Gorenstein Ideals Research Articles

On Posets, Monomial Ideals, Gorenstein Ideals and their Combinatorics

On Posets, Monomial Ideals, Gorenstein Ideals and their Combinatorics

The Tor Algebra of Trimmings of Gorenstein Ideals

The Tor Algebra of Trimmings of Gorenstein Ideals

Quadratically presented grade three Gorenstein ideals

Let J be a quadratically presented grade three Gorenstein ideal in the standard graded polynomial ring R=k[x,y,z], where k is a field. Assume that R/J satisfies the weak Lefschetz property. We give the presentation matrix for J in terms of the coefficients of a Macaulay inverse system for J. (This presentation matrix is an alternating matrix and J is generated by the maximal order Pfaffians of the presentation matrix.) Our formulas are computer friendly; they involve only matrix multiplication; they do not involve multilinear algebra or complicated summations. As an application, we give the presentation matrix for J1=(xn+1,yn+1,zn+1):(x+y+z)n+1, when n is even and the characteristic of k is zero. Generators for J1 had been identified previously; but the presentation matrix for J1 had not previously been known. The first step in our proof is to give improved formulas for the presentation matrix of a linearly presented grade three Gorenstein ideal I in terms of the coefficients of the Macaulay inverse system for I.

Equigenerated Gorenstein ideals of codimension three

We focus on the structure of a homogeneous Gorenstein ideal I of codimension three in a standard polynomial ring \(R={\mathbbm{k}}\,[x_{1},\ldots ,x_{n}]\) over an infinite field \({\mathbbm{k}}\), assuming that I is generated in a fixed degree d. For such an ideal I, there is a simple formula relating this degree, the minimal number of generators of I, and the degree of the entries of the associated skew-symmetric matrix. We give an elementary characteristic-free argument to the effect that, for any such data linked by this formula, there exists a Gorenstein ideal I of codimension three satisfying them. We conjecture that, for arbitrary \(n\ge 2\), an ideal \(I\subset {\mathbbm{k}}[x_1,\ldots ,x_n]\) generated by a general set of \(r\ge n+2\) forms of degree \(d\ge 2\) is Gorenstein if and only if \(d=2\) and \(r= {{n+1}\atopwithdelims ()2}-1\). We prove the ‘only if’ implication of this conjecture when \(n=3\). For arbitrary \(n\ge 2\), we prove that if \(d=2\) and \(r\ge (n+2)(n+1)/6\) then the ideal is Gorenstein if and only if \(r={{n+1}\atopwithdelims ()2}-1\), which settles the ‘if’ assertion of the conjecture for \(n\le 5\). We also elaborate around one of the questions of Fröberg–Lundqvist. In a different direction, we show a connection between the Macaulay inverse and the so-called Newton dual, a matter so far not brought out to our knowledge. Finally, we consider the question as to when the link \((\ell _1^m,\ldots ,\ell _n^m):\mathfrak{f}\) is equigenerated, where \(\ell _1,\ldots ,\ell _n\) are independent linear forms and \(\mathfrak{f}\) is a form. We give a solution in some special cases.

Homogeneous liaison and the sequentially bounded licci property

We consider homogeneously licci ideals in a polynomial ring and focus on the degrees of the forms generating the regular sequences. Using a sequentially bounded condition on these degrees, E. Chong discovered a large class of licci ideals satisfying the Eisenbud–Green–Harris conjecture (among them, grade 3 Gorenstein ideals). He raised the question of whether these sequentially bounded links were possible for all homogeneously licci ideals. We answer his question in the negative, and in doing so answer a question of C. Huneke and B. Ulrich about strongly licci ideals. The structure of certain Betti tables plays a central role in our proof.

A structure result for Gorenstein algebras of odd codimension

The famous structure theorem of Buchsbaum and Eisenbud gives a complete characterization of Gorenstein ideals of codimension 3 and their minimal free resolutions. We generalize the ideas of Buchsbaum and Eisenbud from Gorenstein ideals to Gorenstein algebras and present a description of Gorenstein algebras of any odd codimension. As an application we study the canonical ring of a numerical Godeaux surface.

The dual of an evaluation code

The aim of this work is to study the dual and the algebraic dual of an evaluation code using standard monomials and indicator functions. We show that the dual of an evaluation code is the evaluation code of the algebraic dual. We develop an algorithm for computing a basis for the algebraic dual. Let $$C_1$$ and $$C_2$$ be linear codes spanned by standard monomials. We give a combinatorial condition for the monomial equivalence of $$C_1$$ and the dual $$C_2^\perp $$ . Moreover, we give an explicit description of a generator matrix of $$C_2^\perp $$ in terms of that of $$C_1$$ and coefficients of indicator functions. For Reed–Muller-type codes we give a duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide an explicit duality for Reed–Muller-type codes corresponding to Gorenstein ideals. In addition, when the evaluation code is monomial and the set of evaluation points is a degenerate affine space, we classify when the dual is a monomial code.

DG structure on the length 4 big from small construction

The big from small construction was introduced by Kustin and Miller in [A. Kustin and M. Miller, Constructing big Gorenstein ideals from small ones, J. Algebra 85 (1983) 303–322] and can be used to construct resolutions of tightly double linked Gorenstein ideals. In this paper, we expand on the DG-algebra techniques introduced in [A. Kustin, Use DG methods to build a matrix factorization, preprint (2019), arXiv:1905.11435] and construct a DG [Formula: see text]-algebra structure on the length [Formula: see text] big from small construction. The techniques employed involve the construction of a morphism from a Tate-like complex to an acyclic DG [Formula: see text]-algebra exhibiting Poincaré duality. This induces homomorphisms which, after suitable modifications, satisfy a list of identities that end up perfectly encapsulating the required associativity and DG axioms of the desired product structure for the big from small construction.

Multiplicity of the Saturated Special Fiber Ring of Height Three Gorenstein Ideals

Let R be a polynomial ring over a field and let I ⊂ R be a Gorenstein ideal of height three that is minimally generated by homogeneous polynomials of the same degree. We compute the multiplicity of the saturated special fiber ring of I. The obtained formula depends only on the number of variables of R, the minimal number of generators of I, and the degree of the syzygies of I. Applying results from Busé et al. (Proc. Lond. Math. Soc. 121(4):743–787, 2020) we get a formula for the j-multiplicity of I and an effective method to study a rational map determined by a minimal set of generators of I.

Pullback of the Normal Module of Ideals with Low Codimension

AbstractThe normal module (or sheaf) of an ideal is a celebrated object in commutative algebra and algebraic geometry. In this paper, we prove results about its pullback under the natural projection, focusing on subtle numerical invariants such as, for instance, the reduction number. For certain codimension 2 perfect ideals, we show that the pullback has reduction number two. This is of interest since the determination of this invariant in the context of modules (even for special classes) is a mostly open, difficult problem. The analytic spread is also computed. Finally, for codimension 3 Gorenstein ideals, we determine the depth of the pullback, and we also consider a broader class of ideals provided that the Auslander transpose of the conormal module is almost Cohen–Macaulay.

Mixed multiplicities and projective degrees of rational maps

We consider the notion of mixed multiplicities for multigraded modules by using Hilbert series, and this is later applied to study the projective degrees of rational maps. We use a general framework to determine the projective degrees of a rational map via a computation of the multiplicity of the saturated special fiber ring. As specific applications, we provide explicit formulas for all the projective degrees of rational maps determined by perfect ideals of height two or by Gorenstein ideals of height three.

Quadratic Gorenstein algebras with many surprising properties

Let k be a field of characteristic 0. Using the method of idealization, we show that there is a non-Koszul, quadratic, Artinian, Gorenstein, standard graded k-algebra of regularity 3 and codimension 8, answering a question of Mastroeni, Schenck, and Stillman. We also show that this example is minimal in the sense that no other idealization that is non-Koszul, quadratic, Artinian, Gorenstein algebra, with regularity 3 has smaller codimension. We also construct an infinite family of graded, quadratic, Artinian, Gorenstein algebras $$A_m$$ , indexed by an integer $$m \ge 2$$ , with the following properties: (1) there are minimal first syzygies of the defining ideal in degree $$m+2$$ , (2) for $$m \ge 3$$ , $$A_m$$ is not Koszul, (3) for $$m \ge 7$$ , the Hilbert function of $$A_m$$ is not unimodal, and thus (4) for $$m \ge 7$$ , $$A_m$$ does not satisfy the weak or strong Lefschetz properties. In particular, the subadditivity property fails for quadratic Gorenstein ideals. Finally, we show that the idealization of a construction of Roos yields non-Koszul quadratic Gorenstein algebras such that the residue field k has a linear resolution for precisely $$\alpha $$ steps for any integer $$\alpha \ge 2$$ . Thus there is no finite test for the Koszul property even for quadratic Gorenstein algebras.

On Monomial Ideals and Their Socles

For a finite subset M ⊂ [x1,…, xd] of monomials, we describe how to constructively obtain a monomial ideal $I\subseteq R = K[x_{1},\ldots ,x_{d}]$ such that the set of monomials in Soc(I) ∖ I is precisely M, or such that $\overline {M}\subseteq R/I$ is a K-basis for the the socle of R/I. For a given M we obtain a natural class of monomials ideals I with this property. This is done by using solely the lattice structure of the monoid [x1,…, xd]. We then present some duality results by using anti-isomorphisms between upsets and downsets of the lattice $({\mathbb {Z}}^{d},\preceq )$ . Finally, we define and analyze zero-dimensional monomial ideals of R of type k, where type 1 are exactly the Artinian Gorenstein ideals, and describe the structure of such ideals that correspond to order-generic antichains in ${\mathbb {Z}}^{d}$ .

The equations defining blowup algebras of height three Gorenstein ideals

We find the defining equations of Rees rings of linearly presented height three Gorenstein ideals. To prove our main theorem we use local cohomology techniques to bound the maximum generator degree of the torsion submodule of symmetric powers in order to conclude that the defining equations of the Rees algebra and the special fiber ring have the same image in the symmetric algebra. We show that this image is the unmixed part of the ideal generated by the maximal minors of a matrix of linear forms which is annihilated by a vector of indeterminates, and otherwise has maximal possible grade. An important step of the proof is the calculation of the degree of the variety parametrized by the forms generating the grade three Gorenstein ideal.

STRUCTURE THEOREMS FOR SOME CLASSES OF GRADE FOUR GORENSTEIN IDEALS

The structure theorems [3, 6, 21] for the classes of perfect ideals of grade 3 have been generalized to the structure theorems for the classes of perfect ideals linked to almost complete intersections of grade 3 by a regular sequence [15]. In this paper we obtain structure theorems for two classes of Gorenstein ideals of grade 4 expressed as the sum of a perfect ideal of grade 3 (except a Gorenstein ideal of grade 3) and an almost complete intersection of grade 3 which are geometrically linked by a regular sequence.

A matrix of linear forms which is annihilated by a vector of indeterminates

Let R=k[T1,…,Tf] be a standard graded polynomial ring over the field k and Ψ be an f×g matrix of linear forms from R, where 1≤g<f. Assume [T1⋯Tf]Ψ is 0 and that gradeIg(Ψ) is exactly one short of the maximum possible grade. We resolve R‾=R/Ig(Ψ), prove that R‾ has a g-linear resolution, record explicit formulas for the h-vector and multiplicity of R‾, and prove that if f−g is even, then the ideal Ig(Ψ) is unmixed. Furthermore, if f−g is odd, then we identify an explicit generating set for the unmixed part, Ig(Ψ)unm, of Ig(Ψ), resolve R/Ig(Ψ)unm, and record explicit formulas for the h-vector of R/Ig(Ψ)unm. (The rings R/Ig(Ψ) and R/Ig(Ψ)unm automatically have the same multiplicity.) These results have applications to the study of the blow-up algebras associated to linearly presented grade three Gorenstein ideals.

Extreme rays of Hankel spectrahedra for ternary forms

The cone of sums of squares is one of the central objects in convex algebraic geometry. Its defining linear inequalities correspond to the extreme rays of the dual convex cone. This dual cone is a spectrahedron, which can be explicitly realized as a section of the cone of positive semidefinite matrices with the linear subspace of Hankel (or middle catalecticant) matrices. In this paper we initiate a systematic study of the extreme rays of Hankel spectrahedra for ternary forms. We show that the Zariski closure of the union of extreme rays is the variety of all Hankel matrices of corank at least 4, an irreducible variety of codimension 10 and we determine its degree. We explicitly construct an extreme ray of maximal rank using the Cayley–Bacharach Theorem for plane curves. We apply our results to the study of the algebraic boundary of the cone of sums of squares. Its irreducible components are dual to varieties of Gorenstein ideals with certain Hilbert functions. We determine these Hilbert functions for some cases of small degree. We also observe surprising gaps in the ranks of Hankel matrices of the extreme rays.

An application of liaison theory to the Eisenbud–Green–Harris conjecture

In this paper, we apply liaison theory to the Eisenbud–Green–Harris conjecture and prove that the conjecture holds for a certain subclass of homogeneous ideals in the linkage class of a complete intersection ideal. In the case of three variables, we prove that the conjecture holds for Gorenstein ideals.

On the Canonical Ideals of One-Dimensional Cohen–Macaulay Local Rings

AbstractIn this paper we consider the problem of explicitly finding canonical ideals of one-dimensional Cohen–Macaulay local rings. We show that Gorenstein ideals contained in a high power of the maximal ideal are canonical ideals. In the codimension 2 case, from a Hilbert–Burch resolution, we show how to construct canonical ideals of curve singularities. Finally, we translate the problem of the analytic classification of curve singularities to the classification of local Artin Gorenstein rings with suitable length.

Linear syzygies, flag complexes, and regularity

We show that for every \(r\in \mathbb {Z}_{>0}\) there exist monomial ideals generated in degree two, with linear syzygies, and regularity of the quotient equal to \(r\). For Gorenstein ideals we prove that the regularity of their quotients can not exceed four, thus showing that for \(d > 4\) every triangulation of a \(d\)-manifold has a hollow square or simplex. We also show that for most monomial ideals generated in degree two and with linear syzygies the regularity is \(O(\log (\log (n))\), where \(n\) is the number of variables.