# How to see the geometry and arithmetic of tannakian fundamental groups?

The etale fundamental group is an inverse limit of automorphism groups of finite etale coverings. We can see the geometry of etale fundamental group very well from etale coverings just like topologically fundamental group. But the tannakian fundamental groups are defined as the automorphism of a fiber functor of a tensor category, this is definition abstract for me.

I want ask that how to see the geometry of tannakian fundamental groups?

What is the relation between first etale cohomology and pro-unipotent fundamental group?

I would like to extend my question. If we consider a hyperbolic curve $X$ over a local field $K$ with valuation ring $R$, there is an natural Galois action (i.e., outer Galois action) of $G_{K}$ on the etale fundamental group $\pi_{1}(X)$. From this outer Galois action, we can understand some geometry of $X$ and their reduction. For example, there is a good reduction criterion in terms of pro-$l$ fundamental groups (Oda, Tamagawa).

My question is: Dose there exist some similar Galois actions or criterions for Tannakian fundamental groups? or dose there exist some anabelian type theorems for Tannakian fundamental groups?

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Hi, what do you mean by "tannakian etale fundamental group" in your last line? –  Lars Mar 7 '13 at 14:18
@Lars: I want to say pro-unipotent fundamental group. –  kiseki Mar 7 '13 at 15:16

One perspective is that the tannakian fundamental group is simply a thickening of the ususal fundamental group. It is the inverse limit, over all continuous representations of the fundamental group (lisse sheaves), of the Zariski closure of the image of the fundamental group (automorphism group). The fundamental group, on the other hand, is the inverse limit over all its continuous representations of its image. So the tannakian fundamental group is a sort of Zariski closure.

Edit: On can always recover the etale fundamental group as the maximal profinite quotient of the Tannakian fundamental group. Using the exact same method as one uses algebraically, you can split the Tannakian fundamental group into geometric and arithmetic parts and thereby get a Galois action on the geometric fundamental group. So one recovers all data about the fundamental group used in anabelian geometry from the Tannakian group.

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This is not precise enough I think ; what means "fundamental group" in each occurrence ? Ususal should be corrected of course. It seems better to me to say as Deligne explains that you can reconstruct algebraically the algebraic envelope of the usual topological fundamental group. –  Niels Mar 8 '13 at 8:14
I consider the usual fundamental group to be the etale fundamental group, and the Tannakian fundamental group to be the Tannakian group of the category of $l$-adic lisse sheaves. I actually don't understand how your improvement is different from what I said. –  Will Sawin Mar 8 '13 at 17:34
My comment is just the way Deligne justifies the introduction of tannakian fundamental groups. In your own answer the topological fundamental group does not appear which makes a difference, don't you agree ? Moreover you did not reply to my question: what means fundamental group in occurrences 3, 4, 5 of your answer ? –  Niels Mar 9 '13 at 8:57
Again, it's the etale fundamental group. The topological fundamental group appears indirectly in the case that the variety is over $\mathbb C$. But if one already understands and accepts the etale fundamental group, one can work with it instead, which applies to more cases in algebraic geometry. –  Will Sawin Mar 9 '13 at 17:32

To answer your second question, for any nilpotent neutral Tannakian category $\mathcal{C}$, (i.e. one in which every object is an iterated extension of the unit object $\underline{1}$), with fibre functor $\omega$ and assocaited pro-unipotent group scheme $G=G(\mathcal{C},\omega)$, there is an isomorphism

$\mathrm{Lie}(G^\mathrm{ab})^*\cong \mathrm{Ext}^1_\mathcal{C}(\underline{1},\underline{1})$

So in our case, taking $\mathcal{C}$ to be the category of unipotent lisse $\mathbb{Q}_\ell$-sheaves on some variety over an algebraically closed field of characteristic $\neq\ell$, we can recover the first étale cohomology as the dual of the abelianisation of the pro-unipotent étale fundamental group, exactly as one might expect from the Hurewiz theorem.

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