Normal Local Domain Research Articles

The limiting behavior on the restriction of divisor classes to hypersurfaces

Let A be an excellent local normal domain and { f n } n=1 ∞ a sequence of elements lying in successively higher powers of the maximal ideal, such that each hypersurface A/ f n A satisfies R 1. We investigate the injectivity of the maps Cl( A)→Cl(( A/ f n A)′), where ( A/ f n A)′ represents the integral closure. The first result shows that no non-trivial divisor class can lie in every kernel. Secondly, when A is, in addition, an isolated singularity containing a field of characteristic zero, dim A⩾4, and A has a small Cohen–Macaulay module, then we show that there is an integer N>0 such that if f n∈ m N , then Cl( A)→Cl(( A/ f n A)′) is injective. We substantiate these results with a general construction that provides a large collection of examples.

Geometric domains with specified pseudo-dimensions

This paper pursues the study of the pseudo-supports of a finitely generated module over a finite-dimensional commutative Noetherian ring, and is particularly concerned with the properties of the pseudo-supports of a geometric normal local domain. It is shown that these pseudo-supports have certain properties that can be summarized by the so-called occurrence diagram of the ring, which is a certain subset of N 0× N 0 . Standard properties of local cohomology modules enable one to conclude that there are obvious restrictions on the type of subset of N 0× N 0 that can be an occurrence diagram in this way, and much of the paper is devoted to consideration of whether a subset of N 0× N 0 that meets those obvious restrictions is an occurrence diagram of some geometric normal local domain. These occurrence diagrams are also used to assist with the production of examples that show that a question raised by C. Huneke, about certain sets related to Grothendieck's Finiteness Theorem for local cohomology, has a negative answer.

Some Results on Tight Closure and Completion

We construct two examples of nonexcellent local Noetherian domains which demonstrate that tight closure and completion do not commute. The first example is a local normal domain A with a height one principal prime ideal P such that ( PÁ )* ≠ P * Á . We also construct an example of a complete local normal Gorenstein domain which is not F-regular but is the completion of an F-regular local ring.

Intermediate rings between a local domain and its completion. {II}

We present results connecting flatness of extension rings to the Noetherian property for certain intermediate rings between an excellent normal local domain and its completion. We consider conditions for these rings to have Cohen-Macaulay formal fibers. We also present several examples illustrating these results.

On the normal uniform local domain of attraction for intermediate order statistics

Let X 1, X 2,… be independent observations from the absolutely continuous distribution function F with quantile function Q. The density of the normalized order statistic X n−k n+1 : n converges locally uniformly to the standard normal density for any sequence k n with k n →∞ and k n / n→0 if and only if log ( d/ dx)Q(1−1/x) is slowly oscillating.

Etaleness and Normality

Let A be an excellent normal semilocal ring with strict Henselization Ahs. If A⊆D⊆Ahs is a normal semilocal intermediate ring semilocally dominated by Ahs, then D is a direct limit of semilocal-etale extensions of A. If A is an excellent normal local domain then the integral closure of A in a finite separable field extension L of the quotient field Q(A) is etale over A if and only if there is a Q(A) morphism L→Q(Ahs) and Q(Ahs) contains a normal closure of L. We apply these results to the normalizations of an arbitrary excellent reduced local ring A and its strict Henselization Ahs to obtain a characterization for local-etaleness of a local intermediate ring A⊆D⊆Ahs whose total quotient ring L is finitely generated over Q(A). Specifically, such an intermediate ring D is local-etale over A if and only if the normalization D̃ of D is semilocal and has the proper residual field structure relative to à and to D. We generalize this result to characterize local-etaleness of arbitrary injective reduced local morphisms A→D, where A is excellent, finding that such a morphism is local-etale if and only if the morphism Ã→D̃ is semilocal-etale and there is an isomorphism D̃≅D⊗AÃ.

Note on Birational Extensions in D-Dimension

Let $(R,m)$ be a $d$-dimensional regular local ring with quotient field $K$ and $(S,n)$ be a $d$-dimensional normal local domain birationally dominating $R$ with $l(mS) = d$. In this paper, it is shown that the following three properties hold. $S$ is dominated by the $m$-adic prime divisor of $R$;$n^i \cap R = m^i$, for all $i \ge 1$;$R/m = S/n$.

A Note on the Monomial Conjecture

Several cases of the monomial conjecture are proved. An equivalent form of the direct summand conjecture is discussed. Let (A, m, k) be a noetherian local ring of dimension n, m its maximal ideal and k = A/rn. The Monomial Conjecture (henceforth MC) of Hochster asserts that, given any system of parameters (henceforth s.o.p.) xI, .. ., xn of A, (XIX2 ... Xn) ? Xv***)Xn)bt>O Hochster proved the conjecture in the equicharacteristic case [HI], [H2]. He also established the fact that in the mixed and the positive characteristic cases the Direct Summand Conjecture (henceforth DSC) and hence MC is equivalent to the Canonical Element Conjecture [H12] (henceforth CEC). Thus MC occupies a central position in the study of several homological conjectures. In [I1l] Hochster pointed out that, given any s.o.p. xl,...,xn of A, xt, ... , xt satisfies MC for t sufficiently large. Next Goto [G] proved MC for Buchsbaum rings and Koh proved DSC for degree p extensions [K]. Several special cases of CEC were proved in [DI] when depthA = dimA 1. In [D2], the following result was established: If Ji AnnH;-j(A) and J JiJ2 ... Jr where r dimA depthA > 0, then Xl,... ,xn satisfies MC if J ? (xl,... , xn). This in turn implies that, given any s.o.p. xl, . . , xn in a complete local normal domain A, XI, X2, X3) ... , xt satisfies MC for t > 0. We will have several more applications of this result in Section 2. We recall that there is no loss of generality in assuming A to be a complete local normal domain. In our most recent work we established the validity of CEC over i) A/xA when A is a complete local normal domain and x E mJ1 [D3], ii) any almost complete intersection domain A or any almost complete intersection ring A over which p (= the mixed characteristic) is a non-zero-divisor [D4], iii) rings of the form R/Q, where R is a complete Gorenstein ring such that the complete local normal domain A is a homomorphic image of R, dimR = dimA, and Q is the canonical module of A [D4], and iv) complete local normal domains A for which Q is S3. Thus MC holds in all the above cases. The main results of this paper are arranged in the following way. In Section 1, first we reduce the study of MC to almost complete intersection rings. Recall that, due to the results mentioned earlier, the problem boils down to almost complete intersection rings of depth 0. Next we prove the following: Received by the editors July 20, 1996. 1991 Mathematics Subject Classification. Primary 13D02; Secondary 13H1.0. This work was partially supported by an NSA grant and an NSF grant. (?)1998 American Mathematical Society

Formal fibers and complete homomorphic images

Let ( R , m ) (R,{\mathbf {m}}) be an excellent normal local Henselian domain, and suppose that q {\mathbf {q}} is a prime ideal in R R of height > 1 > 1 . We show that, if R / q R/{\mathbf {q}} is not complete, then there are infinitely many height one prime ideals p ⊆ q R ^ {\mathbf {p}} \subseteq {\mathbf {q}}\hat R of R ^ \hat R with p ∩ R = 0 {\mathbf {p}} \cap R = 0 ; in particular, the dimension of the generic formal fiber of R R is at least one. This result may in fact indicate that a much stronger relationship between maximal ideals in the formal fibers of an excellent Henselian local ring and its complete homomorphic images is possibly satisfied. The second half of the paper is concerned with a property of excellent normal local Henselian domains R R with zero-dimensional formal fibers. We show that for such an R R one has the following good property with respect to intersection: for any field L L such that Q ( R ) ⊆ L ⊆ Q ( R ^ ) \mathcal {Q}(R) \subseteq L \subseteq \mathcal {Q}(\hat R) , the ring L ∩ R ^ L \cap \hat R is a local Noetherian domain which has completion R ^ \hat R .

Extensions of valuations to the completion of a local domain

Let (R, m) be a normal local domain of dimension n > 1. Suppose that R is analytically irreducible, i.e. that ii, the m-adic completion of R, is a domain. Given a valuation domain V birationally dominating R, what can be said about valuation domains W which extend V and birationally dominate Ai? In [l, Lemma 131, Abhyankar shows that V has an extension W which birationally dominates k. Here we discuss the uniqueness of such an extension. For this, we will assume, in addition, that i is normal, i.e., that R is analytically normal and that R satisfies a property, defined below, which is weaker than excellence. Matsumura’s work on dimensions of formal fibers was a motivating influence for the questions we consider on extending valuations from R to ff. In [lo], Matsumura asks if the dimension of the generic formal fiber, i.e., the fiber over 0 in the embedding R+ fi’, can be a positive integer other than 0, y1 2 or n 1. Since extensions of rank 1 valuations to i which grow in rank produce nonzero primes in the generic formal fiber, we initiated a study of these extensions hoping to shed some light on Matsumura’s question.’

Some results in local rings on ramification in low codimension

One way to study the behavior of the algebraic variety Spec A, where A is a normal domain, is to obtain finite maps f: Spec A -+ Spec R, where R is regular, and to investigate properties of Spec A relative to those of Spec R through the mapf. In case A is finitely generated over a field k one can obtain such a map f via Noether normalization (see [19, p. 911). In case A is local, complete, and contains a coefficient field k, one obtains an S by choosing (any) system of parameters xi, . . . . x, for A and then setting R=k[[x,, . . . . x,]] (see [19, pp. 210-2121; the case of unequal characteristic is slightly more delicate). Having constructed f: Spec A -+ Spec R by some means, the map f itself exerts a great deal of influence on the situation. In particular, the ramification which occurs in each fiber plays a central role in the interplay between properties of A and those of R. For example, should the map f be unramilied one concludes that A must be regular and the map f must be &tale. Moreover, the question of whether or not A/R is a ramified extension can be answered by computing ramification indices in codimension one. This is the essence of the theorem on “purity of branch loci” which can be found in [20, Sect. 411 or in [2]. When the regularity assumption on R is removed, then whether or not A/R is ramified cannot simply be detected in codimension one (e.g., see [lo, Example 16.5, p. 851). Nevertheless, one might expect to find “good” properties of R reflected in those of A in such a setting. For example, if R is a complete intersection (or perhaps just Gorenstein) and if A/R is unramified in codimension one, must A at least be Cohen-Macaulay? In Section 1 of this article we obtain sufficient conditions in Theorem 1.4 (and resulting properties) for a complete local normal domain A to be a

On unique and almost unique factorization of complete ideals II

The theory of complete ideals was begun by Zariski in 1938 [Z 1], and further developed in Appendix 5 of [ZS]. Zariski developed a theory of factorization of complete ideals in regular two-dimensional local rings. In this paper we develop the theory of factorization of complete ideals in complete normal two-dimensional local rings. Let (R, m) be a complete normal two-dimensional local ring, and let re(R) be the semi-group of complete m-primary ideals of R. We show how the divisor class group of R, CI(R), is related to factorization in re(R). We generalize results obtained in [C2] where it is assumed that R/m is algebraically closed. Assume that R is any regular local ring of dimension 2. Zariski proved that any complete ideal I of R has unique factorization into products of simple complete ideals. The distinct simple complete ideals occuring in the factorization of I are in 1 1 correspondence with the irreducible components of the exceptional divisor of the blowing up of I. This shows that any proper birational morphism X ~spec(R) with X normal can be decomposed into a sequence of blowups with irreducible exceptional divisors. Now assume that R is a complete normal local domain of dimension 2. It is natural to ask if the condition that m(R) has unique factorization is equivalent to R being a UFD. We show (Theorem 5) that if re(R) has unique factorization then R is a UFD. We also give an example (Example 1) to show that the converse does not hold. However, Lipman has shown in [L] that if R is a UFD, and if R/m is algebraically closed, then re(R) has unique factorization. Recall that R is a UFD if and only if CI(R)=0. G6hner [G] introduced the notion of a semigroup being semi-factorial. This definition is reproduced in definition 3 of this paper. We prove G6hner's conjecture (2 3 of Theorem 4) that re(R) is semifactorial if and only if CI(R) is a torsion group. As a corollary (Corollary to Theorem 4) we obtain that if Rim is algebraically closed of characteristic zero, then re(R) is semi-factorial if and only if R has a rational singularity. Another corollary is that CI(R) is torsion if and only if R satisfies the condition (N) of Muhly and Sakuma [-MS] (1r of Theorem 4). R satisfies condition (N) if any proper birational morphism X ~ spec(R) factors as a sequence of blowups of height 2 ideals with irreducible exceptional divisors.

The divisor class group of surfaces of embedding dimension 3

Using techniques of linear algebra and a theorem of Eisenbud, we prove that k〚x,y,z〛/(x 2 + y 3, z 5), (k any field), R 〚x,y,z〛/(x 2 + y 2, z m), m⩾2 , Z /3 Z 〚x,y,z〛x 2 + y 3 + z 4) and Z /7 Z 〚x,y,z〛/(x 2 + y 3 + 2z 3) are factorial. Moreover applying the Hilbert-Burch Theorem, we prove that the divisor class group Cl( R) of a normal local domain ( R, M) is generated by the classes [ P]of height 1 primes P which are not contained in M e( R) where R is a normal local domain of embedding dimension 3.

Spots Over Analytically Normal Local Domains

A result of Lipman on $2$-dimensional analytically normal local domains is extended to dimension $d > 2$, thereby enlarging the class of local domains known to have certain finiteness properties with respect to birational extensions.

Birational extensions in dimension two and integrally closed ideals

The aim of this paper is to examine, primarily from an algebraic point of view, the structure of a 2-dimensional normal local domain (S, n) which birationally dominates a 2-dimensinal regular local ring (R, m). From the geometric point of view, the sine qua non is Lipman’s paper [Lr] on rational singularities, for it follows from one of the early results in that paper that S has a rational singularity. Using elementary algebraic techniques we are able to recover much information concerning the structure of S. We have particularly focused on the fact (from Lipman [Ll] and Artin [Ar]) that S must have minimal multiplicity. We hope this approach will aid in understanding the algebra of rational singularities and in the exploration of many open questions in dimension two and in higher dimensions. The framework of the paper is set up as follows. In the first section we use basic algebraic techniques such as analytic independence and Zariski’s Main Theorem to prove, for example, that S has a regular height 1 prime if R/m is infinite. If S is not regular, we may assume that R is “maximally regular” in S and when this is the case we show, for instance, that the

Spots over analytically normal local domains

A result of Lipman on 2 2 -dimensional analytically normal local domains is extended to dimension d > 2 d > 2 , thereby enlarging the class of local domains known to have certain finiteness properties with respect to birational extensions.

The fundamental module of a normal local domain of dimension 2

The fundamental module E E of a normal local domain ( R , m ) (R,\,\mathfrak {m}) of dimension 2 2 is defined by the nonsplit exact sequence 0 → K → E → m → 0 0 \to K \to E \to \mathfrak {m} \to 0 , where K K is the canonical module of R R . We prove that, if R R is complete with R / m ≃ C R/\mathfrak {m} \simeq \mathbb {C} , then E E is decomposable if and only if R R is a cyclic quotient singularity. Various other properties of fundamental modules will be discussed.

Cohen-Macaulay normal local domains whose associated graded rings have no depth

Cohen-Macaulay normal local domains whose associated graded rings have no depth

Some multiplicative properties of complete ideals

1. Let L be a noetherian domain that is integrally closed in its quotient field F. To each ideal A of L is assigned an ideal A,a' the integral closure or completion of A, that consists of all elements x E L for which an equation of the form, xn + a1xn + * + an = 0, ai eAr, holds. If Q is the set of all valuations v of F such that the associated valuation ring R, contains L, then Aa coincides with the ideal Ab = {x; v(x) > v(A), V v E Q}, [2; 3]. An ideal A is said to be complete if A = Aa, and the set of all complete ideals is denoted by F(L). If A and B belong to F(L), the product AB may not belong to F(L), but the completion of the product (AB)a does. Hence a binary composition x is defined on F(L) by the condition, A x B = (AB)a. Under this composition F(L) is a commutative semigroup with an identity in which the cancellation law holds [2]. In case L is a two dimensional regular local ring, Zariski has shown [4, Appendix 5] that (1(L), x) is a Gaussian semigroup, and that the composition x is ordinary product. In this paper we study the case in which L is a two dimensional normal local domain which is subject to conditions less stringent than regularity. (See ?2 below.) It is shown that modulo a simple equivalence relation the semigroup (FO(L), x) is Gaussian, where Fo(L) is the subset of F(L) that consists of primary ideals belonging to the maximal ideal of L. However (FO(L), x) is not Gaussian in an absolute sense for in simple examples it is seen that the maximal ideal M of L is an irreducible element of (F(L), x) that is not prime. (Here we are using the semigroup terminology of Jacobson [1, Chapter IV].) Our methods are direct extensions of those of Zariski. In case L is regular, the form ring associated with L and the sequence of powers of the maximal ideal M is a polynomial ring over a field and Zariski's arguments are based in part on the fact that such a ring is a unique factorization domain. In our case the form ring is an integrally closed noetherian domain, and we obtain results analogous to Zariski's by using the Artin theory of factorization in the sense of quasi-equality that is valid in such domains.