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## The puzzle of Galois descent

Suppose we have a system $f_1, f_2, \dots f_n \in k[x_1, x_2, \dots x_m]$ of polynomial equations over a perfect (to keep things simple) field $k$, and we’d like to consider solutions of it over various field extensions $L$ of $k$. Write $V(L)$ for the set of all solutions to this system over $L$.

As it happens, knowing $V(L)$ for any algebraic extension $L$ of $k$ is equivalent to knowing $V(\bar{k})$, where $\bar{k}$ denotes the algebraic closure of $k$, together with the action of the absolute Galois group $G = \text{Gal}(\bar{k}/k)$. After picking an embedding of $L$ into $\bar{k}$, the infinite Galois correspondence says that $L$ is precisely the set of fixed points of the closed subgroup $H$ of $G$ which stabilizes $L$, and it’s not hard to see that this extends to $V(L)$; that is, $G$ naturally acts on $V(\bar{k})$, and we have a natural identification

$\displaystyle V(L) \cong V(\bar{k})^H$.

Now let’s categorify this situation. Before we considered, for each algebraic extension $L$ of $k$, a set $V(L)$. There are many situations in mathematics in which it’s natural to consider instead a category $F(L)$, such that a morphism $L_1 \to L_2$ induces a functor $F(L_1) \to F(L_2)$, and so forth. The basic example is the case that $F(L) = \text{Mod}(L)$ is the category of $L$-vector spaces, and for $f : L_1 \to L_2$ a morphism the corresponding functor is given by extension of scalars

$\displaystyle \text{Mod}(L_1) \ni V \mapsto V \otimes_{L_1} L_2 \in \text{Mod}(L_2)$.

This leads to many other examples coming from equipping vector spaces with extra structure: for example, $F(L)$ might be

• the category of representations of some finite group $G$ over $L$,
• the category of commutative (or associative, or Lie) algebras over $L$, or
• the category of schemes over $L$.

It would be great if understanding all of these categories was in some sense as simple as understanding the category $F(\bar{k})$, which generally tends to be simpler, and the action of the absolute Galois group $G$ on it, whatever that means. For example, the representation theory of finite groups over algebraically closed fields of characteristic zero is well understood, as is, say, the classification of semisimple Lie algebras. The general problem of trying to extract an understanding of $F(L)$ from an understanding of $F(\bar{k})$ is the problem of Galois descent.

We might very optimistically hope that the story here is directly analogous to the story above. This suggests the following puzzle.

Puzzle: In what sense could the statement $F(L) \cong F(\bar{k})^H$ be true for the examples given above?

### 7 Responses

1. […] addition, as mentioned in the first post in this series on Galois descent, this condition should also be regarded as a categorification of […]

2. […] to slightly correct our initial definition). Today, as the next step in our attempt to understand Galois descent, we’ll describe what the fixed points of such a group action […]

3. […] « The puzzle of Galois descent […]

4. In what sense is it true that the representations over $\overline{\mathbb{F}}_p$ of a finite group $G$ (such as $\text{SL}_n\mathbb{F}_p$, for example) are well-understood?

• Fine, I’ll tack on “characteristic zero” to that comment. Happy?

5. Shouldn’t the field be perfect? Otherwise you might have trouble with inseparable extensions…

• Hmm. Yeah, I guess I’m happy to make that restriction to be safe.