**Author**: Eben Matlis

**Publisher:** Lecture Notes in Mathematics

**ISBN:** UOM:39015017317077

**Category:**

**Page:** 178

**View:** 143

*We will say that a ring R is a l-dimensional Cohen–Macaulay ring if it is a Noetherian ring of Krull dimensional l, and if every maximal ideal of R contains a regular element. A Noetherian domain of Krull dimension l is a l-dimensional ...*

**Author**: Eben Matlis

**Publisher:** Springer

**ISBN:** 9783540469230

**Category:**

**Page:** 160

**View:** 876

**One**-**dimensional** local **rings** of finite representation type We have shown in the last chapter that Gorenstein **rings** of finite representation type are simple singularities. This chapter aims at showing that the converse is true for ...

**Author**: Y. Yoshino

**Publisher:** Cambridge University Press

**ISBN:** 9780521356947

**Category:**

**Page:** 191

**View:** 262

The purpose of these notes is to explain in detail some topics on the intersection of commutative algebra, representation theory and singularity theory. They are based on lectures given in Tokyo, but also contain new research. It is the first cohesive account of the area and will provide a useful synthesis of recent research for algebraists.

*(b) e(R) 3 emb dim R—dim R+ 1; if equality holds, then R is said to have minimal multiplicity. ... Let (R,m,k) be a one dimensional Cohen—Macaulay local ring, and x a superficial element of R. (a) Suppose I is an ideal of height 1 in R.*

**Author**: Winfried Bruns

**Publisher:** Cambridge University Press

**ISBN:** 9780521566742

**Category:**

**Page:** 471

**View:** 515

Now in paperback, this advanced text on Cohen-Macaulay rings has been updated and expanded.

*[Rs. [S] 1] [SI 2 [SV1] [SV2 [Sa] [Se] [S] E. Matlis, Some properties of Noetherian domains of dimension one, Canad. J. Math. 13 (1961), 569-586. ., 1-dimensional Cohen–Macaulay rings, Lecture Notes in Math., vol.*

**Author**: Valentina Barucci

**Publisher:** American Mathematical Soc.

**ISBN:** 9780821805442

**Category:**

**Page:** 78

**View:** 828

If $k$ is a field, $T$ an analytic indeterminate over $k$, and $n_1, \ldots, n_h$ are natural numbers, then the semigroup ring $A = k[[T^{n_1}, \ldots, T^{n_h}]]$ is a Noetherian local one-dimensional domain whose integral closure, $k[[T]]$, is a finitely generated $A$-module. There is clearly a close connection between $A$ and the numerical semigroup generated by $n_1, \ldots, n_h$. More generally, let $A$ be a Noetherian local domain which is analytically irreducible and one-dimensional (equivalently, whose integral closure $V$ is a DVR and a finitely generated $A$-module). As noted by Kunz in 1970, some algebraic properties of $A$ such as ``Gorenstein'' can be characterized by using the numerical semigroup of $A$ (i.e., the subset of $N$ consisting of all the images of nonzero elements of $A$ under the valuation associated to $V$ ). This book's main purpose is to deepen the semigroup-theoretic approach in studying rings A of the above kind, thereby enlarging the class of applications well beyond semigroup rings. For this reason, Chapter I is devoted to introducing several new semigroup-theoretic properties which are analogous to various classical ring-theoretic concepts. Then, in Chapter II, the earlier material is applied in systematically studying rings $A$ of the above type. As the authors examine the connections between semigroup-theoretic properties and the correspondingly named ring-theoretic properties, there are some perfect characterizations (symmetric $\Leftrightarrow$ Gorenstein; pseudo-symmetric $\Leftrightarrow$ Kunz, a new class of domains of Cohen-Macaulay type 2). However, some of the semigroup properties (such as ``Arf'' and ``maximal embedding dimension'') do not, by themselves, characterize the corresponding ring properties. To forge such characterizations, one also needs to compare the semigroup- and ring-theoretic notions of ``type''. For this reason, the book introduces and extensively uses ``type sequences'' in both the semigroup and the ring contexts.

*Let (R, m) be a one-dimensional local ring with nilradical N. Then R has finite Cohen-Macaulay type if and only if R/N (which is Cohen-Macaulay) has finite Cohen-Macaulay type and m n N = 0 for some t > 1. For example, let k be a field, ...*

**Author**: Alberto Facchini

**Publisher:** Springer Science & Business Media

**ISBN:** 9789401104432

**Category:**

**Page:** 517

**View:** 817

On the 26th of November 1992 the organizing committee gathered together, at Luigi Salce's invitation, for the first time. The tradition of abelian groups and modules Italian conferences (Rome 77, Udine 85, Bressanone 90) needed to be kept up by one more meeting. Since that first time it was clear to us that our goal was not so easy. In fact the main intended topics of abelian groups, modules over commutative rings and non commutative rings have become so specialized in the last years that it looked really ambitious to fit them into only one meeting. Anyway, since everyone of us shared the same mathematical roots, we did want to emphasize a common link. So we elaborated the long symposium schedule: three days of abelian groups and three days of modules over non commutative rings with a two days' bridge of commutative algebra in between. Many of the most famous names in these fields took part to the meeting. Over 140 participants, both attending and contributing the 18 Main Lectures and 64 Communications (see list on page xv) provided a really wide audience for an Algebra meeting. Now that the meeting is over, we can say that our initial feeling was right.

*R;m;k/ be a one-dimensional standard graded Cohen– Macaulay ring with uncountable residue field k. If R is of graded countable Cohen–Macaulay type then, e.R/ 6 3. Proof. As shown in [20, Corollary 4.5], the possible h-vectors are .1/, ...*

**Author**: Susan M. Cooper

**Publisher:** Springer

**ISBN:** 9781493906260

**Category:**

**Page:** 317

**View:** 109

Commutative algebra, combinatorics, and algebraic geometry are thriving areas of mathematical research with a rich history of interaction. Connections Between Algebra and Geometry contains lecture notes, along with exercises and solutions, from the Workshop on Connections Between Algebra and Geometry held at the University of Regina from May 29-June 1, 2012. It also contains research and survey papers from academics invited to participate in the companion Special Session on Interactions Between Algebraic Geometry and Commutative Algebra, which was part of the CMS Summer Meeting at the University of Regina held June 2–3, 2012, and the meeting Further Connections Between Algebra and Geometry, which was held at the North Dakota State University February 23, 2013. This volume highlights three mini-courses in the areas of commutative algebra and algebraic geometry: differential graded commutative algebra, secant varieties, and fat points and symbolic powers. It will serve as a useful resource for graduate students and researchers who wish to expand their knowledge of commutative algebra, algebraic geometry, combinatorics, and the intricacies of their intersection.

*Chapter II One-Dimensional Semilocal Cohen-Macaulay Rings INTRODUCTION: The local ring of a point on a curve is a one-dimensional local Cohen-Macaulay ring; in this chapter we study this class of rings. After proving Some results on ...*

**Author**: K. Kiyek

**Publisher:** Springer Science & Business Media

**ISBN:** 9781402020292

**Category:**

**Page:** 486

**View:** 617

The Curves The Point of View of Max Noether Probably the oldest references to the problem of resolution of singularities are found in Max Noether's works on plane curves [cf. [148], [149]]. And probably the origin of the problem was to have a formula to compute the genus of a plane curve. The genus is the most useful birational invariant of a curve in classical projective geometry. It was long known that, for a plane curve of degree n having l m ordinary singular points with respective multiplicities ri, i E {1, . . . , m}, the genus p of the curve is given by the formula = (n - l)(n - 2) _ ~ "r. (r. _ 1) P 2 2 L. . ,. •• . Of course, the problem now arises: how to compute the genus of a plane curve having some non-ordinary singularities. This leads to the natural question: can we birationally transform any (singular) plane curve into another one having only ordinary singularities? The answer is positive. Let us give a flavor (without proofs) 2 on how Noether did it • To solve the problem, it is enough to consider a special kind of Cremona trans formations, namely quadratic transformations of the projective plane. Let ~ be a linear system of conics with three non-collinear base points r = {Ao, AI, A }, 2 and take a projective frame of the type {Ao, AI, A ; U}.

*59 ( 1987 ) , 347–374 . ( 9 ] J. Herzog - E. Kunz , Kanonische Modul eines Cohen - Macaulay Rings Springer Lecture Notes in Math . 238 ( 1971 ) . ( 10 ) E. Kunz , The value - semigroup of a one - dimensional Gorestein ring , Proc .*

**Author**: Ayman Badawi

**Publisher:** Nova Publishers

**ISBN:** 1600210651

**Category:**

**Page:** 220

**View:** 281

Focus on Commutative Rings Research

*Finally, he proves the interesting result: If R is a one-dimensional Cohen-Macaulay ring that is analytically unramified and not a DVR, then its conductor C = (R ̄ :R R) is a stable (in our sense) integrally closed ideal and G(C) is ...*

**Author**: Daniel Anderson

**Publisher:** CRC Press

**ISBN:** 9780429530449

**Category:**

**Page:** 449

**View:** 571

Includes current work of 38 renowned contributors that details the diversity of thought in the fields of commutative algebra and multiplicative ideal theory. Summarizes recent findings on classes of going-down domains and the going-down property, emphasizing new characterizations and applications, as well as generalizations for commutative rings wi