Grothendieck says: points are not mere points, but carry Galois group actions

Suppose $k$ is a field, not necessarily algebraically closed. $\text{Spec } k$ fails to behave like a point in many respects. Most basically, its "finite covers" (Specs of finite etale $k$-algebras) can be interesting, and are controlled by its absolute Galois group / etale fundamental group. For example, $\text{Spec } \mathbb{F}_q$, the Spec of a finite field, has the same finite covering theory as $S^1$, which reflects (and is equivalent to) the fact that its absolute Galois group is the profinite integers $\widehat{\mathbb{Z}}$. (So this suggests that one can think of $\text{Spec } \mathbb{F}_q$ itself as behaving like a "profinite circle.")

More generally, suppose you want to classify objects of some kind over $k$ (say, vector spaces, algebras, commutative algebras, Lie algebras, schemes, etc.). A standard way to do this is to instead classify the base changes of those objects to the separable closure $k_s$, then apply Galois descent. The topological picture is that $\text{Spec } k$ behaves like $BG$ where $G$ is the absolute Galois group, $\text{Spec } k_s$ behaves like a point, or if you prefer like $EG$, and the map

$$\text{Spec } k_s \to \text{Spec } k$$

behaves like the map $EG \to BG$. In the topological setting, families of objects over $BG$ are (when descent holds) the same thing as objects equipped with an action of $G$. The analogous fact in algebraic geometry is that objects over $\text{Spec } k$ are (when Galois descent holds) the same thing as objects over $\text{Spec } k_s$ equipped with homotopy fixed point data, which is a generalization of being equipped with a $G$-action which reflects the fact that $k_s$ itself has a $G$-action.

(I need to be a bit careful here about what I mean by "$G$-action" to take into account the fact that $G$ is a profinite group. For simplicity you can pretend that I am instead talking about a finite extension $k \to L$, although I'll continue to write as if I'm talking about the separable closure. Alternatively, pretend I'm talking about $k = \mathbb{R}, k_s = \mathbb{C}$.)

The classification of finite covers is the simplest place to see this: the category of finite covers of $\text{Spec } k_s$ is the category of finite sets, with the trivial $G$-action, so homotopy fixed point data is the data of an action of $G$, and we get that finite covers of $\text{Spec } k$ are classified by finite sets with $G$-action.

But Galois descent holds in much greater generality, and describes a very general sense in which objects over $k$ behave like objects over $k_s$ with a Galois action in a twisted sense.

The points of a Topos have natural transformations between them; restricting to natural isomorphisms you get a groupoid. You can also represent the points of a bounded Topos as principal bundles; I.e. Something with a G -action. Not sure which is being referred to, but you can see this aspect of points without having any rings around.

Question: "I'm trying to slowly digest the last paragraph. As a novice in algebraic geometry I'm always looking for geometric and "philosophical" intuition, so I very much want to understand why Grothendieck was insistent on points having Galois group actions. Why, geometrically (or philosophically?) is it essential and important that points have Galois group actions?"

Example 1: If $k:=\mathbb{R}$ is the field of real numbers and $K:=\mathbb{C}$ the field of complex numbers and $A:=k[x]$ it follows $A$ has maximal ideals on the form $(x-r)$ with $r\in k$, or $(p_z(x))$ where

$$p_z(x):=(x-z)(x-\overline{z})=x^2-2ax+a^2+b^2$$

with $z:=a+ib$ where $b\neq 0$. If $X:=Spec(A)$ is the affine spectrum of $A$ and $f\in \Gamma(X,\mathcal{O}_X)$ is a global section, we may view $f$ as a "function" on the topological space $X$ as follows: Given $x\in X$ we get canonically an element $f(x)\in \kappa(x)$: There is an evaluation map

$$ ev_x: A \rightarrow \kappa(x):=A_{\mathfrak{p}_x}/\mathfrak{p}_x A_{\mathfrak{p}_x}$$

where $\mathfrak{p}_x$ is the prime ideal of $x$, and you define $f(x):=ev_x(f)$. If you want a complex number $f(x)\in K$ (and a complex valued function $f(x)$ in the classical sense) you must choose an explicit isomorphism

$$ \phi_x :\kappa(x)\cong k[x]/(p_z(x)) \cong K$$

for every maximal ideal $(p_z(x))$ corresponding to a non-real complex number $z$. An isomorphism $\phi_x: \kappa(x) \rightarrow K$ has $\phi(x):=z$ or $\phi(x):=\overline{z}$. Hence the isomorphism $\phi_x$ depends on the choice of a root of $p_z(x)$. In this case you may make a consistent choice and choose $z:=a+ib$ satisfying $b>0$ for all $x$ with $\mathfrak{p}_x=(p_z(x))$ and $\kappa(x)\cong K$. With this choice you do get for any global section $f$ a function

$$ f: X \rightarrow K.$$

Hence with this construction a global section $f$ gives rise to a function in the classical sense: It is a map $f: X \rightarrow K$ where $K$ is a "fixed set". Again there are problems involved since everyting is defined "up to isomorphism" and you have for every point $(p_z(x))$ chosen an isomorphism

$$ \phi_x:\kappa(x) \cong K.$$

If you read about this you will find that people that are working with computer algebra calculations involving algebraic geometry encounter such problems.

There is a Galois group involved here: For any point $x\in X$ there is the group $G(x):=G(\kappa(x)/k)$.

"Why is it essential and important that points have Galois group actions?"

It is a fact that for any scheme of finite type over a non-algebraically closed field it follows the residue field $\kappa(x)$ of a closed point is a finite extension of the base field, and hence the group $G(\kappa(x)/x)$ is defined, non-trivial in general and it appears in various formulas.

Example 2. If $K \subseteq L$ are number fields with rings of integers $\mathcal{O}_K, \mathcal{O}_L$ there is for any closed point $\mathfrak{m}\subseteq \mathcal{O}_L$ an "inverse image" ideal $\mathfrak{n}:=\mathfrak{m}\cap \mathcal{O}_K$ and the field extension

$$ \kappa(\mathfrak{n})\subseteq \kappa(\mathfrak{m})$$

is a (non-trivial) extension of finite fields, hence you get the (non-trivial) Galois group $G:=G(\kappa(\mathfrak{m})/\kappa(\mathfrak{n})).$ The book "Algebraic number theory" by Neukirch relates the group $G$ to properties of the field extension $K\subseteq L$ and L-functions.

Example 3. If $f: X\rightarrow Y$ is a map of algebraic varieties of finite type over a non-algebraically closed field $k$ with $x\in X$ a closed point and $y:=f(x)$, it follows there will be an induced field extension

$$ k \subseteq \kappa(y) \subseteq \kappa(x)$$

and the extension $\kappa(y) \subseteq \kappa(x)$ will be finite. Hence there is an induced Galois group $G(\kappa(x)/\kappa(y))$ which is a finite group. For an arbitrary point $x\in X$ with $y:=f(x)$ you still get an extension

$$ \kappa(y) \subseteq \kappa(x) $$

but $\kappa(x)$ and $\kappa(y)$ will be transcendental extensions of $k$ in general. The Galois group $G(\kappa(x)/\kappa(y))$ is still defined but it is larger and not finite in general. But this does not stop people from studying it.

Example 4. If $k$ is a field and $X:=\mathbb{P}^n_k$ is projective $n$-space over $k$ with structure map $\pi: X \rightarrow Spec(k)$ and $\eta\in X$ is the generic point, it follows the induced Galois group

$$G:=G(\kappa(\eta)/k)$$

is the group of $k$-automorphisms of the field $k(x_1,..,x_n)$ (the Cremona group). Not much is known about this group.