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It is my understanding that in dimension 0, the theory of motives should just be Galois theory for fields. I am hoping to find a reference or two to help me get some things straightened out.

One can construct a category $\mathcal{M}$ of 0-dimensional motives over $\mathbb{Q}$. If I'm not mistaken, one possible construction is to start with the category of 0-dimensional varieties over $\mathbb{Q}$, with correspondences as morphisms, and formally adjoin images of idempotents. This $\mathcal{M}$ is a neutral Tannakian category, meaning it is equivalent to the representations of some pro-algebraic group. In this case that group should be the absolute Galois group of $\mathbb{Q}$.

There are several fiber functors $\omega_{dR},\omega_{\ell},\ldots$ from $\mathcal{M}$ to vector spaces (over various fields), given by the different motivic realizations (de Rham, $\ell$-adic,...). Each fiber functor $\omega_\bullet$ has an automorphism group $G_{\mathbb{Q},\bullet}$, which is an affine pro-algebraic group. As I understand it, the groups $G_{\mathbb{Q},\bullet}$ should be viewed as realizations of some motivic absolute Galois group $G_{\mathbb{Q}}$. Each functor $\omega_\bullet$ identifies $\mathcal{M}$ with the representations of $G_{\mathbb{Q},\bullet}$.

What are the groups $G_{\mathbb{Q},\bullet}$, as $\bullet$ ranges over the different realizations on $\mathcal{M}$? Can someone point me to a reference where these groups are described?

I would love to find a reference that also describes the construction of $\mathcal{M}$.

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Motives of $0$-dimensional varieties are usually called Artin motives. The different fiber functors (essentially) all give rise to automorphism group isomorphic to the absolute Galois. There is one difference though: the cohomology theories have different coefficients ($\mathbb{Q}$ for de Rham cohomology, $\mathbb{Q}_\ell$ for $\ell$-adic cohomology, etc). For a finite Galois extension $L/\mathbb{Q}$, the corresponding cohomology will be the group ring of $\operatorname{Gal}(L/\mathbb{Q})$ over the coefficient ring of the cohomology theory with its natural Galois representation. As a result, the automorphism group of the fiber functor will be a pro-algebraic group over the coefficient ring. For Betti cohomology, we get the absolute Galois group as a pro-algebraic group over $\mathbb{Q}$. The comparison isomorphisms then show that the base-change of the pro-algebraic group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ from $\mathbb{Q}$ to $\mathbb{Q}_\ell$ is isomorphic to the automorphism group for the $\ell$-adic realization (and similarly for the other realization functors). The comparison isomorphisms between the different realizations can be interpreted as change-of-basepoint for the automorphism group (the basepoint for the realization functor is defined over the coefficient field of the cohomology theory).

This is discussed in detail in Section 9.4 of the book

  • A. Huber-Klawitter and S. Müller-Stach: Periods and Nori motives.

which can be found here.

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  • $\begingroup$ Fantastic, thanks for the reference. I had seen this book, but only an earlier version without a section on Artin motives. $\endgroup$ – Julian Rosen Dec 18 '16 at 15:04
  • $\begingroup$ You say that the automorphisms of the de Rham realization gives the absolute Galois group as a pro-algebraic group over $\mathbb{Q}$ (= as a pro-(finite constant) group scheme, yes?). I was under the impression that the de Rham realization should give a non-trivial inner form of the absolute Galois group. The reference you gave seems to deal only with the Betti realization. $\endgroup$ – Julian Rosen Dec 18 '16 at 16:00
  • $\begingroup$ Sorry, you are right. For a Galois extension, de Rham cohomology should give the field and Betti cohomology should give the group ring and these become isomorphic after base change to the extension field. The comparison isomorphism (which a priori is with $\mathbb{C}$-coefficients) should be defined over $\overline{\mathbb{Q}}$ because all the periods are all algebraic numbers. Then the de Rham fundamental group is a non-trivial inner form of the Betti fundamental group (in the same way that a Galois extension is a form of the trivial torsor). $\endgroup$ – Matthias Wendt Dec 18 '16 at 16:24

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