Full Tannakian subcategories and surjection of fundamental groups

Question

Let $(\mathcal{T},w)$ be a neutral Tannakian category over a field $k$, with fundamental group $G$, and $w$ a fibre functor.

Let $(\mathcal{S},w|_{\mathcal{S}})$ be a full Tannakian sub-category (i.e. closed under tensor products, direct sums, duals and quotients) as well as taking subobjects/subquotients. Denote its fundamental group by $H$.

In "Deligne, Pierre, and J. S. Milne. "Tannakian categories." Lecture Notes in Matehmatics (2012)", Proposition 2.21, it is said that the natural morphism $G\longrightarrow H$ is faithfully flat (and in particular surjective). Their proof is tautological, and I could not understand the tautology.

The way the authors deduce the claim is by arguing that the induced map on the underlying algebras (taking the duals of $G = \text{Aut}^{\otimes}(w) \longrightarrow \text{Aut}^{\otimes}(w') = H$), the map

$$\text{Aut}^{\otimes}(w')^{\vee}\longrightarrow \text{Aut}^{\otimes}(w)^{\vee}$$ is clearly injective. While I understand how the conclusion follows from this implication, I would love for an explanation of why this is the case, or an alternative argument.

A reference would also be appreciated.

Answer 1

Theorem: A homomorphism $G\rightarrow H$ of affine group schemes over $k$ is faithfully flat if and only the functor $Rep(H)\rightarrow Rep(G)$ is fully faithful and the essential image is closed under taking subobjects.

Recall that a homomorphism of affine group schemes over a field is faithfully flat if and only if the map on coalgebras is injective.

For a $k$-algebra $A$ (not necessarily commutative), we let $M(A,k)$ denote the category of left $A$-modules finite-dimensional over $k$.

Let $f\colon A\rightarrow B$ be a homomorphism of $k$-algebras. Using $f$, we can regard a $B$-module as an $A$-module and $M(B,k)$ as a subcategory of $M(A,k)$.

Lemma: Assume that $B$ is finite-dimensional over $k$. The homomorphism $f\colon A\rightarrow B$ is surjective if and only if $M(B,k)$ is a full subcategory of $M(A,k)$ closed under taking submodules.

Proof: If $f$ is surjective, then the subcategory $M(B,k)$ certainly has the claimed properties. For the converse, let $\bar{A}$ denote the image of $A$ in $B$. Then $\bar{A}$ is an $A$-submodule of $B$, and hence also a $B$-submodule. As it contains the identity element $1$ of $B$, it equals $B$.

Let $f\colon C\rightarrow D$ be a homomorphism of $k$-coalgebras. Using $f$, we can regard a $C$-comodule as a $D$-comodule and $Comodf(C)$ as a subcategory of $Comodf(D)$.

Lemma: The homomorphism $f\colon C\rightarrow D$ is injective if and only if $Comodf(C)$ is a full subcategory of $Comodf(D)$ closed under taking subobjects.

Proof: If $C$ is finite-dimensional over $k$, this follows from the previous lemma applied to $f^{\vee}\colon D^{\vee}\rightarrow C^{\vee}$. In the general case, we can write $C$ as a union $C=\bigcup C_{i}$ of finite-dimensional $k$-subcoalgebras, and correspondingly $Comodf(C)=\bigcup_{i}Comodf(C_{i})$. Now the statement for $C$ follows from the statement for the $C_{i}$.