On Auslander's depth formula

Let [Formula: see text] be a Cohen–Macaulay local ring. Let [Formula: see text] be a finitely generated [Formula: see text]-module and let [Formula: see text] denote the [Formula: see text]-dual of [Formula: see text]. Furthermore, if [Formula: see text] is a maximal Cohen–Macaulay [Formula: see text]-module, then we prove that [Formula: see text], where [Formula: see text] is the cardinality of a minimal generating set of [Formula: see text] as an [Formula: see text]-module and [Formula: see text] is the multiplicity of the local ring [Formula: see text]. Furthermore, if [Formula: see text] is a reflexive [Formula: see text]-module then [Formula: see text]. As an application, we study the bound on the minimal number of generators of specific modules over two-dimensional normal local rings. We also mention some relevant examples.

On Buchsbaum rings

Perfect modules over cohen-macaulay local rings

Let A be a Noetherian local ring with the maximal ideal m, and let I be an m-primary ideal in A. This paper examines the equality on Hilbert coefficients of I first presented by Elias and Valla, but without assuming that A is a Cohen–Macaulay local ring. That equality is related to the Buchsbaumness of the associated graded ring of I.

A commutative Noetherian local ring of finite global dimension is a regular local ring and the structure of such a ring is well described 1n many texts (see §1.9). It is perhaps surprising that very little seems to be known about the corresponding non-commutat1ve rings and it is our aim 1n this thesis to examine some of the ways in which the commutative theory can, and cannot be extended to a right Noetherian local ring of finite right global dimension. Chapter 1 contains basic definitions and, in Chapter 2, results are obtained on the projective and Injective dimensions of modules over right Noetherian local rings. Commutative regular local rings are domains and we begin Chapter 3 by considering the question of when a right Noetherian local ring R of finite right global dimension is a prime ring. An example of Stafford's shows that R is not necessarily prime; however, by examining the Nilpotent radical of R, we are able to extend results of Ramras [57] and Walker [75] and show that R is indeed prime when certain prime ideals are localisable. In Chapter 4 we consider the lattice of prime ideals of R when R is an AR-r1ng and provide theorems which generalise some of the basic results from the theory of commutative regular local rings. Section 4.3 contains examples which not only illustrate points arising in Chapters 3 and 4 but also show that some of the techniques which are mainstays of the commutative theory, fall dramatically in a non-commutat1ve setting. In Chapter 5, we generalise the concepts of regular sequences and Cohen Macaulay rings enabling us to prove, in §5.2 and chapter 6, that a right Noetherian local ring of finite right global dimension, which 1s integral over its centre, is a prime ring and exhibits many properties similar to those enjoyed by commutative regular local rings. Examples are provided which show that some alternative approaches are not applicable to the rings considered in these chapters. Regular local rings are Gorensteln rings and, as such, are the subject of an elegant structure theorem due to Bass [5]. In Chapter 7, we generalise his theorem to the situation of rings Integral over their centres. Each chapter begins with a summary of the results contained in that chapter.

Generalized Loewy length of Cohen-Macaulay local and graded rings

Bass and Betti numbers of A/In

Reiten has demonstrated that the trivial Hochschild extension of a Cohen-Macaulay local ring by a canonical module is a Gorenstein local ring. Here it is proved that any commutative extension of a Cohen-Macaulay local ring by a canonical module is a Gorenstein ring. Also Gorenstein extensions of a local Cohen-Macaulay ring by a module are studied.

Let (R, 𝔪) be a Cohen–Macaulay local ring. If R has a canonical module, then there are some interesting results about duality for this situation. In this article, we show that one can indeed obtain similar results in the case R does not have a canonical module. Also, we give some characterizations of complete big Cohen–Macaulay modules of finite injective dimension, and by using them, some characterizations of Gorenstein modules over the 𝔪-adic completion of R are obtained.

The aim of the Paper is to Obtain information about the flat covers and minimal flat resolutions of Artinian modules over a Noetherian ring. Let R be a commutative Noetherian ring and let A be an Artinian R-module. We prove that the flat cover of a is of the form <TEX>$\prod_{p\epsilonAtt_R(A)}T-p$</TEX>, where <TEX>$Tp$</TEX> is the completion of a free <TEX>R$_{p}$</TEX>-module. Also, we construct a minimal flat resolution for R/xR-module 0: <TEX>$_AX$</TEX> from a given minimal flat resolution of A, when n is a non-unit and non-zero divisor of R such that A = <TEX>$\chiA$</TEX>. This result leads to a description of the structure of a minimal flat resolution for <TEX>${H^n}_{\underline{m}}(R)$</TEX>, nth local cohomology module of R with respect to the ideal <TEX>$\underline{m}$</TEX>, over a local Cohen-Macaulay ring (R, <TEX>$\underline{m}$</TEX>) of dimension n.

In this paper I shall demonstrate certain algebraic properties of ideals of finite homological dimension in local rings.In the first section, I show that no non-zero ideal of finite homological dimension in a local ring can be of zero grade (this is stated by Auslander and Buchsbaum in the appendix to (l), but I cannot find a proof in the literature). Using this result, together with the complex defined by a matrix, which is described by Eagon and Northcott in (2), I prove that an ideal of homological dimension one in a local ring Q may always, for some integer n, be described as the set of determinants of matrices obtained by adjoining to a certain (n–l)× n matrix with elements in the maximal ideal of Q another row with elements arbitrarily chosen in Q. (This result was established in (3), under the additional condition that Q should be a domain.)

Rings whose cyclic modules have restricted injectivity domains

Syzygy Theorems via comparison of order ideals on a hypersurface

Let R be a Cohen–Macaulay local ring. It is shown that under some mild conditions, the Cohen–Macaulay property is preserved under linkage. We also study the connection of the (Sn) locus of a horizontally linked module and the attached primes of certain local cohomology modules of its linked module.

We establish the continuity of Hilbert–Kunz multiplicity and F-signature as functions from a Cohen–Macaulay local ring $(R,\mathfrak{m},k)$ of prime characteristic to the real numbers at reduced parameter elements with respect to the $\mathfrak{m}$-adic topology.