MModel Theory of Proalgebraic Groups
Anand Pillay ∗ and Michael Wibmer † August 28, 2019
Abstract
We lay the foundations for a model theoretic study of proalgebraic groups. Our axiomati-zation is based on the tannakian philosophy. Through a tensor analog of skeletal categories weare able to consider neutral tannakian categories with a fibre functor as many-sorted first orderstructures. The class of diagonalizable proalgebraic groups is analyzed in detail. We show thatthe theory of a diagonalizable proalgebraic group G is determined by the theory of the basefield and the theory of the character group of G . Some initial steps towards a comprehensivestudy of types are also made. Introduction
Our initial inspiration for this paper goes back to the model theoretic treatment of profinite groupsdeveloped by G. Cherlin, L. van den Dries, A. Macintyre and Z. Chatzidakis in the eighties. (See[CvdDM81], [CvdDM82],[Cha84], [Cha98],[Cha87].) To a profinite group G they associate an ω -sorted structure consisting of the cosets gN of the open normal subgroups N of G ; the coset gN is of sort n if [ G : N ] ≤ n . In an appropriate language these structures can be axiomatized bya theory T and there is an anti-equivalence of categories between the category of profinite groupswith epimorphisms as morphisms and the category of models of T with embeddings as morphisms.A certain extension T IP of T is particularly well-behaved. The theory T IP axiomatizes profinitegroups G having the Iwasawa (or embedding) property: Any diagram G (cid:15) (cid:15) (cid:31) (cid:31) B (cid:47) (cid:47) A where B → A is an epimorphism of finite groups and G → A is an epimorphism can be completedto a commutative diagram via an epimorphism G → B , if B is a quotient of G . The theory of aprofinite group having the Iwasawa property is ω -categorical and ω -stable. Moreover, the saturatedmodels of T IP are exactly the free profinite groups.Some parts of the theory of free profinite groups have recently been generalized to proalgebraicgroups ([Wib]). This begs the questions, which aspects, if any, of the model theory of profinite groupshave a proalgebraic counterpart? To begin with, it is a priori rather unclear how to treat proalgebraic ∗ The first author was supported by the NSF grants DMS-1360702, DMS-1665035 and DMS-1760212. † The second author was supported by the NSF grants DMS-1760212, DMS-1760413, DMS-1760448 and the LiseMeitner grant M-2582-N32 of the Austrian Science Fund FWF.
Mathematics Subject Classification Codes:
Key words and phrases :Affine group schemes, proalgebraic groups, Tannakian categories, representation theory. a r X i v : . [ m a t h . L O ] A ug roups as first-order structures. One may envision that the role played by the cardinalities | G/N | ofthe finite quotients of a profinite group G could be replaced by the degrees of defining equations of thealgebraic quotients G/N of a proalgebraic group G . However, a key fact used in the axiomatizationof profinite groups is that if N and N are open normal subgroups of a profinite group G , then | G/ ( N ∩ N ) | is bounded by | G/N | · | G/N | . The degree does not exhibit such a behavior.The main achievement of this paper is the introduction of a many-sorted language that allows usto axiomatize proalgebraic groups. The key idea is based on the tannakian philosophy. Instead ofaxiomatizing proalgebraic groups directly, we axiomatize their categories of representations, i.e., weaxiomatize neutral tannakian categories together with a fibre functor. To implement this approach,certain technical challenges need to be overcome. For example, one cannot directly consider the classof all (finite dimensional, linear) representations of a proalgebraic group as a first order structurebecause this class is too big. Besides the fact that it is a proper class (i.e., not a set) the cardinalityof the first-order structure associated to a proalgebraic group should be something algebraicallymeaningful, like the rank of the profinite group in the profinite setting. Therefore, one has toconsider representations up to isomorphism. In other words, one has to consider skeletons of thecategory of representations of a proalgebraic group. To account for the fact that such a skeletonneed not be closed under the tensor product, we introduce a tensor analog of skeletal categories; anotion that we deem of independent interest in the study of tensor categories.We introduce a first-order theory PROALG in an appropriate many-sorted language such thatthe category of models of PROALG with the homomorphisms as morphisms is equivalent to thecategory of triples ( k, C, ω ), where k is a field, C a neutral tannakian category over k that satisfiesa tensor analog of being skeletal and ω is a fibre functor on C . We also show that the functor( k, C, ω ) (cid:32) ( k, Aut ⊗ ( ω )) to the category of proalgebraic groups (over varying base fields) is full,essentially surjective and induces a bijection on isomorphism classes. Thus, to every proalgebraicgroup G , there is associated a model M of PROALG that is unique up to an isomorphism. We cantherefore unambiguously define the theory of G as the theory of M .Even for algebraic groups as innocuous as the multiplicative group G m it is a non-trivial matterto determine their theory. We show that the theory of the multiplicative group over a field k isdetermined by the theory of k and the theory of its character group, i.e., by the theory of ( Z , +).Indeed, we establish a similar result for any diagonalizable proalgebraic group. If G is a proalgebraicgroup corresponding to a model M of PROALG, then the character group of G is interpretable in M .If G is diagonalizable with character group A there is a converse: The structure M is interpretablein the structure ( k, A ), with the language of fields on k and the language of abelian groups on A .In fact, we show that the theory of all diagonalizable proalgebraic groups is weakly bi-interpretablewith the theory of pairs ( k, A ), where k is a field and A an abelian group.We consider this article to be the first step in a model theoretic treatment of proalgebraic groups.Many, even rather basic questions remain open. However, we do give a flavor of the expressive powerof our theory PROALG by unveiling some of the algebraic information encoded in certain types.There is some thematic overlap between our work and work of M. Kamensky ([Kam15]) in thesense that both articles connect model theory and tannakian categories. However, the approachesand the aims differ: Our theory PROALG axiomatizes neutral tannakian categories together witha fibre functor for the purpose of advancing the theory of proalgebraic groups using model theoretictechniques. M. Kamensky’s theory T C (in a language L C dependent on C ) axiomatizes fibre functorson a fixed neutral tannakian category C for the purpose of reproving the main tannaka reconstructiontheorem using model theoretic techniques. On the other hand, we feel that this article may be seenas a possible answer to the open question 0.1.2 in [Kam15].One of the main motivations for the model theoretic treatment of profinite groups is that it hasapplications in the model theory of fields, in particular the model theory of pseudo algebraicallyclosed fields. This is based on the fact that for a field K , the first-order structure corresponding to the2bsolute Galois group of K is interpretable in the field K . For a differential field ( K, δ ) of characteris-tic zero with algebraically closed constants the absolute differential Galois group ([vdPS03],[BHHW])is a proalgebraic group. It appears that at least some reduct of the structure corresponding to the ab-solute differential Galois group of (
K, δ ) is interpretable in the differential field (
K, δ ). We thereforehope that our model theoretic treatment of proalgebraic groups will eventually lead to applicationsin the model theory of differential fields.Typically model theorists treat algebraic and proalgebraic groups simply as definable respectivelyprodefinable groups in ACF, the theory of algebraically closed fields. Our approach allows us to treatproalgebraic groups as structures in their own right. One advantage of our approach is that we canhandle non-reduced algebraic groups, such as, e.g., the group of p -th roots of unity in characteristic p , without difficulties, whereas the point-set approach dictated by ACF is oblivious to these groups.The article is organized as follows: The first section is purely algebraic, i.e., does not involveany model theory. After recalling the basic definitions and results from the tannakian theory weintroduce tensor skeletal tensor categories and the closely related notion of pointed skeletal neutraltannakian categories. We then proceed to define the category TANN. This category has as objectstripes ( k, C, ω ), where k is a field, C a pointed skeletal neutral tannakian category over k and ω aneutral fibre functor on C . We show that the functor ( k, C, ω ) (cid:32) ( k, Aut ⊗ ( ω )) from the categoryTANN to the category of proalgebraic groups is full, essentially surjective and induces a bijectionon the isomorphism classes.In the second section we present the axioms for PROALG in an appropriate many-sorted lan-guage. We show that the category of models of PROALG is equivalent to the category TANN andwe briefly discuss some elementary classes of proalgebraic groups.In the third section we study the theory of diagonalizable proalgebraic groups. We show that itis weakly bi-interpretable with the theory of pairs ( k, A ), where k is a field and A an abelian group.From this we deduce rather directly a description of the completions of the theory of diagonalizableproalgebraic groups and a characterization of elementary extensions. It also follows that the theoryof a diagonalizable proalgebraic group over an algebraically closed field is stable.In the final section we present some initial results concerning types. The main result is that if arepresentation of a proalgebraic group is considered as an element of a model of PROALG, then itstype over the base field determines the image of the representation. In this section we first recall the main definitions and results from the theory of tannakian cate-gories. Then we introduce a tensor version of skeletal categories and show that the isomorphismclasses of pointed skeletal neutral tannakian categories with a fibre functor are in bijection with theisomorphism classes of proalgebraic groups.
Notation and Conventions:
All rings are assumed to be commutative and unital. Throughout the article k denotes anarbitrary field, usually our “base field”. The category of finite dimensional k -vector spaces is denotedby Vec k . A proalgebraic group over k is, by definition, an affine group scheme over k . An algebraicgroup over k is an affine group scheme of finite type over k . We will often think of a proalgebraicgroup G as the functor R (cid:32) G ( R ) from the category of k -algebras to the category of groups.Conversely, a functor from the category of k -algebras to the category of groups is a proalgebraic It would admittedly be more accurate to use the term “pro-affine algebraic group” or “affine group scheme”instead of “proalgebraic group”. We hope the reader does not object to our choice of brevity over rigor in thisinstance. closed subgroup of a proalgebraic group is a closed subgroupscheme. Some helpful references for the theory of algebraic and proalgebraic groups are [Wat79],[DG70] and [Mil17].For a vector space V over k we denote by GL V the functor from the category of k -algebras tothe category of groups that assigns to any k -algebra R , the group of R -linear automorphisms of V ⊗ k R . A representation of a proalgebraic group G is a pair ( V, φ ), where V is a k -vector spaceand φ : G → GL V is a morphism of functors (also called a natural transformation), i.e., G ( R ) acts,functorially in R , on V ⊗ k R through R -linear automorphisms. A morphism f : ( V, φ ) → ( V (cid:48) , φ (cid:48) ) ofrepresentations of G is a k -linear map f : V → V (cid:48) such that V ⊗ k R f ⊗ R (cid:47) (cid:47) φ ( g ) (cid:15) (cid:15) V (cid:48) ⊗ k R φ (cid:48) ( g ) (cid:15) (cid:15) V ⊗ k R f ⊗ R (cid:47) (cid:47) V (cid:48) ⊗ k R commutes for all k -algebras R and g ∈ G ( R ). All representations are assumed to be finite dimen-sional unless the contrary is explicitly allowed. We denote the category of all finite dimensionalrepresentations of G by Rep( G ). We start by recalling the basic definitions and results from the theory of tannakian categories. See[Del90] and [DM82] for more details.
Definition 1.1. A tensor category is a tuple ( C, ⊗ , Φ , Ψ) , where C is a category, ⊗ : C × C → C is a functor and Φ and Ψ are isomorphisms of functors, called the associativity and commutativityconstraints respectively. More specifically, Φ has components Φ X,Y,Z : X ⊗ ( Y ⊗ Z ) → ( X ⊗ Y ) ⊗ Z and Ψ has components Ψ X,Y : X ⊗ Y → Y ⊗ X for objects X, Y, Z of C such that three commutativediagrams are satisfies. See [DM82, Section 1] for details. It is also required that there exists an identity object ( , u ) . This means that C → C, X (cid:32) ⊗ X is an equivalence of categories and u : → ⊗ is an isomorphism. We will often omit Φ and Ψ from the notation and refer to C or the pair ( C, ⊗ ) as a tensorcategory. If ( , u ) and ( (cid:48) , u (cid:48) ) are identity objects of a tensor category ( C, ⊗ ), then there exists aunique isomorphism a : → (cid:48) such that a (cid:15) (cid:15) u (cid:47) (cid:47) ⊗ a ⊗ a (cid:15) (cid:15) (cid:48) u (cid:48) (cid:47) (cid:47) (cid:48) ⊗ (cid:48) commutes ([DM82, Prop. 1.3, b)]). Moreover, there exists a unique isomorphism of functors l withcomponents l X : X → ⊗ X such that certain diagrams commute ([DM82, Prop. 1.3, a)]. Similarly,for r X : X → X ⊗ . Definition 1.2. A tensor functor from a tensor category ( C, ⊗ ) to a tensor category ( C (cid:48) , ⊗ (cid:48) ) is apair ( T, c ) , where T : C → C (cid:48) is a functor and c is an isomorphism of functors with components c X,Y : T ( X ) ⊗ (cid:48) T ( Y ) → T ( X ⊗ Y ) for objects X, Y of C such that T ( X ) ⊗ (cid:48) ( T ( Y ) ⊗ (cid:48) T ( Z )) id ⊗ (cid:48) c (cid:47) (cid:47) Φ (cid:48) (cid:15) (cid:15) T ( X ) ⊗ (cid:48) ( T ( Y ⊗ T ( Z )) c (cid:47) (cid:47) T ( X ⊗ ( Y ⊗ Z )) T (Φ) (cid:15) (cid:15) ( T ( X ) ⊗ (cid:48) T ( Y )) ⊗ (cid:48) T ( Z ) c ⊗ (cid:48) id (cid:47) (cid:47) T ( X ⊗ Y ) ⊗ (cid:48) T ( Z ) c (cid:47) (cid:47) T (( X ⊗ Y ) ⊗ Z ) commutes for all objects X, Y, Z of C , (ii) T ( X ) ⊗ (cid:48) T ( Y ) Ψ (cid:48) (cid:15) (cid:15) c (cid:47) (cid:47) T ( X ⊗ Y ) T (Ψ) (cid:15) (cid:15) T ( Y ) ⊗ (cid:48) T ( X ) c (cid:47) (cid:47) T ( Y ⊗ X ) commutes for all objects X and Y of C and (iii) if ( , u ) is an identity object of ( C, ⊗ ) , then ( T ( ) , T ( u )) is an identity object of ( C (cid:48) , ⊗ (cid:48) ) .A tensor functor ( T, c ) is strict if c is the identity transformation. In particular, T ( X ) ⊗ (cid:48) T ( Y ) = T ( X ⊗ Y ) for all objects X and Y of C . A tensor functor ( T, c ) is a tensor equivalence if T is anequivalence of categories. We will usually omit the isomorphism of functors c from the notation and refer to T as atensor functor. In Section 2 we will define first order structures corresponding to neutral tannakiancategories. Homomorphisms between such structures correspond to strict tensor functors. Thereforewe are mostly interested in strict tensor functors. We note that for c = id, the above two diagramsreduce to T (Φ X,Y,Z ) = Φ (cid:48) T ( X ) ,T ( Y ) ,T ( Z ) and T (Ψ X,Y ) = Ψ (cid:48) T ( X ) ,T ( Y ) . Definition 1.3.
Let ( T , c ) and ( T , c ) be tensor functors from ( C, ⊗ ) to ( C (cid:48) , ⊗ (cid:48) ) . A morphism α : T → T of functors is a morphism of tensor functors if (i) T ( X ) ⊗ (cid:48) T ( Y ) c (cid:47) (cid:47) α X ⊗ (cid:48) α Y (cid:15) (cid:15) T ( X ⊗ Y ) α X ⊗ Y (cid:15) (cid:15) T ( X ) ⊗ (cid:48) T ( Y ) c (cid:47) (cid:47) T ( X ⊗ Y ) commutes for all objects X, Y of C and (ii) (cid:48) (cid:124) (cid:124) (cid:34) (cid:34) T ( ) α (cid:47) (cid:47) T ( ) commutes, where the downwards morphisms are deduced from the uniqueness of the identityobject in ( C (cid:48) , ⊗ (cid:48) ) . Note that, assuming (i), condition (ii) is equivalent to α being an isomorphism. In particular,if α is an isomorphism of tensor functors, condition (ii) is vacuous. (Cf. [EGNO15, Def. 2.4.8].)5 efinition 1.4. A tensor category ( C, ⊗ ) is rigid, if for every object X of C there exists an object X ∨ (called a dual of X ) of C together with morphisms ev : X ⊗ X ∨ → and δ : → X ∨ ⊗ X suchthat X r −→ X ⊗ id ⊗ δ −−−→ X ⊗ ( X ∨ ⊗ X ) Φ −→ ( X ⊗ X ∨ ) ⊗ X ev ⊗ id −−−−→ ⊗ X l − −−→ X and X ∨ l −→ ⊗ X ∨ δ ⊗ id −−−→ ( X ∨ ⊗ X ) ⊗ X ∨ Φ − −−−→ X ∨ ⊗ ( X ⊗ X ∨ ) id ⊗ ev −−−−→ X ∨ ⊗ r − −−→ X ∨ are the identity morphism. In [Del90, Section 2] it is shown that the above definition of rigid is equivalent to the one usedin [DM82]. In a rigid tensor category the dual X ∨ of an object X is uniquely determined up to anisomorphism. A rigid tensor category ( C, ⊗ ) is abelian if C is an abelian category. In this case ⊗ isautomatically bi-additive ([DM82, Prop. 1.16]).Let k be a field and ( C, ⊗ ) a rigid abelian tensor category. An isomorphism between k and End( )induces the structure of a k -linear category on C such that ⊗ is k -bilinear. (See the discussion afterDef. 1.15 in [DM82].)Let R be a ring. A basic example of a tensor category is the category Mod R of all R -moduleswith the usual tensor product ( M , M ) (cid:32) M ⊗ R M of R -modules. The associativity constraintΦ is given byΦ M ,M ,M : M ⊗ R ( M ⊗ R M ) → ( M ⊗ R M ) ⊗ R M , m ⊗ ( m ⊗ m ) (cid:55)→ ( m ⊗ m ) ⊗ m and the commutativity constraint Ψ is given by Ψ M ,M : M ⊗ R M → M ⊗ R M , m ⊗ m (cid:55)→ m ⊗ m . Any free module U of rank one with an isomorphism u : U → U ⊗ R U is an identity object. Notethat any identity object ( U, u ) can be written in the form U = Ru and u : U → U ⊗ R U, u (cid:55)→ u ⊗ u for some basis element u of U .Let k be a field. Recall that Vec k is the category of all finite dimensional vector spaces over k . As explained above, this is a tensor category. In fact, (Vec k , ⊗ k ) is rigid and abelian. For a k -algebra R , the functor Vec k → Mod R , V (cid:32) V ⊗ k R together with the isomorphism of functors c with components c V,W : ( V ⊗ k R ) ⊗ R ( W ⊗ k R ) (cid:39) ( V ⊗ k W ) ⊗ k R is a tensor functor. Definition 1.5.
Let k be a field. A neutral tannakian category over k is a rigid abelian tensorcategory ( C, ⊗ ) together with an isomorphism k (cid:39) End( ) such that there exists an exact k -lineartensor functor ω : ( C, ⊗ ) → (Vec k , ⊗ k ) . Any such functor is called a (neutral) fibre functor . We note that a fibre functor is faithful by [Del90, Cor. 2.10]. Let ( C, ⊗ ) be a neutral tannakiancategory over the field k and ω : C → Vec k a fibre functor. For a k -algebra R , we denote thecomposition of ω with the tensor functor Vec k → Mod R , V (cid:32) V ⊗ k R by ω R . In particular, ω R ( X ) = ω ( X ) ⊗ k R for ever object X of C . We defineAut ⊗ ( ω )( R ) = Aut ⊗ ( ω R )as the group of tensor automorphisms (i.e., invertible morphisms of tensor functors) of the tensorfunctor ω R : ( C, ⊗ ) → (Mod R , ⊗ R ). This assignment is functorial in R and so Aut ⊗ ( ω ) is a functorfrom the category of k -algebras to the category of groups.Let G be a proalgebraic group over k . Recall (from the beginning of this section) that Rep( G )denotes the category of finite dimensional representations of G . With the associativity and commu-tativity constraint induced from Vec k , Rep( G ) is naturally a neutral tannakian category over k withfibre functor ( V, φ ) (cid:32) V . We are now prepared to state the main tannaka reconstruction theorem.6 heorem 1.6 ([DM82, Theorem 2.11]) . Let k be a field, C a neutral tannakian category over k and ω : C → Vec k a fibre functor. Then G = Aut ⊗ ( ω ) is a proalgebraic group over k and ω definesa tensor equivalence between C and Rep( G ) . Remark 1.7.
If we choose C = Rep( G ) for a some proalgebraic group G over a field k and ω : Rep( G ) → Vec k , ( V, φ ) (cid:32) V in Theorem 1.6, then Aut ⊗ ( ω ) (cid:39) G by [DM82, Prop. 2.8]. We introduce a tensor analog of skeletal categories and show that every tensor category is tensorequivalent to a tensor skeletal tensor category. Besides the cardinality issue already mentioned inthe introduction, there is another reason why it is important to work with skeletons: Two categoriesare ought to be considered the “same” if they are equivalent. However, the eyes of model theoryare conditioned to recognize the stronger notion of isomorphic categories. Since two categories areequivalent if and only if they have isomorphic skeletons, the two points of view can be reconciled byconsidering skeletons. (Cf. the remark at the very end of this section.)
Definition 1.8.
Let ( C, ⊗ ) be a tensor category. An object V of C is tensor irreducible if it is notin the image of ⊗ : C × C → C , i.e., V is not equal to V ⊗ W for objects V and W of C . Thetensor category ( C, ⊗ ) has the unique tensor factorization property if (i) V ⊗ W = V ⊗ W implies V = V and W = W for objects, V , V , W , W of C , i.e., ⊗ isinjective on objects and (ii) every object of C is a finite tensor product of tensor irreducible objects. The following example shows that a tensor category may not have any tensor irreducible objects.It also gives an example of a tensor category that does not satisfy condition (i) of the above definition.
Example 1.9.
Let k be a field and let C be the category whose objects are the k -vector spaces k n ( n ≥ C are all k -linear maps between any two objects of C . For m, n ≥ η m,n : k m ⊗ k k n → k mn .On objects we define ⊗ : C × C → C by k m ⊗ k n = k mn and on morphisms we define ⊗ throughthe isomorphisms η m,n , i.e., such that k m ⊗ k k n f ⊗ k f (cid:47) (cid:47) η m n (cid:15) (cid:15) k m ⊗ k k n η m n (cid:15) (cid:15) k m n f ⊗ f (cid:47) (cid:47) k m n commutes for any f : k m → k n and f : k m → k n . Similarly, we can use the isomorphisms η m,n to define associativity and commutativity constraints. If fact, C then becomes a neutral tannakiancategory over k . Note that no object of C is tensor irreducible. For example, k n = k n ⊗ k for every n ≥ C is a skeleton of the neutral tannakian category Vec k and that some choice(namely the choice of the η m,n ) was involved to define a tensor product on the skeleton C . Usingtensor skeletal tensor categories we will be able to avoid this choice. See Example 1.14 below. Lemma 1.10.
Let ( C, ⊗ ) be a tensor category with the unique tensor factorization property. Thenevery object of C is uniquely the tensor product of a finite completely parenthesized sequence oftensor irreducible objects of C . roof. We only have to establish the uniqueness. Let V be an object of C . If V is tensor irreduciblethe claim is obvious. So we may assume that V is not tensor irreducible. Assume we have twopresentations of V as tensor products of tensor irreducible objects. This yields two presentations V = V ⊗ W = V ⊗ W , where either V or W is tensor irreducible and either V or W aretensor irreducible. Without loss of generality, let us assume that V is tensor irreducible. Then, bycondition (i) of Definition 1.8, we find that V = V and W = W . So V = V is tensor irreducibleand we have two presentations of W = W as tensor products of tensor irreducible objects. Applyingthe same reasoning to W = W and iterating this process, we reach, after finitely many steps, asituation where W = W is tensor irreducible. Definition 1.11.
Let ( C, ⊗ ) be a tensor category with the unique tensor factorization property. The tensor length of an object V of C is the length of the unique completely parenthesized sequence oftensor irreducible objects whose tensor product equals V . In particular, the tensor irreducible objects of C are those of tensor length one. Definition 1.12.
A tensor category ( C, ⊗ ) is tensor skeletal if (i) any two isomorphic tensor irreducible objects are equal, (ii) every object of C is isomorphic to a tensor irreducible object and (iii) C has the unique tensor factorization property. In a tensor skeletal tensor category every isomorphism class contains a unique tensor irreducibleobject and any object is uniquely the tensor product of tensor irreducible objects. In particular, thetensor irreducible objects are a skeleton of the category C . Proposition 1.13.
Let ( C, ⊗ ) be a tensor category. Then there exists a tensor skeletal tensorcategory ( C (cid:48) , ⊗ (cid:48) ) and a strict tensor equivalence ( C (cid:48) , ⊗ (cid:48) ) → ( C, ⊗ ) .Proof. We first choose a skeleton S for the category C , i.e., S is a full subcategory of C such thatevery object of C is isomorphic to a unique object in S . In the next step we close S under thetensor product in a generic fashion. In detail, we define the category C (cid:48) as follows. Let P denote allcompletely parenthesized finite sequences of objects of S . Note that for a finite sequence V , . . . , V n of objects in C there are n (cid:0) n − n − (cid:1) possible ways to completely parenthesize the sequence. (This isthe ( n − n = 4, we have the following possibilities((( V , V ) , V ) , V ) (( V , V ) , ( V , V )) (( V , ( V , V )) , V ) , (( V , ( V , V )) , V ) ( V , (( V , V ) , V )) . To every p ∈ P we associate on object V ( p ) of C by evaluating the parenthesized sequence via ⊗ .For example, for p = ((( V , V ) , V ) , V ), we have V ( p ) = (( V ⊗ V ) ⊗ V ) ⊗ V . The class of objectsof C (cid:48) is defined as all pairs ( p, V ( p )), where p ∈ P . A morphism in C (cid:48) from ( p, V ( p )) to ( q, V ( q )) isa morphism in C from V ( p ) to V ( q ).We define a tensor product ⊗ (cid:48) : C (cid:48) × C (cid:48) → C (cid:48) as follows. On objects we set( p, V ( p )) ⊗ (cid:48) ( q, V ( q )) = ( pq, V ( p ) ⊗ V ( q )) = ( pq, V ( pq )) , where pq denotes the concatenation of two parenthesized sequences. For example, for p = ((( V , V ) , V ) , V )and q = (( W , W ) , W ) we have pq = (((( V , V ) , V ) , V ) , (( W , W ) , W )) . For morphisms f : ( p , V ( p )) → ( p , V ( p )) and g : ( q , V ( q )) → ( q , V ( q )) in C (cid:48) we define f ⊗ (cid:48) g : ( p , V ( p )) ⊗ (cid:48) ( q , V ( q )) = ( p q , V ( p ) ⊗ V ( q )) → ( p q , V ( p ) ⊗ V ( q )) = ( p , V ( p )) ⊗ (cid:48) ( q , V ( q ))8s f ⊗ g . We define an associativity constraint Φ (cid:48) for ⊗ (cid:48) by definingΦ (cid:48) ( p,V ( p )) , ( q,V ( q )) , ( r,V ( r )) : ( p, V ( p )) ⊗ (cid:48) (( q, ( V ( q )) ⊗ (cid:48) ( r, V ( r ))) → (( p, V ( p )) ⊗ (cid:48) ( q, ( V ( q ))) ⊗ (cid:48) ( r, V ( r ))as Φ V ( p ) ,V ( q ) ,V ( r ) : V ( p ) ⊗ ( V ( q ) ⊗ V ( r )) → ( V ( p ) ⊗ V ( q )) ⊗ V ( r ) . Similarly, we define a commutativityconstraint Ψ (cid:48) for ⊗ (cid:48) by definingΨ (cid:48) ( p,V ( p )) , ( q,V ( q )) : ( p, V ( p )) ⊗ (cid:48) ( q, V ( q )) → ( q, V ( q )) ⊗ (cid:48) ( p, V ( p ))as Ψ V ( p ) ,V ( q ) : V ( p ) ⊗ V ( q ) → V ( q ) ⊗ V ( p ). The commutative diagrams for Φ and Ψ yield thecorresponding commutative diagrams for Φ (cid:48) and Ψ (cid:48) . Moreover, if together with u : → ⊗ isan identity object for C , then (( ) , ) together with u (cid:48) : (( ) , ) → (( ) , ) ⊗ (cid:48) (( ) , ), defined as u is an identity object for C (cid:48) . Thus ( C (cid:48) , ⊗ (cid:48) ) is a tensor category.Let us show that ( C (cid:48) , ⊗ (cid:48) ) is tensor skeletal. Clearly, the tensor irreducible objects of C (cid:48) areexactly those of the form (( V ) , V ), where V belongs to S . Since S is a skeleton of C , it follows thatevery object of C (cid:48) is isomorphic to a tensor irreducible object and that any two isomorphic tensorirreducible objects are equal. By construction, every object of C (cid:48) is a tensor product of finitely manytensor irreducible objects. For p , p , q , q ∈ P with p q = p q we have p = p and q = q .It follows that condition (i) of Definition 1.8 is satisfied. Thus ( C (cid:48) , ⊗ (cid:48) ) is a tensor skeletal tensorcategory.The functor T : C (cid:48) → C, ( p, V ( p )) (cid:32) V ( p ) is clearly fully faithful. Since S is a skeleton it is anequivalence of categories. Moreover, T is a strict tensor functor by construction. Example 1.14.
Let S denote the skeleton of C = Vec k consisting of all vector spaces of the form k n , ( n ≥ C (cid:48) whose objectsare in bijection with the completely parenthesized finite sequences of k n ’s. The tensor product ⊗ (cid:48) for C (cid:48) is induced from the tensor product on Vec k . No additional choices an in Example 1.9 arerequired.Let ( C, ⊗ ) be a neutral tannakian category over a field k . Then the tensor product of two objectsof C that are not zero objects is not a zero object (e.g., because this property holds in Rep( G ), forany proalgebraic group G ). It follows that the full subcategory of all objects of C that are not zeroobjects is stable under the tensor product and therefore is naturally a tensor category. Definition 1.15.
A neutral tannakian category ( C, ⊗ ) over a field k is pointed skeletal if it hasexactly one zero object and the full subcategory of all objects that are not zero objects is tensorskeletal. We note that in a pointed skeletal neutral tannakian category ( C, ⊗ ) the zero object is not tensorirreducible. Moreover, an object of C , different from the zero object, is tensor irreducible in C ifand only if it is tensor irreducible in the tensor skeletal tensor category of all objects that are notthe zero object. Every object of C , different from the zero object, is uniquely the tensor productof a finite completely parenthesized sequence of tensor irreducible objects. As before, we call thelength of this sequence the tensor length of the object. The tensor irreducible objects together withthe zero object form a skeleton of C .We need to introduce the above notion for a rather technical reason: If one attempts to work withtensor skeletal neutral tannakian categories one runs into trouble with axiom (20) below, becausefor the zero object, this axiom does not seem to be expressible as a first order statement. We willneed a version of Proposition 1.13 for neutral tannakian categories. Corollary 1.16.
Let ( C, ⊗ ) be a neutral tannakian category over k . Then there exists a pointedskeletal neutral tannakian category ( C (cid:48) , ⊗ (cid:48) ) over k and a k -linear tensor equivalence ( C (cid:48) , ⊗ ) → ( C, ⊗ ) . roof. Let ( D, ⊗ ) denote the full subcategory of ( C, ⊗ ) consisting of all objects that are not zeroobjects. Applying Proposition 1.13 to the tensor category ( D, ⊗ ) yields a tensor category ( D (cid:48) , ⊗ (cid:48) )together with a strict tensor equivalence T : ( D (cid:48) , ⊗ (cid:48) ) → ( D, ⊗ ). Note that D (cid:48) does not have a zeroobject because T is an equivalence of categories and D does not have a zero object.We extend the category D (cid:48) to a category C (cid:48) by adding a zero object . So the objects of C (cid:48) arethe disjoint union of the objects of D (cid:48) with . The morphisms between two objects in C (cid:48) that bothbelong to D (cid:48) are the same as the morphisms in D (cid:48) . For an object V (cid:48) of C (cid:48) there is a unique morphism0 : → V (cid:48) in C (cid:48) . Similarly, there is a unique morphism 0 : V (cid:48) → . Composition of morphisms in C (cid:48) is defined such that composition with a zero morphism always yields a zero morphism. For example,the composition V (cid:48) → → W (cid:48) is the unique f ∈ Hom( V (cid:48) , W (cid:48) ) such that T ( f ) : T ( V (cid:48) ) → T ( W (cid:48) ) isthe zero morphism (in C ).We extend the functor T : D (cid:48) → D to a functor T : C (cid:48) → C by choosing T ( ) to be a zero objectof C and by defining T of a zero morphism to be a zero morphism. Then the functor T : C (cid:48) → C defines an equivalence of categories. Since C is abelian, it follows that also C (cid:48) is abelian.Next we extend ⊗ (cid:48) : D (cid:48) × D (cid:48) → D (cid:48) to a functor ⊗ (cid:48) : C (cid:48) × C (cid:48) → C (cid:48) in the only meaningful way.Namely, ⊗ V (cid:48) = and V (cid:48) ⊗ = for every object V (cid:48) of C (cid:48) . Similarly, f ⊗ ⊗ f = 0for any morphism f in C (cid:48) . (Here 0 denotes an appropriate zero morphism.) The associativity andcommutativity constraints on D (cid:48) extend trivially to associativity and commutativity constraintson C (cid:48) . So ( C (cid:48) , ⊗ (cid:48) ) is a tensor category. Moreover, T : ( C (cid:48) , ⊗ (cid:48) ) → ( C, ⊗ ) is a tensor equivalence.Since ( C, ⊗ ) is rigid, it follows that also ( C (cid:48) , ⊗ (cid:48) ) is rigid. For an identity object (cid:48) of C (cid:48) we haveEnd( (cid:48) ) (cid:39) End( T ( (cid:48) )) (cid:39) k . For the induced k -linear structure on C (cid:48) the functor T is k -linear.Composing T with a fibre functor ω : C → Vec k yields a fibre functor for C (cid:48) . Thus ( C (cid:48) , ⊗ (cid:48) ) is aneutral tannakian category over k . By construction ( C (cid:48) , ⊗ (cid:48) ) is pointed skeletal.The following lemma is needed in the next subsection to define the category TANN. Lemma 1.17.
Let k be a field and C a neutral tannakian category over k . If C is pointed skeletal,then C is small, i.e., the class of objects of C is a set.Proof. Let ω : C → Vec k be a fibre functor and set G = Aut ⊗ ( ω ). According to Theorem 1.6we have an equivalence of categories C → Rep( G ). Every representation of G is isomorphic to arepresentation of G on k n for some n ≥
0. Thus the class of objects of a skeleton of Rep( G ) is aset. Since C is pointed skeletal, the class of all tensor irreducible objects of C together with thezero object is a skeleton of C . Since equivalent categories have isomorphic skeletons, it follows thatthe class of tensor irreducible objects of C is a set. Since every object of C , different from the zeroobject, is a finite tensor product of tensor irreducible objects it follows that the class of objects of C is a set. TANN : The category of neutral tannakian categories
Our goal is to study proalgebraic groups from a model theoretic perspective by axiomatizing theircategories of representations. The models of our theory PROALG will correspond to pointed skeletalneutral tannakian categories with a fibre functor. The models of PROALG together with thehomomorphisms, i.e., the structure preserving maps, form a category that is equivalent to a certaincategory of neutral tannakian categories that we now describe in detail.We define the category TANN as follows. The objects of TANN are triples ( k, C, ω ), where k is a field, C is a pointed skeletal neutral tannakian category over k and ω : C → Vec k is a fibrefunctor. We note that by Lemma 1.17 pointed skeletal neutral tannakian categories are small, sothere is no set theoretic obstruction to forming this category, like the obstruction one encounterswhen attempting to form the category of all categories.10 morphism in TANN from ( k, C, ω ) to ( k (cid:48) , C (cid:48) , ω (cid:48) ) is a pair ( λ, T, α ), where λ : k → k (cid:48) is amorphism of fields, T : C → C (cid:48) is a k -linear strict tensor functor that preserves tensor irreducibleobjects and α : ω k (cid:48) → ω (cid:48) T is an isomorphism of tensor functors. Here ω k (cid:48) : C → Vec k (cid:48) denotesthe tensor functor obtained by composing ω with the tensor functor Vec k → Vec k (cid:48) , V (cid:32) V ⊗ k k (cid:48) induced by λ : k → k (cid:48) .The composition of two morphisms ( λ, T, α ) : ( k, C, ω ) → ( k (cid:48) , C (cid:48) , ω (cid:48) ) and ( λ (cid:48) , T (cid:48) , α (cid:48) ) : ( k (cid:48) , C (cid:48) , ω (cid:48) ) → ( k (cid:48)(cid:48) , C (cid:48)(cid:48) , ω (cid:48)(cid:48) ) in TANN is the pair ( λ (cid:48) λ, T (cid:48) T, γ ) : ( k, C, ω ) → ( k (cid:48)(cid:48) , C (cid:48)(cid:48) , ω (cid:48)(cid:48) ), where γ : ω k (cid:48)(cid:48) → ω (cid:48)(cid:48) T (cid:48) T isgiven by γ V : ω ( V ) ⊗ k k (cid:48)(cid:48) = ( ω ( V ) ⊗ k k (cid:48) ) ⊗ k (cid:48) k (cid:48)(cid:48) α V ⊗ k (cid:48)(cid:48) −−−−−→ ω (cid:48) ( T ( V )) ⊗ k (cid:48) k (cid:48)(cid:48) α (cid:48) T ( V ) −−−−→ ω (cid:48)(cid:48) ( T (cid:48) ( T ( V ))for every object V of C .We also define a category PROALGEBRAIC GROUPS. The objects are pairs ( k, G ), where k isa field and G a proalgebraic group over k . A morphism ( λ, φ ) : ( k, G ) → ( k (cid:48) , G (cid:48) ) inPROALGEBRAIC GROUPS is a pair ( λ, φ ), where λ : k → k (cid:48) is a morphism of fields and φ : G (cid:48) → G k (cid:48) is a morphism of proalgebraic groups over k (cid:48) . Here G k (cid:48) is the base change of G from k to k (cid:48) via λ . The composition ( λ (cid:48)(cid:48) , φ (cid:48)(cid:48) ) : ( k, G ) → ( k (cid:48) , G (cid:48)(cid:48) ) of two morphism ( λ, φ ) : ( k, G ) → ( k (cid:48) , G (cid:48) ) and( λ (cid:48) , φ (cid:48) ) : ( k (cid:48) , G (cid:48) ) → ( k (cid:48)(cid:48) , G (cid:48)(cid:48) ) is defined by λ (cid:48)(cid:48) = λ (cid:48) λ and φ (cid:48)(cid:48) : G (cid:48)(cid:48) φ (cid:48) −→ G (cid:48) k (cid:48)(cid:48) φ k (cid:48)(cid:48) −−→ ( G k (cid:48) ) k (cid:48)(cid:48) = G k (cid:48)(cid:48) .The following proposition is essential for establishing the close relationship between models ofPROALG and proalgebraic groups. Proposition 1.18.
The functor ( k, C, ω ) (cid:32) ( k, Aut ⊗ ( ω )) from the category TANN to the category
PROALGEBRAIC GROUPS is full, essentially surjective and induces a bijection on the isomor-phism classes.Proof.
Let ( k, C, ω ) be an object of TANN. From Theorem 1.6 we know that G = Aut ⊗ ( ω ) is aproalgebraic group over k and so we obtain an object ( k, G ) of PROALGEBRAIC GROUPS. Amorphism ( λ, T, α ) : ( k, C, ω ) → ( k (cid:48) , C (cid:48) , ω (cid:48) ) in TANN defines a morphism ( λ, φ ) : ( k, G ) → ( k (cid:48) , G (cid:48) ) inPROALGEBRAIC GROUPS as follows: Let R (cid:48) be a k (cid:48) -algebra and g (cid:48) ∈ G (cid:48) ( R (cid:48) ) = Aut ⊗ ( ω (cid:48) R (cid:48) ). Sofor every object V (cid:48) of C (cid:48) , we have an R (cid:48) -linear automorphism g (cid:48) V (cid:48) : ω (cid:48) ( V (cid:48) ) ⊗ k (cid:48) R (cid:48) → ω (cid:48) ( V (cid:48) ) ⊗ k (cid:48) R (cid:48) .We define an element φ ( g (cid:48) ) ∈ G k (cid:48) ( R (cid:48) ) = G ( R (cid:48) ) = Aut ⊗ ( ω R (cid:48) ) by φ ( g (cid:48) ) V : ω ( V ) ⊗ k R (cid:48) (cid:39) ( ω ( V ) ⊗ k k (cid:48) ) ⊗ k (cid:48) R (cid:48) α V ⊗ R (cid:48) −−−−−→ ω (cid:48) ( T ( V )) ⊗ k (cid:48) R (cid:48) g (cid:48) T ( V ) −−−−→ ω (cid:48) ( T ( V )) ⊗ k (cid:48) R (cid:48) (cid:39) ω ( V ) ⊗ k R (cid:48) for every object V of C . Then φ R (cid:48) : G (cid:48) ( R (cid:48) ) → G k (cid:48) ( R (cid:48) ) , g (cid:48) (cid:55)→ φ ( g (cid:48) ) is a morphism of groups that isfunctorial in R (cid:48) and therefore defines a morphism φ : G (cid:48) → G k (cid:48) of proalgebraic groups over k (cid:48) . Wethus have a functor from TANN to PROALGEBRAIC GROUPS.Let us show that this functor is full. We assume that a morphism ( λ, φ ) : ( k, G ) → ( k (cid:48) , G (cid:48) ) isgiven. We will define a morphism ( λ, T, α ) : ( k, C, ω ) → ( k (cid:48) , C (cid:48) , ω (cid:48) ) that induces ( λ, φ ). Let us firstexplain the idea for the construction of T : We have tensor functors C ω −→ Rep( G ) → Rep( G k (cid:48) ) → Rep( G (cid:48) ) , (1)where Rep( G ) → Rep( G k (cid:48) ) is given by V (cid:32) V ⊗ k k (cid:48) and Rep( G k (cid:48) ) → Rep( G (cid:48) ) is the restrictionvia φ : G (cid:48) → G k (cid:48) . Composing the functor (1) with a quasi-inverse of the tensor equivalence C (cid:48) ω (cid:48) −→ Rep( G (cid:48) ) yields a functor T : C → C (cid:48) . However, it is a priori not clear that a quasi-inverse canbe chosen in such a way that T is a strict tensor functor that preserves tensor irreducible objects.Moreover, the construction of T is intertwined with the construction of α .To define T and α , consider first a tensor irreducible object V of C . The representation of G = Aut ⊗ ( ω ) on ω ( V ), induces a representation of G k (cid:48) on ω ( V ) ⊗ k k (cid:48) and by restriction via11 : G (cid:48) → G k (cid:48) we obtain a representation of G (cid:48) on ω ( V ) ⊗ k k (cid:48) . By Theorem 1.6 the category C (cid:48) isequivalent (via ω (cid:48) ) to the category of representations of G (cid:48) . Thus there exists an object T ( V ) of C (cid:48) and an isomorphism α V : ω ( V ) ⊗ k k (cid:48) → ω (cid:48) ( T ( V )) of representations of G (cid:48) . In fact, since C (cid:48) is tensorskeletal, we may choose T ( V ) to be tensor irreducible.This defines T on tensor irreducible objects. To define T on an object V of C , different from thezero object and of tensor length n ≥
2, we may assume that T has already been defined on objectsof tensor length less than n . We know from Lemma 1.10 that V is uniquely of the form V = V ⊗ V ,where V and V have tensor length less than n . We can thus define T ( V ) as T ( V ) = T ( V ) ⊗ (cid:48) T ( V ).Finally, we define T of the zero object of C to be the (unique) zero object of C (cid:48) . This completes thedefinition of T on objects. Note that we have T ( V ⊗ W ) = T ( V ) ⊗ (cid:48) T ( W ) for all objects V, W of C .We extend the definition of α in a similar manner: We have already defined α V for tensorirreducible objects V . Let V be an object of C of tensor length n and assume α V has been definedon objects of tensor length less than n . As V is of the form V = V ⊗ V with V and V of tensorlength less then n , we can define α V = α V ⊗ V as the unique map making ω ( V ⊗ V ) ⊗ k k (cid:48) α V ⊗ V (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) ω (cid:48) ( T ( V ⊗ V )) (cid:39) (cid:15) (cid:15) ( ω ( V ) ⊗ k k (cid:48) ) ⊗ k (cid:48) ( ω ( V ) ⊗ k k (cid:48) ) α V ⊗ α V (cid:47) (cid:47) ω (cid:48) ( T ( V )) ⊗ k (cid:48) ( ω (cid:48) ( T ( V )) (2)commutative. For the zero object V , α V is defined as the zero map. Then, by construction, theabove diagram commutes for any pair of objects V and V of C .We next define T on morphisms. Let f : V → W be a morphism in C . We then have a morphism ω ( f ) : ω ( V ) → ω ( W ) of representations of G , that induces a morphism ω ( f ) ⊗ k (cid:48) : ω ( V ) ⊗ k k (cid:48) → ω ( W ) ⊗ k k (cid:48) of representations of G k (cid:48) . This is also a morphism of representations of G (cid:48) . In fact, wehave morphisms of representations of G (cid:48) ω ( V ) ⊗ k k (cid:48) ω ( f ) ⊗ k (cid:48) (cid:47) (cid:47) α V (cid:15) (cid:15) ω ( W ) ⊗ k k (cid:48) α W (cid:15) (cid:15) ω (cid:48) ( T ( V )) ω (cid:48) ( T ( W ))where the vertical maps are isomorphisms. Since ω (cid:48) induces an equivalence of categories, there existsa unique morphism T ( f ) : T ( V ) → T ( W ) in C (cid:48) such that ω ( V ) ⊗ k k (cid:48) ω ( f ) ⊗ k (cid:48) (cid:47) (cid:47) α V (cid:15) (cid:15) ω ( W ) ⊗ k k (cid:48) α W (cid:15) (cid:15) ω (cid:48) ( T ( V )) ω (cid:48) ( T ( f )) (cid:47) (cid:47) ω (cid:48) ( T ( W )) (3)commutes. This completes the definition of ( λ, T, α ). Let us check that T is indeed a strict tensorfunctor. To see that T is compatible with the associativity constraint, let U, V, W be objects of C and Φ U,V,W : U ⊗ ( V ⊗ W ) → ( U ⊗ V ) ⊗ W the corresponding associativity isomorphism. We havethe following commutative diagram: 12 ( U ⊗ ( V ⊗ W )) ⊗ k k (cid:48) ω (Φ U,V,W ) ⊗ k (cid:48) (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) ω (( U ⊗ V ) ⊗ W ) ⊗ k k (cid:48)(cid:39) (cid:15) (cid:15) ( ω ( U ) ⊗ k k (cid:48) ) ⊗ k (cid:48) (cid:0) ( ω ( V ) ⊗ k k (cid:48) ) ⊗ k (cid:48) ( ω ( W ) ⊗ k k (cid:48) ) (cid:1) (cid:47) (cid:47) α U ⊗ ( α V ⊗ α W ) (cid:15) (cid:15) (cid:0) ( ω ( U ) ⊗ k k (cid:48) ) ⊗ k (cid:48) ( ω ( V ) ⊗ k k (cid:48) ) (cid:1) ⊗ k (cid:48) ( ω ( W ) ⊗ k k (cid:48) ) ( α U ⊗ α V ) ⊗ α W (cid:15) (cid:15) ω (cid:48) ( T ( U )) ⊗ k (cid:48) (cid:0) ω (cid:48) ( T ( V )) ⊗ k (cid:48) ω (cid:48) ( T ( W )) (cid:1) (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) (cid:0) ω (cid:48) ( T ( U )) ⊗ k (cid:48) ω (cid:48) ( T ( V )) (cid:1) ⊗ k (cid:48) ω (cid:48) ( T ( W )) (cid:39) (cid:15) (cid:15) ω (cid:48) ( T ( U ) ⊗ ( T ( V ) ⊗ T ( W ))) = (cid:15) (cid:15) ω (cid:48) (Φ (cid:48) T ( U ) ,T ( V ) ,T ( W ) ) (cid:47) (cid:47) ω (cid:48) (( T ( U ) ⊗ T ( V )) ⊗ T ( W )) = (cid:15) (cid:15) ω (cid:48) ( T ( U ⊗ ( V ⊗ W ))) ω (cid:48) ( T (Φ U,V,W )) (cid:47) (cid:47) ω (cid:48) ( T (( U ⊗ V ) ⊗ W ))Thanks to the commutativity of (2), we know that the map from the upper left to the lower leftcorner is α U ⊗ ( V ⊗ W ) . Similarly, the map from the upper right to the lower right corner is α ( U ⊗ V ) ⊗ W .Since, by definition, T (Φ U,V,W ) is the unique morphism such that ω (cid:48) ( T (Φ U,V,W )) makes the outerrectangle of the above diagram commute, we conclude that T (Φ U,V,W ) = Φ (cid:48) T ( U ) ,T ( V ) ,T ( W ) as desired.In a similar fashion one shows that T (Ψ U,V ) = Ψ (cid:48) T ( U ) ,T ( V ) . Since T preserves identity objects weconclude that T is a strict tensor functor. Moreover, the commutativity of (3) shows that T is k -linear and by construction T preserves tensor irreducible objects.The commutativity of (3) also shows that α : ω k (cid:48) → ω (cid:48) T is an isomorphism of functors and thecommutativity of (2) shows that α is an isomorphism of tensor functors. Thus ( λ, T, α ) is indeed amorphism in TANN.As the α V ’s are morphisms of representations of G (cid:48) and G (cid:48) is acting on ω ( V ) ⊗ k k (cid:48) through therestriction via φ : G (cid:48) → G k (cid:48) , it is then clear that the morphism ( λ, T, α ) induces the morphism ( λ, φ )we started with. Thus the functor ( k, C, ω ) (cid:32) ( k, Aut ⊗ ( ω )) is full.We next show that it is essentially surjective. Let G be a proalgebraic group over a field k .Applying Corollary 1.16 to the neutral tannakian category Rep( G ) yields a pointed skeletal neutraltannakian category ( C, ⊗ ) and a k -linear tensor equivalence F : C → Rep( G ). We define a fibrefunctor ω : C → Vec k by composing F with the forgetful functor ω G : Rep( G ) → Vec k . Then( k, C, ω ) is an object of TANN. Moreover, since F is a tensor equivalence, the natural morphismof functors Aut ⊗ ( ω G ) → Aut ⊗ ( ω ) is an isomorphism. Since Aut ⊗ ( ω G ) is isomorphic to G (Remark1.7) we see that ( k, C, ω ) (cid:32) ( k, Aut ⊗ ( ω )) is essentially surjective.Finally, we establish the bijection on isomorphism classes. Since we already proved the essentialsurjectivity, it suffices to show the following: For objects ( k, C, ω ) and ( k (cid:48) , C (cid:48) , ω (cid:48) ) of TANN, if( k, Aut ⊗ ( ω )) and ( k (cid:48) , Aut ⊗ ( ω (cid:48) )) are isomorphic, then ( k, C, ω ) and ( k (cid:48) , C (cid:48) , ω (cid:48) ) are isomorphic. Weabbreviate G = Aut ⊗ ( ω ) and G (cid:48) = Aut ⊗ ( ω (cid:48) ). Let ( λ, φ ) : ( k, G ) → ( k (cid:48) , G (cid:48) ) be an isomorphism inPROALGEBRAIC GROUPS. In the above proof that ( k, C, ω ) (cid:32) ( k, Aut ⊗ ( ω )) is full, we havealready seen how to construct a morphism ( λ, T, α ) : ( k, C, ω ) → ( k (cid:48) , C (cid:48) , ω (cid:48) ) from ( λ, φ ). We claimthat ( λ, T, α ) is an isomorphism.We first show that T is surjective on objects. Let V (cid:48) be a tensor irreducible object of C (cid:48) .Then ω (cid:48) ( V (cid:48) ) is a representation of G (cid:48) . By assumption λ : k → k (cid:48) is an isomorphism of fields and φ : G (cid:48) → G k (cid:48) is an isomorphism of proalgebraic groups. In the sequel we will use λ − : k (cid:48) → k tobase change from k (cid:48) to k . For example, ω (cid:48) ( V (cid:48) ) ⊗ k (cid:48) k is a representation of G (cid:48) k . But G (cid:48) k is isomorphic13o G via G k (cid:48) φ k −→ ( G k (cid:48) ) k (cid:39) G and so we can consider ω (cid:48) ( V (cid:48) ) ⊗ k (cid:48) k to be a representation of G .Since ω : C → Rep( G ) is an equivalence of categories, there exists an object V of C such that ω ( V ) is isomorphic to ω (cid:48) ( V (cid:48) ) ⊗ k (cid:48) k as a representation of G . Moreover, since C is tensor skeletal,we can choose V to be tensor irreducible. It follows that ω ( V ) ⊗ k k (cid:48) is isomorphic to ω (cid:48) ( V (cid:48) ) as arepresentation of G (cid:48) . As T ( V ) is, by definition, the unique tensor irreducible object of C (cid:48) such that ω (cid:48) ( T ( V )) is isomorphic to ω ( V ) ⊗ k k (cid:48) as a representation of G (cid:48) , it follows that T ( V ) = V (cid:48) . So T issurjective on tensor irreducible objects. An arbitrary object V (cid:48) of C (cid:48) , different from the zero object,is a finite tensor product of tensor irreducible objects. We can choose an inverse image under T forall these tensor irreducible objects, form their tensor product in C and then apply the strict tensorfunctor T to see that V (cid:48) is in the image of T .We next show that T is injective on objects. First let V and V be tensor irreducible objects of C such that T ( V ) = T ( V ). Then ω (cid:48) ( T ( V )) = ω (cid:48) ( T ( V )) as representation of G (cid:48) and ω (cid:48) ( T ( V )) ⊗ k (cid:48) k = ω (cid:48) ( T ( V )) ⊗ k (cid:48) k as representation of G . But ω (cid:48) ( T ( V )) ⊗ k (cid:48) k (cid:39) ω ( V ) as representation of G andsimilarly for V . So ω ( V ) and ω ( V ) are isomorphic representations of G . But then V and V mustbe isomorphic objects of C . Since V and V are tensor irreducible, it follows that V = V . Thus T is injective on tensor irreducible objects. From the uniqueness in Lemma 1.10 it then follows that T is injective on objects.Using diagram (3), we see that T is fully faithful. Since T is bijective on objects, T is anisomorphism of categories, i.e., there exists a functor T − : C (cid:48) → C such that T T − = id C (cid:48) and T − T = id C . Since T is a strict tensor functor, also T − is a strict tensor functor. Similarly,as T preserves tensor irreducible objects, also T − preserves tensor irreducible objects. Finally, T − : C (cid:48) → C is k (cid:48) -linear, where the k (cid:48) -linear structure on C is defined via λ − : k (cid:48) → k .For every object V (cid:48) of C (cid:48) we have a k (cid:48) -linear isomorphism α T − ( V (cid:48) ) : ω ( T − ( V (cid:48) )) ⊗ k k (cid:48) → ω (cid:48) ( V (cid:48) ).The base change of this map via λ − : k (cid:48) → k is a k -linear isomorphism α T − ( V (cid:48) ) ⊗ k : ω ( T − ( V (cid:48) )) → ω (cid:48) ( V (cid:48) ) ⊗ k (cid:48) k . We define ( α − ) V (cid:48) = ( α T − ( V (cid:48) ) ⊗ k ) − . Then α − : ω (cid:48) k → ωT − is an isomorphismof functors. Since α is an isomorphism of tensor functors, also α − is an isomorphism of tensorfunctors. Finally, ( λ − , T − , α − ) is an inverse to ( λ, T, α ) in TANN.The functor of Proposition 1.18 is not faithful. This is reflected in the proof of the fullness ofthe functor by the fact that the α V ’s, for V tensor irreducible, can be chosen arbitrarily.We note that for Proposition 1.18 to be valid it is important to consider pointed skeletal neutraltannakian categories. For example, the neutral tannakian categories in Example 1.9 and 1.14 bothcorrespond to the trivial proalgebraic group. However, these two categories are not isomorphic. PROALG : Neutral tannakian categories as first order struc-tures
In this section we define a many-sorted first order theory PROALG such that the isomorphismclasses of models of PROALG are in bijection with the isomorphism classes of proalgebraic groups.The idea is to axiomatize pointed skeletal neutral tannakian categories with a fibre functor.
We define a many-sorted first order theory PROALG as follows:
Sorts:
We have three different types of sorts: The field sort, the objects sorts and the morphisms sorts.The objects sorts and the morphisms sorts split further into the base objects/morphisms sorts and14he total objects/morphisms sorts. We will use the following notation:With k we denote the universe of the field sort. For every pair p = ( m, n ) of integers m, n ≥ B p and the total objects sort withuniverse X p . For every pair p , q , where, as above p = ( m, n ) and q = ( m (cid:48) , n (cid:48) ) with m, n, m (cid:48) , n (cid:48) ≥ B p,q and the total morphismssort with universe X p,q .The idea is that B p , where p = ( m, n ), represents k -vector spaces of tensor length m anddimension n , considered as objects of a category, i.e., every element of B p corresponds to such avector space. On the other hand, X p contains the actual vector spaces.Similarly, for morphisms: B p,q represents morphisms from vector spaces in B p to vector spacesin B q ; every element of B p,q corresponds to a morphism, the actual linear maps are encoded in X p,q . Constant symbols: • We have two constant symbols 0 and 1 for the field-sort.
Relation symbols: • For every p we have a unary relation symbol 0 p on X p . • For every p we have a ternary relation symbol A p on X p . (The “A” is for addition.) • We have a constant symbol 1 in B (1 , . • For every p = ( m, n ) we have an n -ary relation symbol LI p on X p . (“LI” is for linear inde-pendence.) Function symbols: • We have two binary function symbols + and · for the field-sort. • For every p we have a function symbol π p with interpretation π p : X p → B p . • For every pair p, q we have a function symbol π p,q with interpretation π p,q : X p,q → B p,q . • For every p we have a function symbol SM p with interpretation SM p : k × X p → X p . (“SM”is for scalar multiplication.) • For all p, q we have function symbols S Bp,q and T Bp,q with interpretations S Bp,q : B p,q → B p and T Bp,q : B p,q → B q . (“S” is for source and “T” is for target of a morphism.) • For all p, q we have function symbols S Xp,q and T Xp,q with interpretations S Xp,q : X p,q → X p and T Xp,q : X p,q → X q . • For p = ( m, n ) and q = ( m (cid:48) , n (cid:48) ) with m, n, m (cid:48) , n (cid:48) ≥ pq = ( m + m (cid:48) , nn (cid:48) ). We havefunction symbols ⊗ p,q with interpretations ⊗ p,q : X p × X q → X pq .We denote this many-sorted language with L .15 .2 The axioms Rather than stating the axioms explicitly in the above language, we state their mathematical content.It is however clear that all the axioms below can be expressed as a collection of L -sentences.(1) ( k, + , · , ,
1) is a field.(2) For every p , the map π p : X p → B p is surjective. To simplify the notation we set X p ( b ) = π − p ( b ) for b ∈ B p .(3) Existence of zero: For every V = X p ( b ), (where b ∈ B p ), the set V ∩ p has a unique element0 V .(4) Vector space addition: For v , v , v ∈ X p , if A p ( v , v , v ) holds, then π p ( v ) = π p ( v ) = π p ( v ). Moreover, for b = π p ( v ) = π p ( v ) = π p ( v ) ∈ B p , and V = X p ( b ), the set { ( v , v , v ) ∈ V | A p ( v , v , v ) } is the graph of a map + V : V × V → V , that defines on V the structure of an abelian group with identity element 0 V .(5) Scalar multiplication: If λ ∈ k and v ∈ X p ( b ), for some b ∈ B p , then also SM p ( λ, a ) ∈ X p ( b ).Moreover, for every V = X p ( b ), the restriction of SM p to · V : k × V → V defines a scalarmultiplication on V such that V is a vector space over k with addition + V .(6) Dimension: Every X p ( b ) ( b ∈ B p ) is an n -dimensional k -vector space, where p = ( m, n ).(7) The maps π p,q : X p,q → B p,q are surjective. For f ∈ B p,q we set X p,q ( f ) = π − p,q ( f ).(8) The diagram X pπ p (cid:15) (cid:15) X p,qπ p,q (cid:15) (cid:15) T Xp,q (cid:47) (cid:47) S Xp,q (cid:111) (cid:111) X qπ q (cid:15) (cid:15) B p B p,qS Bp,q (cid:111) (cid:111) T Bp,q (cid:47) (cid:47) B q commutes.(9) Morphisms: For every f ∈ B p,q the map S Xp,q : X p,q ( f ) → X p ( S Bp,q ( f )) is bijective. Theimage of ( S Xp,q , T
Xp,q ) : X p,q ( f ) → X p ( S Bp,q ( f )) × X q ( T Bp,q ( f )), is the graph of a k -linear map (cid:101) f : X p ( S Bp,q ( f )) → X q ( T Bp,q ( f )).Moreover, if f, g ∈ B p,q with S Bp,q ( f ) = S Bp,q ( g ) and T Bp,q ( f ) = T Bp,q ( g ) such that (cid:101) f = (cid:101) g , then f = g .(10) Existence of the identity: For all b ∈ B p there exists an f ∈ B p,p such that S Xp,p ( a ) = T Xp,p ( a )for all a ∈ X p,p ( f ).(11) Composition of morphisms: For f ∈ B p,q and g ∈ B q,r with T Bp,q ( f ) = S Bq,r ( g ) there exists h ∈ B p,r such that (cid:101) h = (cid:101) g ◦ (cid:101) f .(12) Linearity: For f, g ∈ B p,q with S Bp,q ( f ) = S Bp,q ( g ) and T Bp,q ( f ) = T Bp,q ( g ), there exists h ∈ B p,q such that (cid:101) f + (cid:101) g = (cid:101) h .Moreover, for every f ∈ B p,q and λ ∈ k , there exists g in B p,q such that λ (cid:101) f = (cid:101) g . (In particular,for λ = 0, we see that the zero morphism is of the form (cid:101) g .)1613) Tensor is compatible with projections: For a ∈ X p and b ∈ X q we write a ⊗ b or a ⊗ p,q b for ⊗ p,q ( a, b ).If a , a ∈ X p with π p ( a ) = π p ( a ) and b , b ∈ X q with π q ( b ) = π q ( b ), then π pq ( a ⊗ b ) = π pq ( a ⊗ b ).(14) Bilinearity of tensor product: If a , a ∈ X p with π p ( a ) = π p ( a ) and b ∈ X q , then ( a + a ) ⊗ b = a ⊗ b + a ⊗ b . Moreover, for λ ∈ k we have λa ⊗ b = λ ( a ⊗ b ). Similarly for left-and right-hand side interchanged.(15) Tensor product: For b ∈ B p and c ∈ B q , let b ⊗ c = π pq ( v ⊗ w ) ∈ B pq , where v ∈ X p ( b ) and w ∈ X q ( c ). (Note that, by axiom 13, b ⊗ c does not depend on the choice of v and w .)The map X p ( b ) ⊗ k X q ( c ) → X pq ( b ⊗ c ) induced by the bilinear map ⊗ p,q : X p ( b ) × X q ( c ) → X pq ( b ⊗ c ) is an isomorphism. We set X p ( b ) ⊗ X q ( c ) = X pq ( b ⊗ c )(16) Functoriality of tensor product: For b ∈ B p , b ∈ B p , c ∈ B q , c ∈ B q , f ∈ B p ,q with S Bp ,q ( f ) = b and T Bp ,q ( f ) = c , and g ∈ B p ,q with S Bp ,q ( g ) = b and T Bp ,q ( g ) = c , thereexists h ∈ B p p ,q q such that (cid:101) h = (cid:101) f ⊗ (cid:101) g : X p p ( b ⊗ b ) = X p ( b ) ⊗ k X p ( b ) → X q ( c ) ⊗ k X q ( c ) = X q q ( c ⊗ c ) . (17) Associativity of the tensor product: For b ∈ B p , c ∈ B q and d ∈ B r , there exists f ∈ B pqr,pqr such that (cid:101) f : X p ( b ) ⊗ ( X q ( c ) ⊗ X r ( d )) → ( X p ( b ) ⊗ X q ( c )) ⊗ X r ( d ) equals the map defined by u ⊗ ( v ⊗ w ) (cid:55)→ ( u ⊗ v ) ⊗ w .(18) Commutativity of the tensor product: For b ∈ B p and c ∈ B q , there exists f ∈ B pq,pq suchthat (cid:101) f : X p ( b ) ⊗ X q ( c ) → X q ( c ) ⊗ X p ( b ) equals the map defined by v ⊗ w (cid:55)→ w ⊗ v .(19) Uniqueness of tensor factorization: If p q = p q , b ∈ B p , c ∈ B q , b ∈ B p and c ∈ B q are such that b ⊗ c = b ⊗ c , then p = p , p = q , b = b and c = c .(20) Existence of tensor factorization: For every b ∈ B ( m,n ) , there exist elements b , . . . , b m with b i ∈ B (1 ,n i ) for some n i with n . . . n m = n and a complete parenthesization of the sequence b , . . . , b m such that the corresponding tensor product of the sequence is equal to b .(21) Tensor skeletal: For b ∈ B ( m,n ) , there exists a c ∈ B (1 ,n ) and f ∈ B (( m,n ) , (1 ,n )) such that (cid:101) f : X ( m,n ) ( b ) → X (1 ,n ) ( c ) is bijective.Moreover, if b, c ∈ B (1 ,n ) are such that there exists an f in B (1 ,n ) , (1 ,n ) with (cid:101) f : X (1 ,n ) ( b ) → X (1 ,n ) ( c ) bijective, then b = c .(22) Existence of the identity object: Recall that 1 is a constant in B (1 , . For every non-zeroelement u of = X (1 , (1) and b ∈ B ( m,n ) , there exists f in B (( m,n ) , ( m +1 ,n ) such that (cid:101) f : X ( m,n ) ( b ) → ⊗ X ( m,n ) ( b ) is the map v (cid:55)→ u ⊗ v .(23) Existence of duals: For every b ∈ B ( m,n ) there exists b ∨ in B (1 ,n ) , f ∈ B ( m +1 ,n ) , (1 , and g ∈ B (1 , , ( m +1 ,n ) with (cid:101) f : V ⊗ V ∨ → and (cid:101) g : → V ∨ ⊗ V , where V = X ( m,n ) ( b ) and V ∨ = X (1 ,n ) ( b ∨ ), such that the maps V → V ⊗ V ⊗ (cid:101) g −−−→ V ⊗ ( V ∨ ⊗ V ) → ( V ⊗ V ∨ ) ⊗ V (cid:101) g ⊗ V −−−→ ⊗ V → VV ∨ → ⊗ V ∨ (cid:101) g ⊗ V ∨ −−−−→ ( V ∨ ⊗ V ) ⊗ V ∨ → V ∨ ⊗ ( V ⊗ V ∨ ) V ∨ ⊗ (cid:101) f −−−−→ V ∨ ⊗ → V ∨ are the identity maps. 1724) Existence of direct sums (biproducts): For b ∈ B ( m,n ) and c ∈ B ( m (cid:48) ,n (cid:48) ) , there exists d ∈ B (1 ,n + n (cid:48) ) , P b ∈ B ((1 ,n + n (cid:48) ) , ( m,n )) , P c ∈ B (1 ,n + n (cid:48) ) , ( m (cid:48) ,n (cid:48) ) , I b ∈ B ( m,n ) , (1 ,n + n (cid:48) ) and I c ∈ B ( m (cid:48) ,n (cid:48) ) , (1 ,n + n (cid:48) ) such that (cid:101) I b ◦ (cid:102) P b + (cid:101) I c ◦ (cid:102) P c = id X (1 ,n + n (cid:48) ) ( d ) , (cid:102) P b ◦ (cid:101) I b = id X ( m,n ) ( b ) , (cid:102) P c ◦ (cid:101) I c = id X ( m (cid:48) ,n (cid:48) ) , (cid:102) P c ◦ (cid:101) I b = 0and (cid:102) P b ◦ (cid:101) I c = 0.(25) Existence of kernels: For every f in B p,q and f (cid:48) in B r,q with (cid:101) f : V → W injective and (cid:101) f (cid:48) : U → W such that (cid:101) f (cid:48) ( U ) ⊆ (cid:101) f ( V ), there exists f (cid:48)(cid:48) ∈ B r,p with (cid:102) f (cid:48)(cid:48) : U → V such that (cid:101) f ◦ (cid:102) f (cid:48)(cid:48) = (cid:101) f (cid:48) .Moreover, for every f ∈ B p,q with (cid:101) f : V → W and dim(ker( (cid:101) f )) = (cid:96) ≥
1, there exists f (cid:48) ∈ B (1 ,(cid:96) ) ,p with (cid:101) f (cid:48) : U → V such that (cid:101) f (cid:48) is injective and (cid:101) f ◦ (cid:101) f (cid:48) = 0.(26) Existence of cokernels: For f ∈ B p,q and f (cid:48) ∈ B p,r with (cid:101) f : V → W surjective and (cid:101) f (cid:48) : V → U such that ker( (cid:101) f ) ⊆ ker( (cid:101) f (cid:48) ), there exists f (cid:48)(cid:48) ∈ B q,r with (cid:102) f (cid:48)(cid:48) : W → U such that (cid:102) f (cid:48)(cid:48) ◦ (cid:101) f = (cid:101) f (cid:48) .Moreover, for every f ∈ B p,q with (cid:101) f : V → W and dim(Im( (cid:101) f )) = (cid:96) < n , where q = ( m, n ),there exists (cid:101) f (cid:48) ∈ B ( q, (1 ,n − (cid:96) )) , with (cid:101) f (cid:48) : W → U , such that (cid:101) f (cid:48) is surjective and (cid:101) f (cid:48) ◦ (cid:101) f = 0.(27) Linear independence: For v , . . . , v n ∈ X p , we have LI p ( v , . . . , v n ), where p = ( m, n ), if andonly if π p ( v i ) = π p ( v j ) for 1 ≤ i, j ≤ n and v , . . . , v n are linearly independent (in X p ( b ),where b = π p ( v i )). Remark 2.1.
The relations LI p are definable from the other symbols of the language. So therelation symbols LI p could in principle be omitted from the language. It is however convenient towork with the LI p ’s, because they imply that a homomorphism of models of PROALG has certaindesirable properties. See Theorem 2.2 and its proof for details. PROALG and
TANN
Let M and M (cid:48) be models of PROALG. Recall that a homomorphism h : M → M (cid:48) is a sequence ofmaps, one for each sort s , that maps the M -universe of the s -sort to the M (cid:48) -universe of the s -sort,such that all constants, relations, and functions are preserved. Theorem 2.2.
The category of models of
PROALG with the homomorphisms as morphisms isequivalent to the category
TANN .Proof.
Let M = ( k, B p , X p , B p,q , X p,q ) be an object of PROALG. We will associate an object( k ( M ) , C ( M ) , ω ( M )) of TANN to M . We set k ( M ) = k (including the field structure) and wedefine a category D = D ( M ) as follows: The set of objects of D is the disjoint union of all B p ’s.For b ∈ B p and c ∈ B q , the set of morphisms from b to c is defined asHom( b, c ) = { f ∈ B p,q | S Bp,q ( f ) = b, T Bp,q ( f ) = c } . To define the composition g ◦ f for f ∈ Hom( b, c ) and g ∈ Hom( c, d ) we us axioms 9 and 11: Wedefine g ◦ f as the unique element of B p,r such that (cid:93) f ◦ g = (cid:101) g ◦ (cid:101) f . As the composition of k -linearmaps is associative, it follows that our composition is also associative. For b ∈ B p we define theidentity id b ∈ Hom( b, b ) to be the unique element of Hom( b, b ) with (cid:102) id b = id X p ( b ) (axiom 10). It isthen clear that D is a category.We define a tensor product ⊗ : D × D → D as follows: On objects, say b ∈ B p and c ∈ B q ,we define b ⊗ c ∈ B pq as in axiom 15. For morphisms f : b → c and g : b → c we define f ⊗ g : b ⊗ b → c ⊗ c as the unique element of Hom( b ⊗ b , c ⊗ c ) with (cid:93) f ⊗ g = (cid:101) f ⊗ (cid:101) g (axiom16). It is then clear that ⊗ is a functor. 18e define an associativity constraint Φ with components Φ b,c,d : b ⊗ ( c ⊗ d ) → ( b ⊗ c ) ⊗ d such that (cid:94) Φ b,c,d corresponds to the usual associativity constraint in Vec k (axiom 17). Similarly, we define acommutativity constraint Ψ using axiom 18. The required diagrams for Φ and Ψ commute becausethe corresponding diagrams commute in Vec k .Let u be any non-zero element of (see axiom 22) and let u be the unique element of Hom(1 , ⊗
1) such that (cid:101) u : → ⊗ , u (cid:55)→ u ⊗ u . We claim that (1 , u ) is an identity object for ( D, ⊗ ).Note that by axiom 25 every f ∈ B p,q such that (cid:101) f is bijective is an isomorphism. So it follows fromaxiom 22 that b (cid:32) ⊗ b is an equivalence of categories. Thus D is a tensor category. Axioms 19, 20and 21 imply that D is tensor skeletal. The tensor irreducible objects are those belonging to some B (1 ,n ) .Note that the category D does not have a zero object, because all the X p ( b ) (cid:48) s are vector spacesof dimension greater or equal to n ≥
1. We now add a zero object to D to form a category C = C ( M ). This is done in a similar fashion as in the proof of Corollary 1.16. So the objects of C are the disjoint union of the objects of D with . The morphisms between two objects in C that bothbelong to D are the same as the morphisms in C . For an object b of D there is a unique morphism0 : → b in C . Similarly, there is a unique morphism 0 : b → . Composition of morphisms in C isdefined in the obvious way. For example, the composition b → → c is the unique f ∈ Hom( b, c )such that (cid:101) f is the zero map (cf. axiom 12).For consistence reasons we extend some of our notation to include zero: We define X ( ) to bethe zero vector space (over k ) and we set (cid:101) ⊗ : D × D → D to a functor ⊗ : C × C → C inthe only meaningful way. The associativity and commutativity constraints on D extend trivially toassociativity and commutativity constraints on C . So ( C, ⊗ ) is a tensor category. It follows fromaxiom 23 that ( C, ⊗ ) is rigid.Let us next show that C is an abelian category. Clearly C has a zero object, namely . Byaxiom 24 the category C has biproducts. It follows from axioms 25 and 26 that C has kernels andcokernels. Let f : b → c be a monomorphism in C . We claim that (cid:101) f : X p ( b ) → X q ( c ) is injective.For a kernel g : a → b of f we have f g = 0 = f g = 0. Since the image of (cid:101) g isthe kernel of (cid:101) f we see that (cid:101) f is injective. It then follows from axiom 26 that f is the kernel of itscokernel. So f is normal. Similarly, we see that also every epimorphism in C is normal. Thus C isan abelian category.For f ∈ Hom( b, c ) and λ ∈ k we define λf as the unique element of Hom( b, c ) such that (cid:102) λf = λ (cid:101) f (axiom 12). Thus C becomes a k -linear category.We now define the fibre functor ω = ω ( M ) from C to Vec k : For b ∈ B p we set ω ( b ) = X p ( b ).For a morphism f : b → c in C we set ω ( f ) = (cid:101) f . By construction f is an exact k -linear functor. Theisomorphism γ : ω ( − ) ⊗ k ω ( − ) → ω ( − ⊗ − ) of functors, with components γ b,c : ω ( b ) ⊗ k ω ( c ) = X p ( b ) ⊗ k X p ( c ) → X pq ( b ⊗ c ) = ω ( b ⊗ c ) , v ⊗ w → v ⊗ p,q w turns ω into a tensor functor (cf. axiom 15). Thus C is a neutral tannakian category over k . Since( D, ⊗ ) is tensor skeletal, we see that, as desired, C is a pointed skeletal neutral tannakian categoryover k . Thus ( k ( M ) , C ( M ) , ω ( M )) is an object of TANN.Let h : M → M (cid:48) be a homomorphism of models of PROALG. Where M = ( k, B p , X p , B p,q , X p,q ), M (cid:48) = ( k (cid:48) , B (cid:48) p , X (cid:48) p , B (cid:48) p,q , X (cid:48) p,q ) and h = ( h field , h Bp , h Xp , h Bp,q , h
Xp,q )We claim that h induces a morphism ( λ ( h ) , T ( h ) , α ( h )) : ( k, C, ω ) → ( k (cid:48) , C (cid:48) , ω (cid:48) ) in TANN. Weset λ ( h ) = h field and consider k (cid:48) as a field extension of k via λ ( h ).We define the functor T = T ( h ) : C → C (cid:48) through the action of h on the base sorts: We set T ( b ) = h Bp ( b ) for b ∈ B p and T ( f ) = h Bp,q ( f ) for f ∈ B p,q . We also set T ( ) = (cid:48) and T (0) = 0. The19ommutativity of B ph p (cid:15) (cid:15) B p,qS Bp,q (cid:111) (cid:111) T Bp,q (cid:47) (cid:47) h p,q (cid:15) (cid:15) B qh q (cid:15) (cid:15) B (cid:48) p B (cid:48) p,qS Bp,q (cid:111) (cid:111) T Bp,q (cid:47) (cid:47) B (cid:48) q shows that T maps Hom( b, c ) into Hom( T ( b ) , T ( c )). The commutativity of X p h Xp (cid:47) (cid:47) π p (cid:15) (cid:15) X (cid:48) pπ p (cid:15) (cid:15) B p h Bp (cid:47) (cid:47) B (cid:48) p shows that h Xp maps X p ( b ) into X (cid:48) p ( T ( b )). Similarly, the commutativity of X p,q h Xp,q (cid:47) (cid:47) π p,q (cid:15) (cid:15) X (cid:48) p,qπ p,q (cid:15) (cid:15) B p,q h Bp,q (cid:47) (cid:47) B (cid:48) p,q shows that h Xp,q maps X p,q ( f ) into X p,q ( T ( f )). For a morphism f : b → c in C the left, right, upperand other squares in X p,q ( f ) T Xp,q (cid:36) (cid:36) h Xp,q (cid:47) (cid:47) S Xp,q (cid:15) (cid:15) X (cid:48) p,q ( T ( f )) S Xp,q (cid:15) (cid:15) T Xp,q (cid:122) (cid:122) X p ( b ) h Xp (cid:47) (cid:47) (cid:101) f (cid:15) (cid:15) X (cid:48) p ( T ( b )) (cid:93) T ( f ) (cid:15) (cid:15) X q ( c ) h Xq (cid:47) (cid:47) X (cid:48) q ( T ( c ))commute. Thus also the lower square X p ( b ) h Xp (cid:47) (cid:47) (cid:101) f (cid:15) (cid:15) X (cid:48) p ( T ( b )) (cid:93) T ( f ) (cid:15) (cid:15) X q ( c ) h Xq (cid:47) (cid:47) X (cid:48) q ( T ( c )) (4)commutes. Since h preserves A p and SM p it follows that the maps h Xp : X p ( b ) → X (cid:48) p ( T ( b )) are k -linear. Furthermore, since h preserves LI p and X p ( b ) and X (cid:48) p ( T ( b )) have the same dimension,we see that the induced map α b : X p ( b ) ⊗ k k (cid:48) → X (cid:48) p ( T ( b )) is an isomorphism of k (cid:48) -vector spaces.Diagram (4) extends to X p ( b ) ⊗ k k (cid:48) α b (cid:47) (cid:47) (cid:101) f ⊗ k (cid:48) (cid:15) (cid:15) X (cid:48) p ( T ( b )) (cid:93) T ( f ) (cid:15) (cid:15) X q ( c ) ⊗ k k (cid:48) α c (cid:47) (cid:47) X (cid:48) q ( T ( c )) (5)20 diagram of k (cid:48) -linear maps.This implies that T (id b ) = id T ( b ) . Moreover, for morphisms f : b → c and g : c → d in C , thecommutativity of the diagram X p ( b ) (cid:102) gf (cid:36) (cid:36) (cid:101) f (cid:47) (cid:47) h Xp (cid:15) (cid:15) X q ( c ) (cid:101) g (cid:47) (cid:47) h Xq (cid:15) (cid:15) X r ( d ) h Xr (cid:15) (cid:15) X (cid:48) p ( b ) (cid:94) T ( g ) T ( f ) (cid:58) (cid:58) (cid:93) T ( f ) (cid:47) (cid:47) X (cid:48) q ( c ) (cid:93) T ( g ) (cid:47) (cid:47) X (cid:48) r ( d )shows that T ( gf ) = T ( g ) T ( f ). Thus T is a functor. We claim that T is a strict tensor functor.Since the diagram X p ( b ) × X q ( c ) h Xp × h Xq (cid:15) (cid:15) ⊗ p,q (cid:47) (cid:47) X pq ( b ⊗ c ) h Xpq (cid:15) (cid:15) X (cid:48) p ( T ( b )) × X (cid:48) q ( T ( c )) ⊗ p,q (cid:47) (cid:47) X (cid:48) pq ( T ( b ) ⊗ T ( c )) (6)commutes, we see that T ( b ⊗ c ) = T ( b ) ⊗ T ( c ). Note that the above diagram can also be expressedas h Xp ⊗ h Xq = h Xpq . The diagram X ( b ⊗ ( c ⊗ d )) (cid:94) Φ b,c,d (cid:47) (cid:47) X (( b ⊗ c ) ⊗ d ) X ( b ) ⊗ ( X ( c ) ⊗ X ( d )) (cid:47) (cid:47) h X ⊗ ( h X ⊗ h X ) (cid:15) (cid:15) ( X ( b ) ⊗ X ( c )) ⊗ X ( d ) ( h X ⊗ h X ) ⊗ h X (cid:15) (cid:15) X (cid:48) ( T ( b )) ⊗ ( X (cid:48) ( T ( c )) ⊗ X (cid:48) ( T ( d ))) (cid:47) (cid:47) ( X (cid:48) ( T ( b )) ⊗ X (cid:48) ( T ( c ))) ⊗ X (cid:48) ( T ( d )) X (cid:48) ( T ( b ⊗ ( c ⊗ d ))) (cid:94) Φ (cid:48) T ( b ) ,T ( c ) ,T ( d ) (cid:47) (cid:47) X (cid:48) ( T (( b ⊗ c ) ⊗ d )))commutes, where for simplicity we have omitted the p, q, r indices. By (6) the map from the upperleft to the lower right corner is h X . Similarly, the map from the upper right corner to the lowerright corner is h X . It thus follows that T (Φ b,c,d ) = Φ (cid:48) T ( b ) ,T ( c ) ,T ( d ) . In a similar fashion one showsthat T (Ψ b,c ) = Ψ (cid:48) T ( b ) ,T ( c ) . Since h B (1 , (1) = 1 (cid:48) we have T (1) = 1 (cid:48) and so T is a strict tensor functor.From diagram (2) it follows that T is k -linear. Clearly T preserves tensor irreducible objects. Insummary, as desired, T is a k -linear strict tensor functor that preserves tensor irreducible objects.To obtain a morphism in TANN, we also need to specify an isomorphism α = α ( h ) : ω k (cid:48) → ω (cid:48) T of tensor functors. But the collection of all α b : X p ( b ) ⊗ k k (cid:48) → X (cid:48) p ( T ( b )) defined above exactly yieldssuch an isomorphism: The commutativity of (5) shows that α is a morphism of functors, whilstthe commutativity of (6) implies that α is an isomorphism of tensor functors. So we indeed have afunctor from PROALG to TANN. 21e next show that the functor M (cid:32) ( k ( M ) , C ( M ) , ω ( M )) is faithful. Let h, g : M → M (cid:48) behomomorphisms such that ( λ ( h ) , T ( h ) , α ( h )) = ( λ ( g ) , T ( g ) , α ( g )). Then h field = λ ( f ) = λ ( g ) = g field . Moreover, h Bp = g Bp and h Bp,q = g Bp,q for all p, q since T ( h ) = T ( g ).On X p ( b ), h Xp agrees with α ( h ) b and g Xp agrees with α ( g ) b . Thus h Xp = g Xp . To show that also h Xp,q = g Xp,q , consider f ∈ B ( p,q ) . The vertical maps in the diagram X p,q ( f ) (cid:47) (cid:47) S Xp,q (cid:15) (cid:15) X (cid:48) p,q ( T ( f )) S Xp,q (cid:15) (cid:15) X p ( b ) h Xp = g Xp (cid:47) (cid:47) X (cid:48) p ( T ( b ))are bijective. Thus there exists a unique map X p,q ( f ) → X (cid:48) p,q ( T ( f )) that makes this diagramcommutative. As the restrictions to X p,q ( f ) of both, h Xp,q and g Xp,q indeed make this diagram com-mutative, we see that h Xp,q = g Xp,q .To show that the functor M (cid:32) ( k ( M ) , C ( M ) , ω ( M )) is full, consider a morphism( λ, T, α ) : ( k ( M ) , C ( M ) , ω ( M )) → ( k ( M (cid:48) ) , C ( M (cid:48) ) , ω ( M (cid:48) ))in TANN. We have to construct a homomorphism h : M → M (cid:48) that induces ( λ, T, α ). We set h field = λ : k → k (cid:48) .Note that for b ∈ B p = B ( m,n ) , the k -vector space ω ( b ) = X ( m,n ) ( b ) has dimension n . Since α b : X p ( b ) ⊗ k k (cid:48) → ω (cid:48) ( T ( b )) is an isomorphism of k (cid:48) -vector spaces we see that ω (cid:48) ( T ( b )) also hasdimension n . Moreover, as T preserves tensor irreducible objects, we see that T ( b ) has tensorlength m . So T ( b ) ∈ B (cid:48) p . Thus T induces maps h Bp : B p → B (cid:48) p . Since B p,q is the set of allmorphism in C with source in B p and target in B q , it then also follows that T induces maps h Bp,q : B p,q → B (cid:48) p,q , f (cid:55)→ T ( f ).We define h Xp : X p → X (cid:48) p by h Xp ( v ) = α b ( v ⊗ b = π p ( v ). To define h Xp,q , consider f ∈ B p,q with f : b → c . We define h Xp,q : X p,q → X (cid:48) p,q to be the unique map whose restriction toany X p,q ( f ) makes X p,q ( f ) (cid:47) (cid:47) S Xp,q (cid:15) (cid:15) X p,q ( T ( f )) S Xp,q (cid:15) (cid:15) X p ( b ) h Xp (cid:47) (cid:47) X (cid:48) p ( T ( b )) (7)commutative. We need to check that h = ( h field , h Bp , h Xp , h Bp,q , h
Xp,q ) is a homomorphism. Clearly f field preserves +, · , 0 and 1. Since the α b are k (cid:48) -linear isomorphisms, 0 p , A p , SM p and LI p arepreserved by h .We note that C = C ( M ) has several identity objects. However, since C is pointed skeletalthere exists a unique tensor irreducible object 1 of C such that (1 , u ) is an identity object, for someisomorphism u : 1 → ⊗
1; similarly for C (cid:48) = C ( M (cid:48) ). Since T preserves identity objects and tensorirreducible objects, we see that T (1) = 1 (cid:48) , i.e., h preserves 1.Since T is a functor, h preserves S Bp,q and T Bp,q . As α b : X p ( b ) ⊗ k k (cid:48) → X (cid:48) p ( T ( b )) we see that h preserves π p . Using diagram (7) we see that h also preserves π p,q . Diagram (7) shows that h S Xp,q . For a morphism f : b → c in C , the diagram X p,q ( f ) h Xp,q (cid:34) (cid:34) T Xp,q ⊗ (cid:38) (cid:38) S Xp,q ⊗ (cid:47) (cid:47) X p ( b ) ⊗ k k (cid:48) (cid:101) f ⊗ k (cid:48) (cid:15) (cid:15) α b (cid:47) (cid:47) X (cid:48) p ( T ( b )) (cid:93) T ( f ) (cid:15) (cid:15) X p,q ( T ( f )) S Xp,q (cid:111) (cid:111) T Xp,q (cid:120) (cid:120) X q ( c ) ⊗ k k (cid:48) α c (cid:47) (cid:47) X (cid:48) q ( T ( c ))shows that f also preserves T Xp,q . As T is strict and α an isomorphism of tensor functors, we havefor equivalence classes b ∈ B p and c ∈ B q a commutative diagram:( X p ( b ) ⊗ X q ( c )) ⊗ k k (cid:48) α b ⊗ c (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) X (cid:48) pq ( T ( b ⊗ c ))( X p ( b ) ⊗ k k (cid:48) ) ⊗ k (cid:48) ( X q ( c ) ⊗ k k (cid:48) ) α b ⊗ α c (cid:47) (cid:47) X (cid:48) p ( T ( b )) ⊗ X (cid:48) q ( T ( c ))Thus for v ∈ X p ( b ) and w ∈ X q ( c ) we obtain h Xpq ( v ⊗ w ) = α b ⊗ c ( v ⊗ w ⊗
1) = α b ( v ⊗ ⊗ α c ( w ⊗
1) = h Xp ( v ) ⊗ h Xq ( w ). Thus h also preserves ⊗ p,q . Therefore f : M → M (cid:48) is a homomorphism of modelsof PROALG. It is then clear that h induces the morphism ( λ, T, α ) in TANN.Finally, we show that the functor M (cid:32) ( k ( M ) , C ( M ) , ω ( M )) is essentially surjective. Let( k, C, ω ) be an object of TANN. We will construct a model M of PROALG such that ( k ( M ) , C ( M ) , ω ( M ))is isomorphic to ( k, C, ω ) in TANN. We define the field sort of M to be k (including the field struc-ture). For p = ( m, n ), with m, n ≥
1, let B p denote the set of all objects b of C of tensor length m and such that ω ( b ) has dimension n . Let B p,q denote the set of all morphisms in C from objects in B p to objects in B q . Let S Bp,q : B p,q → B p denote the map that assigns the source to a morphismand similarly for the target. Let X p denote the disjoint union of the k -vector spaces ω ( b ), b ∈ B p and let π p : X p → B p be the map such that π p ( v ) = b for v ∈ ω ( b ). We define X p,q as the disjointunion of the graphs of the k -linear maps ω ( f ) : ω ( b ) → ω ( c ), where f : b → c is a morphism in C with b ∈ B p and c ∈ B q . The maps π p,q : X p,q → B p,q are defined by π p,q ( a ) = f , if a belongs to thegraph of ω ( f ). The maps S Xp,q : X p,q → X p and T Xp,q : X p,q → X q are defined by S Xp,q (( v, w )) = v and T Xp,q (( v, w )) = w , where ( v, w ) = ( v, ω ( f )( v )) is an element of the graph { ( v, ω ( f )( v )) | v ∈ X p ( b ) } of ω ( f ), for a morphism f : b → c in B p,q . We define 0 p to be the subset of X p consisting of allzero vectors of all vector spaces in X p . We define A p through A p ( v , v , v ) if and only if all threeelements v , v , v of X p belong to the same vector space ω ( b ) and v = v + v , where the + hereis vector space addition in ω ( b ). The map SM p : k × X p → X p is scalar multiplication. For ele-ments v , . . . , v n of X p , where p = ( m, n ), we define LI p through LI p ( v , . . . , v n ) if and only if all π p ( a i ) = π p ( a j ) for all i, j and v , . . . , v n are k -linearly independent (in X p ( b ), where b = π p ( a i )).There exists a unique tensor irreducible object C in C such that ( C , u ) is an identity objectof C for some isomorphism u : C → C ⊗ C . We set 1 = C ∈ B (1 , .Finally, we define ⊗ p,q : X p × X q → X pq by sending ( v, w ) ∈ ω ( b ) × ω ( c ) to the image of v ⊗ w under ω ( b ) ⊗ k ω ( c ) → ω ( b ⊗ c ), where the latter map is part of the functorial isomorphism defining the tensorfunctor ω . (As C is pointed skeletal, ⊗ p,q is well defined.) It is now straight forward to check thatour structure M satisfies all 27 axioms of PROALG. Moreover, ( k ( M ) , C ( M ) , ω ( M )) = ( k, C, ω ).So the isomorphism ( λ, T, α ) can be chosen to be the identity.23or a model M of PROALG we define ( k, C, ω ) = ( k ( M ) , C ( M ) , ω ( M )) as in the proof ofTheorem 2.2 and we call ( k, C, ω ) the object of TANN associated to M . In the sequel, when givena model M = ( k, B p , X p , B p,q , X p,q ) of PROALG we will use this notation without further ado. Forexample, for b ∈ B p we will usually write ω ( b ) instead of X p ( b ).We also set G = G ( M ) = Aut ⊗ ( ω ( M )) . Combining Theorem 2.2 and Proposition 1.18 we obtain:
Corollary 2.3.
The functor M (cid:32) G ( M ) from the category of models of PROALG with homomor-phism as the morphisms to the category
PROALGEBRAIC GROUPS is full, essentially surjectiveand induces a bijection on the isomorphism classes.
Example 2.4.
We describe the object ( k, C, ω ) of TANN that corresponds to the trivial proalgebraicgroup. The objects of C are the zero object together with all completely parenthesized finitesequences of integers greater or equal to 1. For an object b of C corresponding to a completeparenthesization of the sequence ( n , . . . , n m ) we set ω ( b ) = k n ⊗ k . . . ⊗ k k n m . We also define ω ( )to be the zero vector space. For objects b , b of C the set of morphisms Hom( b , b ) is defined asthe set of k -linear maps from ω ( b ) to ω ( b ). The tensor product ⊗ : C × C → C is defined onnon-zero objects as the concatenation of parenthesized sequences. We also set b ⊗ = ⊗ b = forany object b of C . On morphisms ⊗ is defines as the usual tensor product of k -linear maps.Let G be a proalgebraic group. There does not seem to be a canonical way to construct from G a model M of PROALG such that G ( M ) (cid:39) G . However, according to Corollary 2.3, there alwaysexists such a model M . Moreover, if M and M are two models of PROALG such that G ( M )and G ( M ) are isomorphic to G , then M and M are isomorphic. We can therefore safely definethe theory Th( G )of G as Th( M ) for any model M of PROALG such that G ( M ) (cid:39) G . Two proalgebraic groups G and H (not necessarily living over the same base field) are elementarily equivalent if Th( G ) = Th( H ).We may also express this as G ≡ H . In a similar spirit, a class C of proalgebraic groups (potentiallyover varying base fields) is called elementary if the class of all models of PROALG such that theassociated proalgebraic group lies in C is elementary.We conclude this section with a discussion of some elementary classes of proalgebraic groups. Proposition 2.5.
The following classes of proalgebraic groups are elementary: • The class of all diagonalizable proalgebraic groups. • The class of all unipotent proalgebraic groups. • The class of all linearly reductive proalgebraic groups.Proof.
The definition of diagonalizable proalgebraic groups is recalled in the beginning of the nextsection. Let us simply mention here that according to Prop. 1.6, Chapter IV, §
1, in [DG70] aproalgebraic group is diagonalizable if and only if every representation of G is a direct sum ofone-dimensional representations and the latter condition can be axiomatized.A unipotent proalgebraic group can be defined as a proalgebraic group G such that every repre-sentation of G has a fixed vector (cf. [Wat79, Section 8.3]). This condition can be axiomatized bysaying that for every representation V there exists a morphism → V .Recall that a proalgebraic group G is linearly reductive if and only if every representation of G is a direct sum of irreducible representations. This condition can be axiomatized.In the following section we will show that the class of all algebraic groups is not elementary(Corollary 3.5). 24 Diagonalizable proalgebraic groups
In this section we determine the theory of the multiplicative group G m and deduce some basicconsequences for the theory PROALG from this. We show that Th( G m ) is determined by thetheory of the base field and the theory of the abelian group ( Z , +). In fact, we establish a similarresult for any diagonalizable proalgebraic group: The theory of a diagonalizable proalgebraic group G is determined by the theory of the base field and the theory of the character group of G . Indeed,we show that the theory of all diagonalizable proalgebraic groups is weakly bi-interpretable with thetwo-sorted theory of pairs ( k, A ), where k is a field and A an abelian group.Let us first recall some basic facts about diagonalizable proalgebraic groups. See e.g., [Wat79,Section 2.2], [Mil17, Section 12, c] or [DG70, Chapter IV, §
1, 1]. Let k be a field and let A bean abelian group (usually written additively). The proalgebraic group D ( A ) k over k is defined by D ( A ) k ( R ) = Hom( A, R × ) for any k -algebra R , where Hom( A, R × ) denotes the abelian group of allmorphisms of abelian groups from A to R × . (Here, as usual, R × denotes the multiplicative groupof a ring R .) For example, D ( Z ) k (cid:39) G m , or more generally, D ( Z n ) k (cid:39) G nm . A proalgebraic groupis diagonalizable if it is isomorphic to D ( A ) k for some abelian group A . The functor A (cid:32) D ( A ) k isan equivalence of categories from the category of abelian groups to the category of diagonalizableproalgebraic groups over k . The quasi-inverse is the functor that associates the character group toa diagonalizable proalgebraic group. Recall that the character group χ ( G ) of a proalgebraic group G is the abelian group of all morphisms of proalgebraic groups from G to G m . Note that χ ( G )is isomorphic to the abelian group of isomorphism classes of one-dimensional representations of G under the tensor product.As noted before, in general, for a proalgebraic group G , there does not seem to be a canonical way to define a model M of PROALG such that G ( M ) (cid:39) G . (Recall that by Corollary 2.3 such an M always exists and is unique up to an isomorphism.) However, if G = D ( A ) k is diagonalizable,there is a canonical choice which we will now describe. This will be helpful later on (Theorem 3.2)for showing that M is interpretable in the structure ( k, A ).Given a field k and an abelian group A we will now define a model M ( k, A ) = ( k ( k, A ) , B p ( k, A ) , X p ( k, A ) , B p,q ( k, A ) , X p,q ( k, A ))of PROALG such that G ( M ( k, A )) (cid:39) D ( A ) k . We set k ( k, A ) = k (including the field structure).The isomorphism classes of representations of D ( A ) k of dimension n are in bijection with multisets ofelements of A of cardinality n (see e.g., [DG70, Prop. 1.6, Chapter IV, § a ∈ A let k a denote the one-dimensional representation of D ( A ) k given by the morphism χ a : D ( A ) k → G m with χ a ( g ) = g ( a ) for all g ∈ D ( A ) k ( R ) = Hom( A, R × ) and all k -algebras R . Then the map { a , . . . , a n } (cid:55)→ W { a ,...,a n } = k a ⊕ . . . ⊕ k a n induces a bijection between the set of multisets of cardinality n of elements of A and the set ofisomorphism classes of n -dimensional representations of D ( A ) k . So, for n ≥
1, we can define B (1 ,n ) ( k, A ) as the set of multisets of elements from A of cardinality n . In general, for p = ( m, n )we define B p ( k, A ) as the set of all completely parenthesized sequences of m -multisets A , . . . , A m formed from elements from A such that | A | . . . | A m | = n . The representation corresponding tosuch a parenthesized sequence of multisets would be W A ⊗ . . . ⊗ W A m with the correspondingparenthesization of the tensor product, where the W A i ’s are understood to be tensor irreducible.Explicitly, for an element b of B p ( k, A ) determining a parenthesization of A , . . . , A m , we let v b denote the multiset A × . . . × A m , where an element ( a , . . . , a m ) of v b is considered as a parenthesizedsequence with the same parenthesization pattern as the sequence A × . . . × A m . We let V b denotethe n -dimensional k -vector space with basis v b . In other words, V b is the k -vector space of functionsfrom v b to k . 25e define X p ( k, A ) to be the (disjoint) union of the V b ’s and we let π p : X p ( k, A ) → B p ( k, A )denote the map that maps an element in V b to b . We use the vector space structure on the V b ’sto define 0 p , A p , LI p , and SM p . We also define the interpretation of the constant symbol 1 tocorrespond to the neutral element of A , considered as an element of B (1 , ( k, A ).We next want to define ⊗ p, (cid:98) p , where p = ( m, n ) and (cid:98) p = ( (cid:98) m, (cid:98) n ). Note that two elements b ∈ B p ( k, A ) and (cid:98) b ∈ B (cid:98) b ( k, A ) can be concatenated to an element b ⊗ (cid:98) b ∈ B p (cid:98) p ( k, A ). Similarly, twoelements v ∈ v b and (cid:98) v ∈ v (cid:98) b can be concatenated to an element v (cid:98) v ∈ v b ⊗ (cid:98) b . This defines bilinear maps V b × V (cid:98) b → V b ⊗ (cid:98) b that combine to a map ⊗ p, (cid:98) p : X p ( k, A ) × X (cid:98) p ( k, A ) → X p (cid:98) p ( k, A ).We next want to define the morphism sorts. Note that for a , a ∈ A , there is a non-zeromorphism (of D ( A ) k -representations) from k a to k a if and only if a = a . Moreover, for a ∈ A ,every linear map k n a → k n a is a morphism. This yields a description of the morphisms from W A = ⊕ a ∈ A k a to W A = ⊕ a ∈ A k a for finite multisets A and A consisting of elements of A :For a ∈ A and i = 1 , W i,a = (cid:77) a ∈ A a a k a and let Hom k ( W ,a , W ,a ) denote the set of k -linear maps from W ,a to W ,a . Then the set of mor-phisms of D ( A ) k -representations from W A to W A can be identified with (cid:81) a ∈ A Hom k ( W ,a , W ,a ).For b ∈ B p ( k, A ) and v ∈ v b determining a parenthesization of ( a , . . . , a m ) ∈ A m we set | v | = a + . . . + a m . Furthermore, for a ∈ A let V b,a denote the subspace of V b generated by all v ∈ v b such that | v | = a .For b ∈ B p ( k, A ) and (cid:98) b ∈ B (cid:98) p ( k, A ) let H b, (cid:98) b denote (cid:81) a ∈ A Hom k ( V b,a , V (cid:98) b,a ) considered as a subsetof Hom k ( V b , V (cid:98) b ). We set B p, (cid:98) p ( k, A ) = (cid:110) ( b, (cid:98) b, H b, (cid:98) b ) | b ∈ B p ( k, A ) , (cid:98) b ∈ B (cid:98) p ( k, A ) (cid:111) and X p, (cid:98) p ( k, A ) = (cid:110) ( b, (cid:98) b, h, v, h ( v )) | b ∈ B p ( k, A ) , (cid:98) b ∈ B (cid:98) p ( k, A ) , h ∈ H b, (cid:98) b , v ∈ V b (cid:111) . We define π p, (cid:98) p : X p, (cid:98) p ( k, A ) → B p, (cid:98) p ( k, A ) to be the projection onto the first three factors while S Bp, (cid:98) p : B p, (cid:98) p ( k, A ) → B p ( k, A ) and T Bp, (cid:98) p : B p, (cid:98) p ( k, A ) → B (cid:98) p ( k, A ) are defined as the projections onto thefirst and second factor respectively. Similarly, S Xp, (cid:98) p : X p, (cid:98) p ( k, A ) → X p ( k, A ) and T Xp, (cid:98) p : X p, (cid:98) p ( k, A ) → X (cid:98) p ( k, A ) are defined as the projections onto the third and fourth factor respectively.This completes our definition of the L -structure M ( k, A ). It is clear from the construction (andthe proof of Corollary 2.3) that M ( k, A ) is a model of PROALG and that G ( M ( k, A )) (cid:39) D ( A ) k .Since the addition in the character group can be described through the tensor product it is notsurprising that the character group of G ( M ) is interpretable in M for any model M of PROALG: Lemma 3.1.
Let M be a model of PROALG . Then the character group of G ( M ) is definablyinterpreted in M .Proof. Let M = ( k, B p , X p , B p,q , X p,q ). Then B (1 , can be identified with the isomorphism classesof one-dimensional representations of G = G ( M ), i.e., with χ ( G ). The graph of addition in χ ( G ) = B (1 , consists of all ( b , b , b ) ∈ B , such that there exist v ∈ ω ( b ) , v ∈ ω ( b ) and v ∈ ω ( b )and an isomorphism between π (2 , ( v ⊗ v ) and b . This set is ∅ -definable. The identity element of B (1 , is given by the constant symbol 1.To proceed, let us recall the notion of interpretation in the many-sorted context (cf. [Hod93,Chapter 5] for the one-sorted case). Let L and L (cid:48) be many-sorted languages with sorts S and S (cid:48) respectively. An interpretation Ξ of L (cid:48) in L is comprised of the following data:26 For every sort s (cid:48) ∈ S (cid:48) an L -formula ∂ Ξ ,s (cid:48) ( x s , . . . , x s n n ) (the domain formula for the sort s (cid:48) ) inthe free variables x s (cid:48) = ( x s , . . . , x s n n ), where n and s , . . . , s n ∈ S depend on s (cid:48) . (Here thenotation x s means that the variable x belongs to the sort s .) • For every s (cid:48) ∈ S (cid:48) an L -formula = Ξ ,s (cid:48) ( x s (cid:48) , y s (cid:48) ) (the equivalence formula for the sort s (cid:48) ). • For every function symbol f (cid:48) ∈ L (cid:48) that maps sorts s (cid:48) , . . . , s (cid:48) r into sort s (cid:48) r +1 an L -formula f (cid:48) Ξ ( x s (cid:48) , . . . , x s (cid:48) r +1 ). • For every relation symbol R (cid:48) of L (cid:48) between sorts s (cid:48) , . . . , s (cid:48) r an L -formula R (cid:48) Ξ ( x s (cid:48) , . . . , x s (cid:48) r ). • For every constant symbol c (cid:48) in L (cid:48) with sort s (cid:48) an L -formula c (cid:48) Ξ ( x s (cid:48) ).The admissibility conditions of Ξ are the L -sentences expressing that for every L -structure M =( M s ) s ∈ S the following holds: • For all s (cid:48) ∈ S (cid:48) the formula = Ξ ,s (cid:48) defines an equivalence relation on ∂ Ξ ,s (cid:48) ( M ). We will denotethis equivalence relation simply by ∼ (even though it depends on Ξ , s (cid:48) and M ). • For every function symbol f (cid:48) of L (cid:48) that maps sorts s (cid:48) , . . . , s (cid:48) r into sort s (cid:48) r +1 we have – M | = f (cid:48) Ξ ( a s (cid:48) , . . . , a s (cid:48) r +1 ) if and only if M | = f (cid:48) Ξ ( b s (cid:48) , . . . , b s (cid:48) r +1 ) for all tuples a s (cid:48) , . . . , a s (cid:48) r +1 , b s (cid:48) , . . . , b s (cid:48) r +1 from M with a s (cid:48) ∼ b s (cid:48) , . . . , a s (cid:48) r +1 ∼ b s (cid:48) r +1 and – the induced subset of ∂ Ξ ,s (cid:48) ( M ) / ∼ × . . . × ∂ Ξ ,s (cid:48) r +1 ( M ) / ∼ is the graph of a function f (cid:48) ( M ) : ∂ Ξ ,s (cid:48) ( M ) / ∼ × . . . × ∂ Ξ ,s (cid:48) r ( M ) / ∼→ ∂ Ξ ,s (cid:48) r +1 ( M ) / ∼ . • For every relation symbol R (cid:48) of L (cid:48) between sorts s (cid:48) , . . . , s (cid:48) r we have M | = R (cid:48) Ξ ( a s (cid:48) , . . . , a s (cid:48) r ) ifand only if M | = R (cid:48) Ξ ( b s (cid:48) , . . . , b s (cid:48) r ) for all tuples a s (cid:48) , . . . , a s (cid:48) r , b s (cid:48) , . . . , b s (cid:48) r from M with a s (cid:48) ∼ b s (cid:48) , . . . , a s (cid:48) r ∼ b s (cid:48) r . We thus have an induced relation R (cid:48) ( M ) on ∂ Ξ ,s (cid:48) ( M ) / ∼ × . . . × ∂ Ξ ,s (cid:48) r ( M ) / ∼ . • For every constant symbol c (cid:48) in L (cid:48) with sort s (cid:48) the realizations of c (cid:48) Ξ ( x s (cid:48) ) in M are an equiva-lence class c (cid:48) ( M ) in ∂ Ξ ,s (cid:48) ( M ).Note that if M is an L -structure that satisfies the admissibility conditions of Ξ, then Ξ( M ) =( ∂ Ξ ,s (cid:48) ( M ) / ∼ ) s (cid:48) ∈ S (cid:48) is an L (cid:48) -structure.Now let T be an L -theory and let T (cid:48) be an L (cid:48) -theory. An interpretation Ξ of L (cid:48) in L is a lefttotal interpretation of T (cid:48) in T if every model of T satisfies the admissibility conditions of Ξ and iffor every model M of T the L (cid:48) -structure Ξ( M ) is a model of T (cid:48) . Thus, from every model M of T we get a model of Ξ( M ) of T (cid:48) .Finally, following [Vis06, Section 3.3.], the theories T and T (cid:48) are weakly bi-interpretable if thereexists a left total interpretation Ξ of T (cid:48) in T and a left total interpretation Ω of T in T (cid:48) such thatfor every model M of T the L -structures Ω(Ξ( M ) and M are isomorphic and for every model M (cid:48) of T (cid:48) the L (cid:48) -structures Ξ(Ω( M (cid:48) )) and M (cid:48) are isomorphic. (Bi-interpretability is the slightly strongernotion where the above isomorphisms are required to be uniformly definable.)By the theory of diagonalizable proalgebraic groups we mean the set of all L -sentences true in allmodels M of PROALG such that G ( M ) is diagonalizable (cf. Proposition 2.5). Theorem 3.2.
The theory of diagonalizable proalgebraic groups is weakly bi-interpretable with thetwo-sorted theory of pairs ( k, A ) , where k is a field and A an abelian group. roof. Let L (cid:48) denote the two-sorted language with the language of rings on the first sort (the fieldsort) and the language of abelian groups on the second sort (the group sort). Let T (cid:48) denote the L (cid:48) -theory of all pairs ( k, A ) where k is a field and A an abelian group. Moreover, let T denote the L -theory of diagonalizable proalgebraic groups.The interpretation Ξ of L (cid:48) in L is relatively easy to describe (cf. Lemma 3.1): The domainformula for the field sort of L (cid:48) is trivial (i.e., equal to x = x , where x belongs to the field sort of L ) so that it returns the field sort of L . Similarly, the domain formula for the group sort of L (cid:48) istrivial so that it returns the sort B (1 , of L . The two equivalence formulas are also trivial, so thatthe corresponding equivalence relation simply expresses equality of elements. The interpretationof the ring language on the field sort of L (cid:48) is the ring language on the field sort of L . The L (cid:48) -symbol for the identity element of the group is to be interpreted as the L -symbol 1 (correspondingto the trivial representation). Finally, the addition symbol + on the group sort, yields the formula+ Ξ that defines in every model M of T the set of all ( b , b , b ) ∈ B , such that there exist v ∈ ω ( b ) , v ∈ ω ( b ) and v ∈ ω ( b ) and an isomorphism between π (2 , ( v ⊗ v ) and b . ClearlyΞ is a left total interpretation of T (cid:48) in T .We will next construct an interpretation Ω of L in L (cid:48) . The idea for the construction is rathersimple but a little tedious to implement. The formulas for Ω boil down to interpreting the L -structure M ( k, A ) defined above in the L (cid:48) -structure ( k, A ). We begin with the domain formulas ∂ Ω ,s , where s is a sort from L . The domain formula for the field sort of L simply returns the fieldsort of L (cid:48) . Definition of ∂ Ω ,B p : For p = ( m, n ), we consider, for every L (cid:48) -structure ( k, A ), the set P p ( k, A )of completely parenthesized sequences( a , . . . , a n ) , . . . , ( a m , . . . , a mn m )of sequences in A with n . . . n m = n . To describe P p ( k, A ) inside ( k, A ) we can encode the patternassociated with such a parenthesization of a sequence of sequences in a sequence of zero’s and one’sbelonging to k . While there are different ways to do this, for the sake of concreteness, let us fix thefollowing decoding. A sequence of zero’s and one’s always has to be read by blocks of two elementsaccording to the following convention: • A block 10 is to be read as an opening parenthesis “(”. • A block 01 is to be read as a closing parenthesis “)”. • A block 00 is to be interpreted as a place holder for an element of A .We also use parenthesis to separate place holders that correspond to different sequences in A . Forexample, the element (cid:16)(cid:0) ( a , a , a )( a ) (cid:1) ( a , a ) (cid:17) of P (3 , ( k, A ) yields the pattern((( • • • )( • ))( •• ))that is encoded in the sequence10 10 10 00 00 00 01 10 00 01 01 10 00 00 01 01 . Note that different patterns may yield binary sequences of different lengths. Let r = r ( p ) denotethe maximal length of all theses binary sequences and let D p ( k, A ) ⊆ { , } r ⊆ k r denote the set ofall binary sequences that are derived from elements of P p ( k, A ). Here a binary sequence of lengthless than r is extended to a sequence of length r by adding 11-blocks. Let s = s ( p ) denote themaximum number of 00-blocks that occur in any element of D p ( k, A ) and let F p ( k, A ) denote the28ubset of k r × A s consisting of all elements of the form ( d, a , . . . , a t , , . . . , d ∈ D p ( k, A ), t is the number of 00-blocks occurring in d and a , . . . , a t ∈ A . By construction, the set F p ( k, A ) isin bijection with P p ( k, A ).We let ∂ Ω ,B p denote the L (cid:48) -formula that defines in every L (cid:48) -structure ( k, A ) the subset F p ( k, A )of k r × A s . An element f of F p ( k, A ) corresponding to a complete parenthesization of a sequence( a , . . . , a n ) , . . . , ( a m , . . . , a mn m )of sequences in A with n . . . n m = n yields a totally ordered multiset v f of cardinality n consisting ofall sequences of length m in A that are of the form ( a i , . . . , a mi m ) with 1 ≤ i ≤ n , . . . , ≤ i m ≤ n m .Alternatively, we can define v f as the multiset product { a , . . . , a n } × . . . × { a m , . . . , a mn m } . Theorder on v f is obtained by stipulating that a j < a j < . . . < a jn j for j = 1 , . . . , m and then using thelexicographic order. Let V f denote the k -vector space with basis v f . We think of an element f of F p ( k, A ) as determining the pair ( V f , v f ), where v f is an ordered basis of V f . Definition of ∂ Ω ,X p : We let ∂ Ω ,X p denote the formula that defines the set F p ( k, A ) × k n in every L (cid:48) -structure ( k, A ). We think of an element ( f, ξ ) ∈ F p ( k, A ) × k n as determining theelement v f ξ = ξ v f, + . . . + ξ n v f,n of V f , where ξ = ( ξ , . . . , ξ n ) and v f = { v f, , . . . , v f,n } with v f, < v f, < . . . < v f,n . Definition of ∂ Ω ,B p, (cid:98) p : Let p = ( m, n ) and (cid:98) p = ( (cid:98) m, (cid:98) n ). For every L (cid:48) -structure ( k, A ) and( a , . . . , a m ) ∈ A m we set | a | = a + . . . + a m for a fixed but arbitrary parenthesization of this sum .For f ∈ F p ( k, A ) we let Σ( f ) = {| v | | v ∈ v f } denote the multiset of all sums of all tuples in v f .Furthermore, for a ∈ A we let m f ( a ) denote the multiplicity of a in Σ( f ). Of course m f ( a ) = 0 forall but finitely many a ∈ A . For ( f, (cid:98) f ) ∈ F p ( k, A ) × F (cid:98) p ( k, A ) we set r ( f, (cid:98) f ) = (cid:80) a ∈ A m f ( a ) m (cid:98) f ( a ).Let r = max (cid:110) r ( f, (cid:98) f ) (cid:12)(cid:12)(cid:12) ( f, (cid:98) f ) ∈ F p ( k, A ) × F (cid:98) p ( k, A ) (cid:111) . We define H p, (cid:98) p ( k, A ) to be the subset of F p ( k, A ) × F (cid:98) p ( k, A ) × k r consisting of all elements of theform ( f, (cid:98) f , λ ), where f ∈ F (cid:98) p ( k, A ), (cid:98) f ∈ F (cid:98) p ( k, A ), λ = ( λ , . . . , λ r ) ∈ k r and λ i = 0 for i > r ( f, (cid:98) f ).Let ∂ Ω ,B p, (cid:98) p denote the L (cid:48) -formula that defines in every L (cid:48) -structure ( k, A ) the set H p, (cid:98) p ( k, A ).We think of an element ( f, (cid:98) f , λ ) of H p, (cid:98) p ( k, A ) as defining a morphism ψ ( f, (cid:98) f,λ ) : ( V f , v f ) → ( V (cid:98) f , v (cid:98) f )as follows: To simplify the formulas we set m ( a ) = m f ( a ) and (cid:98) m ( a ) = m (cid:98) f ( a ). For each of the finitelymany a ∈ A such that m ( a ) (cid:98) m ( a ) ≥ I a = { i ,a , . . . , i m ( a ) ,a } = { i ∈ { , . . . , n }| | v f,i | = a } and J a = { j ,a , . . . , j (cid:98) m ( a ) ,a } = { j ∈ { , . . . , (cid:98) n }| | v (cid:98) f,j | = a } with v f,i ,a < . . . < v f,i m ( a ) ,a and v (cid:98) f,j ,a < . . . < v (cid:98) f,j (cid:99) m ( a ) ,a . We order the sets of the form I a bycomparing the v f,i ,a . Say I ( a ) < . . . < I ( a s ). For every (cid:96) = 1 , . . . , s let v f,I a(cid:96) denote the (ordered)sequence of elements of v f whose index belongs to I a (cid:96) . We define v (cid:98) f,J a(cid:96) similarly. We define a k -linear map ψ (cid:96) from the subspace of V f generated by v f,I a(cid:96) to the subspace of V (cid:98) f generated by v (cid:98) f,J a(cid:96) by setting ψ (cid:96) ( v f,I a(cid:96) ) = v (cid:98) f,J a(cid:96) M a (cid:96) , where M a (cid:96) is the m ( a (cid:96) ) × (cid:98) m ( a (cid:96) )-matrix obtained by putting, rowby row, the entries of λ = ( λ , . . . , λ r ) that start at index m ( a ) (cid:98) m ( a ) + . . . + m ( a l − ) (cid:98) m ( a l − ) + 1and end at index m ( a ) (cid:98) m ( a ) + . . . + m ( a l ) (cid:98) m ( a l ) into a matrix. Finally, we define the linear map ψ ( f, (cid:98) f,λ ) : V f → V (cid:98) f by ψ ( f, (cid:98) f,λ ) ( v f,i ) = (cid:40) ψ (cid:96) ( v f,i ) if i ∈ I a (cid:96) The parenthesization will ultimately not matter since only the case when A is an abelian group is relevant for us. efinition of ∂ Ω ,X p, (cid:98) p : We define ∂ Ω ,X p, (cid:98) p to be the L (cid:48) -formula that defines in every L (cid:48) -structure( k, A ) the set of all ( f, (cid:98) f , λ, ξ, (cid:98) ξ ) such that ( f, (cid:98) f , λ ) ∈ H p, (cid:98) p ( k, A ), ( f, ξ ) ∈ ∂ Ω ,X p ( k, A ), ( (cid:98) f , (cid:98) ξ ) ∈ ∂ Ω ,X (cid:98) p ( k, A ) and ψ ( f, (cid:98) f,λ ) ( v f ξ ) = v (cid:98) f (cid:98) ξ . This concludes the definition of the domain formulas ∂ Ω ,s forall sorts s of L .We will next define the equivalence formulas = Ω ,s . We define = Ω ,k to be the L (cid:48) -formula x = x ,where x and x are variables from the field sort. (So the equivalence relation on the field sort istrivial.) Definition of = Ω ,B p : We let = Ω ,B p denote the L (cid:48) -formula that defines in every L (cid:48) -structure( k, A ) the following equivalence relation on F p ( k, A ): For ( d, a , . . . , a t , , . . . , , ( d (cid:48) , a (cid:48) , . . . , a (cid:48) t (cid:48) , , . . . , ∈ F p ( k, A ) with d, d (cid:48) ∈ D p ( k, A ) and a , . . . , a t , a (cid:48) , . . . , a (cid:48) t (cid:48) ∈ A we have( d, a , . . . , a t , , . . . , ∼ ( d (cid:48) , a (cid:48) , . . . , a (cid:48) t (cid:48) , , . . . , d = d (cid:48) (so that also t = t (cid:48) ) and a (cid:48) , . . . , a (cid:48) t (cid:48) is a permutation of a , . . . , a t , where elements corre-sponding to the same string of 00-blocks in d = d (cid:48) are permuted among themselves. Note that themap f (cid:55)→ b ( f ) that assigns to an f ∈ F p ( k, A ) corresponding to a complete parenthesization of asequence ( a , . . . , a n ) , . . . , ( a m , . . . , a mn m )of sequences in A the corresponding parenthesization of the sequence { a , . . . , a n } , . . . , { a m , . . . , a mn m } of multisets, induces a bijection between ∂ Ω ,B p ( k, A ) / ∼ and B p ( k, A ).We note that V f only depends on the equivalence class of f ∈ F p ( k, A ). Indeed, the multisetunderlying v f only depends on the equivalence class of f . Only the ordering on the multiset v f depends on f . Moreover, if the equivalence class of f maps to b ∈ B p ( k, A ) under ∂ Ω ,B p ( k, A ) / ∼(cid:39) B p ( k, A ), then, with the notation introduced in the beginning of this section, V f = V b . Definition of = Ω ,X p : Let = Ω ,X p denote the formula that defines on every ∂ Ω ,X p ( k, A ) = F p ( k, A ) × k n the equivalence relation( f, ξ ) ∼ ( f (cid:48) , ξ (cid:48) ) ⇔ f ∼ f (cid:48) and ξ (cid:48) is a permutation of ξ such that v f ξ = v f (cid:48) ξ (cid:48) . We note that the map ( f, ξ ) → v f ξ induces a bijection ∂ Ω ,X p ( k, A ) / ∼→ (cid:93) V f , where the disjointunion is taken over all equivalence classes in F p ( k, A ). In other words, ∂ Ω ,X p ( k, A ) / ∼ is in bijectionwith X p ( k, A ). Definition of = Ω ,B p, (cid:98) p : Let = Ω ,B p, (cid:98) p denote the L (cid:48) -formula that defines on every H p, (cid:98) p ( k, A ) theequivalence relation( f , (cid:98) f , λ ) ∼ ( f , (cid:98) f , λ ) ⇔ f ∼ f , (cid:98) f ∼ (cid:98) f and λ is a permutation of λ such that ψ ( f , (cid:98) f ,λ ) = ψ ( f , (cid:98) f ,λ ) . Then the map ( f, (cid:98) f , λ ) → ( b ( f ) , b ( (cid:98) f ) , ψ ( f, (cid:98) f,λ ) ) induces a bijection between ∂ Ω ,B p, (cid:98) p ( k, A ) / ∼ and B p, (cid:98) p ( k, A ). Definition of = Ω ,X p, (cid:98) p : Let = Ω ,X p, (cid:98) p denote the L (cid:48) -formula that defines on every ∂ Ω ,X p, (cid:98) p ( k, A )the equivalence relation( f , (cid:98) f , λ , ξ , (cid:98) ξ ) ∼ ( f , (cid:98) f , λ , ξ , (cid:98) ξ ) ⇔ ( f , (cid:98) f , λ ) ∼ ( f , (cid:98) f , λ ) , ( f , ξ ) ∼ ( f , ξ ) and ( (cid:98) f , (cid:98) ξ ) ∼ ( (cid:98) f , (cid:98) ξ ) . Again the implicit use of matrix multiplication in this definition need not concern us, since ultimately we areonly interested in the case when k is a field and A an abelian group. f, (cid:98) f , λ, ξ, (cid:98) ξ ) (cid:55)→ ( b ( f ) , b ( (cid:98) f ) , ψ ( f, (cid:98) f,λ ) , v f ξ, v (cid:98) f (cid:98) ξ ) induces a bijection between ∂ Ω ,X p, (cid:98) p ( k, A ) / ∼ and X p, (cid:98) p ( k, A ).This concludes the definition of the equivalence formulas for Ω. Note that for every model( k, A ) of T (cid:48) we have a bijection between ( ∂ Ω ,s ( k, A ) / ∼ ) s ∈ S (where S denotes the set of sorts for L ) and M ( k, A ). Using this bijection, the interpretation of the symbols of L in M ( k, A ) gives riseto an interpretation of the symbols of L in ( ∂ Ω ,s ( k, A ) / ∼ ) s ∈ S . It is now straight forward to checkthat these interpretations can be defined (uniformly in ( k, A )) by appropriate L (cid:48) -formulas. Thiscompletes the definition of Ω. Note that Ω( k, A ) (cid:39) M ( k, A ) for every model ( k, A ) of T (cid:48) .It is clear that Ω is a left total interpretation of T in T (cid:48) . Moreover, Ξ(Ω( k, A )) (cid:39) ( k, A ) for everymodel ( k, A ) of T (cid:48) .For a model M = ( k, B p , X p , B p,q , X p,q ) of T , let us consider B (1 , as an abelian group (via theidentification of B (1 , with the character group of G ( M ) as in the definition of Ξ). Then M (cid:39)M ( k, B (1 , ) because G ( M ) and G ( M ( k, B (1 , )) are both isomorphic to D ( B (1 , ) k . Moreover,Ω(Ξ( M )) = Ω( k, B (1 , ) (cid:39) M ( k, B (1 , ). Thus Ω(Ξ( M )) (cid:39) M as desired.We note that the above isomorphism Ω(Ξ( M )) (cid:39) M is not canonical. For example, on the p -total objects sort we need a bijection between ∂ Ω ,X p ( k, B (1 , ) / ∼ = ( F p ( k, B (1 , ) × k n ) / ∼ and X p . Specifying such a bijection involves the choice of appropriate bases. This is why we have weakbi-interpretability rather than bi-interpretability in Theorem 3.2.In the course of the proof of Theorem 3.2 we have seen the following: Corollary 3.3.
Let M be a model of PROALG such that G ( M ) is diagonalizable. Then M isinterpretable in ( k, A ) , where k is the field sort of M and A the character group of G ( M ) . Based on Theorem 3.2 we can now characterize elementary equivalence and elementary extensionsfor diagonalizable proalgebraic groups.
Corollary 3.4.
Let k be a field and G a diagonalizable proalgebraic group over k . Then a proal-gebraic group G (cid:48) over a field k (cid:48) is elementarily equivalent to G if and only if k (cid:48) is elementarilyequivalent to k , G (cid:48) is diagonalizable and χ ( G (cid:48) ) is elementarily equivalent to χ ( G ) .Proof. First assume that G (cid:48) ≡ G . Then clearly k (cid:48) ≡ k . Moreover, we know from Proposition 2.5that G (cid:48) must be diagonalizable. Let M = ( k, B p , X p , B p,q , X p,q ) and M (cid:48) = ( k (cid:48) , B (cid:48) p , X (cid:48) p , B (cid:48) p,q , X (cid:48) p,q )be models of PROALG such that G ( M ) (cid:39) G and G ( M (cid:48) ) (cid:39) G (cid:48) . Since interpretations preserve ele-mentary equivalence we see that Ξ( M (cid:48) ) ≡ Ξ( M ), where Ξ is defined as in the proof of Theorem 3.2.So ( k (cid:48) , B (cid:48) (1 , ) ≡ ( k, B (1 , ). Since B (cid:48) (1 , and B (1 , are isomorphic with χ ( G (cid:48) ) and χ ( G ) respectively,we see that χ ( G ) ≡ χ ( G (cid:48) ).Conversely, assume now that G (cid:48) is diagonalizable, k (cid:48) ≡ k and χ ( G (cid:48) ) ≡ χ ( G ). Then also( k (cid:48) , χ ( G (cid:48) )) ≡ ( k, χ ( G )). Therefore Ω( k (cid:48) , χ ( G (cid:48) )) ≡ Ω( k, χ ( G )), where Ω is defined as in the proof ofTheorem 3.2. But M (cid:48) (cid:39) Ω( k (cid:48) , χ ( G (cid:48) )) and M (cid:39) Ω( k, χ ( G )). Thus M (cid:48) ≡ M , i.e., G (cid:48) ≡ G .In particular, for a field k , a proalgebraic group G over k is elementarily equivalent to themultiplicative group G m over k if and only if G is isomorphic to D ( A ) k and A is elementarilyequivalent to ( Z , +). Since Th( Z , +) has models that are not finitely generated as abelian groups(e.g., Z ⊕ Q , see [EF72]) and D ( A ) k is algebraic if and only if A is finitely generated we obtain: Corollary 3.5.
The class of all algebraic groups is not elementary.
Corollary 3.6.
Let M be a model of PROALG such that G ( M ) is diagonalizable and let A denotethe character group of G ( M ) . If M (cid:48) (cid:23) M is an elementary extension of M , then G ( M (cid:48) ) (cid:39) D ( A (cid:48) ) k (cid:48) ,where k (cid:48) (cid:23) k and A (cid:48) (cid:23) A .Conversely, if k (cid:48) (cid:23) k and A (cid:48) (cid:23) A are elementary extensions, then there exists an elementaryextension M (cid:48) (cid:23) M such that G ( M (cid:48) ) (cid:39) D ( A (cid:48) ) k (cid:48) . roof. Again, let Ξ and Ω be the interpretations defined in the proof of Theorem 3.2. We observethat G ( M (cid:48) ) is diagonalizable by Proposition 2.5. Let A (cid:48) denote the character group of G ( M (cid:48) ). Sinceinterpretations preserve elementary embeddings (cf. [Hod93, Theorem 5.3.4] for the one-sorted case)we see that Ξ( M (cid:48) ) (cid:23) Ξ( M ), i.e., ( k (cid:48) , A (cid:48) ) (cid:23) ( k, A ). It follows that k (cid:48) (cid:23) k and A (cid:48) (cid:23) A .Conversely, assume we start with elementary extensions k (cid:48) (cid:23) k and A (cid:48) (cid:23) A . Then ( k (cid:48) , A (cid:48) ) (cid:23) ( k, A ) and Ω( k (cid:48) , A (cid:48) ) (cid:23) Ω( k, A ). As Ω( k, A ) (cid:39) M and G (Ω( k (cid:48) , A (cid:48) )) (cid:39) D ( A (cid:48) ) k (cid:48) the claim follows. Corollary 3.7.
Let G be a diagonalizable proalgebraic group over an algebraically closed field k .Then Th( G ) is stable, but not necessarily superstable.Proof. Let M be a model of PROALG such that G ( M ) (cid:39) G and let A denote the character groupof G ( M ). Since Th( k ) is stable and Th( A ) is stable, it follows that also Th( k, A ) is stable. As M can be interpreted in ( k, A ) by Corollary 3.3 it follows that Th( M ) is stable.There are abelian groups whose theory is not superstable (e.g., an infinite direct sum of copiesof Z , see [Hod93, Theorem A.2.13]). Since these can be interpreted in a model M of PROALG with G ( M ) diagonalizable, it follows that Th( M ) cannot be superstable. We postpone a more comprehensive study of types for certain models of PROALG to a futurepublication. Here we establish some initial algebraic results that illustrate the expressive power ofPROALG: • For a model M = ( k, B p , X p , B p,q , X p,q ) of PROALG the type of an element b ∈ B p over theempty set determines the minimal degree of defining equations of the image of the represen-tation G ( M ) → GL ω ( b ) associated with b . • The type of b over k determines the image of G ( M ) → GL ω ( b ) .The key to these results is the fact that the type of b over k knows which subspaces of rep-resentations of G ( M ) obtained from ω ( b ) by forming tensor products, duals and direct sums are G ( M )-stable (i.e., subrepresentations). Moreover, the image of G ( M ) → GL ω ( b ) is determined bythese stable subspaces. The results in this subsection are of a preparatory nature and purely algebraic, i.e., do not involveany model theory. It is well known that any closed subgroup G of GL V , for a finite dimensionalvector space V , is the stabilizer of some subspace of a representation of GL V obtained from V byforming tensor products, duals and direct sums. We will need to understand this result in full detail.In particular, we would like to know how the degree of the polynomials defining the stabilizer isrelated to the constructions (tensor product, duals, direct sums) applied to V .Let G be a closed subgroup of GL n . Then the defining ideal I ( G ) of G is a Hopf ideal of theHopf algebra k [GL n ] = k [ Z, / det( Z )] = k [ Z, Z − ], where Z is an n × n matrix of indeterminates.For every d ≥ k [ Z, Z − ] ≤ d denote the finite dimensional k -subspace of k [ Z, Z − ] consisting ofall elements of the form P ( Z, Z − ), where P is a polynomial over k in 2 n variables of degree atmost d . Lemma 4.1.
The subspace k [ Z, Z − ] ≤ d is a subcoalgebra of k [ Z, Z − ] . roof. Let ∆ : k [ Z, Z − ] → k [ Z, Z − ] ⊗ k k [ Z, Z − ] , Z (cid:55)→ Z ⊗ Z denote the comultiplication. Here Z ⊗ Z is the n × n matrix whose ij -entry is (cid:80) n(cid:96) =1 Z i(cid:96) ⊗ Z (cid:96)j . In other words, Z ⊗ Z is the (matrix)product of the matrices Z ⊗ ⊗ Z , where Z ⊗ Z i,j ⊗ ≤ i,j ≤ n ∈ ( k [ Z, Z − ] ⊗ k k [ Z, Z − ]) n × n and 1 ⊗ Z = (1 ⊗ Z i,j ) ≤ i,j ≤ n ∈ ( k [ Z, Z − ] ⊗ k k [ Z, Z − ]) n × n . We have ∆( Z − ) = ∆( Z ) − = (( Z ⊗ ⊗ Z )) − = (1 ⊗ Z ) − ( Z ⊗ − = (1 ⊗ Z − )( Z − ⊗ Z − )) ij = (cid:80) n(cid:96) =1 ( Z − ) (cid:96)j ⊗ ( Z − ) i(cid:96) .Consequently, if P is a polynomial of degree at most d , then ∆( P ( Z, Z − )) = P (∆( Z ) , ∆( Z − )) ∈ k [ Z, Z − ] ≤ d ⊗ k k [ Z, Z − ] ≤ d . Lemma 4.2.
The ideal of k [GL n ] = k [ Z, Z − ] generated by I ( G ) ∩ k [ Z, Z − ] ≤ d is a Hopf ideal.Proof. If C is a coalgebra with a coideal V and a subcoalgebra D , then V ∩ D is a coideal of D , andso, in particular, a coideal of C . (To see this note that D → C/V is a morphism of coalgebras withkernel V ∩ D and kernels of morphisms of coalgebras are coideals.) It follows, using Lemma 4.1,that I ( G ) ∩ k [ Z, Z − ] ≤ d is a coideal of k [ Z, Z − ].Let I = ( I ( G ) ∩ k [ Z, Z − ] ≤ d ) denote the ideal of k [ Z, Z − ] generated by I ( G ) ∩ k [ Z, Z − ] ≤ d . Inany commutative Hopf algebra, the ideal generated by a coideal is a coideal. It follows that I is acoideal.Thus it only remains to check that I is stable under the antipode S : k [ Z, Z − ] → k [ Z, Z − ] , Z (cid:55)→ Z − . However, since k [ Z, Z − ] ≤ d and I ( G ) are stable under S this is immediate.Let V be a finite dimensional k -vector space. The choice of a basis of V allows us to identifyGL V with GL n and k [GL V ] with k [ Z, Z − ]. We note that k [ Z, Z − ] ≤ d , considered as a subspaceof k [GL V ], does not depend on the choice of the basis and we may therefore safely denote it by k [GL V ] ≤ d .The following notation will be useful: For a polynomial P ∈ N [ X, Y ] in two variables and a finitedimensional k -vector space V we define P ( V, V ∨ ) as the k -vector space obtained from V and P byreplacing X by V , Y by the dual vector space V ∨ of V , addition in P by the direct sum of vectorspaces and multiplication by the tensor product of vector spaces. The constant term of P has tobe interpreted as the appropriate direct sum of copies of k . Note that if V is a representation ofsome proalgebraic group G , then P ( V, V ∨ ) is also naturally a representation of G . (The constantterm has to be interpreted as a trivial representation.) The choice of a basis v = ( v , . . . , v n ) of V determines, for every P ∈ N [ X, Y ], a basis of P ( V, V ∨ ), which we will call the v -canonical basis of P ( V, V ∨ ). It can be defined recursively as follows: • The v -canonical basis of V is v . • The v -canonical basis of V ∨ is the basis v ∨ = ( v ∨ , . . . , v ∨ n ) dual to v . • If w , . . . , w m is the v -canonical basis of W and w (cid:48) , . . . , w (cid:48) m (cid:48) is the v -canonical basis of W (cid:48) ,then w , . . . , w m , w (cid:48) , . . . , w (cid:48) m (cid:48) is the v -canonical basis of W ⊕ W (cid:48) . • If w , . . . , w m is the v -canonical basis of W and w (cid:48) , . . . , w (cid:48) m (cid:48) is the v -canonical basis of W (cid:48) ,then ( w i ⊗ w (cid:48) j ) ≤ i ≤ m, ≤ j ≤ m (cid:48) is the v -canonical basis of W ⊗ k W (cid:48) .Let V be a not necessarily finite dimensional representation of a proalgebraic group G . For a k -algebra R , a subspace W of V is stable under g ∈ G ( R ) (or g stabilizes W ) if g : V ⊗ k R → V ⊗ k R maps W ⊗ k R into W ⊗ k R . If W is stable under all g ∈ G ( R ) for all R , then W is G -stable . Thesubgroup of G consisting of all g that stabilize W is a closed subgroup of G called the stabilizer of W . For n ≥ d ≥ P d = (cid:88) a + b ≤ d (cid:18) a + n − a (cid:19)(cid:18) b + n − b (cid:19) X a Y b ∈ N [ X, Y ] , a, b ) ∈ N with a + b ≤ d . The significance of this polynomialis explained in the following proposition. Proposition 4.3.
Let V be a finite dimensional k -vector space and let G be a closed subgroup of GL V . For d ≥ the following closed subgroups of GL V are equal: (i) The subgroup of GL V defined by the ideal of k [GL V ] generated by I ( G ) ∩ k [GL V ] ≤ d (cf.Lemma 4.2). (ii) The subgroup of GL V consisting of all elements that stabilize all G -stable subspaces of P ( V, V ∨ ) for all P ∈ N [ X, Y ] of degree at most d . (iii) The subgroup of GL V consisting of all elements that stabilize all G -stable subspaces of P d ( V, V ∨ ) .Moreover, there exists a G -stable subspace W of P d ( V, V ∨ ) such that the stabilizer of W (in GL V ),agrees with the group defined in (i), (ii) and (iii).Proof. For j = 1 , , G j denote the group defined in point j above. Clearly G ≤ G . Fixing abasis v = ( v , . . . , v n ) of V , we may identify GL V with GL n .To show that G ≤ G let P ∈ N [ X, Y ] be a polynomial of degree at most d and let W bea subspace of P ( V, V ∨ ). Let u , . . . , u r denote the v -canonical basis of P ( V, V ∨ ). Then thereexist P ij ∈ k [ Z, Z − ] ≤ d such that g ( u i ) = (cid:80) rj =1 P ij ( g ) u j for all g ∈ GL n ( R ) and all k -algebras R . It follows that for any basis w , . . . , w r of P ( V, V ∨ ) there exist Q ij ∈ k [ Z, Z − ] ≤ d such that g ( w i ) = (cid:80) rj =1 Q ij ( g ) w j for all g ∈ GL n ( R ) and all k -algebras R . We may extend a basis w , . . . , w s of W to a basis w , . . . , w s , w s +1 , . . . , w r of P ( V, V ∨ ). Then, using the above notation, an element g ∈ GL n ( R ) stabilizes W if and only if Q ij ( g ) = 0 for all i and j > s . Thus, an element g ∈ GL n ( R )such that Q ( g ) = 0 for all Q ∈ I ( G ) ∩ k [ Z, Z − ] ≤ d , will stabilize every subspace W of P ( V, V ∨ )stabilized by G . Hence G ≤ G .The most difficult part now is to show that G ≤ G . This follows from a detailed analysis ofthe proofs of two basic theorems on representations of algebraic groups (Theorems 4.14 and 4.27 in[Mil17]): We consider the regular representation of GL n on k [ Z, Z − ]. This can be defined as thenot necessarily finite dimensional representation of GL n corresponding to the comodule k [ Z, Z − ]with comodule map ∆ : k [ Z, Z − ] → k [ Z, Z − ] ⊗ k k [GL n ] , Z (cid:55)→ Z ⊗ Z . Explicitly, the action of anelement g ∈ GL n ( R ) on k [ Z, Z − ] ⊗ k R is given by g ( f ( Z )) = f ( Zg ) for f ∈ k [ Z, Z − ]. We note that k [ Z, Z − ] ≤ d is a GL n -stable subspace of k [ Z, Z − ]. For i = 1 , . . . , n let V i denote the k -subspace of k [ Z, Z − ] generated by the i -th row of Z . Then V i is GL n -stable. In fact, V i is isomorphic to V asa GL n -representation, under the isomorphism that identifies Z ij with v j for j = 1 , . . . , n . Similarly,for j = 1 , . . . , n let W j denote the subspace of k [ Z, Z − ] generated by the j -th column of Z − . Then W j is GL n -stable and indeed is isomorphic to V ∨ as a GL n -representation, under the isomorphismthat identifies ( Z − ) ij with v ∨ i for i = 1 , . . . , n . Let Q = (cid:88) d + ... + d n + e + ... + e n ≤ d X d . . . X d n n Y e . . . Y e n n ∈ N [ X , . . . , X n , Y , . . . , Y n ]be the full polynomial of degree d all of whose coefficients are equal to 1. We have a surjective map Q ( V , . . . , V n , W , . . . , W n ) → k [ Z, Z − ] ≤ d of GL n -representations, where Q ( V , . . . , V n , W , . . . , W n )is defined in a fashion similar to the definition of P ( V, V ∨ ) above. Since V i (cid:39) V and W j (cid:39) V ∨ thiscan be interpreted as a surjective map π : P d ( V, V ∨ ) → k [ Z, Z − ] ≤ d of GL n -representations, where P d = Q ( X, . . . , X, Y, . . . , Y ) = (cid:80) a + b ≤ d (cid:0) a + n − a (cid:1)(cid:0) b + n − b (cid:1) X a Y b ∈ N [ X, Y ].Since I ( G ) ∩ k [ Z, Z − ] ≤ d is a G -stable subspace of k [ Z, Z − ] ≤ d , it follows that W = π − ( I ( G ) ∩ k [ Z, Z − ] ≤ d ) is a G -stable subspace of P d ( V, V ∨ ). Thus, by the definition of G , the subspace W is G -stable. Therefore π ( W ) = I ( G ) ∩ k [ Z, Z − ] ≤ d is G -stable. From this we deduce that34 I ( G ) ∩ k [ Z, Z − ] ≤ d ) = I ( G ) is G -stable. However, only elements of G stabilize I ( G ) (cf. [Spr09]Lemma 2.3.8). Therefore G ≤ G as desired.Finally, the G -stable subspace W = π − ( I ( G ) ∩ k [ Z, Z − ] ≤ d ) of P d ( V, V ∨ ) has the propertyrequired in the last statement of the proposition.Let V be a finite dimensional k -vector space and let G be a closed subgroup of GL V . For every d ≥ G ≤ d denote the closed subgroup of GL V characterized in Proposition 4.3. We then have a descendingchain GL V = G ≤ ⊇ G ≤ ⊇ G ≤ ⊇ . . . of closed subgroups of GL V that eventually stabilizes at G . Definition 4.4.
The defining degree of G is the smallest d such that G = G ≤ d . The following lemma will be used in the next subsection. Roughly speaking, it shows thatstabilizers of subspaces of P ( V, V ∨ ) are uniformly definable. Lemma 4.5.
Let P ∈ N [ X, Y ] be a polynomial of degree d , n ≥ and let s denote the di-mension of the vector space P ( V, V ∨ ) , where V is an n -dimensional vector space. Furthermorelet ≤ r ≤ s and ≤ i < . . . < i r ≤ s . Then there exist polynomials Q , . . . , Q ( s − r ) r ∈ Z [ T, / det(( T i (cid:96) ,j ) ≤ (cid:96),j ≤ r ) , Z, Z − ] of degree at most d in Z and Z − , where T = ( T i,j ) ≤ i ≤ s, ≤ j ≤ r and Z = ( Z i,j ) ≤ i,j ≤ n such that the following holds: For every field k , every k -vector space V ofdimension n with basis v = ( v , . . . , v n ) , all matrices A ∈ k s × r such that det( A i (cid:96) ,j ) ≤ (cid:96),j ≤ r (cid:54) = 0 ,all k -algebras R and all g ∈ GL V ( R ) (cid:39) GL n ( R ) (via v ) the k -subspace of P ( V, V ∨ ) generated by uA , where u is the v -canonical basis of P ( V, V ∨ ) , is stable under g if and only if Q i ( A, g ) = 0 for i = 1 , . . . , ( s − r ) r .Proof. Let S = Z [ T, / det(( T i (cid:96) ,j ) ≤ (cid:96),j ≤ r )] and consider a free S -module V S of rank n with basis v S . The definition of P ( V, V ∨ ) and the v -canonical bases of P ( V, V ∨ ) extends from vector spacesto finite rank free modules in a straight forward manner. So let u S denote the v S -canonical basisof P ( V S , V ∨ S ). We extend the matrix T ∈ S s × r to a matrix (cid:101) T ∈ GL s ( S ) by adding the standardbasis vectors e , . . . , e s − r of length s − r in the rows { , . . . , s } (cid:114) { i , . . . , i r } . The group schemeGL n,S = Spec( S [ Z, Z − ]) over S , acts linearly on V S and on P ( V S , V ∨ S ). Moreover, there exista matrix B ∈ Z [ Z, Z − ] s × s with entries of at most degree d such that g ( u S ) = u S B ( g ) for any S -algebra S (cid:48) and g ∈ GL n ( S (cid:48) ). It follows that g ( u S (cid:101) T ) = u S B ( g ) (cid:101) T = u S (cid:101) T ( (cid:101) T − B ( g ) (cid:101) T ) . Thus the submodule of V S with basis u S T is stable under g if and only if the ( s − r ) × r submatrixin the lower left corner of ( (cid:101) T − B ( g ) (cid:101) T ) ∈ S [ Z, Z − ] s × s is zero.We claim that the entries Q , . . . , Q ( s − r ) r of this matrix have the required property. Since theentries of B have degree at most d in Z and Z − , also Q , . . . , Q ( s − r ) r have degree at most d in Z and Z − . The choice of a field k and a matrix A ∈ k s × r with det( A i (cid:96) ,j ) ≤ (cid:96),j ≤ r (cid:54) = 0 defines amorphism of rings S → k , i.e., a k -valued point of Spec( S ). The claim now follows by consideringthe fiber over this point. In more detail: The k -vector space V S ⊗ S k has basis v S ⊗ V → V S ⊗ S k by v (cid:55)→ v S ⊗
1. Similarly, we have an isomorphism P ( V, V ∨ ) → P ( V S , V ∨ S ) ⊗ S k, u (cid:55)→ u S ⊗
1. We extend A to a matrix (cid:101) A in GL s ( k ) is a similar fashionas we did with T . Then, for a k -algebra R and g ∈ GL n ( R ) we have g ( u (cid:101) A ) = u (cid:101) A ( (cid:101) A − B ( g ) (cid:101) A ).Thus the subspace of P ( V, V ∨ ) generated by uA is stable under g if and only if Q i ( A, g ) = 0 for i = 1 , . . . , ( s − r ) r . 35 .2 Types and stable subspaces Let M be a model of PROALG and let ( k, C, ω ) be the associated object of TANN. For an object b of C we let G ( b ) ≤ GL ω ( b ) denote the (scheme-theoretic) image of the representation G ( M ) → GL ω ( b ) defined by b . Note thatan algebraic group is a quotient of G ( M ) if and only if it is isomorphic to some G ( b ). Moreover, G ( M ) is the projective limit of the G ( b )’s. Our main result is that tp( b/k ) determines G ( b ). Webegin by translating the main findings of the previous subsection into a statement about models ofPROALG. The point of the following corollary is that this somewhat clumsy characterization ofwhen G ( b ) ≤ d = G ( b ) ≤ d (cid:48) can be expressed by an L -formula. Corollary 4.6.
Let M be a model of PROALG and ( k, C, ω ) the associated object of TANN . Let b be an object of C and for ≤ d < d (cid:48) let s and s (cid:48) denote the dimensions of P d ( ω ( b ) , ω ( b ) ∨ ) and P d (cid:48) ( ω ( b ) , ω ( b ) ∨ ) respectively. Then G ( b ) ≤ d = G ( b ) ≤ d (cid:48) if and only if the following conditionis satisfied: For all bases v of ω ( b ) , for all r (cid:48) with ≤ r (cid:48) ≤ s (cid:48) , all ≤ i (cid:48) < . . . < i (cid:48) r (cid:48) ≤ s andall A (cid:48) ∈ k s (cid:48) × r (cid:48) with det(( A (cid:48) i (cid:48) (cid:96) ,j ) ≤ (cid:96),j ≤ r (cid:48) ) (cid:54) = 0 such that the subspace of P d (cid:48) ( ω ( b ) , ω ( b ) ∨ ) generatedby u (cid:48) A (cid:48) is G ( M ) -stable, where u (cid:48) is the v -canonical basis of P d (cid:48) ( V, V ∨ ) , there exist ≤ r ≤ s , ≤ i < . . . < i r ≤ s and A ∈ k s × r with det(( A i (cid:96) ,j ) ≤ (cid:96),j ≤ r ) (cid:54) = 0 such that the subspace of P d ( V, V ∨ ) generated by uA is G ( M ) -stable, where u denotes the v -canonical basis of P d ( V, V ∨ ) ,and the polynomials Q (cid:48) ( A (cid:48) , Z ) , . . . , Q (cid:48) ( s (cid:48) − r (cid:48) ) r (cid:48) ( A (cid:48) , Z ) ∈ k [ Z, Z − ] ≤ d (cid:48) lie in the ideal of k [ Z, Z − ] generated by Q ( A, Z ) , . . . , Q ( s − r ) r ( A, Z ) ∈ k [ Z, Z − ] ≤ d . Here the polynomials Q (cid:48) , . . . , Q (cid:48) ( s (cid:48) − r (cid:48) ) r (cid:48) and Q , . . . , Q ( s − r ) r are defined as in Lemma 4.5.Proof. Note that a subspace of some P ( ω ( b ) , ω ( b ) ∨ ) is G ( M )-stable if and only if it is G ( b )-stable.Thus, according to Proposition 4.3 the closed subgroup G ( b ) ≤ d (cid:48) of GL ω ( b ) is the stabilizer of some G ( M )-stable subspace W (cid:48) of P d (cid:48) ( V, V ∨ ). If v is a basis of ω ( b ) and u (cid:48) is the v -canonical basis of P d (cid:48) ( V, V ∨ ), then W has a basis of the form u (cid:48) A (cid:48) , for some A (cid:48) ∈ k s (cid:48) × r (cid:48) with linearly independentcolumns, i.e., det(( A (cid:48) i (cid:48) (cid:96) ,j ) ≤ (cid:96),j ≤ r (cid:48) ) (cid:54) = 0 for an appropriate choice of 1 ≤ i (cid:48) < . . . < i (cid:48) r (cid:48) ≤ s . Sothe defining ideal I ( G ( b ) ≤ d (cid:48) ) of G ( b ) ≤ d (cid:48) is generated by Q (cid:48) ( A (cid:48) , Z ) , . . . , Q (cid:48) ( s (cid:48) − r (cid:48) ) r (cid:48) ( A (cid:48) , Z ) according toLemma 4.5. By construction, the polynomials Q ( A, Z ) , . . . , Q ( s − r ) r ( A, Z ) lie in the ideal I ( G ( b ) ≤ d ).Thus, if Q (cid:48) ( A (cid:48) , Z ) , . . . , Q (cid:48) ( s (cid:48) − r (cid:48) ) r (cid:48) ( A (cid:48) , Z ) lie in the ideal generated by Q ( A, Z ) , . . . , Q ( s − r ) r ( A, Z ),then I ( G ( b ) ≤ d (cid:48) ) ⊆ I ( G ( b ) ≤ d ). So G ( b ) ≤ d = G ( b ) ≤ d (cid:48) as desired.Conversely, assume that G ( b ) ≤ d = G ( b ) ≤ d (cid:48) . Similarly to the above argument, there existsappropriate integers, r, i , . . . , i r and a matrix A ∈ k s × r such that det(( A i (cid:96) ,j ) ≤ (cid:96),j ≤ r ) (cid:54) = 0 and I ( G ( b ) ≤ d ) is generated by Q ( A, Z ) , . . . , Q ( s − r ) r ( A, Z ). Since I ( G ( b ) ≤ d (cid:48) ) = I ( G ( b ) ≤ d ) it follows that Q (cid:48) ( A (cid:48) , Z ) , . . . , Q (cid:48) ( s (cid:48) − r (cid:48) ) r (cid:48) ( A (cid:48) , Z ) lie in the ideal generated by Q ( A, Z ) , . . . , Q ( s − r ) r ( A, Z ).Two verify that the above statement can be expressed by an L -formula we need two lemmas.Roughly speaking, the following lemma shows that we can quantify over the G ( M )-stable subspacesof P ( ω ( b ) , ω ( b ) ∨ ). Lemma 4.7.
Let p = ( m, n ) and P ∈ N [ X, Y ] . Let s be the dimension of the vector space P ( V, V ∨ ) ,where V is an n -dimensional vector space. Then, for every r with ≤ r ≤ s , there exists an L -formula ϕ ( T, y , . . . , y n ) , where T = ( T i,j ) is an s × r matrix of variables from the field sort and y , . . . , y n are variables from the p -total objects sort such that the following holds:For every model M = ( k, B p , X p , B p,q , X p,q ) of PROALG , all b ∈ B p , all bases v , . . . , v n of ω ( b ) and all A ∈ k s × r with linearly independent columns we have M | = ϕ ( A, v , . . . , v n ) if and only ifthe k -subspace of P ( ω ( b ) , ω ( b ) ∨ ) generated by uA is G ( M ) -stable, where u is the v -canonical basisof P ( ω ( b ) , ω ( b ) ∨ ) . roof. Let M = ( k, B p , X p , B p,q , X p,q ) be a model of PROALG and let ( k, C, ω ) be the associatedobject of TANN. Moreover let b ∈ B p and let P ( b, b ∨ ) ∈ B (1 ,s ) denote the unique tensor irreducibleobject of C such that ω ( P ( b, b ∨ )) (cid:39) P ( ω ( b ) , ω ( b ) ∨ ) as G ( M )-representations. A subspace W of P ( ω ( b ) , ω ( b ) ∨ ) of dimension r is G ( M )-stable if and only if there exist b (cid:48) ∈ B ( r, and a morphism h : b (cid:48) → P ( b, b ∨ ) in C such that ω ( h ) : ω ( b (cid:48) ) → ω ( P ( b, b ∨ )) (cid:39) P ( ω ( b ) , ω ( b ) ∨ ) has image W . So, incoordinates, if v is a basis of ω ( b ), and A ∈ k s × r is such that uA is a basis of W , where u is the v -canonical basis of P ( ω ( b ) , ω ( b ) ∨ ), then W is G ( M )-stable if and only if there exists b (cid:48) ∈ B ( r, with a basis v (cid:48) of ω ( b (cid:48) ) and a morphism h : b (cid:48) → P ( b, b ∨ ) in C such that ω ( h ) : ω ( b (cid:48) ) → ω ( P ( b, b ∨ )) (cid:39) P ( ω ( b ) , ω ( b ) ∨ ) maps v (cid:48) to uA . Since the v -canonical basis of P ( ω ( b ) , ω ( b ) ∨ ) can be characterized interms of v by L -formulas, the claim follows.We will also need a classical result of G. Hermann (cf. [Asc04, Theorem 3.4]). Lemma 4.8 (G. Hermann [Her26]) . For every field k , if f, g , . . . , g m ∈ k [ X , . . . , X n ] are poly-nomials of degree at most d such that f lies in the ideal generated by g , . . . , g m , then there existpolynomials f , . . . , f m ∈ k [ X , . . . , X n ] of degree at most (2 d ) n such that f = f g + . . . + f m g m . Combining the above results we see that the set of all b ∈ B p such that G ( b ) ≤ d = G ( b ) ≤ d (cid:48) isdefinable: Proposition 4.9.
For given p and integers ≤ d < d (cid:48) there exists an L -formula ϕ ( x ) in one freevariable x belonging to the p -basic objects sort such that for every model M = ( k, B p , X p , B p,q , X p,q ) of PROALG and every b ∈ B p we have M | = ϕ ( b ) if and only if G ( b ) ≤ d = G ( b ) ≤ d (cid:48) .Proof. It suffices to see that the statement from Corollary 4.6 can be expressed by an L -formula.This is guaranteed by Lemmas 4.7 and 4.8. Corollary 4.10.
Let M = ( k, B p , X p , B p,q , X p,q ) be a model of PROALG . If b, b (cid:48) ∈ B p are suchthat tp( b/ ∅ ) = tp( b (cid:48) / ∅ ) , then the defining degree of G ( b ) agrees with the defining degree of G ( b (cid:48) ) .Proof. Clear from Proposition 4.9.
Theorem 4.11.
Let M = ( k, B p , X p , B p,q , X p,q ) be a model of PROALG and let b, b (cid:48) ∈ B p . If tp( b/k ) = tp( b (cid:48) /k ) , then there exists an isomorphism ω ( b ) → ω ( b (cid:48) ) of k -vector spaces that inducesan isomorphism between G ( b ) and G ( b (cid:48) ) .Proof. According to Corollary 4.10 the defining degree of G ( b ) agrees with the defining degree of G ( b (cid:48) ). Let us denote it with d . Let s denote the dimension of P d ( ω ( b ) , ω ( b ) ∨ ) and let x be avariable from the p -objects sort. Let 1 ≤ r ≤ s and A ∈ k s × r with det(( A i (cid:96) ,j ) ≤ (cid:96),j ≤ r ) (cid:54) = 0 for1 ≤ i < . . . < i r ≤ s . Moreover let Q , . . . , Q ( s − r ) r ∈ Z [ T, / det(( T i (cid:96) ,j ) ≤ (cid:96),j ≤ r ) , Z, Z − ] be definedas in Lemma 4.5. Consider the formula ϕ A ( x ) with parameters from k such that ϕ A ( b ) expressesthe following statement:There exists a basis v of ω ( b ) such that for all r (cid:48) with 1 ≤ r (cid:48) ≤ s , all 1 ≤ i (cid:48) < . . . < i (cid:48) r (cid:48) ≤ s and all A (cid:48) ∈ k s × r (cid:48) with det(( A (cid:48) i (cid:48) (cid:96) ,j ) ≤ (cid:96),j ≤ r (cid:48) ) (cid:54) = 0 such that the subspace of P d ( ω ( b ) , ω ( b ) ∨ ) gener-ated by uA (cid:48) is G ( M )-stable, where u is the v -canonical basis of P d ( ω ( b ) , ω ( b ) ∨ ), the polynomials Q (cid:48) ( A (cid:48) , Z ) , . . . , Q (cid:48) ( s − r (cid:48) ) r (cid:48) ( A (cid:48) , Z ) ∈ k [ Z, Z − ] ≤ d (with Q (cid:48) , . . . , Q (cid:48) ( s − r (cid:48) ) r (cid:48) defined as in Lemma 4.5) liein the ideal of k [ Z, Z − ] generated by Q ( A, Z ) , . . . , Q ( s − r ) r ( A, Z ) ∈ k [ Z, Z − ] ≤ d .If ϕ A ( x ) lies in tp( b/k ) the formula ϕ A ( x ) determines G ( b ) because it shows that I ( G ( b )) = I ( G ( b ) ≤ d ) is generated by Q ( A, Z ) , . . . , Q ( s − r ) r ( A, Z ) ∈ k [ Z, Z − ] ≤ d . (Here GL ω ( b ) (cid:39) GL n via thebasis v deemed to exist by ϕ A ( x ).)On the other hand, ϕ A ( x ) lies in tp( b/k ) for some choice of r , A and i , . . . , i r .37 eferences [Asc04] Matthias Aschenbrenner. Ideal membership in polynomial rings over the integers. J.Amer. Math. Soc. , 17(2):407–441, 2004.[BHHW] Annette Bachmayr, David Harbater, Julia Hartmann, and Michael Wibmer. Free dif-ferential Galois groups. arXiv:1904.07806.[Cha84] Zoe Maria Chatzidakis.
MODEL THEORY OF PROFINITE GROUPS . ProQuestLLC, Ann Arbor, MI, 1984. Thesis (Ph.D.)–Yale University.[Cha87] Zo´e Chatzidakis. Model theory of profinite groups having IP. III. In
Mathematicallogic and theoretical computer science (College Park, Md., 1984–1985) , volume 106 of
Lecture Notes in Pure and Appl. Math. , pages 161–195. Dekker, New York, 1987.[Cha98] Zo´e Chatzidakis. Model theory of profinite groups having the Iwasawa property.
IllinoisJ. Math. , 42(1):70–96, 1998.[CvdDM81] Gregory Cherlin, Lou van den Dries, and Angus Macintyre. Decidability and undecid-ability theorems for PAC-fields.
Bull. Amer. Math. Soc. (N.S.) , 4(1):101–104, 1981.[CvdDM82] Gregory Cherlin, Lou van den Dries, and Angus Macintyre. The elementary theory ofregularly closed fields. preprint , 1982.[Del90] Pierre Deligne. Cat´egories tannakiennes. In
The Grothendieck Festschrift, Vol. II ,volume 87 of
Progr. Math. , pages 111–195. Birkh¨auser Boston, Boston, MA, 1990.[DG70] Michel Demazure and Pierre Gabriel.
Groupes alg´ebriques. Tome I: G´eom´etriealg´ebrique, g´en´eralit´es, groupes commutatifs . Masson & Cie, ´Editeur, Paris; North-Holland Publishing Co., Amsterdam, 1970. Avec un appendice ıt Corps de classes localpar Michiel Hazewinkel.[DM82] Pierre Deligne and James S. Milne. Tannakian categories. In
Hodge cycles, motives,and Shimura varieties , volume 900 of
Lecture Notes in Mathematics , pages 101–228.Springer-Verlag, Berlin, 1982.[EF72] Paul C. Eklof and Edward R. Fischer. The elementary theory of abelian groups.
Ann.Math. Logic , 4:115–171, 1972.[EGNO15] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik.
Tensor categories ,volume 205 of
Mathematical Surveys and Monographs . American Mathematical Society,Providence, RI, 2015.[Her26] Grete Hermann. Die Frage der endlich vielen Schritte in der Theorie der Polynomideale.
Math. Ann. , 95(1):736–788, 1926.[Hod93] Wilfrid Hodges.
Model theory , volume 42 of
Encyclopedia of Mathematics and its Ap-plications . Cambridge University Press, Cambridge, 1993.[Kam15] Moshe Kamensky. Model theory and the Tannakian formalism.
Trans. Amer. Math.Soc. , 367(2):1095–1120, 2015.[Mil17] J. S. Milne.
Algebraic groups , volume 170 of
Cambridge Studies in Advanced Mathe-matics . Cambridge University Press, Cambridge, 2017. The theory of group schemesof finite type over a field. 38Spr09] T. A. Springer.
Linear algebraic groups . Modern Birkh¨auser Classics. Birkh¨auserBoston, Inc., Boston, MA, second edition, 2009.[vdPS03] Marius van der Put and Michael F. Singer.
Galois theory of linear differential equa-tions , volume 328 of
Grundlehren der Mathematischen Wissenschaften [FundamentalPrinciples of Mathematical Sciences] . Springer-Verlag, Berlin, 2003.[Vis06] Albert Visser. Categories of theories and interpretations. In
Logic in Tehran , volume 26of
Lect. Notes Log. , pages 284–341. Assoc. Symbol. Logic, La Jolla, CA, 2006.[Wat79] William C. Waterhouse.
Introduction to affine group schemes , volume 66 of
GraduateTexts in Mathematics . Springer-Verlag, New York, 1979.[Wib] Michael Wibmer. Free proalgebraic groups. arXiv:1904.07455.Author information:Anand Pillay: Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-4618,USA. Email: [email protected]
Michael Wibmer: Institute of Analysis of Number Theory, Graz University of Technology, 8010Graz, Austria. Email: [email protected]@math.tugraz.at