Minimal Free Resolution Research Articles

Minimal free resolutions for homogeneous ideals with Betti numbers 1, n, n,1

We investigate the standard generalized Gorenstein algebras of homological dimension three, giving a structure theorem for their resolutions. Moreover in many cases we are able to give a complete description of their graded Betti numbers.

On curves lying on a rational normal surface scroll

In this paper, we study the minimal free resolution of non-ACM divisors X of a smooth rational normal surface scroll S=S(a1,a2)⊂Pr. Our main result shows that for a2≥2a1−1, there exists a nice decomposition of the Betti table of X as a sum of much simpler Betti tables. As a by-product of our results, we obtain a complete description of the graded Betti numbers of X for the cases where S=S(1,r−2) for all r≥3 and S=S(2,r−3) for all r≥6.

Tor as a module over an exterior algebra

Let S be a regular local ring with residue field k and let M be a finitely generated S -module. Suppose that f_1,\dots ,f_c \in S is a regular sequence that annihilates M , and let E be an exterior algebra over k generated by c elements. The homotopies for the f_i on a free resolution of M induce a natural structure of graded E -module on Tor ^{S}(M,k) . In the case where M is a high syzygy over the complete intersection R:=S/(f_1,\dots,f_c) we describe this E -module structure in detail, including its minimal free resolution over E . Turning to Ext _R(M, k) we show that, when M is a high syzygy over R , the minimal free resolution of Ext _R(M, k) as a module over the ring of CI operators is the Bernstein–Gel’fand–Gel’fand dual of the E -module Tor ^S(M, k) . For the proof we introduce higher CI operators , and give a construction of a (generally non-minimal) resolution of M over S starting from a resolution of M over R and its higher CI operators.

Minimal free resolutions of monomial ideals and of toric rings are supported on posets

We introduce the notion of a \emph{resolution supported on a poset}. When the poset is a CW-poset, i.e. the face poset of a regular CW-complex, we recover the notion of cellular resolution as introduced by Bayer and Sturmfels. Work of Reiner and Welker, and of Velasco, has shown that there are monomial ideals whose minimal free resolutions are not cellular, hence cannot be supported on any CW-poset. We show that for any monomial ideal there is a \emph{homology CW-poset} that supports a minimal free resolution of the ideal. This allows one to extend to every minimal resolution, essentially verbatim, techniques initially developed to study cellular resolutions. As two demonstrations of this process, we show that minimal resolutions of toric rings are supported on what we call toric hcw-posets, and we give a new combinatorial proof of a fundamental result of Miller on the relationship between Artininizations and Alexander duality of monomial ideals.

A∞–resolutions and the Golod property for monomial rings

Let R=S/I be a monomial ring whose minimal free resolution F is rooted. We describe an A-infinity algebra structure on F. Using this structure, we show that R is Golod if and only if the product on Tor^S(R,k) vanishes. Furthermore, we give a necessary and sufficient combinatorial condition for R to be Golod.

Syzygies of Determinantal Thickenings and Representations of the General Linear Lie Superalgebra

We let S denote the ring of polynomial functions on the space of m x n matrices, and consider the action of the group GL = GL_m x GL_n via row and column operations on the matrix entries. For a GL-invariant ideal I in S we show that the linear strands of its minimal free resolution translate via the BGG correspondence to modules over the general linear Lie superalgebra gl(m|n). When I is the ideal generated by the GL-orbit of a highest weight vector, we give a conjectural description of the classes of these gl(m|n)-modules in the Grothendieck group, and prove that our prediction is correct for the first strand of the minimal free resolution.

Betti-linear ideals

We introduce the notion of a Betti-linear monomial ideal, which generalizes the notion of lattice-linear monomial ideal introduced by Clark. We provide a characterization of Betti-linearity in terms of Tchernev's poset construction. As an application we obtain an explicit canonical construction for the minimal free resolutions of monomial ideals having pure resolutions.

Minimal Free Resolution of Monomial Ideals by Iterated Mapping Cone

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 this condition, we obtain the Betti numbers of max-path ideals of rooted trees and ideals containing powers of variables. In particular, we discuss the 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 $${\mathcal {H}}$$ .

The minimal resolution conjecture on a general quartic surface in [formula omitted

Mustaţă has given a conjecture for the graded Betti numbers in the minimal free resolution of the ideal of a general set of points on an irreducible projective algebraic variety. For surfaces in P3 this conjecture has been proven for points on quadric surfaces and on general cubic surfaces. In the latter case, Gorenstein liaison was the main tool. Here we prove the conjecture for general quartic surfaces. Gorenstein liaison continues to be a central tool, but to prove the existence of our links we make use of certain dimension computations. We also discuss the higher degree case, but now the dimension count does not force the existence of our links.

The structure of DGA resolutions of monomial ideals

Let I⊆k[x1,…,xn] be a squarefree monomial ideal in a polynomial ring. In this paper we study multiplications on the minimal free resolution F of k[x1,…,xn]/I. In particular, we characterize the possible vectors of total Betti numbers for such ideals which admit a differential graded algebra (DGA) structure on F. We also show that under these assumptions the maximal shifts of the graded Betti numbers are subadditive.On the other hand, we present an example of a strongly generic monomial ideal which does not admit a DGA structure on its minimal free resolution. In particular, this demonstrates that the Hull resolution and the Lyubeznik resolution do not admit DGA structures in general.Finally, we show that it is enough to modify the last map of F to ensure that it admits the structure of a DG algebra.

The structure of the minimal free resolution of semigroup rings obtained by gluing

We construct a minimal free resolution of the semigroup ring k[C] in terms of minimal resolutions of k[A] and k[B] when 〈C〉 is a numerical semigroup obtained by gluing two numerical semigroups 〈A〉 and 〈B〉. Using our explicit construction, we compute the Betti numbers, graded Betti numbers, regularity and Hilbert series of k[C], and prove that the minimal free resolution of k[C] has a differential graded algebra structure provided the resolutions of k[A] and k[B] possess them. We discuss the consequences of our results in small embedding dimensions. Finally, we give an extension of our main result to semigroups in Nn.

Bouquet algebra of toric ideals

To any toric ideal IA, encoded by an integer matrix A, we associate a matroid structure called the bouquet graph of A and introduce another toric ideal called the bouquet ideal of A. We show how these objects capture the essential combinatorial and algebraic information about IA. Passing from the toric ideal to its bouquet ideal via the graph theoretic properties of the bouquet graph allows us to classify several cases. For example, on the one end of the spectrum, there are ideals that we call stable, for which bouquets capture the complexity of various generating sets as well as the minimal free resolution. On the other end of the spectrum lie toric ideals whose various bases (e.g., minimal generating sets, Gröbner, Graver bases) coincide. Apart from allowing for classification-type results, bouquets provide a new way to construct families of examples of toric ideals with various interesting properties, such as robustness, genericity, and unimodularity. The new bouquet framework can be used to provide a characterization of toric ideals whose Graver basis, the universal Gröbner basis, any reduced Gröbner basis and any minimal generating set coincide.

CW-resolutions of monomial ideals that are supported on face posets

ABSTRACTGiven a monomial ideal I with minimal free resolution ℱ supported in characteristic p>0 on a CW-complex X with regular 2-skeleton, in general it is not the case that the face poset of X, P(X), also supports ℱ in the sense of Clark and Tchernev. We construct a (not necessarily regular) CW-complex Y that also supports ℱ and such that the face poset P(Y) also supports ℱ.

On minimal free resolution of the associated graded rings of certain monomial curves : New proofs in A⁴

In this article, even if it is known for general case in sharifan-nahandi , we give the explicit minimal free resolution of the associated graded ring of certain affine monomial curves in affine 4-space based on the standard basis theory. As a result, we give the minimal graded free resolution and the Hilbert function of the tangent cone of these families in A⁴ in the simple form according to sharifan-nahandi .

A minimal free complex associated to the minors of a matrix

This paper describes a construction of a minimal free resolution of a generic ideal defined by determinants in characteristic zero. It produces not only the free modules in the resolution, but it also defines the maps between them explicitly and in detail in terms of idempotents in the group algebra of the symmetric group.

Powers of edge ideals with linear resolutions

ABSTRACTWe show that if G is a gap-free and diamond-free graph, then I(G)s has a linear minimal free resolution for every s≥2.

Prime splittings of determinantal ideals

ABSTRACTWe consider determinantal ideals, where the generating minors are encoded in a hypergraph. We study when the generating minors form a Gröbner basis. In this case, the ideal is radical, and we can describe algebraic and numerical invariants of these ideals in terms of combinatorial data of their hypergraphs, such as the clique decomposition. In particular, we can construct a minimal free resolution as a tensor product of the minimal free resolution of their cliques. For several classes of hypergraphs we find a combinatorial description of the minimal primes in terms of a prime splitting. That is, we write the determinantal ideal as a sum of smaller determinantal ideals such that each minimal prime is a sum of minimal primes of the summands.

On minimal free resolutions of sub-permanents and other ideals arising in complexity theory

We compute the linear strand of the minimal free resolution of the ideal generated by k×k sub-permanents of an n×n generic matrix and of the ideal generated by square-free monomials of degree k. The latter calculation gives the full minimal free resolution by [1]. Our motivation is to lay groundwork for the use of commutative algebra in algebraic complexity theory. We also compute several Hilbert functions relevant for complexity theory.

An integer sequence and standard monomials

For an (oriented) graph [Formula: see text] on the vertex set [Formula: see text] (rooted at [Formula: see text]), Postnikov and Shapiro (Trans. Amer. Math. Soc. 356 (2004) 3109–3142) associated a monomial ideal [Formula: see text] in the polynomial ring [Formula: see text] over a field [Formula: see text] such that the number of standard monomials of [Formula: see text] equals the number of (oriented) spanning trees of [Formula: see text] and hence, [Formula: see text], where [Formula: see text] is the truncated Laplace matrix of [Formula: see text]. The standard monomials of [Formula: see text] correspond bijectively to the [Formula: see text]-parking functions. In this paper, we study a monomial ideal [Formula: see text] in [Formula: see text] having rich combinatorial properties. We show that the minimal free resolution of the monomial ideal [Formula: see text] is the cellular resolution supported on a subcomplex of the first barycentric subdivision [Formula: see text] of an [Formula: see text] simplex [Formula: see text]. The integer sequence [Formula: see text] has many interesting properties. In particular, we obtain a formula, [Formula: see text], with [Formula: see text] for [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text], similar to [Formula: see text].

Minimal free resolutions of lattice ideals of digraphs

Based upon a previous work of Manjunath and Sturmfels for a finite, complete, undirected graph, and a refined algorithm by Erocal, Motsak, Schreyer and Steenpas for computing syzygies, we display a free resolution of the lattice ideal associated to a finite, strongly connected, weighted, directed graph. Moreover, the resolution is minimal precisely when the digraph is strongly complete.