# Intersecting valuation rings in the Zariski-Riemann space of a field

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1 Intersecting valuation rings in the Zariski-Riemann space of a field Bruce Olberding Department of Mathematical Sciences New Mexico State University November, 2015

2 Motivation from Birational Algebra Problem: Find a framework for classifying/describing/studying integrally closed domains when viewed as intersections of valuation rings. Outline of talk: (a) The Zariski-Riemann space as a locally ringed spectral space (b) Affine subsets of the Zariski-Riemann space and Prüfer domains (c) Geometric criteria for Prüfer intersections (d) Patch topology and intersection representations (e) Quadratic transforms of regular local rings (f) Overrings of two-dimensional Noetherian domains

3 The Zariski-Riemann space Let F be a field and D be a subring of F (e.g., D is prime subring of F ). X = projective limit of the projective models of F /D. X is the Zariski-Riemann space of valuation rings of F containing D with the topology inherited from the projective limit Basis of the topology is given by sets of form U(x 1,..., x n ) = {V X : x 1,..., x n V }. X is quasicompact (Zariski, 1944). Why Zariski cared: finite resolving system can replace an infinite one. Theorem (Dobbs-Federer-Fontana, Heubo-Kwegna, Kuhlmann,...) X is a spectral space. Proof: X prime spectrum of the Kronecker function ring of F /D.

4 Why the name? Nagata, 1962: The name of Riemann is added because Zariski called this space Riemann manifold in the case of a projective variety, though this is not a Riemann manifold in the usual sense in differential geometry. The writer believes that the motivation for the terminology came from the case of a curve. Anyway, the notion has nearly nothing to do with Riemann, hence the name Zariski space is seemingly preferable. But, unfortunately, the term Zariski space has been used in a different meaning [e.g., a Noetherian topological space for which every nonempty closed irreducible subset has a unique generic point]. Therefore we are proposing the name Zariski-Riemann space.

5 A subset of X is qcpt and open iff it is a finite union of sets of form U(x 1,..., x n ) = {V X : x 1,..., x n V }. Inverse topology: closed basis of qcpt open sets...also called the dual topology. Patch topology: basis of qcpt opens and their complements...compact, Hausdorff and zero-dimensional. Every cpct Hausdorff zero-dim l space having an isolated point arises as the patch space of a Zariski-Riemann space (even an affine scheme).

6 Sheaf structures Intersection presheaf: O(U) := V U V (U open in X). Zariski topology O is a sheaf. Inverse or patch topologies: must sheafify. Zariski topology: Main virtue: compatible with morphisms into schemes. Patch topology: Result is a Pierce sheaf: ringed Stone space with indecomposable stalks. O(X) is a complicated ring with many idempotents. The geometry here is really algebra is disguise: The global sections functor is exact, hence cohomologically trivial. Inverse topology: Ring of global sections can contain idempotents. Disadvantage: Stalks need not be valuation rings (but are a kind of pullback ring).

7 Affine subsets of X X = Zariski-Riemann space of F /D with the Zariski topology. Non-degenerate case: When is Z Spec( V Z V ) an isomorphism? I.e., which subspaces of X are affine schemes? Proposition. Z X is an affine scheme iff Z is inverse closed and A = V Z V is a Prüfer domain with q.f. F. An integral domain A is a Prüfer domain if A M is a valuation domain for each maximal ideal M of A. So to detect when an intersection of valuation rings is Prüfer is the same as detecting when a subspace of X is an affine scheme.

8 Prüfer domains Prüfer domains are a fundamental object of study in non-noetherian commutative ring theory and multiplicative ideal theory. 100 characterizations (ideal-theoretic, module-theoretic, homological) Examples: Finite intersection of valuation rings Ring of entire functions The ring of integer-valued polynomials Int(Z) Real holomorphy rings

9 Criterion for affiness D = subring of the field F. Recall: A = V Z V is a Prüfer domain iff Z affine scheme in X Theorem. (O., 2014) A = V Z V is Prüfer image of each D-morphism Z P1 D with quotient field F = is in a distinguished affine open and torsion Picard group subset of P 1 D P 1 D is covered by many affine open subsets. What conditions guarantee Z P 1 D lands in one of them?

10 Applications Three classical independent results about Prüfer intersections can now be reduced to prime avoidance arguments... Corollary. (Nagata) A = V Z V is Prüfer when Z is finite. Proof. Let φ : Z P 1 D be a morphism. Its image is finite. Prime Avoidance f D[T 0, T 1 ] not in any prime ideal in Im φ. {P P 1 D : f P} is an affine open set containing Im φ. So by the theorem, A is Prüfer. (In fact, f can be chosen to be linear and this implies that A is Bézout.)

11 Corollary. (Dress, Gilmer, Loper, Roquette, Rush) A = V Z V is Prüfer when there exists a nonconstant monic polynomial f A[T ] that has no root in residue field of any V Z. Proof. Let φ : Z P 1 D be a morphism. Let f be the homogenization of f. Then {P P 1 D : f P} is an affine open set containing Im φ. So by the theorem, A is Prüfer with torsion Picard group. Example: Use f (X ) = X to show the real holomorphy ring is Prüfer.

12 Corollary. (Roitman) A = V Z V is Bézout when A contains a field of cardinality > Z. Proof. Let φ : Z P 1 D be a morphism. Use the fact that there are more units in A than valuation rings in Z to construct a homogeneous f D[T 0, T 1 ] that is not contained in any prime ideal in the image of φ. Then {P P 1 D : f P} is an affine open set containing Im φ. So by the theorem, A is Prüfer. (In fact, f can be chosen to be linear, so A is a Bézout domain.)

13 Local uniformization Corollary D = quasi-excellent local Noetherian domain with quotient field F. Z = valuation rings dominating D that don t admit local uniformization. If Z, then V Z V is a Prüfer domain with torsion Picard group. So if nonempty, Z lies in an affine scheme in X.

14 Patch topology Patch topology: qcpt opens and their complements as a basis Conrad-Temkin, Favre-Jonsson, Finnocchiaro-Fontana-Loper, Huber-Knebusch, Knaf-Kuhlmann, Kuhlmann, O.-, Prestel-Schwartz,... Patch density: useful for replacing a valuation with a better one that behaves the same on a finite set of data Suppose Z is patch dense in X and V X. x 1,..., x n V, y 1,..., y m M V = W Z with same property F.-V. Kuhlmann has proved a number of deep theorems for patch density in the space of valuations on a function field. Patch density is useful for understanding the ideal theory of real holomorphy rings (O-, 2005).

15 Patch limit points Theorem. (Finnocchiaro, Fontana, Loper, 2013) V is a patch limit point of Z X iff V = {x F : {V Z : x V } F}, where F is a nonprincipal ultrafilter on Z. Theorem (O-, 2014) Suppose Spec(D) is a Noetherian space. V {patch closure of Z} iff in every projective model X of F /D, V maps to a generic point for a subset of the image of Z in X. So patch limit points arise from generic points in the projective models.

16 Example: Accounting for all valuations A non-constructive construction of all valuations in a function field... Theorem (Kuhlmann, 2004) F /k = function field. The set of DVRs in X whose residue fields are finite over k is patch dense. = every valuation ring in X is an ultrafilter limit of such DVRs. These DVRs arise from prime ideals in the generic formal fiber of local rings of closed points in projective models of (Heinzer-Rotthaus-Sally, 1993). Taking completions then taking ultrafilter limits give all valuations.

17 Application: Intersection representations A subset Z of X represents a ring R if R = V Z V. Theorem (O-, 2015) Every patch closed rep. of a ring contains a minimal patch closed rep. isolated point irredundant member of the representation (= consequences for uniqueness and existence of representations). Theorem (A, M) = integrally closed local domain with End(M) = A. = A is a val n domain or perfect dominating representation of A. Corollary A = completely integrally closed local domain dominating rep n with countably many limit points = valuation ring.

18 More Prüfer criteria Theorem (D, m) = local subring of F that is not a field. Z = set of dominating, rank 1 valuations rings. {limit points of Z} < ℵ 0 D/m = V is a Prüfer domain. Corollary. {limit points of Z} = finite = Application: Order holomorphy ring V Z V Z V is a Prüfer domain. Let R = RLR, X = blow-up of Spec(R) at the maximal ideal. V x = order valuation ring of O X,x (x = closed point) = x V x is a Prüfer domain. (What information does it contain?)

19 Application: Quadratic Transforms Theorem. (Heinzer, Loper, O-, Schoutens, Toenskoetter) R = RLR of dimension > 1; {R i } = sequence of local quadratic transforms. = i R i = V T, T = smallest Noetherian overring (it s a localization of one of the R i ) V = unique patch limit point of the order valuation rings of the R i s. Theorem. (Heinzer, O.-, Toenskoetter) an explicit asymptotic description of V (in particular, rank V = 2). Application of Prüfer criterion: V is a localization of the intersection of the order valuation rings. Reason: The intersection of the order valuation rings is a Prüfer domain.

20 Application: Overrings of two-dim l Noetherian domains Suppose D is a two-dimensional Noetherian domain with q.f. F. Goal: Describe the integrally closed rings V Z V between D and F. Special case: morphism Z P 1 D with small fibers. This is in keeping with the philosophy of understanding intersections of valuation rings when there are not too many of them. Theorem. Suppose morphism Z P 1 D with Noetherian fibers. Then (1) unique strongly irredundant representation of O(Z) = V Z V. (2) local classification of the ring O(Z) (somewhat involved). Note: This includes all integrally closed Noetherian domains.

21 Application to Rees valuations Theorem (O-, Tartarone, 2013) Suppose D = two-dimensional regular local ring with regular parameter f. D is equicharacteristic or has mixed characteristic with f a prime integer. A = integral closure of a finitely generated D-subalgebra of D f. Then distinct height one prime ideals of A lying over the maximal ideal of D are comaximal. Thus A is essentially one-fibered. Question: If f is not a regular parameter, is there a bound on the number of height one primes of A lying over the maximal ideal of D and contained in the same maximal ideal of A? I.e., is each A essentially n-fibered? Yes nice consequences for not-necessarily-noetherian overrings of D.

22 Thank you

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