[I already posted this question on stackexchange a while ago, but did not get any response: http://math.stackexchange.com/questions/93437/ideal-class-groups-and-extension-of-number-fields]


Let $(X, \mathcal{O}\_X)$, $(Y, \mathcal{O}\_Y)$ be schemes and $ f:X \to Y$ be a morphism. Suppose $f^\#:\mathcal{O}\_Y \to f\_\*\mathcal{O}\_X $is injective. Then so is the restriction $\mathcal{O}\_Y^\* \to f\_\*\mathcal{O}\_X^\*$ and we can complete to an exact sequence $1 \to \mathcal{O}\_Y^\* \to f\_\*\mathcal{O}\_X^\* \to \mathcal{Q} \to 1$. From this we get an exact sequence in cohomology starting 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}) $.


**Question(s)**


Consider the case where $L/K$ is an extension of number fields with rings of integers $\mathcal{O}\_L$, $\mathcal{O}\_K$, $X = Spec(\mathcal{O}\_L)$, $Y = Spec(\mathcal{O}\_K)$ and $f$ induced from the inclusion $\mathcal{O}\_K \to \mathcal{O}\_L$.


1. How do $H^1(f\_\*\mathcal{O}\_X^\*)$ and $H^1(\mathcal{O}\_X^\*) = Pic(X)$ relate? (In particular, are they equal?)
2. Can we describe $\mathcal{Q}$ explicitely? In particular, is there a good expression for $Q = H^0(\mathcal{Q})$? Does $H^1(\mathcal{Q}) = 0$?.


**Examples/Motivation**


The motivation for considering the above is to end up with an interesting exact sequence relating the Picard and unit groups of $X$ and $Y$ via $Q$. Nice results are obtainable 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.


**A start on question 1**


This is probably not the most elegant method, but the five-term exact sequence from the Leray spectral sequence starts as $ 1 \to H^1(f\_\*\mathcal{O}\_X^\*) \to H^1(\mathcal{O}\_X) \to \Gamma(Y, R^1f\_\*\mathcal{O}\_X^\*) $. A sufficient condition for $ Pic(X) = H^1(f\_\*\mathcal{O}\_X^\*) $ is thus that the stalks of the first higher direct image are trivial, which is easily seen to be equivalent to $Pic((f\_\*\mathcal{O}\_X)\_p) = 0$ for all $p \in Y$. This does hold for an open immersion of number rings, but it is not clear to me if it applies to a normalisation or to an extension of number fields.


It is also easy to see that $H^1(\mathcal{F}) = H^1(f\_\* \mathcal{F})$ for all $\mathcal{F}$ if $f\_\*$ is exact, for example a closed immersion. This does not seem applicable here either, though.



Ideal class groups and extension of number fields
if your aim is really to relate $Pic(Y)$ to $Pic (X)$ it is probably a good idea to pybass the use of $\mathcal Q$ and to consider instead the more powerful technique of introducing the relative Picard scheme (functor) $Pic\_{X/Y}$. There are plenty of references, among them :


Néron models.


Bosch; Lütkebohmert ; Raynaud, 


Ergebnisse der Mathematik und ihrer Grenzgebiete Springer-Verlag, Berlin, 1990.


for instance :


chapter 8 page 204 proposition 4
if $f:X\to Y$ is quasi-compact and quasi-separated and $f\_\*(\mathcal O\_X)=\mathcal O\_Y$ then there is an exact sequence :


$$ 0 \to Pic (Y) \to Pic (X) \to Pic\_{X/Y}(Y) \to Br(Y) \to Br(X)$$ 


If you are interested, the standard is FGA (Grothendieck's Fondations de la Géométrie Algébrique), and the modern version in


The Picard scheme 


Steven L. Kleiman


<http://arxiv.org/abs/math/0504020>


