scholarly journals Betti Numbers of Cut Ideals of Trees

2013 ◽  
Vol 4 (1) ◽  
Author(s):  
Samu Potka ◽  
Camilo Sarmiento
Keyword(s):  
Betti Numbers ◽  
Upper Bounds ◽  
Free Resolutions ◽  
Combinatorial Topology ◽  
Induced Subgraphs ◽  
Tree Graphs ◽  
Minimal Free Resolutions

Cut ideals, introduced by Sturmfels and Sullivant, are used in phylogenetics and algebraicstatistics. We study the minimal free resolutions of cut ideals of tree graphs. By employingbasic methods from combinatorial topology, we obtain upper bounds for the Betti numbers of thistype of ideals. These take the form of simple formulas on the number of vertices, which arise fromthe enumeration of induced subgraphs of certain incomparability graphs associated to the edgesets of trees.

scholarly journals Bundles on with vanishing lower cohomologies

2020 ◽  
pp. 1-20
Author(s):  
Mengyuan Zhang
Keyword(s):  
Hilbert Function ◽  
Betti Numbers ◽  
Projective Spaces ◽  
Hilbert Functions ◽  
Free Resolutions ◽  
Rational Varieties ◽  
Minimal Free Resolutions ◽  
Graded Lattice

Abstract We study bundles on projective spaces that have vanishing lower cohomologies using their short minimal free resolutions. We partition the moduli $\mathcal{M}$ according to the Hilbert function H and classify all possible Hilbert functions H of such bundles. For each H, we describe a stratification of $\mathcal{M}_H$ by quotients of rational varieties. We show that the closed strata form a graded lattice given by the Betti numbers.

scholarly journals Nearly extremal Cohen–Macaulay and Gorenstein algebras

2007 ◽  
Vol 75 (2) ◽  
pp. 211-220 ◽  
Author(s):  
Chanchal Kumar ◽  
Pavinder Singh ◽  
Ashok Kumar
Keyword(s):  
Betti Numbers ◽  
Free Resolutions ◽  
Paper Study ◽  
Gorenstein Algebras ◽  
Graded Betti Numbers ◽  
Minimal Free Resolutions ◽  
Explicit Bounds

This paper study nearly extremal Cohen–Macaulay and Gorenstein algebras and characterise them in terms of their minimal free resolutions. Explicit bounds on their graded Betti numbers and their multiplicities are obtained.

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 RESOLUTIONS AND COHOMOLOGIES OF TORIC SHEAVES: THE AFFINE CASE

2013 ◽  
Vol 24 (09) ◽  
pp. 1350069
Author(s):  
MARKUS PERLING
Keyword(s):  
Polynomial Ring ◽  
Local Cohomology ◽  
Hyperplane Arrangement ◽  
Betti Numbers ◽  
Vector Spaces ◽  
Free Resolutions ◽  
Graded Betti Numbers ◽  
Higher Rank ◽  
Minimal Free Resolutions ◽  
Reflexive Modules

We study equivariant resolutions and local cohomologies of toric sheaves for affine toric varieties, where our focus is on the construction of new examples of indecomposable maximal Cohen–Macaulay modules of higher rank. A result of Klyachko states that the category of reflexive toric sheaves is equivalent to the category of vector spaces together with a certain family of filtrations. Within this setting, we develop machinery which facilitates the construction of minimal free resolutions for the smooth case as well as resolutions which are acyclic with respect to local cohomology functors for the general case. We give two main applications. First, over the polynomial ring, we determine in explicit combinatorial terms the ℤn-graded Betti numbers and local cohomology of reflexive modules whose associated filtrations form a hyperplane arrangement. Second, for the nonsmooth, simplicial case in dimension d ≥ 3, we construct new examples of indecomposable maximal Cohen–Macaulay modules of rank d – 1.

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 Sharp upper bounds for the Betti numbers of a given Hilbert polynomial

2013 ◽  
Vol 7 (5) ◽  
pp. 1019-1064 ◽  
Author(s):  
Giulio Caviglia ◽  
Satoshi Murai
Keyword(s):  
Betti Numbers ◽  
Upper Bounds ◽  
Hilbert Polynomial
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