Quasi-compact maps in Number Theory Can someone give me an example of a non-quasi-compact morphism of schemes which arises naturally in the field of Algebraic Number Theory?
 A: Rex, about your scheme-theoretic image problem, I don't know how "contrived" the following example is (I am afraid it is not particularly related to number theory).
Notations: $R$  is a discrete valuation ring, $t$ a uniformizer, $R_n:=R/(t^{n+1})$ ($n\in\mathbb{N}$), $X_n=\mathrm{Spec}\,R_n$, $A=\prod_n R_n$.
Take $X:=\coprod_n X_n$ and $Y:=\mathrm{Spec}\,A$. There is a natural open immersion $f:X\to Y$ since each $X_n$ embeds in $Y$ as an open and closed subscheme.
The scheme-theoretic image of $f$ is $Y$: since $Y$ is affine, it just means that each $x\in A$ vanishing on each $X_n$ is zero, which is obvious. (In fact, $A=\Gamma(X,\mathcal{O}_X)$).
But $X$ is not topologically dense in $Y$: indeed, consider $x=(t,t,\dots)\in A$. Then $x$ is locally nilpotent on $X$ but not nilpotent on $Y$, hence the open set $D(x)\subset Y$ is nonempty and disjoint from $X$.
A: You might see non-quasi-compact maps in the context of universal covers of maximally degenerate pointed curves, and depending on who you ask, this might be called algebraic number theory.  Specifically, a maximally degenerate positive genus (proper) curve $X$ has the form of a connected graph made out of finitely many projective lines intersecting transversely, where each line has exactly 3 special points (namely intersections and markings).  If $x$ is a marked point, then $\pi_1^{geom}(X,x)$ is a finitely generated free group.  One then has a universal cover $(\tilde{X},\tilde{x}) \to (X,x)$, where $\tilde{X}$ is a tree of projective lines.  The covering map is étale but not quasi-compact.
Gerritzen and van der Put wrote a book (Schottky Groups and Mumford Curves, Springer LNM 817) describing some number-theoretic data like theta functions on these objects.  Some brief web searching suggests that there seem to be some more modern treatments using rigid analytic techniques (that I don't really understand).
I'm afraid this doesn't answer the more focused question about scheme-theoretic image that you posed in the comments.
A: 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.
