scholarly journals Linear Maps in Minimal Free Resolutions of Stanley-Reisner Rings

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 605
Author(s):  
Lukas Katthän
Keyword(s):  
Simplicial Complex ◽  
Linear Part ◽  
Monomial Ideal ◽  
Short Note ◽  
Free Resolution ◽  
Free Resolutions ◽  
Restriction Maps ◽  
Minimal Free Resolution ◽  
Linear Maps ◽  
Minimal Free Resolutions

In this short note we give an elementary description of the linear part of the minimal free resolution of a Stanley-Reisner ring of a simplicial complex Δ . Indeed, the differentials in the linear part are simply a compilation of restriction maps in the simplicial cohomology of induced subcomplexes of Δ . Along the way, we also show that if a monomial ideal has at least one generator of degree 2, then the linear strand of its minimal free resolution can be written using only ± 1 coefficients.

scholarly journals Minimal free resolutions of 2 × n domino tilings

2019 ◽  
Vol 18 (06) ◽  
pp. 1950118
Author(s):  
Rachelle R. Bouchat ◽  
Tricia Muldoon Brown
Keyword(s):  
Monomial Ideal ◽  
Betti Numbers ◽  
Projective Dimension ◽  
Free Resolution ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Domino Tilings ◽  
Minimal Free Resolutions ◽  
The Ideal

We introduce a squarefree monomial ideal associated to the set of domino tilings of a [Formula: see text] rectangle and proceed to study the associated minimal free resolution. In this paper, we use results of Dalili and Kummini to show that the Betti numbers of the ideal are independent of the underlying characteristic of the field, and apply a natural splitting to explicitly determine the projective dimension and Castelnuovo–Mumford regularity of the ideal.

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′.

scholarly journals Linear syzygies of Stanley-Reisner ideals

2001 ◽  
Vol 89 (1) ◽  
pp. 117 ◽  
Author(s):  
V Reiner ◽  
V Welker
Keyword(s):  
Simplicial Complex ◽  
Lower Bounds ◽  
Monomial Ideal ◽  
Independent Sets ◽  
Free Resolution ◽  
Monomial Ideals ◽  
Minimal Free Resolution ◽  
Linear Syzygies ◽  
Alexander Dual ◽  
Syzygy Modules

We give an elementary description of the maps in the linear strand of the minimal free resolution of a square-free monomial ideal, that is, the Stanley-Reisner ideal associated to a simplicial complex $\Delta$. The description is in terms of the homology of the canonical Alexander dual complex $\Delta^*$. As applications we are able to prove for monomial ideals and $j=1$ a conjecture of J. Herzog giving lower bounds on the number of $i$-syzygies in the linear strand of $j^{th}$-syzygy modules show that the maps in the linear strand can be written using only $\pm 1$ coefficients if $\Delta^*$ is a pseudomanifold exhibit an example where multigraded maps in the linear strand cannot be written using only $\pm 1$ coefficients compute the entire resolution explicitly when $\Delta^*$ is the complex of independent sets of a matroid

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 Projective covers and minimal free resolutions

1996 ◽  
Vol 19 (1) ◽  
pp. 185-192
Author(s):  
Mark A. Goddard
Keyword(s):  
Sufficient Conditions ◽  
Free Resolution ◽  
Projective Cover ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Direct Sum ◽  
Minimal Free Resolutions ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Projective Covers

Using a generalization of the definition of the projective cover of a module, a special type of surjective free resolution, known as the projective cover of a complex, may be defined. The projective cover is shown to be a direct summand of every surjective free resolution and to be the direct sum of the minimal free resolution and an exact complex. Necessary and sufficient conditions for the projective cover and minimal free resolution to be identical are discussed.

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 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.

On multigraded resolutions

1995 ◽  
Vol 118 (2) ◽  
pp. 245-257 ◽  
Author(s):  
Winfried Bruns ◽  
Jürgen Herzog
Keyword(s):  
Monomial Ideal ◽  
Free Resolution ◽  
Suitable Choice ◽  
Minimal Free Resolution ◽  
Common Multiple ◽  
Least Common Multiple ◽  
The Ideal

This paper was initiated by a question of Eisenbud who asked whether the entries of the matrices in a minimal free resolution of a monomial ideal (which, after a suitable choice of bases, are monomials) divide the least common multiple of the generators of the ideal. We will see that this is indeed the case, and prove it by lifting the multigraded resolution of an ideal, or more generally of a multigraded module, keeping track of how the shifts ‘deform’' in such a lifting; see Theorem 2·1 and Corollary 2·2.

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