Can someone give me an example of a non-quasi-compact morphism of schemes which arises naturally in the field of Algebraic Number Theory?



Quasi-compact maps in Number Theory
A typical non-Noetherian ring that would arise in algebraic number theory would be the
ring $\mathbb Z\_p \otimes\_{\mathbb Z\_{(p)}} \mathbb Z\_p$, where I am writing $\mathbb Z\_{(p)}$ to denote the localization of $\mathbb Z$ at the prime ideal $(p)$, and $\mathbb Z\_p$ to denote its completion (the usual ring of $p$-adic integers).


Such tensor products come up in considerations related to faithfully flat descent, and can arise for example in justfiying (in certain situations) the passage from working over the localization $\mathbb Z\_{(p)}$ to its completion $\mathbb Z\_p$. (The kind of thing I have in mind is studying finite flat group schemes over $\mathbb Z$, say, by working over $\mathbb Z[1/p]$ and $\mathbb Z\_p$ separately. It is not hard to justify working over
$\mathbb Z[1/p]$ and $\mathbb Z\_{(p)}$ separately, but to justify the replacement of
$\mathbb Z\_{(p)}$ by $\mathbb Z\_p$, one needs to make (or at least, might naturally find oneself making) a descent argument,
in which the tensor product written above could play a role.)


Another (perhaps simpler) example of a non-Noetherian ring that naturally appears in algebraic number theory is the ring of all algebraic integers.


