Trimming complexes and applications to resolutions of determinantal facet ideals
TL;DR: In this article, a family of complexes called trimming complexes is proposed and used to deduce the Betti table for the minimal free resolution problem. But the complexity of trimming complex is not fixed.
Abstract: We produce a family of complexes called trimming complexes and explore its applications. We demonstrate how trimming complexes can be used to deduce the Betti table for the minimal free resolution ...
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TL;DR: In this paper, it was shown that all ideals of the above form are resolved by an iterated trimming complex, such that R / I is a ring of Tor algebra class G ( r ) for some fixed r ⩾ 2, and R/I may be chosen to have arbitrarily large type.
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TL;DR: In this paper, the authors define and study the notion of determinantal facet ideal (DFI) which is generated by a subset of maximal minors of a generic matrix indexed by the facets of a simplicial complex.
Abstract: A determinantal facet ideal (DFI) is generated by a subset of the maximal minors of a generic $n\times m$ matrix indexed by the facets of a simplicial complex. We consider the more general notion of an $r$-DFI, which is generated by a subset of $r$-minors of a generic matrix indexed by the facets of a simplicial complex for some $1\leq r\leq n$. We define and study so-called lcm-closed and interval DFIs, and show that the minors parametrized by the facets of $\Delta$ form a reduced Grobner basis with respect to any diagonal term order in both of these cases. We also see that being lcm-closed generalizes conditions previously introduced in the literature, and conjecture that in the case $r=n$, lcm-closedness is necessary for being a Grobner basis. We also give conditions on the maximal cliques of $\Delta$ ensuring that lcm-closed and interval DFIs are Cohen-Macaulay. Finally, we conclude with a variant of a conjecture of Ene, Herzog, and Hibi on the Betti numbers of certain types of $r$-DFIs, and provide a proof of this conjecture for Cohen-Macaulay interval DFIs.
4 citations
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TL;DR: In this article, it was shown that all ideals of the above form are resolved by an iterated trimming complex, which is a new conjecture of Avramov not already constructed by Christensen, Veliche, and Weyman.
Abstract: Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$ denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$ defining compressed rings with socle $k(-s)^{\ell} \oplus k(-2s+1)$, where $s \geq3$ and $\ell \geq 1$ are integers. The case for $\ell =1$ was studied in a previous paper by the author; a generically minimal resolution was constructed for all such ideals. More recently, this resolution is generalized in the guise of (iterated) trimming complexes. In this paper, we show that all ideals of the above form are resolved by an iterated trimming complex. Moreover, we apply this machinery to construct ideals $I$ such that $R/I$ is a ring of Tor algebra class $G (r)$ for some fixed $r \geq2$, and $R/I$ may be chosen to have arbitrarily large type. In particular, this provides a new class of counterexamples to a conjecture of Avramov not already constructed by Christensen, Veliche, and Weyman.
3 citations
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TL;DR: In this article, the iterated trimming complex associated to data yielding a complex of length $3 was considered, and an explicit algebra structure in this complex was computed in terms of the algebra structures of the associated input data.
Abstract: In this paper, we consider the iterated trimming complex associated to data yielding a complex of length $3$. We compute an explicit algebra structure in this complex in terms of the algebra structures of the associated input data. Moreover, it is shown that many of these products become trivial after descending to homology. We apply these results to the problem of realizability for Tor-algebras of grade $3$ perfect ideals, and show that under mild hypotheses, the process of "trimming" an ideal preserves Tor-algebra class. In particular, we construct new classes of ideals in arbitrary regular local rings defining rings realizing Tor-algebra classes $G(r)$ and $H(p,q)$ for a prescribed set of homological data.
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30 Mar 1995
TL;DR: In this article, the authors define basic constructions and dimension theory, and apply them to the problem of homological methods for combinatorial problem solving in the context of homology.
Abstract: Introduction.- Elementary Definitions.- I Basic Constructions.- II Dimension Theory.- III Homological Methods.- Appendices.- Hints and Solutions for Selected Exercises.- References.- Index of Notation.- Index.
5,674 citations
671 citations
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.
244 citations
TL;DR: In this article, the authors give a description of the minimal primes of the ideal generated by the 2×2 adjacent minors of a generic matrix and show that the ideals which appear as minimal prime ideals are, in fact, prime ideals.
53 citations
TL;DR: In this paper, it was shown that there is a real number γ > 1, such that μ R d + i ≥ γ μ Rd + i − 1 holds for all i ≥ 0, except for i = 2 in two explicitly described cases.
33 citations