Linear truncations package for Macaulay2

Journal of Symbolic Computation, 2019

In this paper, we present an algorithm for computing the minimal reductions of m-primary ideals of Cohen-Macaulay local rings. Using this algorithm, we are able to compute the Hilbert-Samuel multiplicities and solve the membership problem for the integral closure of m-primary ideals.

Pacific Journal of Mathematics, 1973

The first three theorems concern localizations of a Noetherian ring such that the localization is a Macaulay ring, and the other two theorems give some necessary and sufficient conditions for certain Rees rings and form rings of a Noetherian ring to be locally Macaulay. Numerous consequences of the theorems are proved.

2012

Let (R;m) be a Noetherian local ring which is a homomorphic image of a local Gorenstein ring and let M be a nitely generated R-module of dimension d > 0. According to Schenzel (Sc1), M is called a Cohen-Macaulay canonical module (CMC module for short) if the canonical module K(M) of M is Cohen-Macaulay. We give another characterization of CMC modules. We describe the non Cohen-Macaulay canonical locus nCMC(M) of M. If d6 4 then nCMC(M) is closed in Spec(R). For each d 5 there are reduced geometric local rings R of dimension d such that nCMC(R) is not stable under specialization 1 .

Mathematische Zeitschrift, 2010

In this paper we completely classify all the special Cohen-Macaulay (=CM) modules corresponding to the exceptional curves in the dual graph of the minimal resolutions of all two dimensional quotient singularities. In every case we exhibit the specials explicitly in a combinatorial way. Our result relies on realizing the specials as those CM modules whose first Ext group vanishes against the ring R, thus reducing the problem to combinatorics on the AR quiver; such possible AR quivers were classified by Auslander and Reiten. We also give some general homological properties of the special CM modules and their corresponding reconstruction algebras. Lemma 2.3. We have the following commutative diagram whose rows are equivalences and columns are dualities:

Journal of Software for Algebra and Geometry

We describe a Macaulay2 package for computations in prime characteristic commutative algebra. This includes those for Frobenius powers and roots, p −e-linear and p e-linear maps, singularities defined in terms of these maps, different types of test ideals and modules, and ideals compatible with a given p −e-linear map.

2019

A finitely generated module M over a commutative Noetherian ring R is called an I-Cohen Macaulay module, if (I,M) + (M/IM)= (M), where I is a proper ideal of R. The aim of this paper is to study the structure of this class of modules. It is discovered that I-Cohen Macaulay modules enjoy many interesting properties which are analogous to those of Cohen Macaulay modules. Also, various characterizations of I-Cohen Macaulay modules are presented here.

Journal of Algebra, 2009

Numerical invariants of a minimal free resolution of a module M over a regular local ring (R, n) can be studied by taking advantage of the rich literature on the graded case. The key is to fix suitable n-stable filtrations M of M and to compare the Betti numbers of M with those of the associated graded module gr M (M ). This approach has the advantage that the same module M can be detected by using different filtrations on it. It provides interesting upper bounds for the Betti numbers and we study the modules for which the extremal values are attained. Among others, the Koszul modules have this behavior. As a consequence of the main result, we extend some results by Aramova, Conca, Herzog and Hibi on the rigidity of the resolution of standard graded algebras to the local setting. * 2000 Mathematics Subject Classification: Primary 13H05, Secondary 13D02.

Journal of Pure and Applied Algebra, 1984

Communications, Faculty Of Science, University of Ankara Series A1Mathematics and Statistics, 2009

Let be an endomorphism of an arbitrary ring R with identity. In this note, we concern the relations between polynomial and power series extensions of a reduced module. Among others we prove that a ring R isreduced if and only if every ‡at right R-module is-reduced, and for a module M , M [x] is-reduced if and only if M [x; x 1 ] is-reduced.

2015

A generalization of the notion of depth of an ideal on a module is introduced by applying the concept of local cohomology modules with respect to a pair of ideals . The concept of (I;J)- Cohen{Macaulay modules is also introduced as a generalization of the concept of Cohen{Macaulay modules . This kind of modules is dierent from the Cohen{Macaulay modules, as shown in an example. Also an Artinian result is given for such modules.

Journal of Algebra, 2016

Let M be a finitely generated module over a Noetherian local ring R. In this paper we introduce the notion of sequential polynomial type of M , which is denoted by sp(M), in order to measure how far M is different from the sequential Cohen-Macaulayness. We study the sequential polynomial type under localization and m-adic completion. We investigate an ascentdescent property of sequential polynomial type between M and M/xM for certain parameter x of M. When R is a quotient of a Gorenstein local ring, we describe sp(M) in term of the deficiency modules of M .

Pacific Journal of Mathematics, 1968

This paper shows some instances where properties of a local ring are closely connected with the homological properties of a single module. Particular stress is placed on conditions implying the regularity or the Cohen-Macaulay property of the ring.

Journal of Algebra, 1976

It is well known that the reduced ring, Crea , of a Cohen-Macaulay local ring C, of finite characteristic, need not be Cohen-Macaulay (for example, see [l, p. 1251). In fact, in this case, even if C is a complete intersection Crea need not be Cohen-Macaulay. In this note, we give an example (see-proposition and the remarks following the proposition) in characteristic zero, of a Cohen-Macaulay ring C, such that the reduced ring is not Cohen-Macaulay.

arXiv: Commutative Algebra, 2016

Let $(R, \frak m)$ be a homomorphic image of a Cohen-Macaulay local ring and $M$ a finitely generated $R$-module. We use the splitting of local cohomology to shed a new light on the structure of non-Cohen-Macaulay modules. Namely, we show that every finitely generated $R$-module $M$ is associated by a sequence of invariant modules. This modules sequence expresses the deviation of $M$ with the Cohen-Macaulay property. This result generalizes the unmixed theorem of Cohen-Macaulayness for any finitely generated $R$-module. As an application we construct a new extended degree in sense of W. Vasconcelos.