω f o : Comod o ( L ′ ) −→ Comod o ( L ) will denote the restriction of ω f to the subcategory offinite projective comodules. Proposition 3.2.1.
Let f : L ′ −→ L be homomorphism of flat coalgebras over a Dedekind ring R .Then (i) f is injective if and only if the functor ω f o is fully faithful and its image in Comod f ( L ) isclosed under taking special subobjects. (ii) f is injective and special if and only if the natural functor ω f o is fully faithful and its imagein Comod f ( L ) is closed under taking subobjects. In this case, the functor ω f is also fullyfaithful and its image is closed under taking subobjects. ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 19
Proof.
We shall show the “only if” implication for both claims at once. First notice that thefull faithfulness claim in (i) is essentially proved in Lemma 1.1.7. For claim (ii) the proof isalmost the same: Assume that f is injective and special. Then the functor ω f : Comod f ( L ′ ) −→ Comod f ( L ) can be considered as the identity functor on the underlying module category. Henceit is obviously faithful (and so is ω f o ).As mentioned in the proof of Lemma 1.1.7, the condition for a map ϕ : M −→ N to be L ′ -comodules reads as follows: ρ ′ N ϕ − ( ϕ ⊗ id ) ρ ′ M : M −→ N ⊗ L ′ is the zero map: M ρ ′ M / / ϕ (cid:15) (cid:15) M ⊗ L ′ id ⊗ f / / ϕ ⊗ id (cid:15) (cid:15) M ⊗ L ϕ ⊗ id (cid:15) (cid:15) N ρ ′ N / / N ⊗ L ′ id ⊗ f / / N ⊗ L Now, if
L/L ′ is flat over R , the horizontal map id ⊗ f : N ⊗ L ′ −→ N ⊗ L in the above diagramis injective. Hence the map ρ ′ N ϕ − ( ϕ ⊗ id ) ρ ′ M is zero if and only if its composition with id ⊗ f ,which is ρ N ϕ − ( ϕ ⊗ id ) ρ M : M −→ N ⊗ L , is zero. Thus ω f is full.We show the closedness under taking (special) subquotients. For ( M , ρ ′ ) ∈ Comod f ( L ′ ) , itimage under ω f is denoted by ( M , ρ ) . Let ( N , ρ N ) be a sub L -comodule of ( M , ρ ) . Thus we havecommutative diagram M ρ ′ / / ρ ❍❍❍❍❍❍❍❍❍ M ⊗ L ′ id ⊗ f (cid:15) (cid:15) N ?(cid:31) O O ρ N ❍❍❍❍❍❍❍❍❍ M ⊗ LN ⊗ L ?(cid:31) O O To show that ρ N comes from a coaction of L ′ on Y amounts to showing that ρ N ( N ) ⊂ N ⊗ f ( L ′ ) .The above diagram shows that ρ N ( N ) ⊂ N ⊗ L ∩ M ⊗ f ( L ′ ) . If either M , N , M/N or L/f ( L ′ ) areflat over R , according to Lemma 3.1.4, one has equality N ⊗ L ∩ M ⊗ f ( L ′ ) = N ⊗ f ( L ′ ) .(4)Thus ρ N ( N ) ⊂ N ⊗ f ( L ′ ) , that is, ρ ′ restricts to a coaction of L ′ on N .Conversely, assume that the functor ω f o : Comod ( L ′ ) o −→ Comod ( L ) o is fully faithful andits image is closed under taking special subobjects. By the flatness of K over R , the functor Comod f ( L ′ ) K −→ Comod f ( L ) K is fully faithful and preserves mono- and epimorphisms. There-fore, by the equivalence in 1.3.2 the functor Comod ( L ′ ⊗ K ) −→ Comod ( L ⊗ K ) satisfies conditionsof [5, Thm 2.21], hence L ′ ⊗ K −→ L ⊗ K is injective. Consequently the map f : L ′ −→ L isinjective, (i) is proved.Now assume that the image of ω f o is closed under taking subobjects. The discussion aboveshows that f is injective. We will identify L ′ with a subcoalgebra of L . For a non-invertible0 = π ∈ R , denote C := πL ∩ L ′ ⊃ πL ′ . By means of Lemma 3.1.4 we see that C is also an L -subcomodule of L ′ . Since an L -comodule is the union of its finite subcomodule, the assumption on ω f o implies that C is in fact an L ′ -subcomodule of L ′ : ∆ ( C ) ⊂ C ⊗ L ′ ⊂ πL ⊗ L ′ .Thus for any c ∈ C we have ∆ ( c ) = X πa i ⊗ b i , a i ∈ L , b i ∈ L ′ .Hence c = P i πε ( a i ) b i ∈ πL ′ . That is C ⊂ πL ′ , consequently πL ∩ L ′ = πL ′ . The last equationholds for any π ∈ R , it follows that L/L ′ is torsion free over R , hence flat, as R is a Dedekindring. (cid:3) Remarks 3.2.2.
The proof of Proposition 3.2.1 is based on the fact that, when R is a field, themap Coend ( ω | h X i ) −→ L is injective. The reader is referred to [27, Thm 6.4.4] for the detailedproof of this fact or to [16, Lem 1.2] for a short proof. In Saavedra’s proof of this fact [21,II.2.6.2.1], the proof of implication d) ⇒ a) is incomplete.We now express the local finiteness in terms of Tannakian duality. First we have the followingcharacterization of finite coalgebras. Proposition 3.2.3.
Let L be a flat coalgebra over a Dedekind ring R . Then L is finite over R if andonly if Comod f ( L ) has a projective generator which is R -projective.Proof. Assume that L is finite. Then L is projective over R and there is an equivalence between Comod ( L ) and Mod ( L ∨ ) , where L ∨ is the dual R -module to L . Now L ∨ is a projective generatorin Comod f ( L ) .Conversely, assume that Comod f ( L ) has a projective generator, say P , which is R -projective.Then this category is equivalent to Mod f ( A ) , where A = End L ( P ) . As End R ( P ) is R -finite pro-jective and A is an R -submodule of End R ( P ) , it is also projective. Therefore A ∨ is a finite flat R -coalgebra and Mod ( A ) is equivalent to Comod ( A ∨ ) . By Tannakian duality we conclude that A ∨ ∼ = L , whence L is finite. (cid:3) Recall that for a comodule X of L , h X i denotes the full subcategory of Comod f ( L ) consisting ofsubquotients of finite direct sums of copies of X . If X is R -finite projective then h X i satisfies theconditions of Definition 1.2.1 (since a submodule of a finite projective R -module is again finiteprojective, we can take for h X i o the full subcategory of subobjects of finite direct sums of copiesof X ). Hence, by means of Theorem 1.2.2, we obtain a coalgebra Coend ( ω | h X i ) , the comodulecategory of which is equivalent to h X i . Proposition 3.2.4.
Let L be a flat coalgebra over a Dedekind ring R and let ω denote the forgetfulfunctor Comod f ( L ) −→ Mod ( R ) . (i) For each R -finite projective comodule X of L the natural map Coend ( ω | h X i ) −→ Coend ( ω ) = L is injective and special. If L is specially locally finite then Coend ( ω | h X i ) is finite over R . (ii) If for any R -finite projective comodule X , Coend ( ω | h X i ) is finite, then L is specially locallyfinite.Proof. (i) The first claim follows from Tannakian duality Theorem 1.2.2 and Proposition 3.2.1.Assume now that L is specially locally finite. Let M be an R -finite projective comodule of L . Then by assumption on L , there exists a special subcoalgebra C of L , which is finite over ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 21 R and such that the coaction of L on M factors though that of C . According Proposition 3.2.1(ii), Comod f ( C ) is a full subcategory of Comod f ( L ) closed under taking subobjects. Since M ∈ Comod f ( C ) , we have h M i ⊂ Comod f ( C ) . Hence we have a factorization Coend ( ω | h M i ) −→ C −→ L .But C is R -finite, whence the claim.(ii) According to (i), we can cover L by its finite special subcoalgebras L = [ i L i , where { L i } isa cofinal directed system, i.e. any two coalgebras L i , L j are contained in some L k . Hence, if C isa finite subcoalgebra of L , then C is contained in some L i . Since L i is saturated in L , C sat ⊂ L i ,hence is finite. (cid:3) Finally we provide a condition for the surjectivity of a colagebra homomorphism.
Proposition 3.2.5.
Let f : L ′ −→ L be a morphism of flat coalgebras over R . Then f is surjective ifand only if the induced functor ω f o : Comod o ( L ′ ) −→ Comod o ( L ) satisfies the following condition:each M ∈ Comod o ( L ) is a special subquotient of ω f ( N ) for some N ∈ Comod o ( L ′ ) .Proof. Assume that f is surjective. Let M be an R -finite projective L -comodule. Then M is aspecial subcomodule of L ⊕ r for some r > M with a subcomoduleof L ⊕ r and choose a generating set { m i } of M . Let m ′ i ∈ L ′ ⊕ r be such that f ( m ′ i ) = m i . Theset { m ′ i } is contained in some finite module N of L ′ ⊕ r . The homomorphic image of N in L ⊕ r isdenoted by N , this is an L -subcomodule of L ⊕ r which contains M . Since M is special in L ⊕ r , itis also special in N . Thus M is a special subcomodule of the quotient N of ω f ( N ) , where N isan L ′ -comodule.Conversely, assume that ω f o has the stated property. It suffices to show that any R -finite sub-coalgebra C of L is in the image of f . Consider such a C as a (right) L -comodule. By assumption,there exists an R -finite projective L ′ -comodule N , such that C is a special subquotient of ω f ( N ) in Comod f ( L ) . Since C = Cf ( C ) , and C is a special subquotient of N , according to subsection1.1.5 we conclude that C ⊂ Cf ( N ) . On the other hand, it follows from the construction that Cf ( N ) is the image of Cf L ′ ( N ) - the coefficient space of N considered as an L ′ -comodule. Thusthe map L ′ −→ L is surjective. (cid:3)
4. T
ANNAKIAN DESCRIPTION OF GROUP SCHEME HOMOMORPHISMS
Let R be a Dedekind ring with fraction field K . Let f : G −→ G ′ be a homomorphism offlat affine group schemes over R . We say that f is surjective or a quotient homomorphism if itis faithfully flat. In the first part of this chapter we give a necessary and sufficient conditionfor the faithful flatness of f , in terms of Tannakian duality. Then we will give a condition for f to be a closed immersion. Finally we give a criterion for the exactness of a sequence of grouphomomorphisms.4.1. Faithfully flat homomorphisms and closed embedding.
The following theorem is a gen-eralization of the well-known faithful flatness theorem for Hopf algebras: a (commutative) Hopf algebra is faithfully flat over any Hopf subalgebra , see, e.g.,[28, Thm 14.1]. The proof is devel-oped from an idea of J.C. Moore [20]. The advantage of this proof is that we don’t need toassume the Hopf algebras involved to be of finite type.
Theorem 4.1.1.
Let L be a flat commutative Hopf algebras over R and L ′ be a Hopf subalgebras.Then L is faithfully flat over L ′ if and only if L/L ′ is R -flat, i.e., L ′ is a special subcoalgebra of L .Proof. Assume that L is faithfully flat over L ′ . Consider the tensor product of the exact sequence0 / / L ′ / / L / / Q / / L over L ′ we get an exact sequence0 / / L / / L ⊗ L ′ L / / L ⊗ L ′ Q / / L ⊗ L ′ L −→ L splits this sequence, hence L ⊗ L ′ Q is flat over L . Since L isfaithfully flat over L ′ , Q is flat over L ′ and therefore it is flat over R .Conversely, assume that L/L ′ is R -flat. We first show that L is flat over L ′ , i.e., for any L ′ -module M , Tor L ′ ( M , L ) =
0. The claim holds if R is a field.First assume that M is R -flat. Choose P ∗ , a projective resolution of L as an L ′ -module. Then P ∗ ⊗ k is a projective resolution of L ⊗ k over L ′ ⊗ k , where k is a residue field or the fractionfield of R . We have ( M ⊗ k ) ⊗ ( L ′ ⊗ k ) ( P ∗ ⊗ k ) ∼ = ( M ⊗ L ′ P ∗ ) ⊗ k ,implying H i (( M ⊗ L ′ P ∗ ) ⊗ k ) ∼ = Tor L ′ ⊗ ki ( M ⊗ k , L ⊗ k ) , for all i > M ⊗ L ′ P ∗ is flat over R , a Dedekind ring, the universal coefficient theorem, (see, e.g., [30,Thm 3.1.6]), applies. Thus, for each i >
1, we have an exact sequence0 −→ H i ( M ⊗ L ′ P ∗ ) ⊗ k −→ H i (( M ⊗ L ′ P ∗ ) ⊗ k ) −→ Tor R ( H i − ( M ⊗ L ′ P ∗ ) , k ) −→ i > −→ Tor L ′ i ( M , L ) ⊗ k −→ Tor L ′ ⊗ ki ( M ⊗ k , L ⊗ k ) −→ Tor R ( Tor L ′ i − ( M , L ) , k ) −→ L/L ′ is flat, the map L ′ ⊗ k −→ L ⊗ k is injective, hence flat. Therefore Tor L ′ ⊗ ki ( M ⊗ k , L ⊗ k ) =
0, for all i >
1. ConsequentlyTor R ( Tor L ′ i − ( M , L ) , k ) = Tor L ′ i ( M , L ) ⊗ k =
0, for all i > R , hence Tor L ′ ( M , L ) is flat over R andTor L ′ i ( M , L ) = i > M be an arbitrary L ′ -module. Then the R -torsion submodule M τ of M is also an L ′ -submodule. The quotient module M/M τ is then R -flat. As we have the exact sequenceTor L ′ ( M τ , L ) −→ Tor L ′ ( M , L ) −→ Tor L ′ ( M/M τ , L ) −→ . . .it suffices to show Tor L ′ ( M , L ) = M being R -torsion.For each non-zero ideal p ⊂ R , the submodule M p of elements annihilated by p , is also an L ′ -submodule. As M is torsion, it is the direct limit of M p . Since the Tor-functor commuteswith direct limits, one can replace M by some M p . Since R is a Dedekind ring, each non-zeroideal p is a product of finitely many prime ideals. Therefore each M p has a filtration, each ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 23 grade module of which is annihilated by a certain non-zero prime ideal. Thus using inductionwe can reduce to the case M is annihilated by a prime ideal p . In this case M = M ⊗ k p is an L ′ ⊗ k p -module, where k p := R/p and we have M ⊗ L ′ P ∗ = M ⊗ L ′ ⊗ k p ( P ∗ ⊗ k p ) .Since P ∗ ⊗ R k p is an L ′ ⊗ R k p -projective resolution of L ⊗ R k p , we see thatTor L ′ i ( M , L ) = Tor L ′ ⊗ k p i ( M , L ⊗ k p ) = L ⊗ k p is flat over L ′ ⊗ k p .Finally, we show that L is faithfully flat over L ′ .Let M be an L ′ -module, such that M ⊗ L ′ L =
0. Then we have M k ⊗ L ′ k L k ∼ = ( M ⊗ L ′ L ) ⊗ R k = k or the fraction field K of R . Since L ′ k −→ L k is faithfully flat, we have M k = M K =
0. If M is finite over L ′ , this implies that M =
0, according to [19, Thm 4.9].In the general case, M always contains a non-zero finite submodule and since L is flat over L ′ ,we see that M = M ⊗ L ′ L =
0. Thus L is faithfully flat over L ′ . (cid:3) As a corollary of Theorem 4.1.1 and Propositions 3.2.1, 3.2.5 we have the following theorem.
Theorem 4.1.2.
Let f : G −→ G ′ be a homomorphism of affine flat groups over R , and ω o f be thecorresponding functor Rep o ( G ′ ) −→ Rep o ( G ) .(i) f is faithfully flat if and only if ω f o : Rep o ( G ′ ) −→ Rep o ( G ) is fully faithful and its imageis closed under taking subobjects. (ii) f is a closed immersion if and only if every object of Rep o ( G ) is isomorphic to a specialsubquotient of an object of the form ω f ( X ′ ) , X ′ ∈ Rep o ( G ′ ) . Remarks 4.1.3. (i) For (commutative) Hopf algebras over a field, any injective homomorphism L ′ −→ L is automatically faithfully flat. This is not the case for Hopf algebras over a Dedekindring. Take for example R = k [ x ] and L = R [ T ] = k [ x , T ] with the R -coalgebra structure given by ∆ ( T ) = T ⊗ + ⊗ T , ε ( T ) = S ( T ) = − T (i.e. L = R [ G a ] ), and consider the map of Hopf R -algebras f : L −→ L , T xT .The quotient L/f ( L ) is not flat over R , hence Theorem 4.1.2 implies that L is not faithfully flatover f ( L ) .(ii) Theorem 4.1.2 implies that, if f : L ′ −→ L is a homomorphism of flat Hopf R -algebras,such that L/f ( L ′ ) is flat over R , then L is faithfully flat over f ( L ′ ) .(iii) The claim (ii) of Theorem 4.1.2 generalizes a result of dos Santos, [23, Prop. 12]. Questions 4.1.4.
Theorem 4.1.2 suggests the following question: find a criterion of a Tannakiancategory C such that its Tannakian group G is pro-algebraic in the sense that G = lim ←− α G α where each group scheme G α is of finite type and each structure map G β −→ G α is faith-fully flat. Similarly, find a criterion such that G is smooth in the sense that the G α above are smooth. It seems that the local finiteness mentioned in the previous section closely relates tothese problems.4.2. Tannakian description of exact sequences.
Recall that the right regular representationof G on its coordinate ring L , ( g , h ) gh : ( gh )( x ) = h ( xg ) , g ∈ G , h ∈ L correspondsto the right coaction of L on itself by the coproduct ∆ . The left regular action of G on L , ( g , h ) gh : ( gh )( x ) = h ( g − x ) corresponds to the following (right) coation of L on itself: a X i a ′ i ⊗ S ( a i ) , where ∆ ( a ) = X i a i ⊗ a ′ i , and S denotes the antipode.Let G −→ A be a homomorphism of affine groups schemes over R . Let I A be the kernel ofcounit ǫ : R [ A ] −→ R , i.e. the augmentation ideal of R [ A ] , and let I A R [ G ] be the ideal generatedby the image of I A in R [ G ] . Then the kernel of G −→ A is the closed subscheme of G withcoordinate ring R [ G ] /I A R [ G ] . A sequence1 / / H q / / G p / / A / / p is a quotient map with kernel H . We will provide a criterion for theexactness in terms of the functors Rep o ( A ) p ∗ / / Rep o ( G ) q ∗ / / Rep o ( H ) .(7)We first need a lemma relating the coordinate rings R [ A ] , R [ G ] and R [ H ] . Lemma 4.2.1.
Let G −→ A be a quotient map with kernel H .(i) If M is a G -module, then M H , the H -trivial submodule of M , is stable under the action of G , i.e. it is a G -submodule of M .(ii) R [ A ] is equal to R [ G ] H as G -modules.Proof. (i) This follows immediately from the normality of A in G , see [17, I.3.2] for a proof.(ii) The proof is based on the fact that R [ G ] is faithfully flat over R [ A ] and follows closely theproof for group schemes over fields, cf. [28, Sect. 15.4].There is an isomorphism H × G ≃ −→ G × A G ; ( h , g ) −→ ( hg , g ) , which precisely means that G −→ A is a principal bundle under H . In terms of the coordinate rings this isomorphism hasthe form(8) ϕ : R [ G ] ⊗ R [ A ] R [ G ] ∼ = R [ H ] ⊗ R R [ G ] , a ⊗ b X i q ( a i ) ⊗ a ′ i b ,where q : R [ G ] −→ R [ H ] denotes the quotient map and ∆ ( a ) = P i a i ⊗ a ′ i . The inverse map isgiven by q ( a ) ⊗ b X i a i ⊗ S ( a ′ i ) b , ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 25 where S denotes the antipode of R [ G ] . One checks that this assignment depends on q ( a ) ∈ R [ H ] but does not depend on the choice of a ∈ R [ G ] . Now consider the following diagram: R [ G ] H / / R [ G ] d / / d ′ & & ◆◆◆◆◆◆◆◆◆◆◆◆ R [ G ] ⊗ R [ A ] R [ G ] ϕ (cid:15) (cid:15) R [ H ] ⊗ R R [ G ] where d ( a ) := a ⊗ − ⊗ a . Then d ′ is computed as follows: d ′ ( a ) = X i q ( a i ) ⊗ a ′ i − ⊗ a , where ∆ ( a ) = X i a i ⊗ a ′ i .Thus R [ G ] H is precisely the kernel of d ′ , hence is also the kernel of d as ϕ is an isomorphism.On the other hand, as R [ G ] is faithfully flat over R [ A ] , R [ A ] is the kernel of d . We conclude that R [ A ] = R [ G ] H . (cid:3) Theorem 4.2.2.
Let us be given a sequence H q / / G p / / A with q a closed immersion and p faithfully flat. Then this sequence is exact if and only if thefollowing conditions are fulfilled: (a) For an object V ∈ Rep o ( G ) , q ∗ ( V ) in Rep o ( H ) is trivial if and only if V ∼ = p ∗ U for some U ∈ Rep o ( A ) .(b) Let W be the maximal trivial subobject of q ∗ ( V ) in Rep o ( H ) . Then there exists V ⊂ V ∈ Rep o ( G ) , such that q ∗ ( V ) ∼ = W .(c) Any W ∈ Rep o ( H ) is a quotient in (hence, by taking duals, a subobject of) q ∗ ( V ) for some V ∈ Rep o ( G ) . Proof.
Assume that q : H −→ G is the kernel of p : G −→ A . Then (a) and (b) follow from 4.2.1(i) and (ii). We prove (c).Let Ind : Rep ( H ) −→ Rep ( G ) be the induced representation functor, it is the right adjointfunctor to the restriction functor Res : Rep ( G ) −→ Rep ( H ) that is Hom G ( V , Ind ( W )) ∼ = / / Hom H ( Res ( V ) , W ) .(9)One has Ind ( W ) ∼ = ( W ⊗ R R [ G ]) H , where H acts on R [ G ] by the left regular action, [17, I.3.4].Notice that the subspace ( W ⊗ R R [ G ]) H ⊂ W ⊗ R R [ G ] is invariant under the action of R [ A ] , i.e.it is an R [ A ] submodule.Notice that the isomorphism (8) R [ G ] ⊗ R [ A ] R [ G ] ≃ −→ R [ H ] ⊗ R R [ G ] is a map of G -modules where G acts on the second tensor terms by the right regular action.Since R [ G ] is faithfully flat over its subalgebra R [ A ] , taking the tensor product with R [ G ] over R [ A ] commutes with taking H -invariants, hence we have Ind ( W ) ⊗ R [ A ] R [ G ] ≃ ( W ⊗ R R [ G ]) H ⊗ R [ A ] R [ G ] ≃ W ⊗ R R [ G ] . This in turns implies that the functor
Ind is faithfully exact.Setting V = Ind ( W ) in (9), one obtains a canonical map u W : Ind ( W ) −→ W in Rep ( H ) whichgives back the isomorphism in (9) as follows: Hom G ( V , Ind ( W )) ∋ h u W ◦ h ∈ Hom H ( Res ( V ) , W ) .The map u W is non-zero whenever W is non-zero. Indeed, since Ind is exact and faithful,
Ind ( W ) is non-zero whenever W is non-zero. Thus if were u W =
0, then (9) were the zero mapfor any V . On the other hand, for V = Ind ( W ) , the right hand side contains the identity map. Acontradiction, which shows that u W cannot vanish.We show now that u W is always surjective. Let U = im ( u W ) and T = W/U ∈ Rep f ( H ) . Wehave the following diagram0 / / Ind ( U ) u U (cid:15) (cid:15) / / Ind ( W ) u W (cid:15) (cid:15) / / Ind ( T ) / / u T (cid:15) (cid:15) / / U / / W / / T / / Ind ( W ) ։ Ind ( T ) −→ T is 0, therefore Ind ( T ) −→ T is a zeromap, implying T = W is finitely generated projective over R . Then Ind ( W ) is torsion free. Hence Ind ( W ) is the union of its finitely generated R -projective modules, we can find a finitely gener-ated G − submodule W ( W ) of Ind ( W ) which still maps surjectively on W . In order to obtain thestatement on the embedding of W , we dualize the map W ( W ∨ ) ։ W ∨ .Assume now that (a), (b), (c) are satisfied. Then it follows from (a) that for U ∈ Rep o ( A ) , q ∗ p ∗ ( U ) ∈ Rep o ( H ) is trivial. Hence pq : H −→ A is the trivial homomorphism. Recall that byassumption, q is injective, p is surjective. Let q : H −→ G be the kernel of p . Then we havecommutative diagram H q (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ i / / H q (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ G ←→ Rep o ( H ) Rep o ( H ) i ∗ o o Rep o ( G ) . q ∗ f f ▲▲▲▲▲▲▲▲▲▲ q ∗ rrrrrrrrrr It remains to show that i is surjective. We use Proposition 3.2.1. We first show the functor i ∗ is fully faithful. The faithfulness is obvious, we show the fullness. Let W , W be objects in Rep o ( H ) , and ϕ : W = i ∗ ( W ) −→ W = i ∗ ( W ) be a morphism of H -modules. Since H isthe kernel of p , the first part of this proof shows that there exist surjection q ∗ ( V ) ։ W andinjection W ֒ → q ∗ ( V ) where V , V are objects of Rep o ( G ) . Thus ϕ combined with these mapsyields a map b ϕ : q ∗ ( V ) −→ q ∗ ( V ) . b ϕ corresponds to an element of ( q ∗ ( V ) ⊗ q ∗ ( V ) ∨ ) H . Byconditions (a), (b) for H and by the fact that H also satisfies (a), (b), there exists ψ : q ∗ ( V ) −→ q ∗ ( V ) such that b ϕ = i ∗ ( ψ ) : q ∗ ( V ) ։ W ϕ −→ W ֒ → q ∗ ( V ) .This implies that ϕ = i ∗ ( ϕ ) for some ϕ : W −→ W . Thus i ∗ is full. ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 27
For any W ∈ Rep o ( H ) , by (c) there exist V , V in Rep o ( G ) and ϕ : q ∗ ( V ) −→ q ∗ ( V ) suchthat W = im ϕ . Since i ∗ is full, ϕ = i ∗ ϕ , hence W ∼ = i ∗ ( im ϕ ) . Thus we have proved that anyobject in Rep o ( H ) is isomorphic to the image under i ∗ of an object in Rep o ( H ) . Together with thediscussion above this implies that H ∼ = H . (cid:3)
5. S
TRATIFIED SHEAVES ON A SMOOTH SCHEME OVER A D EDEKIND RING
Stratified sheaves on smooth schemes over a Dedekind ring.
Let R be a Dedekind ring,denote S := Spec ( R ) . We shall assume that the residue fields of R are all perfect . Let f : X −→ S be a smooth morphism with geometrically connected fibers. Consider the category str ( X/S ) of O X -coherent modules over the sheaf D ( X/S ) of algebras of differential operators on X/S . Thisis an abelian tensor category with the unit object being O X equipped with the usual K¨ahlerdifferential. We call objects of str ( X/S ) stratified sheaves.If R is a perfect field then objects of str ( X/S ) are locally free as sheaves on X (see 5.1.5). Forthe proof of this fact in characteristic 0 see [18] and in positive characteristic see [22].We assume that f admits a section ξ : S −→ X . The pull-back along ξ provides a functor ξ ∗ : str ( X/S ) −→ Mod ( R ) . The following results are similar to those of dos Santos [23, Sect 4.2]. Proposition 5.1.1.
The following claims hold: (i)
An object of str ( X/S ) , which is R -torsion-free, is flat (and hence is locally free) as an O X -module. Consequently, the subcategory of O X -locally free objects is closed under takingsubobjects. (ii) The functor ξ ∗ : str ( X/S ) −→ Mod ( R ) is faithful and exact.Proof. Since these are local properties, we can assume that R is a discrete valuation ring with auniformizer t and X is an affine scheme over R , X = Spec A . Then the proof of [23, Lem. 19,Cor. 20] can be used.(i). By the local flatness criterion, an object M of str ( X/S ) is flat over O X if Tor R ( M , O X ) = M := M/tM is flat over X = X × Spec ( R ) Spec ( R/tR ) . The second condition is triviallysatisfied as X is a scheme over a field R/tR . The first condition just means M is t -torsion free.(ii) We show that ξ ∗ is left exact. Let m be the kernel of ξ : A −→ R then ξ ∗ is the functortensoring with A/m = R . Shrinking X if necessarily, we can assume the existence of a regularsequence of generators of m , say x , x , . . . , x n . It suffices to check that Tor A ( M , A/m ) vanishes for any M ∈ str ( X/S ) . Let M τ be the R -torsion part of M and M f := M/M τ then M f is R -torsion free hence is O X -flat by (i). Thus one is led to check the claim for those M , whichare R -torsion. Such an M has a filtration (which is finite as M is coherent)0 = M ⊂ M ⊂ . . . ⊂ M m = M ,where M i /M i − is killed by t , so that M i /M i − is supported on X . Using the long exactsequence for Tor, one reduces the problem to the case M is supported on X . Thus M is locallyfree as a sheaf on X . Let T j := Tor A ( M , A/ ( x , . . . , x j )) . We will show by induction that T j = for j =
1, 2, . . . , n . To see that T = −→ A · x −−→ A −→ A/ ( x ) −→ T is the kernel of the multiplication by x on M , which is 0 as M is locallyfree over O X . For the induction step, assuming that T j − = −→ A j − · x j −−→−→ A j − −→ A j −→ A j := A/ ( x , x , . . . , x j ) we obtain a short exact sequence0 −→ T j −→ A j − ⊗ A M · x j −−→−→ A j − ⊗ A M .Now A j − ⊗ A M is a stratified module on the smooth schemeSpec A/ ( t , x , . . . , x j − ) ,hence, as above, we conclude that T j = ω is faithful, it suffices to see that ξ ∗ ( M ) = M = X . We follow again the argument above: the torsion-free part is restrictedto the generic fiber and the torsion part is filtered with sucessive quotients supported on closedfiber. (cid:3) A locally free stratified sheaf is also called stratified bundle . Let str o ( X/S ) be the subcategoryof stratified bundles. This is a rigid tensor category, and by the above proposition it is closedunder taking subobjects. Lemma 5.1.2.
Let K be quotient field of R . Then the category obtained by scalar extension R −→ K , str o ( X/S ) K , is abelian.Proof. It is to define quotients in str o ( X/S ) K . We first show the following: for M ⊂ N in str o ( X/S ) , let M sat denote the saturation of M in N , i.e. the minimal extension of M in N such that N/M sat is R -torsion free, then M and M sat are isomorphic as objects in str o ( X/S ) K .Indeed, since M sat /M is R -torsion, as in the proof of Proposition 5.1.1 (ii), there exists a finitefiltration for M sat /M such that each successive quotient is killed by a non-zero element of R .Hence M sat /M is killed by a single non-zero element a ∈ R . Thus M sat /M ⊗ R K =
0, that is, M ⊗ R K ∼ = M sat ⊗ R K .Let now f : M −→ N be a morphism in str o ( X/S ) K , then there exists a ∈ R such that af is amorphism in str o ( X/S ) . The kernel of f is defined to be the kernel of af , since M is R -torsionfree, the definition is independent of the choice of a . On the other hand, the co-kernel of f isdefined to be the quotient of N by the saturation of Im ( af ) . The discussion above shows thatthis definition is independent of the choice of a . Indeed, the saturations of Im ( af ) and Im ( baf ) in N are the same for any b ∈ R . (cid:3) Based on results of section 2 we make the following definition.
Definition 5.1.3.
The fiber functor ξ ∗ makes str o ( X/R ) a Tannakian lattice, its Tannakian group,denoted by π ( X/S , ξ ) , is called the relative fundamental group scheme of X/S . ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 29
Let M ∈ str o ( X/S ) . Consider the full subcategory h M i ⊗ s of str o ( X/S ) , consisting of specialsubquotients of direct sums of tensors powers of the form T a , b ( M ) := M ⊗ a ⊗ M ∨ ⊗ b , a , b ∈ N .Then h M i ⊗ s is also a Tannakian lattice. Its Tannakian group scheme G ( M ) is called the differen-tial Galois group scheme of M . This group was first studied in [23].5.1.4. We don’t know if str ( X/S ) is a Tannakian category over R , be cause we cannot check ifany stratified sheaf is representable as a quotient of a stratified bundle. However we can definethe abelian envelope of str o ( X/S ) , denoted by C = C ( X/S ) , as the full subcategory of str ( X/S ) consisting of stratified sheaves which can be represented as quotients of stratified bundles. Ac-cording to Proposition 5.1.1, C is a (neutral) Tannakian category over R . We conclude that C isequivalent to the representation category of π ( X/S , ξ ) by means of the fiber functor ξ ∗ . Thus C is the abelian envelope of str o ( X/S ) in the sense of 2.4.1.5.1.5. Let s be a closed point of S , k := k s = R/p s – the residue field of s , and let X s denotethe fiber of f at s . Consider the category str ( X s /k ) . Its objects are automatically locally free as O X s -modules, cf. 5.1. Thus str ( X s /k ) is an abelian rigid tensor category over k .The fiber of str ( X/S ) at s ∈ S is defined as in Remark 1.3.4, and is denoted by str ( X/S ) s . Thisfull subcategory of str ( X/S ) is identified with the category of stratified bundle on X s . Indeed,the restriction functor str ( X/S ) −→ str ( X s /k ) (given by pulling-back along the closed immersion X s −→ X ) can be identified with the functor which associates to each object M of str ( X/S ) thequotient M/p s M . In particular, str ( X s /k ) is naturally a subcategory of str ( X/S ) , consisting ofthose objects which are annihilated by p s . Lemma 5.1.6. C ( X/S ) s is a full subcategory of str ( X s /k ) , closed under taking subquotients.Proof. This is obvious by definition (see 1.3.4). A stratified sheaf M is an object of C ( X/S ) s if itis annihilated by p s and can be presented as a quotient of a stratified bundle, say M . Thus anyquotient of M is again in C ( X/S ) s . On the other hand, if N is a subobject of M then takingthe pull-back of N along the projection map M −→ M we get a subobject N of M whichsurjects onto N . Since N itself is locally free, we conclude that N is in C ( X/S ) s . (cid:3) Remarks 5.1.7.
In general we don’t know if M can be presented in the from M = M/πM ,that is if M is the fiber at s of some stratified bundle M . The difference between C ( X/S ) and str ( X/S ) consists essentially in those stratified sheaves supported in X s and not representable asa quotient of a stratified bundle. See 5.2 below.5.1.8. In the rest of this subsection we shall assume that R contains a field k which is mappedisomorphically onto any residue field. Thus R can be see as the coordinated ring on the affinecurve S on k (recall that k is assumed to be perfect).Consider X as a scheme over k . A stratified sheaf on X/k is automatically locally free on X and is called an absolute stratified bundle. There is a natural “inflation” functor from str ( X/k ) to str o ( X/S ) : consider an (absolute) stratified bundle on X/k as a (relative) stratified bundleon
X/S . In the other direction, pulling-back along f yields the functor ω f = f ∗ : str ( S/k ) −→ str ( X/k ) . Thus, to summarize, we have the following sequence of tensor functors: str ( S/k ) ω f / / str ( X/k ) infl / / C ( X/S ) res / / C ( X/S ) s / / str ( X s /k ) .Let x be a k -rational point of X/k . The fiber functor at x makes str ( X/k ) a neutral Tan-nakian category. Its Tannakian group is denoted by π ( X/k , x ) and called the fundamental groupscheme of X at x . Let s = f ( x ) . The functor ω f is compatible with the fiber functors at s and x . Thus, according to A.1.11, we have a homomorphism of fundamental group schemes f ∗ : π ( X/k , x ) −→ π ( S/k , s ) . The restriction functor str ( X/k ) −→ str ( X s /k ) is also a ten-sor functor and is compatible with the fiber functors at x , hence induces a homomorphism π ( X s /k , x ) −→ π ( X/k , x ) .Assume now that x = ξ ( s ) : X f (cid:15) (cid:15) Spec ( k ) s / / x ; ; ✇✇✇✇✇✇✇✇✇ S ξ Z Z Then the fiber functors x ∗ , ξ ∗ and f ∗ s are compatible in the sense that the following diagram iscommutative: str ( X/k ) / / x ∗ (cid:31) (cid:31) ❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃ C ( X/S ) s / / ξ ∗ ⊗ R k (cid:15) (cid:15) str ( X s /k ) f ∗ s (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ Vect ( k ) Hence it yields a sequence of group scheme homomorphism π ( X s /k , x ) −→ π ( X/S , ξ ) s −→ π ( X/k , x ) −→ π ( S/k , s ) . Proposition 5.1.9.
The homomorphism π ( X s /k , x ) −→ π ( X/S , ξ ) s is surjective.Proof. Here, the group schemes are defined over a field. Hence we can use the criterion forsurjectivity of Deligne-Milne [5, Thm. 2.21]. We show that for each object M ∈ str ( X/S ) s ,when considered as object in str ( X s /k , x ) all its subobjects will be an object in str ( X/S ) s . This isobvious from the fact that the category str o ( X/S ) is closed under taking subobjects. There existsby assumption an X ∈ str o ( X/S ) which surjects on M . We have the following pull-back diagram X / / / / MY ?(cid:31) O O / / / / N ?(cid:31) O O Since Y is a subobject of X , it is itself locally free. (cid:3) ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 31 π ( X s , x ) −→ π ( X , x ) −→ π ( S , s ) −→ f is a proper map. Hence in this case we also have an exact sequence π ( X/S , ξ ) s −→ π ( X/k , x ) −→ π ( S/k , s ) −→ ξ ∗ and the categories str ( X/k ) and str ( S/k ) yieldsthe fundamental groupoid schemes Π ( X/k , ξ ) and Π ( S/k , ξ ) , and the functor ω f : str ( S/k ) −→ str ( X/k ) yields a surjective homomorphism f ∗ : Π ( X , ξ ) −→ Π ( S , ξ ) . The kernel of this groupoidhomomorphism, is by definition L := S × Π ( S , ξ ) Π ( X , ξ ) ,where the map S −→ Π ( X , ξ ) is given by the unit element. This is a flat group scheme over S . Onother hand, the inflation functor str ( X/k ) −→ str ( X/S ) induces a homomorphism π ( X/S , ξ ) −→ Π ( X/k , ξ ) . The following question is motivated by dos Santos’ result mentioned above. Questions 5.1.11.
Assume that f : X −→ S is a smooth, proper map with connected fibers. Isthe following sequence exact π ( X/S , ξ ) −→ Π ( X/k , ξ ) −→ Π ( S/k , ξ ) −→ The case of a complete discrete valuation ring.
Let A = k [[ t ]] where k is a perfect field,with quotient field K = k (( t )) . Let X be a smooth connected formal affine scheme over Spf A . Let X be the special fiber of X and let X be the generic fiber. Assume that X admits an A -rationalpoint ξ . Our aim is to show that the category str (X /A ) is Tannakian.We will show that str o (X /A ) is subcategory of definition in str (X /A ) , that is, any stratifiedsheaf in str (X /A ) is a quotient of a stratified bundle. First we need the following. Proposition 5.2.1.
The restriction functor from str (X /k ) to str ( X /k ) is an equivalence. In partic-ular, any exact sequence in str ( X /k ) can be lifted to an exact sequence in str (X /k ) .Proof. If k is of positive characteristic, this is a result of Gieseker [13, Lemma 1.5]. He con-structed an explicit lift of a stratified bundle on X /k to a stratified bundle on X /k and showedthat this lift yields a functor which is quasi-inverse to the restriction functor, thus giving theequivalence.The case of zero characteristic can be proved using the method of Katz in the proof of [18,Prop. 8.8]. Let M be a stratified module over D (X /k ) . The action of this algebra on M will bedenoted as usual by ∇ . Since O X contains the field k , D (X /k ) is generated by the derivations,that is a stratification is nothing but a flat connection. We first show that it is locally free. Bymeans of Proposition 5.1.1 it suffices to show that M is t -torsion free. This is a local propertyon X .Let ( x , . . . , x r , t ) be local coordinates on X . Thus ∂ x i ’s commute each other and commutewith ∂ t , and we have ∂ x i ( x j ) = δ ij ; ∂ x i ( t ) = ∂ t ( x i ) = ∂ t ( t ) = One considers the k -linear operator P := ∞ X i = (− t ) i i ! ∂ it . It has the following properties, cf. [18,Sect. 8]: P = P ; P ( m ) = m (mod tM ) ; P ( fm ) = f ( ) P ( m ) , f ∈ A , m ∈ M .Hence, setting M ∇ t := ker ∇ ( ∂ t ) , we have(11) M ∇ t = im P ; M = M ∇ t ⊕ tM ; M ∇ t ∼ = M := M/tM .Further the map A ⊗ M ∇ t −→ M , f ⊗ m fm is injective, in particular, M ∇ t is t -torsion free.Assume that tm = m ∈ M . If m = m = t k − ( tm + m ) , where k > m ∈ M ∇ t (this is due to the completeness of the t -adic topologyon O X ). Then we have 0 = ∇ ( ∂ t ) k ( tm ) = ∇ ( ∂ t ) k ( t k + m ) + k ! m ,(since ∇ ( ∂ t )( m ) = m ∈ tM , which implies m =
0, contradiction. Hence m = M is t -torsion free, hence locally free over O X and consequently the restriction functor M M = M/tM is exact. It also implies that str (X /k ) is an abelian rigid tensor category over k . A section of M (as an object of str (X /k ) ) is horizontal iff (locally) it lies in M ∇ t ∼ = M and isannihilated by ∇ ( ∂ x i ) , and hence iff its image in M is a horizontal section of M as an objectof str ( X /k ) . We conclude that the restriction functor str (X /k ) −→ str ( X /k ) is faithful.Conversely, the third isomorphism in (11) shows that on an open U of X (which topologicallyhomeomorphic to X ), small enough so that local coordinates on it exist, a horizontal section of M | U can be uniquely lifted to a horizontal section of M | U . Let now s be a horizontal sectionof M ∈ str ( X /k ) . Consider an open covering ( U α ) of X such that on each U α there exist localcoordinates ( x i , t ) . Let s α be the restriction of s on U α . Lift s α to a horizontal section s α of M on U α . The restrictions of s α and s β on U αβ agree as they are liftings of the same section.Hence the s α ’s glue together to give a horizontal section of M on X . Thus the restriction functoris full.It remains to show that this functor is essentially surjective, that is, each stratified bundle M on X /k can be lifted to a stratified bundle on X /k . We first assume that on X there exist globalcoordinates ( x , . . . , x n , t ) and that M is free over O X with basis ( e i ) . Given this, we will showthat a flat connection on M can be lifted to M = h e i i O X .Consider the action of the operator P defined above on the algebra O X . We have ( D := ∇ ( ∂ t ) ) P ( ab ) = X i (− t ) i i ! D i ( ab )= X i (− t ) i i ! X j (cid:18) ij (cid:19) D j ( a ) D i − j ( b )= X j (− t ) j j ! D j ( a ) X i > j (− t ) i − j i ! D i − j ( b )= P ( a ) P ( b ) . ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 33
Hence the isomorphism ϕ : O ∇ t X −→ O X , induced by P , is an isomorphism of algebras. Noticethat, since [ ∂ x i , ∂ t ] = [ ∂ x i , D ] =
0, for all i . Thus ϕ commutes with the action of ∂ x i .Let ψ be the inverse of ϕ . Assume that the actions of ∇ ( ∂ x i ) on the basis ( e j ) is given by aset of matrices a i = ( a kij ) : ∇ ( ∂ x i )( e j ) = X k a kij e k .The flatness of ∇ is expressed in terms of the Maurer-Cartan equation involving the matices ( a kij ) and their partial derivatives in x i ’s. This equation is preserved by ψ , which means we canlift them to a set of matrices A j = ( A kij ) such that the equations: ∇ ( ∂ x i )( e j ) = X k A kij e k defines a flat connection on X /A .Finally we simply set ∇ ( ∂ t )( e i ) =
0. It is straightforward to check that ∇ ( ∂ t ) commuteswith ∇ ( ∂ x i ) using the fact that ∇ ( ∂ t )( A kij ) = A ∇ t . Thus, we haveconstructed a flat connection on M .In the general case we consider an open covering of X , such that on each open, the connec-tion M is free and local coordinates exist. Then on each open we can lift M . As the lift oneach open is unique, they glue together to give a lift of M on the whole X . (cid:3) Notice that if a stratified sheaf E on X /A is annihilated by t then it can be considered as asheaf on X /k and hence can be lifted to a stratified bundle on X /k , say E and we have an exactsequence(12) 0 −→ E [ t ] −→ E −→ E −→ [ t ] denotes the map multiplying by t . Thus in this case E is a quotient of E (consideredas stratified sheaf on X /A ). Proposition 5.2.2.
Each object of str (X /A ) is a quotient of an object of str (X /A ) . Consequently, str (X /A ) is a Tannakian category.Proof. Let E be an object of str (X /A ) . Then the subsheaf E t consisting of sections annihilated bysome power of t is invariant under the stratification. We have an exact sequence(13) 0 −→ E t −→ E −→ E fr −→ E fr a t -torsion free stratified sheaf (hence is locally free by the lemma above). There existsa least integer r such that E t is annihilated by t r . We will use induction on r .For r =
1, the subsheaf tE ⊂ E is t -torsion free. Indeed, if a section ts in tE is torsion then s is itself torsion, hence is annihilated by t . Consider the exact sequence0 −→ tE −→ E −→ E/tE −→ E/tE is in str ( X /k ) , hence can be lifted to a stratified bundle F on X /k : F −→ E/tE .Pull back the above sequence along this map (considered as morphism in str (X /A ) ), we get the following commutative diagram0 / / tE / / E / / E/tE / / / / tE / / = O O b E / / O O F / / O O O O b E −→ E is surjective. But b E is torsion free as the two sheaves tE and F arelocally free. Thus b E is the needed stratified bundle on X /A .Let now E be such that E t , the subsheaf of torsion sections, is annihilated by t n , n >
1. Let E be the subsheaf of E of section annihilated by t . Then we have exact sequence0 −→ E −→ E −→ E ′ −→ E ′ its torsion part E ′ t is annihilated by t n − . By induction we can lift E ′ to F ′ andhence, by pulling-back, we can lift E :0 / / E / / E / / E ′ / / / / E / / = O O F / / O O F ′ / / O O F ′ is locally free, E is the torsion subsheaf of F and we can lift F . (cid:3) Remarks 5.2.3.
Y. Andre has given in [1, 3.2.1.5] an example showing that the differentialGalois group of a connection in str o (X /A ) may be of infinite type over A .A PPENDIX
A. T
ANNAKIAN DUALITY FOR FLAT COALGEBRAS OVER D EDEKIND RINGS
In this appendix we give a quick, complete and self-contained proof of Theorem 1.2.2. First wewill recall the notion ind-category of an abelian category. The two equivalent descriptions of theind-category will play a crucial role in Saavedra’s proof. A category I is called a filtered categoryif to every pair i , j of objects in I there exists an object k such that Hom ( i , k ) and Hom ( j , k ) areboth not empty, and for every pair u , v : i −→ j , there exists a morphism w : j −→ k such that wu = wv . Definition A.1.1.
Ind-categories.
Let C be an abelian category. The category Ind ( C ) consists offunctors X : I −→ C , where I is a filtering category. We usually denote X i for X ( i ) , i ∈ I , an write X = lim −→ i ∈ I X i .For two objects X = lim −→ i ∈ I X i and Y = lim −→ j ∈ J Y j their hom-set is defined to be Hom ( X , Y ) := lim ←− i ∈ I lim −→ j ∈ J Hom ( X i , Y j ) . ✷ Let ω : C −→ D be a functor. The extension of ω , Ind ( ω ) : Ind ( C ) −→ Ind ( D ) is defined by Ind ( ω )( lim −→ i X i ) := lim −→ i ω ( X i ) . ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 35
There is an alternative description of
Ind ( C ) . Denote Lex ( C op , S ets ) the category of left exactfunctors from C op to the category of sets. For X = lim −→ i X i we define functorlim −→ i h X i (−) := lim −→ i Hom (− , X i ) ∈ Lex ( C op , S ets ) .This yields a functor Ind ( C ) −→ Lex ( C op , S ets ) which is an equivalence (cf.[2], I.8.3.3). Recallthat the Hom-sets for objects of Lex ( C op , S ets ) are by definition the sets of natural transforma-tions. For simplicity, we shall use the notation Hom ( F , G ) instead of Nat ( F , G ) for objects of thiscategory.A.1.2. Suppose that C is an R -linear Noetherian abelian category. Let
Lex R ( C op , Mod ( R )) becategory of R -linear left exact functors from C op to the category of modules Mod ( R ) . Then thenatural functor Lex R ( C op , Mod ( R )) ≃ −−→ Lex ( C op , S ets ) is an equivalence (cf. Gabriel [12, II]). Thus, for an R -linear Noetherian abelian category wehave an equivalence Ind ( C ) ≃ Lex R ( C op , Mod ( R )) , X = lim −→ i X i lim −→ i h X i (−) .Further the category Ind ( C ) is locally Noetherian and the inclusion C −→ Ind ( C ) identifies C withthe full subcategory of Noetherian objects in Ind ( C ) , [12, II,4, Thm.1].The following are our main examples. Example A.1.3.
The category
Mod f ( R ) of finitely generated R -modules, where R is a Noetherianring, is a Noetherian category. Its Ind category is precisely the category Mod ( R ) of all R -modules.This is obvious. Example A.1.4.
Let L be a coalgebra over a commutative ring R . Denote by Comod ( L ) thecategory of right L -comodules and by Comod f ( L ) the subcategory of comodules which are finitelygenerated as R -module. Then:(i) If L is flat over R then Comod ( L ) is an abelian category. In fact, the flatness of L impliesthat the kernel of a homomorphism of L -comodules is equipped with a natural coactionof L . In particular, the forgetful functor from Comod ( L ) to Mod ( R ) is exact. The converseis also true: if the forgetful functor preserves kernels then L is flat over R .(ii) Assume that L is flat over R and R is Noetherian. According to Serre [25, Cor. 2] each L -comodule is the union of its R -finite subcomodules. Consequently, Comod ( L ) is locallyNoetherian and Comod f ( L ) is the full subcategory of Noetherian objects.Let C be an R -linear abelian category, and ω : C −→ Mod f ( R ) be an R -linear exact faithfulfunctor. Suppose that there exists a full subcategory of definition C o in C . Our aim is to showthat there exists a flat R -coalgebra L such that ω induces an equivalence between Comod f ( L ) and C , and between Comod ( L ) and Ind ( C ) .The functor ω induces a functor Ind ( C ) −→ Mod ( R ) , which we, by abuse of language, willdenote simply by ω . Recall that we identify Ind ( C ) with Lex ( C op , Mod ( R )) , the category of leftexact functors on C op with values in Mod ( R ) . The key technique is to use alternatively these twoequivalent descriptions of one category. A.1.5. For any R -algebra A , we define functor F A : C op −→ Mod ( A ) , X Hom ( ω ( X ) , A ) .Then F A is an object of Lex ( C op , Mod ( R )) . Set F := F R . There is a natural A -linear transformation A ⊗ F −→ F A : θ X : A ⊗ Hom ( ω ( X ) , R ) −→ Hom ( ω ( X ) , A ) , a ⊗ f af . Lemma A.1.6.
The A -linear transformation θ : A ⊗ F −→ F A given above is an isomorphism.Proof. For any K , G ∈ Lex ( C op , Mod ( R )) we denote K o , G o their restrictions to ( C o ) op , respec-tively. We claim that Hom ( K , G ) ≃ Hom ( K o , G o ) .(14)Indeed, let θ ∈ Hom ( K o , G o ) , that is we have a family θ X : K o ( X ) −→ G o for X ∈ C o commutingwith morphism in C o . Since each object of C can be represented as a cokernel of a morphism X −→ X in C o , we see that θ extends uniquely to a natural transformation K −→ G (as thesefunctors are left exact on C op ).For X ∈ C o , ω ( X ) is finite projective over R , hence F A ( X ) = Hom ( ω ( X ) , A ) ≃ Hom ( ω ( X ) , R ) ⊗ A = A ⊗ F ( X ) .Therefore, for any G ∈ Lex ( C op , Mod ( R )) , we have Hom (( F A ) o , G o ) = Hom (( A ⊗ F ) o , G o ) (15)and (14) yields Hom ( F A , G ) = Hom ( A ⊗ F , G ) .(16)So we have F A ≃ A ⊗ F . (cid:3) We will show that L := ω ( F ) is the coalgebra to be found. To show this, first we will need Lemma A.1.7.
For any X ∈ Lex ( C op , Mod ( R )) and R -algebra A we have the following A -linearisomorphism: Hom ( X , F A ) ≃ Hom A ( A ⊗ ω ( X ) , A ) = Hom R ( ω ( X ) , A ) .(17) Proof.
Every X ∈ Lex ( C op , Mod ( R )) can be represented as X = lim −→ i h X i ( X i ∈ C ) , where h X i is afunctor over C , defined by h X i (−) := Hom C (− , X i ) . Hence we have Hom ( X , F A ) = Hom ( lim −→ h X i , F A )= lim ←− Hom ( h X i , F A )= lim ←− F A ( X i )= Hom R ( lim −→ ω ( X i ) , A )= Hom R ( ω ( lim −→ h X i ) , A )= Hom R ( ω ( X ) , A ) .It is easy to see that all isomorphisms are A -linear. (cid:3) ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 37
Isomorphism (17) for A = R and X = F reads Hom ( F , F ) ≃ Hom R ( ω ( F ) , R ) . We denote L := ω ( F ) and let ε : L −→ R be the map on the right hand side that corresponds to the identitytransformation on the left hand side of this isomorphism. The next lemma shows that one canreplace the algebra A in (17) by any R -module M to get R -linear isomorphisms. Lemma A.1.8.
There exists a natural R -linear isomorphism extending (17)(18) Φ X , M : Hom ( X , M ⊗ F ) ≃ Hom R ( ω ( X ) , M ) , which is given explicitly by Φ X , M ( f ) = ( id M ⊗ ε ) ◦ ω ( f ) . Proof.
For any R -module M , we can make R ⊕ M into an R -algebra by letting M be an ideal withsquare null. Hence the isomorphism (18) is a direct consequence of (17). By definition Φ F , R isgiven by Φ F , R ( f ) = ε ◦ ω ( f ) .Each R -linear map ι : R −→ M induces by functoriality the commutative diagram Hom ( F , F ) ( ι ⊗ id F ) ◦ − (cid:15) (cid:15) ε ◦ ω (−) / / Hom R ( ω ( F ) , R ) ι ◦ − (cid:15) (cid:15) Hom ( F , M ⊗ F ) Φ F , M / / Hom A ( ω ( F ) , M ) Now, the identity on F yields the equality: ι ◦ ε = Φ F , M ( ι ⊗ id ω ( F ) ) : ω ( F ) −→ M .Hence, for m = ι ( ) , we have Φ F , M ( l ) = ε ( l ) m , l ∈ ω ( F ) . Thus the claim holds for X = F . Sincethe ω and Hom-functor in the first variant commute with direct limits we conclude that theclaim hold of X = N ⊗ F for any R -module N . Now the general case follows from the followingdiagram Hom ( M ⊗ F , M ⊗ F ) Φ F , M ⊗ F / / (−) ◦ f (cid:15) (cid:15) Hom ( M ⊗ ω ( F ) , M ) (−) ◦ ω ( f ) (cid:15) (cid:15) Hom ( X , M ⊗ F ) Φ X , M / / Hom ( ω ( X ) , M ) applied for the identity of M ⊗ F : Φ X , M ( f ) = Φ F , M ( id ) ◦ ω ( f ) = ( id M ⊗ ε ) ◦ ω ( f ) . (cid:3) Proposition A.1.9.
Let L := ω ( F ) . Then it is a coalgebra with ε being the counit and ω factorsthough a functor Ind ( C ) −→ Comod ( L ) . Proof.
Choose M = ω ( X ) in (18) we have a morphism σ X : X −→ ω ( X ) ⊗ F which correspondsto the identity element id ω ( X ) under the isomorphism Φ X , ω ( X ) of Lemma A.1.8, thus we have(19) ( id ω ( X ) ⊗ ε ) ◦ ω ( σ X ) = id ω ( X ) . For any morphism λ : X −→ Y in Ind ( C ) , according to A.1.8 we have the following equalities: Φ X , ω ( Y ) (( ω ( λ ) ⊗ id F ) ◦ σ X ) = ω ( λ ) , Φ X , ω ( Y ) ( σ Y ◦ λ ) = ω ( λ ) .Thus ( ω ( λ ) ⊗ id F ) ◦ σ X = σ Y ◦ λ , i.e, the following diagram commutes:(20) X λ / / σ X (cid:15) (cid:15) Y σ Y (cid:15) (cid:15) ω ( X ) ⊗ F ω ( λ ) ⊗ id F / / ω ( Y ) ⊗ F .For Y = ω ( X ) ⊗ F and λ = σ X , we get(21) X σ X (cid:15) (cid:15) σ X / / ω ( X ) ⊗ F id ω ( X ) ⊗ σ F (cid:15) (cid:15) ω ( X ) ⊗ F ω ( σ X ) ⊗ id F / / ω ( X ) ⊗ L ⊗ F Applying ω on this diagram we obtain a commutative diagram in Mod ( R ) :(22) ω ( X ) ω ( σ X ) (cid:15) (cid:15) ω ( σ X ) / / ω ( X ) ⊗ L id ⊗ ∆ (cid:15) (cid:15) ω ( X ) ⊗ L ω ( σ X ) ⊗ id L / / ω ( X ) ⊗ L ⊗ L ,where ∆ := ω ( σ F ) . Together with (19), this diagram for X = F gives a coalgebra structure on L with ∆ being the coproduct and hence, for any X , it gives a comodule structure of L on ω ( X ) . (cid:3) Proof. (of Theorem 1.2.2) Let L be defined as in Proposition A.1.9. We consider ω as a functor C −→ Comod f ( L ) . It is to show that ω is an equivalence of category. By definition it is faithful. Tosee the fullness, suppose X , Y ∈ C and α : ω ( X ) −→ ω ( Y ) is a homomorphism of L -comodules,i.e., we have ( α ⊗ id ) ◦ ω ( σ X ) = ω ( σ Y ) ◦ α : ω ( X ) −→ ω ( Y ) ⊗ L .Then ω ( X ) α / / ω ( Y ) ω ( σ Y ) / / ω ( Y ) ⊗ L is the image under ω of the morphism X σ X / / ω ( X ) ⊗ F α ⊗ id F / / ω ( Y ) ⊗ F .Notice that (22) (for X replaced by Y ) yields a split exact sequence(23) 0 / / ω ( Y ) ω ( σ Y ) / / ω ( Y ) ⊗ L δ / / ω ( Y ) ⊗ L ⊗ L ,where the second homomorphism is δ = id ⊗ ∆ − ω ( σ X ) ⊗ id , and the splitting is given by id ⊗ ε : ω ( Y ) ⊗ L −→ ω ( Y ) . This sequence is the similar image under ω of the sequence comingfrom (21): 0 −→ Y −→ ω ( Y ) ⊗ F −→ ω ( Y ) ⊗ L ⊗ F . ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 39
Hence the latter sequence is also exact. On the other hand, it follows from the faithfulness of ω that the composed map X σ X / / ω ( X ) ⊗ F α ⊗ id / / ω ( Y ) ⊗ F / / ω ( Y ) ⊗ L ⊗ F is the zero morphism (since its image under ω is zero by means of (22) and the fact that α is a ho-momorphism of L -comodules). Consequently, the morphism X σ X / / ω ( X ) ⊗ F α ⊗ id / / ω ( Y ) ⊗ F factor through a morphism f : X −→ Y and the morphism σ Y . Applying ω on the compositionof these maps we conclude ω ( f ) = α , as ω ( σ Y ) is injective. Thus ω is full.It remains to show that ϕ is essentially surjective. For any L -comodule ( E , ρ E ) let E o ∈ C besuch that the sequence 0 −→ E o −→ E ⊗ F δ −−→ E ⊗ L ⊗ F .is exact, where δ = ρ E ⊗ id − id ⊗ σ F . Applying ω to this sequence and comparing with (23) weconclude that ω ( E o ) = E .Thus ω : C −→ Comod f ( L ) is an equivalence of categories. Thus the forgetful functor Comod f ( L ) −→ Mod ( R ) is exact, hence L is flat over R . (cid:3) Remarks A.1.10. (i) Under the equivalence of Theorem 1.2.2, L , with the right coaction of itselfgiven by the coproduct, corresponds to F . Indeed, this follows from the natural isomorphism Hom L ( E , L ) ≃ Hom R ( E , R ) , f ε ◦ f .(ii) There is another way to determine L from the category of its comodules as follows. Weclaim that there is a natural isomorphism(24) Nat ( ω , ω ⊗ M ) ≃ Hom R ( L , M ) ,for any R -module M . Indeed, we have Hom R ( L , M ) ≃ Hom ( F , F ⊗ M ) ≃ Hom ( Hom ( ω , R ) , Hom ( ω , R ) ⊗ M ) .By means of (14), it suffices to show the isomorphism Nat ( ω ( X ) , ω ( X ) ⊗ M ) ≃ Hom ( Hom ( ω ( X ) , R ) , Hom ( ω ( X ) , R ) ⊗ M ) for any X ∈ C o . Since for such X , ω ( X ) is finitely generated projective over R , the last isomor-phism is obvious. L is usually referred to as the Coend of ω , denoted Coend ( ω ) .(iii) If C = Comod f ( L ) and ω is the forgetful functor from C to Mod ( R ) , then the isomorphism(24) implies that Coend ( ω ) ≃ L . Thus a flat coalgebra over R can be reconstructed from thecategory of its comodules. ✷ Remarks A.1.11.
Let ( C , ω ) and ( C ′ , ω ′ ) be two categories satisfying the condition of Theorem1.2.2 and let η : C −→ C ′ be an R -linear functor such that ω ′ η = ω . Then η induces a coalgebrahomomorphism f : L −→ L ′ . This can be seen from (24) as follows. The coaction of L ′ on ω ′ ( X ′ ) defines a natural transformation δ ′ : ω ′ −→ ω ′ ⊗ L ′ . Combine this with η we obtain anatural transformation δ : ω −→ ω ⊗ L ′ . Thus (24) yields a linear map L −→ L ′ , which satisfies the following commutative diagram: ω ( X ) δ / / δ ′ % % ❑❑❑❑❑❑❑❑❑❑ ω ( X ) ⊗ L id ⊗ f (cid:15) (cid:15) ω ( X ) ⊗ L ′ ✷ A CKNOWLEDGMENT
The second named author would like to thank H. Esnault and J.P. dos Santos for their interestsin the work and very helpful discussions. He would also like to express his gratitude to J.-P. Serrefor explaining him about flat coalgebras. R
EFERENCES [1] Y. Andr´e,
Diff´erentielles non commutatives et Th´eorie de Galois diff´erentielle ou aux diff´erences.
Ann. Scient. Ec.Norm. Sup., 4 serie, t. 34, (2001), p. 685-739.[2] M. Artin, A. Grothendieck and J. L. Verdier,
Th´eorie des topos et cohomologie ´etale des sch´emas , Lecture Note inMath., 269, 270, 305, Springer-Verlag.[3] A. Brugui`eres,
On a Tannakian theorem due to Nori
Algebre commutative , Chapt. 1-4, Springer 2006.[5] P. Deligne, J. Milne,
Tannakian Categories , Lecture Notes in Mathematics 900, p. 101-228, Springer Verlag(1982).[6] P. Deligne,
Cat´egories tannakiennes , The Grothendieck Festschrift, Vol. II, p. 111-195, Progr. Math. 87, Birkh¨auser(1990).[7] M. Demazure and A. Grothendieck,
Propri´et´es Generales des Schemas en Groupes (SGA III), Expose VI B , LectureNotes in Mathematics, 151, Springer-Verlag (1970).[8] N.D Duong, P.H. Hai and J.P.P. dos Santos, On the structure of affine flat groups schemes over discrete valuationrings , preprint 2015, 46 pages, https://arxiv.org/abs/1701.06518.[9] H. Esnault and P. H. Hai,
The Gauß-Manin connection and Tannaka duality , IMRN, vol. 2006, Article ID. 93978,p. 1-35 (2006).[10] H. Esnault and P.H. Hai,
The fundamental group schemes and applications , Ann. Inst. Fourier, , 7 (2008), p.2381-2412.[11] H. Esnault, P. H. Hai and X. Sun, On Nori’s fundamental group scheme . Geometry and dynamics of groups andspaces, 377398, Progr. Math., 265, Birkh¨auser, Basel (2008).[12] P. Gabriel,
Des cat´egories ab´eliennes , Bulletin de la S.M.F., 90 (1962), p.323-448.[13] D. Gieseker,
Flat vector bundles and the fundamental group in non-zero characteristics , Ann. Sc. Norm. Super.Pisa Cl. Sci. (4) 2 (1) (1975), p. 1-31.[14] M. Hashimoto,
Auslander-Buchweitz approximations of equivariant modules , London Mathematical Society Lec-ture Note Series, 282. Cambridge University Press, Cambridge (2000), xvi+281 pp.[15] P.H. Hai, Gauss-Manin stratification and stratified fundamental group schemes., Ann. Inst. Fourier, , 6(2013), p. 2267-2285.[16] P.H. Hai, On an injectivity lemma in the proof of Tannakian duality , J. Alg. and its Appl., , 9 (2016), 1650167(9 pages).[17] J.C. Jantzen, Representations of algebraic groups , Pure and Applied Mathematics, 131. Academic, Inc., Boston(1987).[18] N. Katz,
Nilpotent connections and the monodromy theorem: applications of a result of Turrittin , Publ. Math.IHES, 39 (1970), p. 175-232.[19] H. Matsumura,
Commutative ring theory , Cambridge University Press (1986).[20] J. C. Moore,
Compl´ements sur les alg´ebres de Hopf , S´eminaire H. Cartan, 12, No. 1 (1959-1960), exp. 4, p. 1-12.
ANNAKIAN DUALITY OVER DEDEKIND RINGS AND APPLICATIONS 41 [21] N. Saavedra Rivano,
Cat´egories Tannakiennes , Lecture Notes in Mathematics, 265, Springer-Verlag, Berlin,(1972).[22] J. P. P. dos Santos,
Fundamental group schemes for stratified sheaves , J. Algebra, 317, p. 691-713 (2007).[23] J. P. P. dos Santos,
The behavior of the differential Galois group on the generic and special bres: A Tannakianapproach , J. reine angew. Math. 637, p. 63-98 (2009).[24] J. P. P. dos Santos,
The homotopy exact sequence for the fundamental group schemes and infinitesimal equivalencerelations , Algebraic Geometry, 2 (5) (2015), pp 535-590.[25] J-P. Serre,
Groupe de Grothendieck des sch´emas en groupes r´eductifs d´eploy´es , Publ. Math. 34 (1968), p.37-52.[26] The Stacks Project Authors,
Stacks Project , http://stacks.math.columbia.edu , 2015.[27] T. Szamuely, Galois groups and fundamental groups.
Cambridge Studies in Advanced Mathematics, 117 (2009).[28] W. C. Waterhouse,
Introduction to affine group schemes , Springer-Verlag (1979).[29] T. Wedhorn,
On Tannakian duality over valuation rings , J. Algebra 282 (2004), 575-609.[30] C. Weibel,
An introduction to homological algebra , Cambridge studies in advanced mathematics, 38.CambridgeUniversity Press, Cambridge, 1994.
E-mail address : [email protected] E-mail address : [email protected] (Nguyen Dai Duong & Ph`ung Hˆo Hai) I NSTITUTE OF M ATHEMATICS , V
IETNAM A CADEMY OF S CIENCE AND T ECH - NOLOGY , H
ANOI , V, V