Let X be an integral scheme that is separated (say over an affine scheme). Define a Weil divisor as a finite integral combination of height 1 points of X, where the height of a point of X is the dimension of its local ring. Let Z be a closed subscheme of X, Z not equal to X. 


Is there an example of such an X and Z where Z contains infinitely many Weil divisors?



Weil divisors on non Noetherian schemes
Yes, there are such examples (with *X* quasi-compact, otherwise it is trivial). There is a general result due to Hochster ("Prime Ideal structures in commutative rings", his thesis) which says that spectra of rings are exactly those topological spaces *X* such that:


1) *X* is Kolmogorov (*T*\_0).


2) *X* is quasi-compact.


3) The quasi-compact open subsets form an open basis.


4) *X* is quasi-separated, i.e., quasi-compact open subsets are closed under finite intersections.


5) Every non-empty irreducible closed subset has a generic point.


We will construct an irreducible topological space *X* satisfying 1-5 with underlying set 2^**N** U {*x*} such that 2^**N** is closed in *X* and totally disconnected and *X* has generic point *x*. If *X*=Spec(*A*), then the closed subvariety 2^**N** is a union of infinitely many Weil divisors. (*X* has dimension 1)


The topology on 2^**N** will be the product topology, i.e., that of the 2-adic integers (or if you prefer, the Cantor set). An open basis for this topology are cylinders, i.e., sets where a finite number of components are fixed. The cylinders are also closed. The space 2^**N** is Hausdorff, compact and totally disconnected, hence satisfies 1-5.


An open basis for the topology of *X* = 2^**N** U {*x*} is given by sets of the form *W* U {*x*} where *W* is a cylinder or the empty set. The quasi-compact open subsets of *X* are the finite unions of such sets and *X* satisfies 1-4. To see 2-4 note that if *V* is an open subset of *X* then


*V* is quasi-compact <=> The intersection of *V* and 2^**N** is clopen


Finally, the non-empty irreducible closed subsets of *X* are the singleton sets of 2^**N** and *X* itself and these all admit generic points so *X* satisfies 5.


**Remark**: *X* seems to be closely related to the spectrum of the Tate-algebra **Q**\_p<*x*>. The *canonical* topology on the closed points are exactly the *p*-adic integers. But the *Zariski* topology is completely different (**Q**\_p<*x*> is noetherian and regular of dimension 1).


