I'm having trouble finding a reference for something that I'm guessing the experts worked out long ago. Let's take a local or global field $F$ for this post, and fix a separable algebraic closure $\bar F / F$. If $F \subset F' \subset \bar F$ and $F' / F$ is finite, then write $Gal[F']$ for $Gal(\bar F / F')$. Let $k$ be a field of characteristic zero.

An Artin motive over $F'$ (with coefficients in $k$) is simply a continuous Galois representation $\rho : Gal[F'] \rightarrow GL(V)$. If one wants to avoid the choice of $\bar F$, an Artin motive is just a local system on $F_{et}'$ of $k$-vector spaces.

A "Weil motive" over $F'$, with coefficients in $k = \bar Q_\ell$ is a continuous representation $\rho$ of the Weil group of $F'$. Or one can work with $\ell$-adic sheaves if one wishes to avoid the choice of $\bar F$.

Now, both Artin motives (I'm sure) and Weil motives (I'm pretty sure) obey Galois descent. The slightly fancy way of saying this is the following: to give an Artin/Weil motive over $F$ is equivalent to giving the following data:

  1. For each sufficiently large finite separable extension $F' / F$, an Artin/Weil motive $V[F']$ over $F'$.

  2. For any $F$-morphism of such extensions $F_1' \rightarrow F_2'$, a $k$-linear map $V[F_1'] \rightarrow V[F_2']$.

The only condition is that the $k$-linear maps are functorial in the obvious way for morphisms $F_1' \rightarrow F_2' \rightarrow F_3'$ (and identity morphisms $F' \rightarrow F'$ I guess).

So, I guess that this means we have the following: for each such $F' / F$ a Tannakian category (of Artin motives, or Weil motives) over $k$. These Tannakian categories satisfy a gluing condition on $F_{et}$. So... does this form a "stack of Tannakian categories over $F_{et}$"? Has such a thing appeared in the literature?

The reason I ask is this -- I suspect the Galois descent property has some Tannakian interpretation as follows: consider $Gal[F]$ as a group scheme over $k$, and take an extension of group schemes over $k$:

$$1 \rightarrow P_0 \rightarrow P \rightarrow Gal[F] \rightarrow 1.$$

Define $P[F']$ to be the preimage of $Gal[F']$ in $P$. Define a $P$-motive over $F'$ to be a $k$-representation of $P[F']$.

Is there a simple condition (e.g. something with Galois cohomology and $P_0$?) which dictates whether such $P$-motives obey Galois descent?

  • 1
    see Theorem 6.16 in the following reference for one instance of notion of a stack of Tannakian categories in the literature:ams.org/journals/jams/2004-17-03/S0894-0347-04-00458-8/… – guest Mar 19 '17 at 3:03
  • 2
    see the articles of Schappacher, Panchiskin, Milne and Breen in the Motives (1994) volume 1 of the AMS. Especially the first should answer your questions. – guest Mar 22 '17 at 2:26

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