This is a very quick introduction to Galois descent for schemes defined over fields. It is a very special case of faithfully flat descent and other topos-descent theorems, which I won't go into at all. Typically, if you look up descent in an algebraic geometry text you will quickly run into all sorts of diagrams and descent data. In my opinion, that is a very counterintuitive way to present the basic idea.

## What is the descent theorem?

Here is the main topic of this post:

**Theorem.**Let $E/F$ be a finite Galois extension of fields with Galois group $G$. Then the functor

$$\begin{align*}

\{\text{quasprojective $F$-schemes}\}&\longrightarrow \{\text{quasprojective $E$ schemes with compatible $G$ action} \} \\

X&\longmapsto X\otimes_F E\end{align*}$$

where $X\otimes_F E$ is given an Galois action via the canonical action on $E$, is an equivalence of categories.

This is the basic theorem of Galois descent. What does it mean, and how does it work? First, I have to tell you what a compatible Galois action is. Well, if $X$ is an $E$-scheme, then there is a map $X\to{\rm Spec}(E)$, and there is the usual action of $G$ on ${\rm Spec}(E)$. Compatible just means that for each $\sigma\in G$, the square

commutes.

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