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Linear Strands Supported on Cell Complexes
TLDR
In this article, Almousa, Fl{\o}ystad, and Lohne showed that a certain class of rainbow monomial ideals always have linear strand supported on a cell complex, including any initial ideal of the ideal of maximal minors of a generic matrix.Abstract:
In this paper, we study ideals $I$ whose linear strand can be supported on a
polyhedral cell complex We provide a sufficient condition for the linear
strand of an arbitrary subideal of $I$ to remain supported on an easily
described subcomplex In particular, we prove that a certain class of rainbow
monomial ideals always have linear strand supported on a cell complex,
including any initial ideal of the ideal of maximal minors of a generic matrix
This follows from a general statement on the cellularity of complexes whose
associated poset forms a meet semilattice We also provide a sufficient
condition for these ideals to have linear resolution, which is also an
equivalence under mild assumptions We employ a result of Almousa, Fl{\o}ystad,
and Lohne to apply these results to polarizations of Artinian monomial ideals
We conclude with further questions relating to cellularity of certain classes
of squarefree monomial ideals and the relationship between initial ideals of
maximal minors and algebra structures on certain resolutionsread more
References
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Book
Combinatorial Commutative Algebra
Ezra Miller,Bernd Sturmfels +1 more
TL;DR: In this paper, the authors present a set of monomial ideals for three-dimensional staircases and cellular resolutions, including two-dimensional lattice ideals, and a threedimensional staircase with cellular resolutions.
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TL;DR: It follows that all binomial edge ideals are radical ideals, and the results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones.
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Graphs and Ideals Generated by Some 2-Minors
TL;DR: In this article, the Grobner basis of I G of S generated by 2-minors [i, j] of X which correspond to edges of G was constructed.
Book ChapterDOI
Lattice Walks and Primary Decomposition
TL;DR: It is shown how primary decompositions of an ideal can give useful descriptions of components of a graph arising in problems from combinatorics, statistics, and operations research.