WORKSHOP on TANNAKIAN CATEGORIES 1. Motivaton A

WORKSHOP ON TANNAKIAN CATEGORIES

1. Motivaton A neutral Tannakian category is a special kind of monoidal category together with some extra structure relative to a field k. The central result about Tannakian categories is that every Tannakian category is equivalent to the category of finite dimensional representations of group scheme over k, which is unique up to isomorphism. Historically, the theory of Tannakian categories derives from Tannaka–Krein duality for compact topological groups, which I am going to explain next. Let G be a compact haussdorff topological group. Tannaka’s theorem (1939) provides a way to reconstruct G from the monoidal category Rep (G) of its finite dimensional complex representations C and the forgetful functor F : Rep(G) −−→ VectC, which is a faithful monoidal functor. Roughly this works as follows: One puts a topology on the set(!) End⊗(F ) of monoidal endomorphisms τ : F −−→ F by defining it to be the coarsest topology for which all projections

End⊗(F ) −−→ End(V ) are continuous. We say that τ is self–conjugate if τ = τ where the bar denotes complex conjugation. Then the set T (G) of all monoidal, self–conjugate endomorphisms of F is a a compact topological group under the operation of composition of functors (the nontrivial statement here is that there exist inverse elements). Every element x of G gives rise to a monoidal self–conjugate endomorphism via multiplication by x on each representation, and hence one has a map G −−→T (G). Tannaka’s theorem then says that this map is an isomorphism.

Let C be a C–linear monoidal category with internal homomorphisms and let F : C −−→ VectC be a C–linear faithful monoidal functor from C to the category of finite dimensional complex vector spaces. Less formally, C is a category whose objects are finite dimensional complex vector spaces with some extra structure. For any two such objects X and Y there should be given naturally the same extra structure on the vector space X ⊗C Y , and the sets of morphisms HomC(X,Y ) in C form a C–linear subspace of HomC(X,Y ). Krein’s theorem (1949) states that such a category C is equivalent to the category of finite dimensional representations of a compact topological group G (which will be unique up to isomorphism by Tannaka’s theorem) if and only if the category C is semisimple, and if X and Y are simple objects then HomC(X,Y ) is either 1–dimensional (when X and Y are isomorphic) or trivial (if not). Altogether, the theorems of Tannaka and Krein constitute what nowadays is known as Tannaka– Krein duality. In the case of commutative compact groups, Tannaka–Krein duality specialises to Pontrjagin duality, isomorphism classes of simple objects in the category Rep (G) corresponding C to elements of the discrete commutative group of characters of G. After some decades of neglect, Tannaka–Krein duality has found new interest, especially from physicists because it has become an important tool in the study of quantum groups. We now come to the formal definition of Tannakian categories due to Grothendieck (letters to Serre in the 1960’s, the standard reference is LNM 265 by N. Saavedra Rivano 1972). The basic idea is that we want to give an algebraic analog of Tannaka–Krein duality. Let k be a field. A neutralised Tannakian category over k is an abelian k–linear monoidal rigid category C with End(1) = k, together with a monoidal faithful exact functor F : C −−→ Vectk from C to the category of finite dimensional k vector spaces, called fibre functor. Here, monoidal means 1 2 WORKSHOP ON TANNAKIAN CATEGORIES that C is equipped with a tensor product and an internal homomorphism functor ⊗ : C × C −−→C and Hom : C × C −−→C which come together with several natural transforms of functors ensuring that tensor products and internal homomorphisms behave as they should. There is also specified a neutral object 1 for the tensor product. The adjective rigid means that for all objects A, A1,...,An,B1,...,Bn of C the natural maps n n n O O O A 7−→ Hom(Hom(A, 1), 1)) and Hom(Ai,Bi) −−→Hom Ai, Bi i=1 i=1 i=1 (which exist), are isomorphisms. The adjective neutralised refers to the fact that a fibre functor F is part of our data. If we only demand the existence of a fibre functor but do not choose one, we speak of a neutral Tannakian category (the terminology differs from source to source). There is also a more general notion of not-neutral Tannakian categories. Examples of neutralised Tannakian categories are quickly at hand: Any category whose objects are finite dimensional k–vector spaces with some extra structure is likely to be neutral Tannakian, the fibre functor F being the forgetful functor. For instance:

(1) The category of finite dimensional k–vector spaces Vectk itself, with F the identity functor. More generally, for any field extension k|k0, the category of finite dimensional k0–vector spaces, with F the functor − ⊗ k. (2) Let k|k0 be a field extension. The category of finite dimensional reduced k0–algebras, with

F the functor that associates with a k0–algebra A the k vector space A ⊗k0 k. (3) The category of finite dimensional Z–graded k–vector spaces, with F the forgetful functor. (4) The category of finite dimensional k–representations of a linear group G over k (or just of an abstract group), with F the forgetful functor. (5) If k is a topological field, the category of finite dimensional continuous k–representations of a topological group G, with F the forgetful functor. (6) The category of rational or real mixed Hodge structures and F the forgetful functor to

VectQ or VectR respectively. (A rational Hodge structure is a finite dimensional Q–vector space V together with a filtration on V and a filtration on V ⊗ C, morphisms of rational Hodge structures are Q–linear maps compatible with these filtrations. If X is a complex i variety, the Betti cohomology groups H (X, Q) carry naturally a mixed Hodge structure). (7) Let (X, x) be a pointed topological space. The category of locally constant sheaves of finite dimensional k–vector spaces on X, with F the functor that associates with such a sheaf it’s fibre over x (hence the name “fibre functor”). (8) Given a neutral Tannakian category C with fibre functor F and an object C of C we can consider the abelian tensor subcategory of C generated by C. This subcategory, together with the restriction of F to it, is again neutral Tannakian.

Let C be a neutralised Tannakian category with fibre functor F : C −−→ Vectk. The Main Theorem of Tannakian formalism states that the functor End⊗(F ) of k–algebras is representable by a group scheme G over k (or equivalently a proalgebraic group, which is like profinite, just that the quotients are not finite but algebraic. One can obtain these algebraic quotients via example (8)). This group scheme G is called the fundamental group of C. If we choose another fibre functor F 0 of C we obtain another fundamental group G0. The groups G and G0 are isomorphic, canonically up to an inner automorphism, so the choice of a fibre functor for a neutral Tannakian category is much like the choice of a base point on a topological space. This is more than just an analogy: If in example (7) we choose some reasonable path connected pointed space (X, x), we know that locally constant sheaves of finite dimensional k–vector spaces on X are determined up to isomorphism by the monodromy action of the fundamental group π1(X, x). The group G is then the proalgebraic WORKSHOP ON TANNAKIAN CATEGORIES 3 completion of π1(X, x) (again, formally like profinite completion), and the choice of another base point x0 essentially yields the same fundamental group up to an inner automorphism. Similarly, for the algebraist, the choice of a fibre functor for a neutral Tannakian category is much like the choice of an algebraic closure K of a field K, where also Gal(K|K) is independent of K up to an inner automorphism. That this also is more than just a mere analogy is shown by example (2), where in the case k|k0 is Galois we obtain for G to be the Galois group of this extension, as a profinite group. We can thus say that Tannakian formalism encompasses fundamental groups, as well as Galois groups. In example (6) the resulting group G is called the absolute Hodge–group. Its algebraic quotient obtained as in example (8) from a single Hodge structure H is called the Mumford–Tate group of H, the main protagonist of the Mumford–Tate conjecture. Finally, I must mention that Grothendieck’s category of mixed motives, which is still an object of speculation, if it exists, should be a Tannakian category (k = Q). Its fundamental group is to be called the motivic Galois group.

2. Programme We will agree that rings always have a unity. We agree that monoidal category and tensor category as well as monoidal functor and tensor functor are synonyms.

MONDAY. Goal of the day: To define neutral Tannakian categories, and to give many examples. We use [Del80] as our main reference. Lecture 0: Introductory Lecture. Lecture 1. Tensor categories, tensor functors. Introduce monoidal categories and monoidal functors without loosing too much time. Discuss morphisms of monoidal functors. Introduce internal homomorphisms and define rigid monoidal categories. Show that if C and C0 are rigid, then every morphism between monoidal functors F,G : C −−→C0 is an isomorphism (that’s only two lines, see [Del80], Proposition 1.13). Lecture 2. Abelian and linear tensor categories. Give the compatibility constraints between additive (linear) and monoidal structure on a category. Discuss endomorphisms of the unity object ([Del80], pp. 118ff). Discuss the full abelian tensor subcategory generated by a single object. Lecture 3. Tannakian categories. Formally introduce the notion of a neutral Tannakian category. Give many examples and non–examples (see what you can do with the examples of Deligne: [Del80], pp. 122ff).

THUESDAY. Goal of the day: To introduce affine group schemes and representations, and to prove that the category Repk(G) of finite dimensional k–representations of a group scheme G over a field k is Tannakian. We use [Del80] and [Sza09] as sources. For additional material on affine group schemes I recommend [Wat79]. Lecture 4. Affine schemes and the functor of points. Let R be a commutative ring. We introduce affine schemes over R in the following way: an affine scheme is a representable covariant functor from the category of commutative R–algebras to the category of sets. For an R–algebra 4 WORKSHOP ON TANNAKIAN CATEGORIES

A we write spec A = HomR(A, −) for the functor represented by A. Discuss some examples where R = k is a field. Discuss the example S 7−→ V ⊗k S where V is a k–vector space. Lecture 5. Affine group schemes and Hopf algebras. Define affine group schemes over R to be group objects in the category of affine schemes over R. Show that these are the same as representable functors from R–algebras to groups. Show that affine group schemes are represented by Hopf–algebras. Discuss the following examples: The functor Ga associating to an R–algebra its underlying additive group, and the functor Gm associating to an R–algebra its group of unities. The functor µn of n–th roots of unity, constant functors. In the case R = k is a field, discuss the functor GLV : S 7−→ GL(V ⊗k S) for a k–vectorspace V . Lecture 6. Representations and comodules. Let G = spec A be an affine group scheme over a field k. Introduce representations of G in the functorial setting. Show that they correspond to A–comodules ([Del80], Proposition 2.2). Prove the Propositions 2.3 and 2.6 of loc.cit. and their corollaries. Lecture 7. Categories of finite dimensional representations are Tannakian. Let G = spec A be an affine group scheme over a field k. Define direct sums and tensor products of k–linear representations of G. Show that the category Repk(G) of finite dimensional representations of G together with the forgetful functor is a neutralised Tannakian category. State how properties of G are reflected in the category Repk(G).

WEDNESDAY. Goal of the day: To state and prove the Main Theorem of Tannakian formalism, i.e. the Tannaka–Krein Duality Theorem in the algebraic setting. Our main souce is now [Sza09] (the proof in [Del80] is outdated). Lecture 8. Recovering a group scheme from its representations. This is Tannakas theorem for group schemes. Prove Theorem 6.3.4 and Proposition 6.3.5 of [Sza09], (this is Proposition 2.8 in [Del80]). Lecture 9. The Main Theorem. Let C be a neutralised Tannakian category with ﬁbre functor F . State and explain: The group valued functor End⊗(F ) is representable by a group scheme G over k, and F induces an equivalence of monoidal categories C −−→ Repk(G). Draw an overview of the proof given in section 6.5 of [Sza09]. Lecture 10. Proof. Details of the proof: Prove Gabber’s Lemma (Proposition 6.5.5 of [Sza09]). Lecture 11. Proof. Details of the proof: Conclusion, following [Sza09]. Maybe give an illustration by means of an example, such as Proposition 6.5.15 in [Sza09].

THURSDAY. Example-day.

Lecture 12. Example 1: Graded vector spaces. The caterory of finite dimensional Z– Z graded vector spaces Vectk together with the forgetful functor is a neutralised Tannakian category. Z Show that its fundamental group is Gm. Give explicitly the equivalences between Vectk and Repk(Gm) in both directions. Lecture 13. Example 2: Fibre bundles and fundamental groups. Let (X, x) be a good connected pointed topological space. Recall what locally constant sheaves of finite dimensional k–vector spaces on X are and how they determine and are detemined by the monodromy action of π1(X, x). Show that these sheaves together with the functor F that associates with a sheaf it’s fibre over x is Tannakian. Show that the associated Tannakian fundamental group G is the algebraic WORKSHOP ON TANNAKIAN CATEGORIES 5 closure of the fundamental group and determine the image of G in the endomorphisms of a sheaf (have look at Proposition 6.5.15 in [Sza09]). Lecture 14. Example 3: Galois groups. Lecture 15. Example 4: ?. TBA

FRIDAY. More on the connection between Tannakian formalism and Galois theory. We show what Galois groups have to do with fundamental groups and sketch the construction of Nori’s fundamental (aka Galois) group. Lecture 16, Friday. Galois theory according to Grothendieck. Let R be an integral commutative ring, let K be an algebraically closed field and let r : R −−→ K be a morphism of rings. For every finite étale R–algebra A (étalemeans flat and unramified – think of A as a finite product of finite Galois extensions of R), its pushforward K ⊗R A is a finite étale K–algebra, hence just a finite sum of copies of K, in fact, one copy for each prime ideal of K ⊗R A. We consider the functor ω from the category of finite étale R–algebras to the category of finite sets given by

ωr : A 7−→ Prime ideals of K ⊗R A ét The automorphism group of the functor ωr is a profinite group. We denote it by π1 (R, r) and call it the étalefundamental group of R. The main theorem of Grothendiecks Galois theory states that the functor ωr induces an equivalence between the category of finite étale R algebra and finite ét π1 (R, r)–sets. We show that in the case where R is a field one recovers classical Galois theory. Lecture 17, Friday. Galois groups and fundamental groups. Give interesting examples for Lecture 16. Give Grothendieck’s fundamental group of a k–variety as an extension of the Galois group of k by the geometric fundamental group. Lecture 18, Friday. Nori’s fundamental group. Construct Nori’s fundamental group scheme following [Sza09], section 6.7. Lecture 19, Friday. We are all exhausted and thirsty. Fortunately there is a remedy.

References [Del80] P. Deligne, Tannakian categories, Lecture notes in math. 100, Springer, 1980. [Sza09] T. Szamuely, Galois groups and fundamental groups, Cambridge studies in adv. math. vol. 117, Cambridge University Press, 2009. [Wat79] W.C. Waterhouse, Introduction to haﬃne group schemes, Graduate texts in math. vol. 66, 1979.

Peter Jossen FakultätfürMathematik UniversitätRegensburg Universitätsstr.31 93040 Regensburg, GERMANY [email protected]