arXiv subjects
Publications and source records attributed to William Heinzer.
Interval rings comprise a class of one-dimensional integrally closed local integral domains which are overrings of a two-dimensional regular local ring. We construct them by intersecting carefully chosen valuation rings and prove that they have various nice properties. Our interest in these rings is that they represent a major stepping stone toward classifying all integrally closed overrings of a two-dimensional regular local ring.
Let $D$ be a two-dimensional regular local ring. We prove there is a one-to-one correspondence between closed connected sets in the space of valuation overrings of $D$ that dominate $D$ and the integrally closed local overrings of $D$ that are not essential valuation rings or divisorial valuation rings of $D$.
Let $F$ be a field, let $D$ be a local subring of $F$, and let Val$_F(D)$ be the space of valuation rings of $F$ that dominate $D$. We lift Zariski's connectedness theorem for fibers of a projective morphism to the Zariski-Riemann space of valuation rings of $F$ by proving that a subring $R$ of $F$ dominating $D$ is local, residually algebraic over $D$ and integrally closed in $F$ if and only if there is a closed and connected subspace $Z$ of Val$_F(D)$ such that $R$ is the intersection of the rings in $Z$. Consequently, the intersection of the rings in any closed and connected subset of Val$_F(D)$ is a local ring. In proving this, we also prove a converse to Zariski's connectedness theorem. Our results do not require the rings involved to be Noetherian.
Let $R$ be a normal Noetherian local domain of Krull dimension two. We examine intersections of rank one discrete valuation rings that birationally dominate $R$. We restrict to the class of prime divisors that dominate $R$ and show that if a collection of such prime divisors is taken below a certain ``level,'' then the intersection is an almost Dedekind domain having the property that every nonzero ideal can be represented uniquely as an irredundant intersection of powers of maximal ideals.
Let $D$ be a 2-dimensional regular local ring and let $Q(D)$ denote the quadratic tree of 2-dimensional regular local overrings of $D$. We explore the topology of the tree $Q(D)$ and the family ${\mathcal{R}}(D)$ of rings obtained as intersections of rings in $Q(D)$. If $A$ is a finite intersection of rings in $ Q(D)$, then $A$ is Noetherian and the structure of $A$ is well understood. However, other rings in ${\mathcal{R}}(D)$ need not be Noetherian. The two main goals of this paper are to examine topological properties of the quadratic tree $Q(D)$, and to examine the structure of rings in the set ${\mathcal{R}}(D)$.
Let $D$ be a 2-dimensional regular local ring and let $Q(D)$ denote the quadratic tree of 2-dimensional regular local overrings of $D$. We examine the Noetherian rings that are intersections of rings in $Q(D)$. To do so, we describe the desingularization of projective models over $D$ both algebraically in terms of the saturation of complete ideals and order-theoretically in terms of the quadratic tree $Q(D)$.
We discuss the work of Abhyankar on dicritical divisors with a special focus on the algebraic aspects of this work. We also discuss related work on local quadratic transforms, infinitely near points and Rees valuation rings of an ideal.
Let $\{ R_n, {\mathfrak m}_n \}_{n \ge 0}$ be an infinite sequence of regular local rings with $R_{n+1}$ birationally dominating $R_n$ and ${\mathfrak m}_nR_{n+1}$ a principal ideal of $R_{n+1}$ for each $n$. We examine properties of the integrally closed local domain $S = \bigcup_{n \ge 0}R_n$.
Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). We consider singularities of the normalization of the blow-up Proj R[It] of I. A theorem of Lipman implies that the ideal I has a unique factorization as a star-product of special star-simple complete ideals with possibly negative exponents for some of the factors. If the normalization of the projective model Proj R[It] is regular, we prove that it is the regular model obtained by blowing up the finite set of base points of I. Extending work of Lipman and Huneke-Sally in dimension 2, we prove that every local ring S on the normalization of Proj R[It] that is a unique factorization domain is regular. Moreover, if dim S is at least 2 and S dominates R, then S is an infinitely near point to R, that is, S is obtained from R by a finite sequence of local quadratic transforms.
We consider the directed union S of an infinite sequence {(R_n, m_n)} of successive local quadratic transforms of a regular local ring (R, m). If dim R = 2, Abhyankar proves that S is a valuation ring. If dim R > 2, Shannon gives necessary and sufficient conditions for S to be a rank 1 valuation domain and Granja gives necessary and sufficient conditions that S be a rank 2 rational rank 2 valuation domain. Granja observes that these are the only cases where S is a valuation domain. If the sequence is along a rank 1 valuation ring V with valuation v, Granja, Martinez, and Rodriguez show that if the infinite sum of the values v(m_n) diverges, then S = V. We prove that this infinite sum is finite if V has rational rank at least 2. We present an example of a sequence whose union S is a rank 2 valuation domain, but whose value group is not Z^2. We also consider sequences of monomial local quadratic transforms and give necessary and sufficient conditions that the union be a rank 1 valuation domain. If it is, it has rational rank d. We string together finite sequences of monomial local quadratic transforms to construct examples where S is a rank 1 valuation domain with rational rank < d.
We consider infinite sequences {R_n} of successive local quadratic transforms of a regular local ring. Let S denote the directed union of the sequence of regular local rings R_n. We previously showed the existence of a unique limit point V of the family of order valuation rings of the sequence. In this paper, we examine asymptotic properties of this family of order valuations. We link this asymptotic behavior to ring-theoretic properties of S, namely whether S is archimedean and whether S is completely integrally closed. We construct examples of such S that are archimedean and completely integrally closed but not valuation domains. We give an explicit description of V, where the description depends on whether S is archimedean or non-archimedean.
Let R* be an ideal-adic completion of a Noetherian integral domain R and let L be a subfield of the total quotient ring of R* such that L contains R. Let A denote the intersection of L with R*. The integral domain A sometimes inherits nice properties from R* such as the Noetherian property. For certain fields L it is possible to approximate A using a localzation B of a nested union of polynomial rings over R associated to A; if B is Noetherian, then B = A. If B is not Noetherian, we can sometimes identify the prime ideals of B that are not finitely generated. We have obtained in this way, for each positive integer s, a 3-dimensional local unique factorization domain B such that the maximal ideal of B is 2-generated, B has precisely s prime ideals of height 2, each prime ideal of B of height 2 is not finitely generated and all the other prime ideals of B are finitely generated. We examine the map Spec A to Spec B for this example. We also present a generalization of this example to dimension 4. We describe a 4-dimensional local non-Noetherian UFD B such that the maximal ideal of B is 3-generated, there exists precisely one prime ideal Q of B of height 3, the prime ideal Q is not finitely generated. We consider the question of whether Q is the only prime ideal of B that is not finitely generated, but have not answered this question.
Let (R,m) be a Noetherian local domain of dimension n that is essentially finitely generated over a field and let R^ denote the m-adic completion of R. Matsumura has shown that n-1 is the maximal height possible for prime ideals of R^ in the generic formal fiber of R. In this article we prove that every prime ideal of R^ that is maximal in the generic formal fiber of R has height n-1. We also present a related result concerning the generic formal fibers of certain extensions of mixed polynomial-power series rings.
We consider properties of extensions of Krull domains such as flatness that involve behavior of extensions and contractions of prime ideals. Let (R,m) be an excellent normal local domain with field of fractions K, let y be a nonzero element in m, and let R* denote the (y)-adic completion of R. For a finite set w of elements of yR* that are algebraically independent over R, we construct two Krull domains: an intersection domain A that is the intersection of R* with the field of fractions of K[w], and an approximation domain B to A. If R is countable with dim R at least 2, we prove that there exist sets w as above such that the extension R[w] to R*[1/y] is flat. In this case B = A is Noetherian, but may fail to be excellent as we demonstrate with examples. We present several theorems involving the construction. These theorems yield examples where B is properly contained in A and A is Noetherian while B is not Noetherian, and other examples where B = A is not Noetherian.
Let I be a complete m-primary ideal of a regular local ring (R,m). In the case where R has dimension two, the beautiful theory developed by Zariski implies that I factors uniquely as a product of powers of simple complete ideals and each of the simple complete factors of I has a unique Rees valuation. In the higher dimensional case, a simple complete ideal of R often has more than one Rees valuation, and a complete m-primary ideal I may have finitely many or infinitely many base points. For the ideals having finitely many base points, Lipman proves a unique factorization involving special star-simple complete ideals with possibly negative exponents of the factors. Let T be an infinitely near point to R with dim R = dim T and T having residue field equal to R/m. We prove that the special star simple complete ideal associated with the sequence from R to T has a unique Rees valuation if and only if either dim R = 2 or there is no change of direction in the unique finite sequence of local quadratic transforms from R to T. We also examine conditions for a complete ideal to be projectively full.
Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). A theorem of Lipman implies that I has a unique factorization as a *-product of special *-simple complete ideals with possibly negative exponents for some of the factors. The existence of negative exponents occurs if the dimension of R is at least 3 because of the existence of finitely supported *-simple ideals that are not special. We consider properties of special *-simple complete ideals such as their Rees valuations and point basis. Let (R, m) be a d-dimensional equicharacterstic regular local ring with m = (x_1, ..., x_d)R. We define monomial quadratic transforms of R and consider transforms and inverse transforms of monomial ideals. For a large class of monomial ideals I that includes complete inverse transforms, we prove that the minimal number of generators of I is completely determined by the order of I. We give necessary and sufficient conditions for the complete inverse transform of a *-product of monomial ideals to be the *-product of the complete inverse transforms of the factors. This yields examples of finitely supported *-simple monomial ideals that are not special. We prove that a finitely supported *-simple monomial ideal with linearly ordered base points is special *-simple.
Let Q be a parameter ideal of a Noetherian local ring (R,m). The Goto number g(Q) of Q is the largest integer g such that Q:m^g is integral over Q. We examine the values of g(Q) as Q varies over the parameter ideals of R. We concentrate mainly on the case where dim R = 1, and many of our results concern parameter ideals of a numerical semigroup ring.
We discuss projective equivalence of ideals in Noetherian rings and the existence or failure of existence of projectively full ideals. We describe connections with the Rees valuations and Rees integers of an ideal, and consider the question of whether improvements can be made by passing to an integral extension ring.