scholarly journals Betti numbers and shifts in minimal graded free resolutions

2010 ◽  
Vol 54 (2) ◽  
pp. 449-467 ◽  
Author(s):  
Tim Römer
Keyword(s):  
Betti Numbers ◽  
Free Resolutions ◽  
Graded Free Resolutions ◽  
And Shifts

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 Minimal graded free resolutions for monomial curves defined by arithmetic sequences

2013 ◽  
Vol 388 ◽  
pp. 294-310 ◽  
Author(s):  
Philippe Gimenez ◽  
Indranath Sengupta ◽  
Hema Srinivasan
Keyword(s):  
Free Resolutions ◽  
Monomial Curves ◽  
Arithmetic Sequences ◽  
Graded Free Resolutions

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.

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.

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.

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