Minimal Free Resolution Research Articles

Sampling Algebra Structures on Minimal Free Resolutions of Length 3

Ideals in the ring of power series in three variables over a field can be classified based on algebra structures on their minimal free resolutions. The classification is incomplete in the sense that it remains open which algebra structures actually occur; this realizability question was formally raised by Avramov in 2012. We discuss the outcomes of an experiment performed to shed light on Avramov’s question: Using the computer algebra system Macaulay2, we classify a billion randomly generated ideals and build a database with examples of ideals of all classes realized in the experiment. Based on the outcomes, we discuss the status of conjectures that relate to the realizability question.

Mapping free resolutions of length three II - Module formats

Let M be a perfect module of projective dimension 3 over a Gorenstein, local or graded ring R. We denote by F the minimal free resolution of M. Using the generic ring associated to the format of F we define higher structure maps, according to the theory developed by Weyman in [26]. We introduce a generalization of classical linkage for R-module using the Buchsbaum–Rim complex, and study the behavior of structure maps under this Buchsbaum–Rim linkage. In particular, for certain formats we obtain criteria for these R-modules to lie in the Buchsbaum–Rim linkage class of a Buchsbaum–Rim complex of length 3.

Minimal free resolution of generalized repunit algebras

Let k be an arbitrary field and let b > 1 , n > 1 and a be three positive integers. In this paper we explicitly describe a minimal S-graded free resolution of the semigroup algebra k [ S ] when S is a generalized repunit numerical semigroup, that is, when S is the submonoid of N generated by { a 1 , a 2 , … , a n } where a 1 = ∑ j = 0 n − 1 b j and a i − a i − 1 = a b i − 2 , i = 2 , … , n , with gcd ( a , a 1 ) = 1 .

Products and powers of principal symmetric ideals

Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case.

Linear strand of edge ideals of zero divisor graphs of the ring ℤn

For a simple graph G with edge ideal I(G), we study the N -graded Betti numbers in the linear strand of the minimal free resolution of I ( Γ ( Z n ) ) , where Γ ( Z n ) is the zero divisor graph of the ring Z n . We present sharp bounds for the Betti numbers of Γ ( Z n ) and characterize the graphs attaining these bounds, thereby establishing the correct equality case for one of the results of the earlier published paper (Theorem 4.5, S. Pirzada and S. Ahmad, On the linear strand of edge ideals of some zero divisor graphs, Commun. Algebra 51(2) (2023) 620–632). Also, we present homological invariants of the edge rings of Γ ( Z n ) for n = p 2 q and pqr, with primes p < q < r .

A minimal free resolution of the duplication of a monomial ideal

This article explores a novel operation of a monomial ideal, termed duplication, which produces a monomial ideal I⋄\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I^\\diamond $$\\end{document} from another I. This operation is inspired by and generalizes how vertex duplication affects the edge ideal of a graph. The main result describes a multigraded minimal free resolution of I⋄\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I^\\diamond $$\\end{document} as long as we know a resolution for I. Consequently, we obtain formulas for the projective dimension and depth of I⋄\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I^\\diamond $$\\end{document} in terms of the corresponding invariants of I.

Perazzo n-folds and the weak Lefschetz property

In this paper, we determine the maximum hmax\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_{max}$$\\end{document} and the minimum hmin\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_{min}$$\\end{document} of the Hilbert vectors of Perazzo algebras AF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_F$$\\end{document}, where F is a Perazzo polynomial of degree d in n+m+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n+m+1$$\\end{document} variables. These algebras always fail the Strong Lefschetz Property. We determine the integers n, m, d such that hmax\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_{max}$$\\end{document} (resp. hmin\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_{min}$$\\end{document}) is unimodal, and we prove that AF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_F$$\\end{document} always fails the Weak Lefschetz Property if its Hilbert vector is maximum, while it satisfies the Weak Lefschetz Property if it is minimum, unimodal, and satisfies an additional mild condition. We determine the minimal free resolution of Perazzo algebras associated to Perazzo threefolds in P4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}^4$$\\end{document} with minimum Hilbert vectors. Finally we pose some open problems in this context.

Cohomology operations for moment-angle complexes and resolutions of Stanley–Reisner rings

A fundamental result in toric topology identifies the cohomology ring of the moment-angle complex Z K \mathcal {Z}_K associated to a simplicial complex K K with the Koszul homology of the Stanley–Reisner ring of K K . By studying cohomology operations induced by the standard torus action on the moment-angle complex, we extend this to a topological interpretation of the minimal free resolution of the Stanley–Reisner ring. The exterior algebra module structure in cohomology induced by the torus action recovers the linear part of the minimal free resolution, and we show that higher cohomology operations induced by the action (in the sense of Goresky–Kottwitz–MacPherson [Invent. Math. 131 (1998), pp. 25–83]) can be assembled into an explicit differential on the resolution. Describing these operations in terms of Hochster’s formula, we recover and extend a result due to Katthän [Mathematics 7 (2019), no. 7, p. 605]. We then apply all of this to study the equivariant formality of torus actions on moment-angle complexes. For these spaces, we obtain complete algebraic and combinatorial characterisations of which subtori of the naturally acting torus act equivariantly formally.

Iterated Mapping Cones on the Koszul Complex and Their Application to Complete Intersection Rings

Iterated Mapping Cones on the Koszul Complex and Their Application to Complete Intersection Rings

Ideals of submaximal minors of sparse symmetric matrices

We study the ideal of submaximal minors of a sparse generic symmetric matrix. This ideal is generated by all (n−1)-minors of a symmetric n×n matrix whose entries in the upper triangle are distinct variables or zeros, and the zeros are only allowed at off-diagonal places. The surviving off-diagonal entries are encoded in a simple graph G with n vertices. We prove that the minimal free resolution of this ideal is obtained from the case without any zeros via a simple pruning procedure, extending methods of Boocher. This allows us to compute all graded Betti numbers in terms of n and a single invariant of G. Moreover, it turns out that these ideals are always radical and have Cohen–Macaulay quotients if and only if G is either connected or has no edges at all. The key input are some new Gröbner basis results with respect to non-diagonal term orders associated to G.

The Scarf complex and betti numbers of powers of extremal ideals

This paper is concerned with finding bounds on betti numbers and describing combinatorially and topologically (minimal) free resolutions of powers of ideals generated by a fixed number q of square-free monomials. Among such ideals, we focus on a specific ideal Eq, which we call extremal, and which has the property that for each r≥1 the betti numbers of Eqr are an upper bound for the betti numbers of Ir for any ideal I generated by q square-free monomials (in any number of variables). We study the Scarf complex of the ideals Eqr and use this simplicial complex to extract information on minimal free resolutions. In particular, we show that Eqr has a minimal free resolution supported on its Scarf complex when q≤4 or when r≤2, and we describe explicitly this complex. For any q and r, we also show that β1(Eqr) is the smallest possible, or in other words equal to the number of edges in the Scarf complex. These results lead to effective bounds on the betti numbers of Ir, with I as above. For example, we obtain that pd(Ir)≤5 for all ideals I generated by 4 square-free monomials and any r≥1.

The regularity of almost all edge ideals

A fruitful contemporary paradigm in graph theory is that almost all graphs that do not contain a certain subgraph have common structural characteristics. An example is the well-known result saying that almost all triangle-free graphs are bipartite. The “almost” is crucial, without it such theorems do not hold. In this paper we transfer this paradigm to commutative algebra and make use of deep graph theoretic results. A key tool is the critical graphs introduced relatively recently by Balogh and Butterfield, who proved that almost all graphs not containing a critical subgraph have common structural characteristics analogous to being bipartite.For a graph G, let IG denote its edge ideal, the monomial ideal generated by xixj for every edge ij of G. In this paper we study the graded Betti numbers of IG, which are combinatorial invariants that measure the complexity of a minimal free resolution of IG. The Betti numbers of the form βi,2i+2 constitute the “main diagonal” of the Betti table. It is well known that for edge ideals any Betti number to the left of this diagonal is always zero. We identify a certain “parabola” inside the Betti table and call parabolic Betti numbers the entries of the Betti table bounded on the left by the main diagonal and on the right by this parabola. Let βi,j be a parabolic Betti number on the r-th row of the Betti table, for r≥3. We prove that almost all graphs G with βi,j(IG)=0 can be partitioned into r−2 cliques and one independent set. In particular, for almost all graphs G with βi,j(IG)=0, the regularity of IG is r−1.

Linear strand of edge ideals of comaximal graphs of commutative rings

For an edge ideal I(G) of a simple graph G , we study the N -graded Betti numbers that appear in the linear strand of the minimal free resolution of I ( Γ ( Z n ) ) , where Γ ( Z n ) is the comaximal graph of the integral modulo ring Z n . We show that the extremal Betti number of I ( Γ ( Z n ) ) is ϕ ( n ) , where ϕ ( n ) is the Euler’s totient function and thereby we obtain a large class of edge ideals with even extremal Betti numbers. We find the regularity (Castelnuovo-Mumford) and the projective dimension of these ideals. Moreover, we exhibit explicit formulae that determine all the N -graded Betti numbers in the linear strand of the minimal free resolution of I ( Γ ( Z n ) ) for certain values of n .

Artinian Gorenstein algebras of embedding dimension four and socle degree three

We prove that in the polynomial ring Q=k[x,y,z,w], with k an algebraically closed field of characteristic zero, all Gorenstein homogeneous ideals I satisfying the inclusions (x,y,z,w)4⊆I⊆(x,y,z,w)2 can be obtained by doubling from a grade three perfect ideal J⊂I such that Q/J is a locally Gorenstein ring. Moreover, a graded minimal free resolution of the Q-module Q/I can be completely described in terms of a graded minimal free resolution of the Q-module Q/J and a homogeneous embedding of a shift of the canonical module ωQ/J into Q/J.

On constructions related to the generalized Taylor complex

In this paper, we extend constructions and results for the Taylor complex to the generalized Taylor complex constructed by Herzog. We construct an explicit DG-algebra structure on the generalized Taylor complex and extend a result of Katthän on quotients of the Taylor complex by DG-ideals. We introduce a generalization of the Scarf complex for families of monomial ideals, and show that this complex is always a subcomplex of the minimal free resolution of the sum of these ideals. We also give an example of an ideal where the generalized Scarf complex strictly contains the standard Scarf complex. Moreover, we introduce the notion of quasitransverse monomial ideals, and prove a list of results relating to Golodness, Koszul homology, and other homological properties for such ideals.

Free resolutions and Lefschetz properties of some Artin Gorenstein rings of codimension four

In (Stanley, 1978), Stanley constructs an example of an Artinian Gorenstein (AG) ring A with non-unimodal H-vector (1,13,12,13,1). Migliore-Zanello show in (Migliore and Zanello, 2017) that for regularity r=4, Stanley's example has the smallest possible codimension c for an AG ring with non-unimodal H-vector.The weak Lefschetz property (WLP) has been much studied for AG rings; it is easy to show that an AG ring with non-unimodal H-vector fails to have WLP. In codimension c=3 it is conjectured that all AG rings have WLP. For c=4, Gondim shows in (Gondim, 2017) that WLP always holds for r≤4 and gives a family where WLP fails for any r≥7, building on Ikeda's example (Ikeda, 1996) of failure for r=5. In this note we study the minimal free resolution of A and relation to Lefschetz properties (both weak and strong) and Jordan type for c=4 and r≤6.

Elementary construction of the minimal free resolution of the Specht ideal of shape (n − d,d)

Let K be a field with char(K)=0. For a partition λ of n∈N, let IλSp be the ideal of R=K[x1,…,xn] generated by all Specht polynomials of shape λ. These ideals have been studied from several points of view (and under several names). Using advanced tools of the representation theory, Berkesch Zamaere et al. [1] constructed a minimal free resolution of I(n−d,d)Sp except differential maps. The present paper constructs the differential maps, and also gives an elementary proof of the result of [1].

Free resolutions for the tangent cones of some homogeneous pseudo symmetric monomial curves

In this article, we study minimal graded free resolutions of Cohen-Macaulay tangent cones of some monomial curves associated to 4-generated pseudo symmetric numerical semigroups. We explicitly give the matrices in these minimal free resolutions.

Tri-linear birational maps in dimension three

A tri-linear rational map in dimension three is a rational map ϕ : ( P C 1 ) 3 ⇢ P C 3 \phi : (\mathbb {P}_\mathbb {C}^1)^3 \dashrightarrow \mathbb {P}_\mathbb {C}^3 defined by four tri-linear polynomials without a common factor. If ϕ \phi admits an inverse rational map ϕ − 1 \phi ^{-1} , it is a tri-linear birational map. In this paper, we address computational and geometric aspects about these transformations. We give a characterization of birationality based on the first syzygies of the entries. More generally, we describe all the possible minimal graded free resolutions of the ideal generated by these entries. With respect to geometry, we show that the set B i r ( 1 , 1 , 1 ) \mathfrak {Bir}_{(1,1,1)} of tri-linear birational maps, up to composition with an automorphism of P C 3 \mathbb {P}_\mathbb {C}^3 , is a locally closed algebraic subset of the Grassmannian of 4 4 -dimensional subspaces in the vector space of tri-linear polynomials, and has eight irreducible components. Additionally, the group action on B i r ( 1 , 1 , 1 ) \mathfrak {Bir}_{(1,1,1)} given by composition with automorphisms of ( P C 1 ) 3 (\mathbb {P}_\mathbb {C}^1)^3 defines 19 orbits, and each of these orbits determines an isomorphism class of the base loci of these transformations.

Scrollar invariants, syzygies and representations of the symmetric group

Abstract We give an explicit minimal graded free resolution, in terms of representations of the symmetric group S d {S_{d}} , of a Galois-theoretic configuration of d points in 𝐏 d - 2 {\mathbf{P}^{d-2}} that was studied by Bhargava in the context of ring parametrizations. When applied to the geometric generic fiber of a simply branched degree d cover of 𝐏 1 {\mathbf{P}^{1}} by a relatively canonically embedded curve C, our construction gives a new interpretation for the splitting types of the syzygy bundles appearing in its relative minimal resolution. Concretely, our work implies that all these splitting types consist of scrollar invariants of resolvent covers. This vastly generalizes a prior observation due to Casnati, namely that the first syzygy bundle of a degree 4 cover splits according to the scrollar invariants of its cubic resolvent. Our work also shows that the splitting types of the syzygy bundles, together with the multi-set of scrollar invariants, belong to a much larger class of multi-sets of invariants that can be attached to C → 𝐏 1 {C\to\mathbf{P}^{1}} : one for each irreducible representation of S d {S_{d}} , i.e., one for each partition of d.