Nagata Ring Research Articles

PrÜfer-Like Domains and the Nagata Ring of Integral Domains

A subring A of a Prüfer domain B is a globalized pseudo-valuation domain (GPVD) if (i) A↪B is a unibranched extension and (ii) there exists a nonzero radical ideal I, common to A and B such that each prime ideal of A (resp., B) containing I is maximal in A (resp., B). Let D be an integral domain, X be an indeterminate over D, c(f) be the ideal of D generated by the coefficients of a polynomial f ∈ D[X], N = {f ∈ D[X] | c(f) = D}, and N v = {f ∈ D[X] | c(f)−1 = D}. In this article, we study when the Nagata ring D[X] N (more generally, D[X] N v ) is a GPVD. To do this, we first use the so-called t-operation to introduce the notion of t-globalized pseudo-valuation domains (t-GPVDs). We then prove that D[X] N v is a GPVD if and only if D is a t-GPVD and D[X] N v has Prüfer integral closure, if and only if D[X] is a t-GPVD, if and only if each overring of D[X] N v is a GPVD. As a corollary, we have that D[X] N is a GPVD if and only if D is a GPVD and D has Prüfer integral closure. We also give several examples of integral domains D such that D[X] N v is a GPVD.

About the spectrum of Nagata rings

In this paper, we deal with the Nagata ring R(n) in case R is obtained by a (T, I, D) construction. We characterize when R(n) is a strong S-domain and catenarian. This study allows us to provide several interesting applications and examples.

Sagbi bases of Cox–Nagata rings

We degenerate Cox–Nagata rings to toric algebras by means of sagbi bases induced by configurations over the rational function field. For del Pezzo surfaces, this degeneration implies the Batyrev–Popov conjecture that these rings are presented by ideals of quadrics. For the blow-up of projective n -space at n + 3 points, sagbi bases of Cox–Nagata rings establish a link between the Verlinde formula and phylogenetic algebraic geometry, and we use this to answer questions due to D’Cruz–Iarrobino and Buczyńska–Wiśniewski. Inspired by the zonotopal algebras of Holtz and n s Ron, our study emphasizes explicit computations, and offers a new approach to Hilbert functions of fat points.

Über Die Assoziierten Primideale des Bidualen

Let (R, 𝔪) be a commutative, noetherian, local ring, E the injective hull of the residue field R/𝔪, and M ○○ = Hom R (Hom R (M, E), E) the bidual of an R-module M. We investigate the elements of Ass(M ○○) as well as those of Coatt(M) = {𝔭 ∈ Spec(R)|𝔭 = Ann R (Ann M (𝔭))} and provide criteria for equality in one of the two inclusions Ass(M) ⊂ Ass(M ○○) ⊂ Coatt(M). If R is a Nagata ring and M a minimax module, i.e., an extension of a finitely generated R-module by an artinian R-module, we show that Ass(M ○○) = Ass(M) ∪ {𝔭 ∈ Coatt(M)| R/𝔭 is incomplete}.

GOING-DOWN AND SEMISTAR OPERATIONS

If D ⊆ T is an extension of (commutative integral) domains and ⋆ (resp., ⋆′) is a semistar operation on D (resp., T), we define what it means for D ⊆ T to satisfy the (⋆,⋆′)-GD property. Sufficient conditions are given for (⋆,⋆′)-GD, generalizing classical sufficient conditions for GD such as flatness, openness of the contraction map of spectra and the hypotheses of the classical going-down theorem. If ⋆ is a semistar operation on a domain D, we define what it means for D to be a ⋆-GD domain, generalizing the notion of a going-down domain. In determining whether a domain D is a [Formula: see text] domain, the domain extensions T of D for which [Formula: see text] is tested can be the [Formula: see text]-valuation overrings of D, the simple overrings of D, or all T. P ⋆ MD s are characterized as the [Formula: see text]-treed (resp., [Formula: see text]) domains D which are [Formula: see text]-finite conductor domains such that [Formula: see text] is integrally closed. Several characterizations are given of the [Formula: see text]-Noetherian domains D of [Formula: see text]-dimension 1 in terms of the behavior of the (⋆,⋆′)-linked overrings of D and the ⋆-Nagata rings Na(D,⋆).

Note on localizing systems and Kronecker function rings of semistar operations

We study the results of M. Fontana and J. Huckaba [FHu] on localizing systems and semistar operations, and give a couple of remarks for them. After M. Fontana and K.A. Loper [FL3], we study also Nagata rings, Kronecker function rings, and related semistar operations on semigroups.

Displays and formal p-divisible groups

Over p-adic Nagata rings, formal p-divisible groups are classified by nilpotent displays according to T. Zink. We extend this result to arbitrary p-adic rings. The proof uses the Grothendieck–Illusie deformation theory of truncated p-divisible groups.

Asymptotic Behavior of Integral Closures in Modules

Let R be a commutative Noetherian Nagata ring, let M be a non-zero finitely generated R-module, and let I be an ideal of R such that height MI > 0. In this paper, there is a definition of the integral closure Na for any submodule N of M extending Rees' definition for the case of a domain. As the main results, it is shown that the operation N → Na on the set of submodules N of M is a semi-prime operation, and for any submodule N of M, the sequences Ass R M/(InN)a and Ass R (InM)a/(InN)a(n=1,2,…) of associated prime ideals are increasing and ultimately constant for large n.

A generalization of Kronecker function rings and Nagata rings

Let D be an integral domain with quotient field K. The Nagata ring D(X) and the Kronecker function ring Kr(D) are both subrings of the field of rational functions K(X) containing as a subring the ring D[X] of polynomials in the variable X. Both of these function rings have been extensively studied and generalized. The principal interest in these two extensions of D lies in the reflection of various algebraic and spectral properties of D and Spec(D) in algebraic and spectral properties of the function rings. Despite the obvious similarities in definitions and properties, these two kinds of domains of rational functions have been classically treated independently, when D is not a Prüfer domain. The purpose of this note is to study two different unified approaches to the Nagata rings and the Kronecker function rings, which yield these rings and their classical generalizations as special cases.

Abhyankar places admit local uniformization in any characteristic

We prove that every place P of an algebraic function field F | K of arbitrary characteristic admits local uniformization, provided that the sum of the rational rank of its value group and the transcendence degree of its residue field FP over K is equal to the transcendence degree of F | K , and the extension F P | K is separable. We generalize this result to the case where P dominates a regular local Nagata ring R ⊆ K of Krull dimension dim R ⩽ 2 , assuming that the valued field ( K , v P ) is defectless, the factor group v P F / v P K is torsion-free and the extension of residue fields F P | K P is separable. The results also include a form of monomialization.

On the FIP Property for Extensions of Commutative Rings #

ABSTRACT A (unital) extension R ⊆ T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ⊂ S ⊂ T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ⊂ T is a (module-) finite minimal ring extension, then R(X)⊂T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ⊆ T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ⊆ R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ⊆ R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ℤ which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (ℤ:R)≠0. Some results are also given for the residually FIP property.

Localization of tight closure in two-dimensional rings

It is shown that tight closure commutes with localization in any two-dimensional ringR of prime characteristic if eitherR is a Nagata ring orR possesses a weak test element. Moreover, it is proved that tight closure commutes with localization at height one prime ideals in any ring of prime characteristic.

Local–Global Properties for Semistar Operations#

We study the “local” behavior of several relevant properties concerning semistar operations, like finite type, stable, spectral, e.a.b. and a.b. We deal with the “global” problem of building a new semistar operation on a given integral domain, by “gluing” a given homogeneous family of semistar operations defined on a set of localizations. We apply these results for studying the local–global behavior of the semistar Nagata ring and the semistar Kronecker function ring. We prove that an integral domain D is a Prüfer ⋆-multiplication domain if and only if all its localizations D P are Prüfer ⋆ P -multiplication domains.

PRÜFER ⋆-MULTIPLICATION DOMAINS AND SEMISTAR OPERATIONS

Starting from the notion of semistar operation, introduced in 1994 by Okabe and Matsuda [49], which generalizes the classical concept of star operation (cf. Gilmer's book [27]) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer, P. Lorenzen and P. Jaffard (cf. Halter–Koch's book [32]), in this paper we outline a general approach to the theory of Prüfer ⋆-multiplication domains (or P⋆MDs), where ⋆ is a semistar operation. This approach leads to relax the classical restriction on the base domain, which is not necessarily integrally closed in the semistar case, and to determine a semistar invariant character for this important class of multiplicative domains (cf. also J. M. García, P. Jara and E. Santos [25]). We give a characterization theorem of these domains in terms of Kronecker function rings and Nagata rings associated naturally to the given semistar operation, generalizing previous results by J. Arnold and J. Brewer ]10] and B. G. Kang [39]. We prove a characterization of a P⋆MD, when ⋆ is a semistar operation, in terms of polynomials (by using the classical characterization of Prüfer domains, in terms of polynomials given by R. Gilmer and J. Hoffman [28], as a model), extending a result proved in the star case by E. Houston, S. J. Malik and J. Mott [36]. We also deal with the preservation of the P⋆MD property by ascent and descent in case of field extensions. In this context, we generalize to the P⋆MD case some classical results concerning Prüfer domains and PvMDs. In particular, we reobtain as a particular case a result due to H. Prüfer [51] and W. Krull [41] (cf. also F. Lucius [43] and F. Halter-Koch [34]). Finally, we develop several examples and applications when ⋆ is a (semi)star given explicitly (e.g. we consider the case of the standardv-, t-, b-, w-operations or the case of semistar operations associated to appropriate families of overrings).

Nagata Rings, Kronecker Function Rings, and Related Semistar Operations

In 1994, Matsuda and Okabe introduced the notion of semistar operation. This concept extends the classical concept of star operation (cf. for instance, Gilmer's book (Gilmer, R. (1972). Multiplicative Ideal Theory. New York: Marcel Dekker) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer and P. Lorenzen from 1930's. Fontana and Loper investigated properties of the Kronecker function rings which arise from arbitrary semistar operations on an integral domain D (Fontana M., Loper K. A. (2001a). Kronecker function rings: a general approach. In Anderson, D. D., Papick, I. J., eds. Ideal Theoretic Methods in Commutative Algebra. Lecture Notes Pure Appl. Math. 220, Marcel Dekker, pp. 189–205 and Fontana, M., Loper, K. A. (2001b). A Krull-type theorem for the semistar integral closure of an integral domain. ASJE Theme Issue “Commutative Algebra” 26:89–95). In this paper we extend that study and also generalize Kang's notion of a star Nagata ring to the semistar setting (Kang, B. G. (1987). ⋆-Operations on Integral Domains. Ph.D. dissertation, Univ. Iowa and Kang, B. G. (1989). Prüfer v-multiplication domains and the ring R[X] N v , J. Algebra 123: 151–171). Our principal focuses are the similarities between the ideal structure of the Nagata and Kronecker semistar rings and between the natural semistar operations that these two types of function rings give rise to on D.

Nagata rings and directed unions of Artinian subrings

We investigate when a Nagata ring R(X) can be written as a directed union of Artinian subrings. For a family of zero‐dimensional rings , we show that ∏α∈ARα(Xα) is not a directed sum of Artinian subrings.

On Strong Going-Between, Going-Down, And Their Universalizations, II

We consider analogies between the logically independent properties of strong going-between (SGB) and going-down (GD), as well as analogies between the universalizations of these properties. Transfer results are obtained for the (universally) SGB property relative to pullbacks and Nagata ring constructions. It is shown that if A C B are domains such that A is an LFD universally going-down domain and B is algebraic over A, then the inclusion map A[X 1 , ..., X n ] → B[X 1 ,..., X n ] satisfies GB for each n ≥ 0. However, for any nonzero ring A and indeterminate X over A, the inclusion map A → A[X] is not universally (S)GB.

About a conjecture on Nagata rings

About a conjecture on Nagata rings

About a conjecture on Nagata rings

This paper provides a counterexample to a conjecture on Nagata rings: a two-dimensional quasi-local domain such that, for every n, the Nagata ring A ( n) is strong S (and also catenary) but for n ≥ 1, the polynomial ring A[ n] is not strong S (nor catenary).

Valuative Heights and Infinite Nagata Rings

We prove here that if the ring R has locally finite valuative dimension then the Nagata ring with infinite indeterminates R(∞) is a stably strong S-ring. We also study the catenarity of R(∞)