Hom(X,F0⊗F Y1)
0 //Hom(V0X,V0Y) //Hom(V0X,VY0) //Hom(V0X,VY1) By what we have already proven, the last two vertical arrows are isomorphisms, so by the Five Lemma so is the first, and we have shown that V0is fully faithful.
(c): If V0 is semisimple, then V is semisimple as a composition of the semi-simple functors V0and F0⊗F−, the latter being semisimple by Theorem 23.19(b).
Conversely, assume that V0 restricted toA is semisimple. Let X be a semisimple object ofA ⊗F F0, we must show that V0(X) is semisimple. There exists an ob-ject X0 ofA, and an epimorphism F0 ⊗F X0 → X. As in the proof of Theorem 23.19(b), we may assume that X0 itself is semisimple, since our given epimor-phism factors through F0⊗F (X0/rad X0). Hence V0(X), being a quotient of the
semisimple object V(X0), is semisimple. ∴
Proposition 25.9. LetB be a rigid abelian F0-linear tensor category, and con-sider an exact F0-linear tensor functor V0 : A0 → B. Let η : V0 ⇒ V0 be an automorphism of functors. Thenηis a tensor automorphism of V if and only if its restriction to V is a tensor automorphism.
Proof. Again, as in Theorems 25.2(a) and 25.5(a), this is a matter of checking that certain natural transformations are equal, and we suppress it. ∴
26 Tannakian categories
In this section, we use the results of the previous sections in order to discuss non-neutral Tannakian categories using only the neutral flavour of Tannakian cat-egories – groups, not groupoids.
Let F be a field extension.
Definition 26.1. (a) A pre-Tannakian category over F is a finite rigid abelian tensor category over F.
(b) A subcategory S of a pre-Tannakian category T over F is a strictly full pre-Tannakian subcategory if it is a full subcategory closed under tensor products, duals, and all subquotients inT.
(c) Given a set S of objects of a pre-Tannakian categoryT over F, we let ((S ))⊗
denote the smallest strictly full pre-Tannakian subcategory ofT containing all objects of S .
(d) A fibre functor over some field extension F0 ⊃ F of a pre-Tannakian cate-goryT is an F-linear exact faithful tensor functor onT with values in the category of finite-dimensional F0-vector spaces.
(e) A Tannakian category over F is a pre-Tannakian category over F for which there exists a fibre functor over some field extension F0 of F.
(f) A Tannakian category over F is neutral if there exists a fibre functor over F itself.
For the rest of this section, we fix a Tannakian categoryT over F.
Definition 26.2. The monodromy group ofT with respect to a given fibre functor ωofT over a field extension F0 is the functor
Gω(T ) : F0-Algebras−→Groups
mapping an F0-algebra R0 to the group of tensor automorphisms of the tensor functor R0⊗F0ω(−) which maps X∈T to the R0-module R0⊗F0ω(X).
The monodromy group Gω(X) of an object X of T is the monodromy group of the strictly full Tannakian subcategory ((X))⊗ofT with respect toω(cf. Defi-nitions 1.6 and 26.1).
From the literature on Tannakian categories, we use (only) the following two theorems:
Theorem 26.3. Let G be an algebraic group over F. The monodromy group of RepF(G) with respect to the forgetful functor RepF(G)→VecF is G.
Proof. [Del82, Theorem 2.8]. ∴
Theorem 26.4. Assume thatT is neutral, and fix a fibre functorωover F.
(a) Gω(T) is an affine group scheme over F. It is of finite type if and only ifT is finitely generated.
(b) ωinduces an equivalence of categoriesT −→RepF(Gω(T )).
Proof. [Saa72] or [Del82, Theorem 2.11]. ∴
Remark 26.5. In the situation of Theorem 26.4, if the Tannakian categoryT is finitely generated then for every M ∈ T with ((M))⊗ = T the vector spaceω(M) gives rise to a faithful representation of Gω(T).
We complement it in the non-neutral case by the following:
Theorem 26.6. Fix a fibre functorωover some field extension F0 of F.
(a) Gω(T) is an affine group scheme over F0. It is of finite type if and only if T is finitely generated.
(b) ωinduces an equivalence of categoriesT ⊗F F0 −→RepF0(Gω(T )).
Remark 26.7. The general theory of Tannaka categories associates to a pair (T , ω) – consisting of a Tannakian categoryT over F and a fibre functorωover F0– an affine groupoid schemeGω(T ), the definition of which we suppress, and shows that ω induces an equivalence of categories from T to the category of finite-dimensional representations of the groupoid schemeGω(T ).
Note that the original reference [Saa72] is faulty in the non-neutral case. For this, [Del90] is the correct place to look. For even further generality, see [Del02].
Proof of Theorem 26.6. The categoryT ⊗F F0 is pre-Tannakian over F0 by the results of Sections 23 and 25. Using Corollary 24.11, we may choose an extension ω0 : T ⊗F F0 → VecF0 ofω, which is a fibre functor ofT ⊗F F0over F0 by the results of Sections 24 and 25. SoT ⊗F F0 is a neutral Tannakian category, and Theorem 26.6 applies to it.
It remains to show that Gω(T) and Gω0(T ⊗F F0) coincide. But given an F0-algebra R, Theorem 24.1(b) shows that the restriction map
Aut (R⊗F0 −)◦ω0−→Aut ((R⊗F0 −)◦ω) is a bijection, which implies by Proposition 25.9 that its restriction
Gω0(T ⊗F F0)(R)= Aut⊗ (R⊗F0−)◦ω0→Aut⊗((R⊗F0 −)◦ω)=Gω(T)(R) to tensor automorphisms is a bijection, so we are done. ∴ Proposition 26.8. Let S be a Tannakian category over F, let T be a neutral Tannakian category over F, and let V : S → T be an exact fully faithful F-linear tensor functor. Then V is semisimple if and only if the essential image of V is closed under subquotients inT .
Proof. If the essential image of V is subquotient-closed inT, then V is semisim-ple: This has been proven more generally in Proposition 3.6.
Conversely, let us assume that V is semisimple. We show first that the essential image of V is closed under subobjects inT , that is, for every object X ofS and every subobject Y0 ⊂ V(X) there exists an object X0ofS with Y0 V(X0). Since T is a neutral Tannakian category over F, it is equivalent to the category of finite-dimensional representations of a group scheme over F by Theorem 26.4, and the usual rules of the machinery of exterior algebra apply.
In particular, in the situation Y0 ⊂ V(X) there is a well-defined rank r := rk(Y0)≥ 1 of Y0, and Y0 coincides with the kernel of the homomorphism
V(X) −→Hom ΛrY0,Λr+1V(X), v7→(x7→ v∧x), as in the proof of Theorem 21.1(a).
Now ΛrY0 has rank 1, so it is simple, and so there exists a projection of the semisimplification ΛrV(X)ss
of ΛrV(X) onto ΛrY0. Since V is semisimple, we may identify ΛrV(X)ss
withΛrV(Xss). Therefore, the displayed homomorphism of the previous paragraph induces a homomorphism
g : V(X)−→Hom
That the essential image of V is closed under quotients inT follows formally from the above: If V(X) →Y00 →0 is an exact sequence inT with X ∈S, then the above applied to the dual exact sequence 0→(Y00)∨→ V(X∨) gives an object X0 ∈S with (Y00)∨ V(X0), and so Y00 (Y00)∨,∨ V(X0,∨). ∴ Theorem 26.9. Let F0/F be a separable field extension, let T be a Tannakian category over F, and letT0 be a neutral Tannakian category over F0.
Assume that V : T → T 0 is an exact F-linear tensor functor which is both F0/F-fully faithful and semisimple. Then V induces an equivalence of Tannakian categories
V0 : T ⊗F F0 −→((VT ))⊗,
where ((VT ))⊗ denotes the strictly full pre-Tannakian subcategory ofT 0 gener-ated by the image ofT under V.
Proof. By Theorem 25.6, the exact functor V0 : T ⊗F F0 → T 0 induced by V is fully faithful and semisimple. We must show that its essential image coincides with the strictly full Tannakian subcategory ofT 0generated by VT .
On the one hand, we have V0(T ⊗F F0) ⊂ ((VT))⊗, since every object of T ⊗FF0has a presentation by objects arising fromT, V0 extends V and is exact.
On the other hand, we must show that ((VT ))⊗ ⊂ V0(T ⊗F F0). Clearly, the essential image of V0 is closed under direct sums, and also under tensor products and duals since V0 is a tensor functor by 25.5. We need to show that this essential image is closed under subquotients. And this follows from Proposition 26.8. ∴ Proposition 26.10. In the situation of Theorem 26.9, let ω0 be a fibre functor of T 0 over F0. For every object X of T, the monodromy groups Gω0◦V(X) and Gω0(V X) coincide.
Proof. By Theorems 26.3 and 26.6, the monodromy group Gω0◦V(X) coincides with the monodromy group of F0⊗F X as calculated inT ⊗F F0.
Applying Theorem 26.9 to the Tannakian categoriesTf:= ((X))⊗ and Tf0 := ((V X))⊗, we obtain an equivalence of categories Tf⊗F F0 Tf0, which clearly implies that the monodromy group of F0⊗F X as calculated inT ⊗F F0coincides with the monodromy group of V(X).
Taken together, the two previous paragraphs prove the statement of this
Propo-sition. ∴