Open accessJournal Article

# Trimming complexes and applications to resolutions of determinantal facet ideals

04 Mar 2021-Communications in Algebra (Informa UK Limited)-Vol. 49, Iss: 3, pp 1017-1036
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 ...

Topics: Facet (geometry) (55%), Trimming (53%)
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Open accessJournal Article
15 Nov 2021-Journal of Algebra
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 ⊆ R defining compressed rings with socle k ( − s ) l ⊕ k ( − 2 s + 1 ) , where s ⩾ 3 and l ⩾ 1 are integers. The case for l = 1 was previously studied in [8] ; a generically minimal resolution was constructed for all such ideals. The paper [7] generalizes this resolution 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 ⩾ 2 , 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 in [5] .

Topics: Polynomial ring (56%), Ring (mathematics) (53%), Resolution (algebra) (51%) ... show more

4 Citations

Open accessPosted Content
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.

Topics: Simplicial complex (56%), Betti number (55%), Ideal (ring theory) (51%)

4 Citations

Open accessPosted Content
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.

Topics: Polynomial ring (55%), Type (model theory) (54%), Ring (mathematics) (52%) ... show more

3 Citations

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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.

Topics: Local ring (56%), Algebraic structure (55%), Realizability (52%) ... show more

1 Citations

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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 resolutions

Topics: Ideal (set theory) (57%), Monomial (57%), Semilattice (51%)

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10 results found

Open accessBook
30 Mar 1995-
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.

5,404 Citations

Open accessJournal Article
Abstract: We introduce binomial edge ideals attached to a simple graph G and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Grobner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and claw-free. We give a reduced squarefree Grobner basis for general G. It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of G. We provide sufficient conditions for Cohen-Macaulayness for closed and nonclosed graphs. Binomial edge ideals arise naturally in the study of conditional independence ideals. Our 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. In this case the primary decomposition has a natural statistical interpretation.

202 Citations

Open accessJournal Article
Serkan Hosten1, Seth Sullivant2Institutions (2)
15 Jul 2004-Journal of Algebra
Abstract: We give a description of the minimal primes of the ideal generated by the 2×2 adjacent minors of a generic matrix. We also compute the complete prime decomposition of the ideal of adjacent m×m minors of an m×n generic matrix when the characteristic of the ground field is zero. A key intermediate result is the proof that the ideals which appear as minimal primes are, in fact, prime ideals. This introduces a large new class of mixed determinantal ideals that are prime.

47 Citations

Open accessJournal Article
Luchezar L. Avramov1Institutions (1)
Abstract: The generating series of the Bass numbers μ R i = rank k Ext R i ( k , R ) of local rings R with residue field k are computed in closed rational form, in case the embedding dimension e of R and its depth d satisfy e − d ≤ 3 . For each such R it is proved that there is a real number γ > 1 , such that μ R d + i ≥ γ μ R d + i − 1 holds for all i ≥ 0 , except for i = 2 in two explicitly described cases, where μ R d + 2 = μ R d + 1 = 2 . New restrictions are obtained on the multiplicative structures of minimal free resolutions of length 3 over regular local rings.

Topics: , Residue field (52%), Local ring (50%)

31 Citations

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