# Ideal class groups and extension of number fields

Let $(X, \mathcal{O}_X)$ and $(Y, \mathcal{O}_Y)$ be schemes and $f: X \to Y$ a morphism. By restricting the structure morphism, we get a morphism of sheaves $\mathcal{O}_Y^* \to f_*\mathcal{O}_X^*$. Under certain favorable conditions, this is an injection (1) and we can complete to an exact sequence $1 \to \mathcal{O}_Y^* \to f_*\mathcal{O}_X^* \to \mathcal{Q} \to 1$. Here $\mathcal{Q}$ denotes the quotient sheaf. From this we get a long exact sequence in cohomology which starts as

$1 \to H^0(\mathcal{O}_Y^*) \to H^0(f_*\mathcal{O}_X^*) \to H^0(\mathcal{Q}) \to H^1(\mathcal{O}_Y^*) \to H^1(f_*\mathcal{O}_X^*) \to H^1(\mathcal{Q}).$

Here we know the first two terms, and $H^1(\mathcal{O}_Y^*) = Pic(Y)$. Under certain favorable circumstances, moreover:

• (2) $H^1(f_*\mathcal{O}_X^*) = Pic(X)$
• (3) $H^1(\mathcal{Q}) = 0$
• (4) $\mathcal{Q}$ and in particular $Q=H^0(\mathcal{Q})$ have a fairly explicit description.

In this case we end up with an interesting exact sequence relating the Picard and unit groups of $X$ and $Y$ via $Q$. This happens for example if $Y$ is the spectrum of a Dedekind domain and $X$ is an open subset or if $Y$ is the spectrum of a one-dimensional integral domain, $X$ its normalisation, and some finiteness conditions hold. These two cases are worked out "by hand" in Neukirch, algebraic number theory, sections 1.11 and 1.12.

Now is probably the time to admit that my knowledge of algebraic geometry (and also algebraic number theory) is rather tiny. In full generality, my question would be: what can be said about the "favorable conditions" that make (1), (2) and (3) hold.

Since this seems a bit excessive, I would like to consider the case where $L/K$ is an extension of number fields and $X$, $Y$ are the spectra of the respective rings of integers. Then (1) obviously holds. I would be interested to know if (or when) (2) holds, or if not how $H^1(f_*\mathcal{O}_L^*)$ and $Pic(X)=Cl(\mathcal{O}_L)$ are related. Finally I would like to know if we can give a good description of $\mathcal{Q}$.

[I know this is more than one question but I did not see a way to meaningfully separate them, except perhaps by removing all references to (4) and hence most of my motivation. It also seems to me that this is a fairly natural question so that answers should be available; however I could not find anything.]

-