Also derived functors have a rather simple description in the model category setting: So suppose you have a model category $C$, then you can define the associated homotopy category $\operatorname{Ho}(C)$ together with a canonical functor $C \to \operatorname{Ho}(C)$, then for a functor $F: C \to D$ to any category $D$, a derivied functor is just a functor $\operatorname{Ho}(C) \to D$, such that the obvious triangle with $C$, $D$ and $\operatorname{Ho}(C)$ to commutes
which is "universal" with this property (formally, a Kan extension)

(Beware: abstract algebra: if you have a local homorphism between local commutative rings $f:R \to S$, then you say that $f$ is unramified if the induced map $R/\mathfrak{m} \to S/\mathfrak{m}S$ is a finite separable field extension.
So when you have a morphism of locally ringed spaces $(X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ is ramified at $x \in X$, if the induced map on stalks $\mathcal{O}_{Y,f(x)} \to \mathcal{O}_{X,x}$ is unramified.)

The point is, if you have a holomorphic map between Riemann surfaces, then this induces a map of locally ringed spaces where you take the sheaf of ho…

random analogies between Riemann surfaces and ANT: the fact that $\Bbb P^1(\Bbb C)$ is simply connected, so it has no unramified cover, is the analog of the classical result due to Minkowski that $\Bbb Q$ has no unramified extensions which is also equivalent to the fact that $\pi_1^{ét}(\operatorname{Spec}(\Bbb Z))$ is trivial.
You can also phrase that in a very elementary way: take any irreducible monic polynomial in $\Bbb Z[x]$. Then there exists at least one prime $p$ such that $f$ has a multiple root mod $p$

@BalarkaSen sorry, if I'm annoying you with my ANT/ramification stuff, you can just say it.
There's another way to characterize ramification in the Galois/normal setting.
Suppose $L/K$ is a Galois extension of number fields (you can actually show that this is equivalent to the the action of $\operatorname{Aut}_K(L)$ on the fibers of the induced map $p:\operatorname{Spec}(\mathcal{O}_L) \to \operatorname{Spec}(\mathcal{O}_K)$ being transitive, just like for covering spaces)
Then for every $\mathfrak{p} \in \operatorname{Spec}(\mathcal{O}_K)$, we have a (transitive) action of $\operatorname{G…

The map $\mathcal{O}_K/\mathfrak{p} \to \mathcal{O}_L/\mathfrak{q}$ is "almost" the map from the definition of ramification:
When we take the map that tells us about ramification, i.e. $(\mathcal{O}_K)_{\mathfrak{p}}/\mathfrak{p}(\mathcal{O}_K)_{\mathfrak{p}} \to (\mathcal{O}_L)_{\mathfrak{q}}/\mathfrak{p}(\mathcal{O}_L)_{\mathfrak{q}}$ and we quotient out all the nilpotents, then the residue field extension $\mathcal{O}_K/\mathfrak{p} \to \mathcal{O}_L/\mathfrak{q}$ is what we get.
So I guess in some sense this difference between ANT and Riemann surfaces, geometrically speaking is that whe…

If $K$ is a number field and $L$ is the Hilbert class field, then we have an exact sequence
$ 1 \to (\mathcal{O}_K)^\times \to K^\times \to \operatorname{I}(K) \to \operatorname{Gal}(L/K) \to 1$, where $\operatorname{I}(K)$ is the group of fractional ideals in $K$. The last map is the Artin map from class field theory.

I found an exact sequence in these notes, that looks reall similar:

$0 \to \Bbb C^\times \to \mathcal{M}^*(X) \to \operatorname{Div}_0(X) \to \operatorname{Jac}(X) \to 0$, where $X$ is a compact Riemann surface