scholarly journals Free resolutions of Dynkin format and the licci property of grade $3$ perfect ideals

2019 ◽  
Vol 125 (2) ◽  
pp. 163-178
Author(s):  
Lars Winther Christensen ◽  
Oana Veliche ◽  
Jerzy Weyman
Keyword(s):  
Recent Work ◽  
Local Ring ◽  
Complete Intersection ◽  
Dynkin Diagram ◽  
Free Resolution ◽  
Free Resolutions ◽  
Regular Local Ring ◽  
Minimal Free Resolution ◽  
Grade 3

Recent work on generic free resolutions of length $3$ attaches to every resolution a graph and suggests that resolutions whose associated graph is a Dynkin diagram are distinguished. We conjecture that in a regular local ring, every grade $3$ perfect ideal whose minimal free resolution is distinguished in this way is in the linkage class of a complete intersection.

scholarly journals ON THE ASYMPTOTIC BEHAVIOR OF THE LINEARITY DEFECT

2017 ◽  
Vol 230 ◽  
pp. 35-47 ◽  
Author(s):  
HOP D. NGUYEN ◽  
THANH VU
Keyword(s):  
Asymptotic Behavior ◽  
Local Ring ◽  
General Result ◽  
Free Resolution ◽  
Graded Algebra ◽  
Regular Local Ring ◽  
Finitely Generated ◽  
Minimal Free Resolution ◽  
Noetherian Local Ring ◽  
Graded Module

This work concerns the linearity defect of a module $M$ over a Noetherian local ring $R$, introduced by Herzog and Iyengar in 2005, and denoted $\text{ld}_{R}M$. Roughly speaking, $\text{ld}_{R}M$ is the homological degree beyond which the minimal free resolution of $M$ is linear. It is proved that for any ideal $I$ in a regular local ring $R$ and for any finitely generated $R$-module $M$, each of the sequences $(\text{ld}_{R}(I^{n}M))_{n}$ and $(\text{ld}_{R}(M/I^{n}M))_{n}$ is eventually constant. The first statement follows from a more general result about the eventual constancy of the sequence $(\text{ld}_{R}C_{n})_{n}$ where $C$ is a finitely generated graded module over a standard graded algebra over $R$.

scholarly journals Gluing semigroups and strongly indispensable free resolutions

2019 ◽  
Vol 29 (02) ◽  
pp. 263-278
Author(s):  
Mesut Şahi̇n ◽  
Leah Gold Stella
Keyword(s):  
Complete Intersection ◽  
Special Property ◽  
Free Resolution ◽  
Numerical Semigroups ◽  
Free Resolutions ◽  
Semigroup Rings ◽  
Minimal Free Resolution ◽  
Minimal Free Resolutions ◽  
Symmetric Numerical Semigroups

We study strong indispensability of minimal free resolutions of semigroup rings focusing on the operation of gluing used in the literature to take examples with a special property and produce new ones. We give a naive condition to determine whether gluing of two semigroup rings has a strongly indispensable minimal free resolution. As applications, we determine simple gluings of [Formula: see text]-generated non-symmetric, [Formula: see text]-generated symmetric and pseudo symmetric numerical semigroups as well as obtain infinitely many new complete intersection semigroups of any embedding dimensions, having strongly indispensable minimal free resolutions.

scholarly journals Stably free resolutions of lattices over finite groups

1990 ◽  
Vol 49 (3) ◽  
pp. 364-385
Author(s):  
K. W. Gruenberg
Keyword(s):  
Euler Characteristic ◽  
Finite Groups ◽  
Class Group ◽  
Free Resolution ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Residue Classes ◽  
Projective Class

AbstractFor a ZG-lattice A, the nth partial free Euler characteristic εn(A) is defined as the infimum of all where F* varies over all free resolutions of A. It is shown that there exists a stably free resolution E* of A which realises εn(A) for all n≥0 and that the function n → εn(A) is ultimately polynomial no residue classes. The existence of E* is established with the help of new invariants σn(A) of A. These are elements in certain image groups of the projective class group of ZG. When ZG allows cancellation, E* is a minimal free resolution and is essentially unique. When A is periodic, E* is ultimately periodic of period a multiple of the projective period of A.

scholarly journals Determinantal ideals without minimal free resolutions

1990 ◽  
Vol 118 ◽  
pp. 203-216 ◽  
Author(s):  
Mitsuyasu Hashimoto
Keyword(s):  
Free Resolution ◽  
Unit Element ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Determinantal Ideals ◽  
Arbitrary Base ◽  
Generic Matrix ◽  
Base Ring ◽  
Minimal Free Resolutions ◽  
The Ideal

Let R be a Noetherian commutative ring with, unit element, and Xij be variables with 1 ≤ i ≤ m and 1 ≤ j ≤ n. Let S = R[xij] be the polynomial ring over R, and It be the ideal in S, generated by the t × t minors of the generic matrix (xij) ∈ Mm, n(S). For many years there has been considerable interest in finding a minimal free resolution of S/It, over arbitrary base ring R. If we have a minimal free resolution P. over R = Z, the ring of integers, then R′ ⊗z P. is a resolution of S/It over the base ring R′.

The Minimal Free Resolution of Fat Almost Complete Intersections in ℙ1 × ℙ1

2017 ◽  
Vol 69 (6) ◽  
pp. 1274-1291 ◽  
Author(s):  
Giuseppe Favacchio ◽  
Elena Guardo
Keyword(s):  
Complete Intersection ◽  
Free Resolution ◽  
Complete Intersections ◽  
Minimal Free Resolution ◽  
Symbolic Powers ◽  
Research Theme ◽  
Homogeneous Set ◽  
Triple Points ◽  
Set Of Points ◽  
A Current

AbstractA current research theme is to compare symbolic powers of an ideal I with the regular powers of I. In this paper, we focus on the case where I = IX is an ideal deûning an almost complete intersection (ACI) set of points X in ℙ1 × ℙ1. In particular, we describe a minimal free bigraded resolution of a non-arithmetically Cohen-Macaulay (also non-homogeneous) set 𝒵 of fat points whose support is an ACI, generalizing an earlier result of Cooper et al. for homogeneous sets of triple points. We call 𝒵 a fat ACI.We also show that its symbolic and ordinary powers are equal, i.e, .

𝐿-dimension for modules over a local ring

2021 ◽  
pp. 83-91
Author(s):  
Courtney Gibbons ◽  
David Jorgensen ◽  
Janet Striuli
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Residue Field ◽  
Free Resolution ◽  
Homological Dimension ◽  
Finitely Generated ◽  
Content Type ◽  
Commutative Local Ring ◽  
Finitely Generated Modules

We introduce a new homological dimension for finitely generated modules over a commutative local ring R R , which is based on a complex derived from a free resolution L L of the residue field of R R , and called L L -dimension. We prove several properties of L L -dimension, give some applications, and compare L L -dimension to complete intersection dimension.

Minimal Free Resolutions of Zero-dimensional Schemes in ℙ1 × ℙ1

2015 ◽  
Vol 22 (01) ◽  
pp. 97-108 ◽  
Author(s):  
Paola Bonacini ◽  
Lucia Marino
Keyword(s):  
Free Resolution ◽  
Hilbert Functions ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Minimal Set ◽  
Minimal Free Resolutions ◽  
The Ideal

Let X be a zero-dimensional scheme in ℙ1 × ℙ1. Then X has a minimal free resolution of length 2 if and only if X is ACM. In this paper we determine a class of reduced schemes whose resolutions, similarly to the ACM case, can be obtained by their Hilbert functions and depend only on their distributions of points in a grid of lines. Moreover, a minimal set of generators of the ideal of these schemes is given by curves split into the union of lines.

Multi-graded Betti numbers of path ideals of trees

2017 ◽  
Vol 16 (01) ◽  
pp. 1750018 ◽  
Author(s):  
Rachelle R. Bouchat ◽  
Tricia Muldoon Brown
Keyword(s):  
Betti Numbers ◽  
Free Resolution ◽  
Explicit Description ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Generating Set ◽  
Graded Betti Numbers ◽  
Minimal Free Resolutions ◽  
Vertex Sets ◽  
Minimal Generating Set

A path ideal of a tree is an ideal whose minimal generating set corresponds to paths of a specified length in a tree. We provide a description of a collection of induced subtrees whose vertex sets correspond to the multi-graded Betti numbers on the linear strand in the corresponding minimal free resolution of the path ideal. For two classes of path ideals, we give an explicit description of a collection of induced subforests whose vertex sets correspond to the multi-graded Betti numbers in the corresponding minimal free resolutions. Lastly, in both classes of path ideals considered, the graded Betti numbers are explicitly computed for [Formula: see text]-ary trees.

scholarly journals Self-linked curve singularities

1990 ◽  
Vol 120 ◽  
pp. 129-153 ◽  
Author(s):  
Jürgen Herzog ◽  
Bernd Ulrich
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Complete Intersection ◽  
Three Dimensional ◽  
Regular Local Ring ◽  
Rees Algebra ◽  
Curve Singularities ◽  
Noetherian Property

Let S be a three-dimensional regular local ring and let I be a prime ideal in S of height two. This paper is motivated by the question of when I is a set-theoretic complete intersection and when the symbolic Rees algebra S(I) = ⊕n≥0I(n)tn is Noetherian. The connection between the two problems is given by a result of Cowsik which says that the Noetherian property of S(I) implies that I is a set-theoretic complete intersection ([1]).

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