In this paper we study the resolution of a facet ideal associated with a special class of simplicial complexes introduced by S. Faridi. These simplicial complexes are called trees, and are a generalization (to higher dimensions) of the concept of a tree in graph theory. We show that the Koszul homology of the facet ideal I of a tree is generated by the homology classes of monomial cycles, determine the projective dimension and the regularity of I if the tree is 1-dimensional, show that the graded Betti numbers of I satisfy an alternating sum property if the tree is connected in codimension 1, and classify all trees whose facet ideal has a linear resolution.
In this paper we describe the irreducible decomposition of the facet ideal $\F(Δ_{m,n})$ of the chessboard complex $Δ_{m,n}$ with $n\geq m$. We also provide some lower bounds for depth and regularity of the facet ideal $\F(Δ_{m,n})$. When $m\leq 3$, we prove that these lower bounds can be obtained.
We give a sufficient condition for a monomial ideal to have a nonzero Betti number in each multidegree. In the case of facet ideals of simplicial forests, this condition becomes a necessary one and it allows us to characterize Betti numbers, projective dimension and regularity of such ideals combinatorially. Our condition is expressed in terms of minimal facet covers of simplicial complexes.
We consider the face ideal associated to a line-type simplicial complex. We compute the \texttt{depth} and the \texttt{sdepth} for its quotient ring. In particular, the facet ideal and its quotient ring satisfy the Stanley inequality.
We produce a family of complexes called trimming complexes and explore applications. We study how trimming complexes can be used to deduce the Betti table for the minimal free resolution of the ideal generated by subsets of a generating set for an arbitrary ideal $I$. In particular, we compute the Betti table for removing an arbitrary generator from the ideal of submaximal pfaffians of a generic skew symmetric matrix $M$. We also explicitly compute the Betti table for the ideal generated by certain subsets of the generating set of the ideal of maximal minors of a generic $n \times m$ matrix. S…
A determinantal facet ideal (DFI) is an ideal $J_Δ$ generated by maximal minors of a generic matrix parametrized by an associated simplicial complex $Δ$. In this paper, we construct an explicit linear strand for the initial ideal with respect to any diagonal term order $<$ of an arbitrary DFI. In particular, we show that if $Δ$ has no \emph{1-nonfaces}, then the Betti numbers of the linear strand of $J_Δ$ and its initial ideal coincide. We apply this result to prove a conjecture of Ene, Herzog, and Hibi on Betti numbers of closed binomial edge ideals in the case that the associated graph has a…
We consider ideals generated by general sets of $m$-minors of an $m\times n$-matrix of indeterminates. The generators are identified with the facets of an $(m-1)$-dimensional pure simplicial complex. The ideal generated by the minors corresponding to the facets of such a complex is called a determinantal facet ideal. Given a pure simplicial complex $Δ$, we discuss the question when the generating minors of its determinantal facet ideal $J_Δ$ form a Gröbner basis and when $J_Δ$ is a prime ideal.
We survey some recent results on the minimal graded free resolution of a square-free monomial ideal. The theme uniting these results is the point-of-view that the generators of a monomial ideal correspond to the maximal faces (the facets) of a simplicial complex Δ.
Let $L_n$ be a line graph with $n$ edges and $\F(L_n)$ the facet ideal of its matching complex. In this paper, we provide the irreducible decomposition of $\F(L_n)$ and some exact formulas for the projective dimension and the regularity of $\F(L_n)$.
For a simplicial complex $Δ$, the affect of the expansion functor on combinatorial properties of $Δ$ and algebraic properties of its Stanley-Reisner ring has been studied in some previous papers. In this paper, we consider the facet ideal $I(Δ)$ and its Alexander dual which we denote by $J_Δ$ to see how the expansion functor alter the algebraic properties of these ideals. It is shown that for any expansion $Δ^α$ the ideals $J_Δ$ and $J_{Δ^α}$ have the same total Betti numbers and their Cohen-Macaulayness are equivalent, which implies that the regularities of the ideals $I(Δ)$ and $I(Δ^α)$ are …
A determinantal facet ideal (DFI) is generated by a subset of the maximal minors of a generic $n\times m$ matrix where $n\leq m$ indexed by the facets of a simplicial complex $Δ$. 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 $Δ$ for some $1\leq r\leq n$. We define and study so-called lcm-closed and unit interval $r$-DFIs, and show that the minors parametrized by the facets of $Δ$ form a reduced Gröbner basis with respect to \emph{any} term order for an lcm-closed $r$-DFI. We also see that being lcm-…
To a simplicial complex, we associate a square-free monomial ideal in the polynomial ring generated by its vertex set over a field. We study algebraic properties of this ideal via combinatorial properties of the simplicial complex. By generalizing the notion of a tree from graphs to simplicial complexes, we show that ideals associated to trees satisfy sliding depth condition, and therefore have normal and Cohen-Macaulay Rees rings. We also discuss connections with the theory of Stanley-Reisner rings.
Let $X$ be an $(m\times n)$-matrix of indeterminates, and let $J$ be the ideal generated by a set $\mathcal{S}$ of maximal minors of $X$. We construct the linear strand of the resolution of $J$. This linear strand is determined by the clique complex of the $m$-clutter corresponding to the set $\mathcal{S}$. As a consequence one obtains explicit formulas for the graded Betti numbers $β_{i,i+m}(J)$ for all $i\geq 0$. We also determine all sets $\mathcal{S}$ for which $J$ has a linear resolution.
To a matroid M with n edges, we associate the so-called facet ideal F(M) generated by monomials corresponding to bases of M. We show that the Betti numbers related to an N-graded minimal free resolution of F(M) are determined by the Betti numbers related to the blocks of M. Similarly, we show that the higher weight hierarchy of M is determined by the weight hierarchies of the blocks, as well. Drawing on these results, we show that when M is the cycle matroid of a cactus graph, the Betti numbers determine the higher weight hierarchy -- and vice versa. Finally, we demonstrate by way of counterex…
We study d-dimensional generalizations of three mutually related topics in graph theory: Hamiltonian paths, (unit) interval graphs, and binomial edge ideals. We provide partial high-dimensional generalizations of Ore and Posa's sufficient conditions for a graph to be Hamiltonian. We introduce a hierarchy of combinatorial properties for simplicial complexes that generalize unit-interval, interval, and co-comparability graphs. We connect these properties to the already existing notions of determinantal facet ideals and Hamiltonian paths in simplicial complexes. Some important consequences of our…
In this short note we prove that the projective dimension of a sequentially Cohen-Macaulay square-free monomial ideal is equal to the maximal height of its minimal primes (also known as the big height), or equivalently, the maximal cardinality of a minimal vertex cover of its facet complex. This in particular gives a formula for the projective dimension of facet ideals of these classes of ideals, which are known to be sequentially Cohen-Macaulay: graph trees and simplicial trees, chordal graphs and some cycles, chordal clutters and graphs, and some path ideals to mention a few. Since polarizat…
Using the existence of a good leaf in every simplicial tree, we order the facets of a simplicial tree in order to find combinatorial information about the Betti numbers of its facet ideal. Applications include an Eliahou-Kervaire splitting of the ideal, as well as a refinement of a recursive formula of Hà and Van Tuyl for computing the graded Betti numbers of simplicial trees.
In this paper we study simplicial complexes as higher dimensional graphs in order to produce algebraic statements about their facet ideals. We introduce a large class of square-free monomial ideals with Cohen-Macaulay quotients, and a criterion for the Cohen-Macaulayness of facet ideals of simplicial trees. Along the way, we generalize several concepts from graph theory to simplicial complexes.
We outline a generalization of the Groebner fan of a homogeneous ideal with maximal cells parametrizing truncated Groebner bases. This "truncated" Groebner fan is usually much smaller than the full Groebner fan and offers the natural framework for conversion between truncated Groebner bases. The generic Groebner walk generalizes naturally to this setting by using the Buchberger algorithm with truncation on facets. We specialize to the setting of lattice ideals. Here facets along the generic walk are given by unique (facet) binomials. This along with the representation of binomials as integer v…
This paper uses dualities between facet ideal theory and Stanley-Reisner theory to show that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay. The proof involves showing that the Alexander dual (or the cover dual, as we call it here) of a simplicial tree is a componentwise linear ideal. We conclude with additional combinatorial properties of simplicial trees.
We construct the minimal resolutions of three classes of monomial ideals: dominant, 1-semidominant, and 2-semidominant ideals. The families of dominant and 1-semidominant ideals extend those of complete and almost complete intersections. We show that dominant ideals give a precise characterization of when the Taylor resolution is minimal, 1-semidominant ideals are Scarf, and the minimal resolutions of 2-semidominant ideals can be obtained from their Taylor resolutions by eliminating faces and facets of equal multidegree, in arbitrary order. We study the combinatorial properties of these classe…
We prove the subadditivity property for the maximal degrees of the syzygies of facet ideals simplicial forests. For such an ideal $I$, if the $i$-th Betti number is nonzero and $i=a+b$, we show that there are monomials in the lcm lattice of $I$ that are complements in part of the lattice, each supporting a nonvanishing $a$-th and $b$-th Betti numbers. The subadditivity formula follows from this observation.
We study when Taylor resolutions of monomial ideals are minimal. We consider monomial ideals with linear quotients. In particular, we determine precisely the stable ideals and the monomial ideals with linear resolutions having the miminal Taylor resolutions.
We consider powers of lexsegment ideals with a linear resolution (equivalently, with linear quotients) which are not completely lexsegment ideals. We give a complete description of their minimal graded free resolution.
A translation into English of Winfried Bruns' article: "Jede" endliche freie Auflösung ist freie Auflösung eines von drei Elementen erzeugten Ideals' (published in J. Algebra 39, 429-439 (1976))
For positive integers $d<n$, let $[n]_d=\{A\in 2^{[n]}\mid |A|=d\}$ where $[n]=:\{1,2,\ldots, n\}$. For a pure $f$-simplicial complex $Δ$ such that ${\rm dim}(Δ)={\rm dim}(Δ^c)$ and $\mathcal{F}(Δ)\cap \mathcal{F}(Δ^c)=\emptyset$, we prove that the facet ideal $I(Δ)$ is Cohen-Macaulay if and only if it has linear resolution. For a $d$-dimensional pure $f$-simplicial complex $Δ$ such that $Δ'=:\langle F\mid F\in [n]_d\smallsetminus \mathcal F(Δ)\rangle$ is an $f$-simplicial complex, we prove that $I(Δ^c)$ is Cohen-Macaulay if and only if $I(Δ')$ has linear resolution.
We give a necessary and sufficient condition on a homogeneous polynomial ideal for its Taylor complex to be exact. Then we give a combinatorial construction of a minimal resolution for ideals satisfying the above condition (in particular for monomial ideals).
For a monomial ideal $I$, let $G(I)$ be its minimal set of monomial generators. If there is a total order on $G(I)$ such that the corresponding Lyubeznik resolution of $I$ is a minimal free resolution of $I$, then $I$ is called a Lyubeznik ideal. In this paper, we characterize the Lyubeznik ideals, and we discover some classes of Lyubeznik ideals.
We study the WLP and SLP of artinian monomial ideals in $S=\mathbb{K}[x_1,\dots ,x_n]$ via studying their minimal free resolutions. We study the Lefschetz properties of such ideals where the minimal free resolution of $S/I$ is linear for at least $n-2$ steps. We give an affirmative answer to a conjecture of Eisenbud, Huneke and Ulrich for artinian monomial ideals with almost linear resolutions.
We use the theory of poset resolutions to construct the minimal free resolution of an arbitrary stable monomial ideal in the polynomial ring whose coefficients are from a field. This resolution is recovered by utilizing a poset of Eliahou-Kervaire admissible symbols associated to a stable ideal. The structure of the poset under consideration is quite rich and in related analysis, we exhibit a regular CW complex which supports a minimal cellular resolution of a stable monomial ideal.
We use the lcm-lattice of a monomial ideal to study its minimal free resolutions. A new concept called a Taylor basis of a minimal free resolution is introduced and then used throughout the paper. We give a method of constructing minimal free resolutions of a monomial ideal from its lcm-lattice, which is called the atomic lattice resolution theory. Some applications of this theory is given. As the main application, we rewrite the theory of poset resolutions, and we obtain an approximation formula for minimal free resolutions of all monomial ideals.
In this paper we study minimal free resolutions of some classes of monomial ideals. we first give a sufficient condition to check the minimality of the resolution obtained by the mapping cone. Using it, we obtain the Betti numbers of max-path ideals of rooted trees and ideals containing powers of variables. In particular, we discuss about resolutions of ideals of the form $J_{\mathcal{H}}+(x_{i_1}^2,\ldots, x_{i_m}^2)$ where $J_{\mathcal{H}}$ is the edge ideal of a hypergraph $\mathcal{H}$.
We will explore some properties of minimal graded free resolutions of $R/I$, where $R$ is a trivariate polynomial ring over a field and $I$ is a monomial ideal. Our focus will be to consider a specific form of the resolutions when $I$ is primary to the homogeneous maximal ideal. We will identify certain characteristics of the last matrix of these resolutions, and observe differences in the resolutions for generic ideals in comparison to non-generic ideals. Finally, we learn how to identify whether $I$ is generic by knowing the structure of the last matrix in the minimal free resolution of $R/I…
We provide a characterisation of all graphs whose parity binomial edge ideals have pure resolutions. In particular, we show that the minimal free resolution of a parity binomial edge ideal is pure if and only if the corresponding graph is a complete bipartite graph, or a disjoint union of paths and odd cycles.
We show that the minimal resolution of the quotient of the polynomial algebra over a field by a cointerval edge ideal can be given the structure of a DG-algebra.
We use the natural homeomorphism between a regular CW-complex X and its face poset P_X to establish a canonical isomorphism between the cellular chain complex of X and the result of applying the poset construction of arXiv:0806.4532v1 to P_X. For a monomial ideal whose free resolution is supported on a regular CW-complex, this isomorphism allows the free resolution of the ideal to be realized as a CW-poset resolution. Conversely, any CW-poset resolution of a monomial ideal gives rise to a resolution supported on a regular CW-complex.
We introduce the concept of quasi-linearity and prove it is necessary for a monomial ideal to have a linear resolution and identify all the quasi-linear quadratic monomial ideals. We define a strongly linear monomial for a monomial ideal $I$ and prove that if $\mathbf{u}$ is a strongly linear monomial over $I$ then $I$ has a linear resolution (resp: is quasi-linear) if and only if $I+\mathbf{u}\mathfrak{p}$ has a linear resolution (resp: is quasi-linear). Here $\mathfrak{p}$ is any monomial prime ideal.
We introduce the class of lattice-linear monomial ideals and use the LCM-lattice to give an explicit construction for their minimal free resolution. The class of lattice-linear ideals includes (among others) the class of monomial ideals with linear free resolution and the class of Scarf monomial ideals. Our main tool is a new construction by Tchernev that produces from a map of posets $η:P\lra\mbb{N}^n$ a sequence of multigraded modules and maps.
All powers of lexsegment ideals with linear resolution (equivalently, with linear quotients) have linear quotients with respect to suitable orders of the minimal monomial generators. For a large subclass of the lexsegment ideals the corresponding Rees algebra has a quadratic Gröbner basis, thus it is Koszul. We also find other classes of monomial ideals with linear quotients whose powers have linear quotients too.
We consider vector-spread Borel ideals. We show that these ideals have linear quotients and thereby we determine the graded Betti numbers and the bigraded Poincaré series. A characterization of the extremal Betti numbers of such a class of ideals is given. Finally, we classify all Cohen-Macaulay vector-spread Borel ideals.
Let $I$ be a graded ideal of $K[x_1,\ldots,x_n]$ generated by homogeneous polynomials of a same degree $d$, and assume that $I$ has linear quotients. In this note, we use Horseshoe Lemma to give a relatively direct inductive construction of a minimal free resolution of $I$, which is called a $d$-linear resolution.
We give a formula to compute all the top degree graded Betti numbers of the path ideal of a cycle. Also we will find a criterion to determine when Betti numbers of this ideal are non zero and give a formula to compute its projective dimension and regularity.
Let $S=K[x_1,\dots,x_n]$ be a polynomial ring in $n$ variables with coefficients over a field $K$. A $t$-spread lexsegment ideal $I$ of $S$ is a monomial ideal generated by a $t$-spread lexsegment set. We determine all $t$-spread lexsegment ideals with linear resolution by means of Betti splittings. As applications we provide formulas for the Betti numbers of such a class of ideals and furthermore we characterize all incompletely $t$-spread lexsegment ideals with linear quotients.
Recent work on generic free resolutions of length 3 attaches to every resolution a graph and suggests that resolutions whose associated graph is a Dynkin diagram are distinguished. We conjecture that in a regular local ring, every grade 3 perfect ideal whose minimal free resolution is distinguished in this way is in the linkage class of a complete intersection.
We introduce the class of vector-spread monomial ideals. This notion generalizes that of $t$-spread ideals introduced by Ene, Herzog and Qureshi. In particular, we focus on vector-spread strongly stable ideals, we compute their Koszul cycles and describe their minimal free resolution. As a consequence the graded Betti numbers and the Poincaré series are determined. Finally, we consider a generalization of algebraic shifting theory for such a class of ideals.
In this paper we primarily study monomial ideals and their minimal free resolutions by studying their associated LCM lattices. In particular, we formally define the notion of coordinatizing a finite atomic lattice P to produce a monomial ideal whose LCM lattice is P, and we give a complete characterization of all such coordinatizations. We prove that all relations in the lattice L(n) of all finite atomic lattices with n ordered atoms can be realized as deformations of exponents of monomial ideals. We also give structural results for L(n). Moreover, we prove that the cellular structure of a min…
