Splittable ideals and the resolutions of monomial ideals

DOI10.1016/j.jalgebra.2006.08.022zbMath1151.13017arXivmath/0503203OpenAlexW2069797376MaRDI QIDQ875092

Adam Van Tuyl, Huy Tái Hà

Publication date: 11 April 2007

Published in: Journal of Algebra (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0503203

zbMATH Keywords

regularitygraphfree resolutionsimplicial complexBetti numberforestedge idealmonomial ideals

Mathematics Subject Classification ID

Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes (13F55) Structure, classification theorems for modules and ideals in commutative rings (13C05) Syzygies, resolutions, complexes and commutative rings (13D02)

Related Items (38)

Betti splitting via componentwise linear ideals ⋮ ARITHMETICAL RANK OF THE CYCLIC AND BICYCLIC GRAPHS ⋮ Stanley-Reisner Rings ⋮ Splittings of monomial ideals ⋮ Bounds on the regularity and projective dimension of ideals associated to graphs ⋮ Associated primes of monomial ideals and odd holes in graphs ⋮ Koszul homology of quotients by edge ideals ⋮ Multi-graded Betti numbers of path ideals of trees ⋮ Betti numbers of monomial ideals via facet covers ⋮ Subadditivity, strand connectivity and multigraded Betti numbers of monomial ideals ⋮ Regularity of powers of forests and cycles ⋮ Linear resolutions of t-spread lexsegment ideals via Betti splittings ⋮ Path ideals of rooted trees and their graded Betti numbers ⋮ Multigraded Betti numbers of simplicial forests ⋮ Tetrahedral curves via graphs and Alexander duality ⋮ Betti splitting from a topological point of view ⋮ The reconstruction conjecture and edge ideals ⋮ Whiskers and sequentially Cohen-Macaulay graphs ⋮ Cellular Structure on the Minimal Resolution of the Edge Ideal of the Complement of then-Cycle ⋮ Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers ⋮ Algebraic Properties of the Path Ideal of a Tree ⋮ The regularity of powers of edge ideals ⋮ Bass numbers of local cohomology of cover ideals of graphs ⋮ In the shadows of a hypergraph: looking for associated primes of powers of square-free monomial ideals ⋮ Lattice complements and the subadditivity of syzygies of simplicial forests ⋮ Nonvanishing of Betti Numbers of Edge Ideals and Complete Bipartite Subgraphs ⋮ Monomial and toric ideals associated to Ferrers graphs ⋮ Linearity defect of edge ideals and Fröberg's theorem ⋮ Bounding Betti numbers of bipartite graph ideals ⋮ Projective dimension and regularity of powers of edge ideals of vertex-weighted rooted forests ⋮ Betti numbers of graphs with an application to anomaly detection ⋮ Algebraic properties of edge ideals of some vertex-weighted oriented \(m\)-partite graphs ⋮ Betti Numbers of Hypergraphs ⋮ Betti numbers of some circulant graphs ⋮ Projective dimension and regularity of edge ideals of some weighted oriented graphs ⋮ Some algebraic properties of hypergraphs ⋮ Pruned cellular free resolutions of monomial ideals ⋮ On the Linear Strand of an Edge Ideal

Uses Software

• CoCoA

Cites Work

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• Koszul cohomology and the geometry of projective varieties. Appendix: The nonvanishing of certain Koszul cohomology groups (by Mark Green and Robert Lazarsfeld)
• Minimal resolutions of some monomial ideals
• Resolutions of Stanley-Reisner rings and Alexander duality
• On the ideal theory of graphs
• Cohen-Macaulay graphs
• The facet ideal of a simplicial complex
• Simplicial trees are sequentially Cohen-Macaulay
• Dirac's theorem on chordal graphs and Alexander duality
• Characteristic-independence of Betti numbers of graph ideals
• Whiskers and sequentially Cohen-Macaulay graphs
• Cohen--Macaulay chordal graphs
• Koszul modules
• Resolutions of Facet Ideals
• Restricting linear syzygies: algebra and geometry
• Rees algebras of edge ideals
• On the resolution of ideals of fat points

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