Sylvia Wiegand

Stable isomorphism of modules over one-dimensional rings

Abstract Two R-modules M and N are said to be stably isomorphic provided M+R( n )≅N+R( n for some n ⩾ 1. In this paper R is always a commutative, reduced, one-dimensional Noetherian ring with finite normalization R, and M and N are torsionfree. All modules are assumed to be finitely generated. There is a natural action of ( R c )∗, the group of units of R c (where c is the conductor of R in R ) on torsionfree R-modules; and the main theorem of this paper is that the orbits of the induced action of ( R c ∗ are exactly the stable isomorphism classes. As applications, we show that stably isomorphic modules are actually isomorphic if R is a domain finitely generated as an R -algebra, with at most one singular real maximal ideal. We give several examples of modules that are stably isomorphic but not isomorphic, illustrating that these hypotheses cannot be significantly weakened. We are grateful to L. S. Levy and the referee for several suggestions that have improved the exposition of this work.

One-Dimensional Rings of Finite Representation Type

Let R be a commutative, semilocal, Noetherian domain, not a field. We say that R has finite representation type provided R has, up to isomorphism, only finitely many indecomposable finitely generated torsion-free modules. A special case (0.6) of our main theorem states that R has finite representation type if and only if

Bounds for one-dimensional rings of finite cohen-macaulay type

Abstract Wiegand, R. and S. Wiegand, Bounds for one-dimensional rings of finite Cohen-Macaulay type, Journal of Pure and Applied Algebra 93 (1994) 311-342. Let R be an integral domain finitely generated as an algebra over a field of characteristic not equal to 2 (or the localization of such a ring at some multiplicatively closed set); and assume that, for each maximal ideal , there is a bound on the ranks of the indecomposable finitely generated torsion-free R -modules. We show that the only possible ranks for such indecomposable modules over R are 1, 2, 3, 4, 5, 6, 8, 9 and 12. An example having indecomposables of each of these ranks is constructed over the field of rational numbers. Furthermore, over a broader class of reduced one-dimensional rings, the only possible ranks for indecomposable finitely generated torsion-free modules of constant rank are also 1, 2, 3, 4, 5, 6, 8, 9 and 12.

Mixed polynomial/power series rings and relatinos among their spectra

C ↪→ D1 := k[x] [[y/x]] ↪→ · · · ↪→ Dn := k[x] [[y/x]] ↪→ · · · ↪→ E. (2) With regard to Equation 2, for n a positive integer, the map C ↪→ Dn is not flat, but Dn ↪→ E is a localization followed by an adic completion of a Noetherian ring and therefore is flat. We discuss the spectra of these rings and consider the maps induced on the spectra by the inclusion maps on the rings. For example, we determine whether there exist nonzero primes of one of the larger rings that intersect a smaller ring in zero. We were led to consider these rings by questions that came up in two contexts. The first motivation is from the introduction to the paper [AJL] by AlonzoTarrio, Jeremias-Lopez and Lipman: If a map between Noetherian formal schemes can be factored as a closed immersion followed by an open one, can this map also be factored as an open immersion followed by a closed one? This is not true in general. As mentioned in [AJL], Brian Conrad observed that a counterexample can be constructed for every triple (R, x, p), where

Block diagonalization and 2-unit sums of matrices over Prüfer domains

We show that matrices over a large class of Prüfer domains are equivalent to “almost diagonal” matrices, that is, to matrices with all the nonzero entries congregated in blocks along the diagonal, where both dimensions of the diagonal blocks are bounded by the size of the class group of the Prüfer domain. This result, a generalization of a 1972 result of L. S. Levy for Dedekind domains, implies that, for n sufficiently large, every n× n matrix is a sum of two invertible matrices. We also generalize from Dedekind to certain Prüfer domains a number of results concerning the presentation of modules and the equivalence of matrices presenting them, and we uncover some connections to combinatorics.

Projective Lines Over One-Dimensional Semilocal Domains and Spectra of Birational Extensions

In [7], Nashier asked if the condition on a one-dimensional local domain R that each maximal ideal of the Laurent polynomial ring R[y, y -1] contracts to a maximal ideal in R[y] or in R[y -1] implies that R is Henselian. Motivated by this question, we consider the structure of the projective line Proj(R[s, t]) over a one-dimensional semilocal domain R (the projective line regarded as a topological space, or equivalently as a partially ordered set). In particular, we give an affirmative answer to Nashier’s question. (Nashier has also independently answered his question [9].) Nashier has also studied implications on the prime spectrum of the Henselian property in [8] as well as in the papers cited above.

On the compositum of two power series rings

This paper concerns subrings of the bivariate power series ring over a field. In this note, we study subrings of the power series ring k[[X, Y]] in two variables over a field k. In particular, in ? 1, we exhibit an element z in k[[X, Y]] which is transcendental over the compositum A = k[k[[X]], k[[Y]] ] of the power series rings k[[X]] and k[[Y]]. In fact we show that the power series ring k[[X, Y]] has uncountable transcendence degree over the compositum A. Stephen McAdam raised a question, discussed in [HW], concerning the existence of a non-Henselian Noetherian two-dimensional local domain S such that S/P is Henselian for each height one prime P. In ?2 of this article, we construct such a domain S, lying between k[[X, Y]] and the compositum A, and containing an element transcendental over A, such as z. In ?3, some questions are raised concerning various subrings of k[[X, Y]]. Also we include an argument due to Kunz that the compositum is not Noetherian. We would like to thank Ray Heitmann for a careful reading of an earlier draft of this paper and for providing us with an improved proof (given in ? 1) of the transcendence of the element z mentioned above. Also Heitmann has communicated to us another proof that the compositum A is not Noetherian. We thank the referee for pointing out that the transcendence degree of k[[X, Y]] over A is uncountable.

Prime ideals in polynomial rings over one-dimensional domains

Let R be a one-dimensional integral domain with only finitely many maximal ideals and let x be an indeterminate over R. We study the prime spectrum of the polynomial ring R[x] as a partially ordered set. In the case where R is countable we classify Spec(R[x]) in terms of splitting properties of the maximal ideals m of R and the valuative dimension of Rm . Let R be as in the abstract. Since Spec(R) is finite, Spec(R[x]) is Noetherian; Ohm and Pendleton show that every finitely generated algebra over a ring with Noetherian spectrum again has Noetherian spectrum [OP, Corollary 2.6, page 634]. Thus in our setting, the partial order on Spec(R[x]) uniquely determines Spec(R[x]) as a topological space with the Zariski topology. In the case where R is a countable one-dimensional local Noetherian domain, we show in [HW, Theorem 2.7] that there are precisely two possibilities for Spec(R[x]), one of which occurs when R is Henselian and the other when R is not Henselian. We also show that if R is a countable one-dimensional semilocal Noetherian domain having more than one maximal ideal, then the spectrum of R[x] is uniquely determined up to isomorphism by the number of maximal ideals of R. (In this latter case, R cannot be Henselian.) An important concept related to our work here and in [HW] is the n-split property introduced by McAdam in [Mc] for a prime ideal P of an integral domain D: If Da is the integral closure of D in an algebraic closure of the quotient field of D, then P is said to be n-split if there are exactly n primes in Da which lie over P. (Possibly n = oo.) We show in [HW, Theorem 1.1] that every prime ideal of a Noetherian domain D is either 1-split or oo-split and if P is a nonzero prime ideal of D that is 1-split, then D is local with maximal ideal P. It is noted in [HW, Example 1.6] that for every positive integer n, there exists a one-dimensional non-Noetherian local integral domain with n-split maximal ideal. Using the more varied behavior of the splitting of prime ideals in a non-Noetherian domain, we show in this article that the possible spectra of the polynomial ring R[x] over a one-dimensional non-Noetherian local domain R are considerably more varied than in the Noetherian case. Theorem 3.1 demonstrates that the n-split property in R yields a distinguishing characteristic of Spec(R[x]). Received by the editors July 12, 1993; originally communicated to the Proceedings of the AMS by Wolmer V. Vasconcelos. 1991 Mathematics Subject Classification. Primary 13E05, 13F20, 13G05, 13H99, 13J15. ? 1995 American Mathematical Society 0002-9947/95

Catenary Local Rings with Geometrically Normal Formal Fibers

1.00 +

Prime Ideals in Polynomial and Power Series Rings over Noetherian Domains

.25 per page