Roger Wiegand

Tensor products of modules and the rigidity of Tor.

Let R be a hypersurface domain, that is, a ring of the form S/(f), where S is a regular local ring and f is a prime element. Suppose M and N are finitely generated R- modules. We prove two rigidity theorems on the vanishing of Tor. In the first theorem we assume that the regular local ring S is unramified, that M R N has finite length, and that dim(M)+dim(N) dim(R). With these assumptions, if Tor R (M,N) = 0 for some j 0, then Tor R (M,N) = 0 for all i j. The second rigidity theorem states that if M R N is reflexive, then Tor R (M,N) = 0 for all i 1. We use these theorems to prove the following theorem (valid even if S is ramified): If M R N is a maximal Cohen-Macaulay R-module, then both M and N are maximal Cohen-Macaulay modules, and at least one of them is free.

Cohen-Macaulay Representations

This book is a comprehensive treatment of the representation theory of maximal Cohen-Macaulay (MCM) modules over local rings. This topic is at the intersection of commutative algebra, singularity theory, and representations of groups and algebras. Two introductory chapters treat the Krull-Remak-Schmidt Theorem on uniqueness of direct-sum decompositions and its failure for modules over local rings. Chapters 3-10 study the central problem of classifying the rings with only finitely many indecomposable MCM modules up to isomorphism, i.e., rings of finite CM type. The fundamental material--ADE/simple singularities, the double branched cover, Auslander-Reiten theory, and the Brauer-Thrall conjectures--is covered clearly and completely. Much of the content has never before appeared in book form. Examples include the representation theory of Artinian pairs and Burban-Drozds related construction in dimension two, an introduction to the McKay correspondence from the point of view of maximal Cohen-Macaulay modules, Auslander-Buchweitzs MCM approximation theory, and a careful treatment of nonzero characteristic. The remaining seven chapters present results on bounded and countable CM type and on the representation theory of totally reflexive modules.

Cancellation over commutative rings of dimension one and two

All rings in this paper are commutative and Noetherian, and all modules are finitely generated. We will study the question: If A @ C z B @ C, is A z B? (Here A, B and C are modules over a ring R.) In 1962 Chase [4] gave an affirmative answer for R = k[X, Y], the polynomial ring in two variables over an algebraically closed field k, provided A and B are torsionfree and char(k)trank(A). In the first section of this paper we extend Chase’s result to include all two-dimensional regular affine k-domains. Most of the paper deals with one-dimensional rings. In Section 2 we prove some general results, which we hope will eventally lead to some sort of structure theory for torsionfree modules over one-dimensional reduced rings with finite normalization. In this context we show that one can cancel from projectives (that is, the answer is “yes” when A and B are projective) if and only if Pit R = Pit z, where g is the normalization. When A and B are assumed to be merely torsionfree, the problem is more subtle, but we have some partial results. The last three sections deal with three naturally occurring examples of one-dimensional rings: coordinate rings of curves, quadratic orders and integral group rings of finite abelian groups. In each of these cases we are able to give a fairly complete answer to the cancellation problem for torsionfree modules. Many of the ideas in this paper are the direct or indirect results of innumerable conversations with Raymond Heitmann and Lawrence Levy. In particular, Levy discovered a serious error in the proof of the main theorem of an earlier version, and Heitmann showed by example that the statement itself was incorrect. I am extremely grateful to both of them for their interest and insight.

Noetherian Rings of Bounded Representation Type

Let R be a reduced Noetherian ring with minimal prime ideals P 1,…, P m. The rank of the torsion-free R-module M is the m-tuple ρ(M) = (r 1,…, r m ), where r i is the dimension of M Pi (as a vector space over R Pi . We say R has bounded representation type provided there is an integer N such that each indecomposable finitely generated torsion-free R-module has rank less than or equal to (N,…, N). The least such N will be denoted by β(R), and if no such N exists we write β(R) = ∞.

Rings whose finitely generated modules are direct sums of cyclics

All rings in this discussion are commutative with unity, and all modules are unital. The purpose of this paper is to study the structure of those rings whose modules satisfy the conclusion of a classical theorem well known to every beginning student of algebra, namely: Every finitely generated module over a principal ideal domain is a direct sum of cyclic submodules. Let us canonize this theorem with the following terminology: A ring is said to be an FGC-ring (or to have FGC) if every finitely generated module over the ring is a direct sum of cyclic submodules. What can be said about such rings ? In particular, how far are they from being principal ideal rings ? As a point of departure, we review the known results on FGC-rings. (The reader is referred to Section 1 for the relevant definitions.) A major step beyond the classical theorem was taken by Kaplansky, who proved in [l] and [2] that a local domain has FGC if and only if it is an almost maximal valuation ring. Subsequently, Matlis [3] generalized this theorem by showing that an h-local domain has FGC if and only if it is Bezout and every localization is almost maximal. An example of such a ring, neither a local ring nor a principal ideal ring, was provided by Osofsky [4]. Recently Gill [5] and Lafon [6] completely disposed of the local problem by generalizing Kaplansky’s theorem to arbitrary local rings. In another direction, Pierce [7] has characterized the von Neumann regular FGC-rings as finite direct products of fields. It is interesting to observe (see Section 1 for details) that all known examples of FGC-rings satisfy a module-theoretic version of another classical theorem, namely the elementary divisor theorem for matrices over a principal ideal domain. To be precise, we define a canonical form for an R-module ilP to be a decomposition ME R/I, @ ... @ R/In , where I1 _C I, C ... _C I, f R. A ring R is then called a CF-ring (or said to have CF) if every direct sum of

Factorization Theory and Decompositions of Modules

Abstract Let R be a commutative ring with identity. It often happens that M1 ⊕ ⋯ ⊕ Ms ≅ N1 ⊕ ⋯ ⊕ Nt for indecomposable R-modules M1, …, Ms and N1, …, Nt with s ≠ t. This behavior can be captured by studying the commutative monoid {[M] ❘ M is an R-module} of isomorphism classes of R-modules with operation given by [M] + [N] = [M ⊕ N]. In this mostly self-contained exposition, we introduce the reader to the interplay between the the study of direct-sum decompositions of modules and the study of factorizations in integral domains.

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.

THE PRIME SPECTRUM OF A TWO-DIMENSIONAL AFFINE DOMAIN

From a distance, the two partially ordered sets spec Z[x] and spec Q[x, y] appear rather similar: They each have a unique minimal element and denumerable sets of elements of heights one and two. Moreover, there are several restrictions placed on the orderings by the fact that both rings are Noetherian. We will show that the partial ordering of spec7/[x] is much simpler than that of specQ[x,y]. In fact, U= spec Z[x] is characterized among countable partially ordered sets by the follow

Galois groups and the multiplicative structure of field extensions

Let K/k be a finite Galois field extension, and assume k is not an algebraic extension of a finite field. Let K. be the multiplicative group of K, and let Θ(K/k) be the product of the multiplicative groups of the proper intermediate fields. The condition that the quotient group Γ=K./Θ(K/k) be torsion is shown to depend only on the Galois group G. For algebraic number fields and function fields, we give a complete classification of those G for which Γ is nontrivial

Direct-sum decompositions over one-dimensional Cohen-Macaulay local rings

1 Alberto Facchini, Dipartimento di Matematica Pura e Applicata, Universita di Padova, Via Belzoni 7, I-35131 Padova, Italy, facchini@math.unipd.it 2 Wolfgang Hassler, Institut fur Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universitat Graz, Heinrichstrase 36/IV, A-8010 Graz, Austria, wolfgang.hassler@uni-graz.at 3 Lee Klingler, Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431-6498, klingler@fau.edu 4 Roger Wiegand, Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0323, rwiegand@math.unl.edu