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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?

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up vote 12 down vote accepted

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).

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Let R be the integral closure of $\mathbb{Z}$ in the algebraic closure of $\mathbb{Q}$. Then $\dim (R)=1$ and there are infinitely many prime ideals lying above a prime ideal $p\mathbb{Z}$. Hence the closed set $V(pR)$ has the required property.

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Take X to be A^\infty and Z to be the union of the coordinate hyperplanes.

Edit: Jonathan pointed out to me that the infinite union of hyperplanes in A^\infty is not a closed subscheme: any closed subscheme is contained in the pre-image of a closed subscheme from A^n for some n. So I withdraw my example.

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