The Multiplicity Conjecture is a deep problem relating the multiplicity (or degree) of a Cohen-Macaulay standard graded algebra with certain extremal graded Betti numbers in its minimal free resolution. In the case of level algebras of codimension three, Zanello has proposed a stronger conjecture. We prove this conjecture for the case of codimension three graded Gorenstein algebras.
In this note, we calculate the Koszul homology of the codimension 3 Gorenstein ideals. We find filtrations for the Koszul homology in terms of modules with pure free resolutions and completely describe these resolutions. We also consider the Huneke-Ulrich deviation 2 ideals.
In this note we supply an elementary proof of the following well-known theorem of R. Stanley: the $h$-vectors of Gorenstein algebras of codimension 3 are SI-sequences, i.e. are symmetric and the first difference of their first half is an $O$-sequence.
We study the weak Lefschetz property of a class of graded Artinian Gorenstein algebras of codimension three associated in a natural way to the Apéry set of a numerical semigroup generated by four natural numbers. We show that these algebras have the weak Lefschetz property whenever the initial degree of their defining ideal is small.
We study Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. We introduce rank matrices of linear forms for such algebras that represent the ranks of multiplication maps in various degrees. We show that there is a 1-1 correspondence between rank matrices and Jordan degree types. For Artinian Gorenstein algebras with codimension three we classify all rank matrices that occur for linear forms with vanishing third power. As a consequence, we show for such algebras that the possible Jordan types with parts of length at most four are uniquely determine…
We prove that if M, N are finite modules over a Gorenstein local ring R of codimension at most 4, then the vanishing of Ext^n_R(M,N) for n\gg 0 is equivalent to the vanishing of Ext^n_R(N,M) for n\gg 0. Furthermore, if the completion of $R$ has no embedded deformation, then such vanishing occurs if and only if M or N has finite projective dimension.
The main goal of this paper is to characterize the Hilbert functions of all (artinian) codimension 4 Gorenstein algebras that have at least two independent relations of degree four. This includes all codimension 4 Gorenstein algebras whose initial relation is of degree at most 3. Our result shows that those Hilbert functions are exactly the so-called {\em SI-sequences} starting with (1,4,h_2,h_3,...), where $h_4 \leq 33$. In particular, these Hilbert functions are all unimodal. We also establish a more general unimodality result, which relies on the values of the Hilbert function not being too…
We analyze the structure of spinor coordinates on resolutions of Gorenstein ideals of codimension four. As an application we produce a family of such ideals with seven generators which are not specializations of Kustin-Miller model.
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.
We prove upper bounds for the Hilbert-Samuel multiplicity of standard graded Gorenstein algebras. The main tool that we use is Boij-Söderberg theory to obtain a decomposition of the Betti table of a Gorenstein algebra as the sum of rational multiples of symmetrized pure tables. Our bound agrees with the one in the quasi-pure case obtained by Srinivasan [J. Algebra, vol.~208, no.~2, (1998)].
We prove that the Hilbert functions of codimension four graded Gorenstein Artin algebras R/I are unimodal provided I has a minimal generator in degree less than five. It is an open question whether all Gorenstein h-vectors in codimension four are unimodal. In this paper, we prove that Hilbert functions of all artinian codimension four Gorenstein algebras starting with (1,4,10, 20, h_4,...), where h_4\leq 34 are unimodal. Combining this with the previously known results, we obtain that all Gorenstein h-vectors (1, h_1, h_2,h_3, h_4, ....) are unimodal if h_4\leq 3.
Let $R$ be a local, Gorenstein ring with algebraically closed residue field $k$ of characteristic 0 and let $P_R(z):=\sum_{p=0}^{\infty}\dim_k(\tor_p^R(k,k))z^p$ be its Poincaré series. We compute $P_R$ when $R$ belongs to a particular class defined in the introduction, proving its rationality. As a by--product we prove the rationality of $P_R$ for all local, Gorenstein rings of multiplicity at most 10.
We determine new bounds on the entries of Gorenstein Hilbert functions, both in any fixed codimension and asymptotically. Our first main theorem is a lower bound for the degree $i+1$ entry of a Gorenstein $h$-vector, in terms of its entry in degree $i$. This result carries interesting applications concerning unimodality: indeed, an important consequence is that, given $r$ and $i$, all Gorenstein $h$-vectors of codimension $r$ and socle degree $e\geq e_0=e_0(r,i)$ (this function being explicitly computed) are unimodal up to degree $i+1$. This immediately gives a new proof of a theorem of Stanle…
We study Gorenstein ideals of codimension $4$ derived from generic doublings of almost complete intersection perfect ideals of codimension $3$. We also investigate spinor coordinates of such Gorenstein ideals with $8$ and $9$ generators. For an ideal $J$ of commutative ring $R$, the $R/J$ module $J/J^2$ is called conormal module and $R/J$-dual of $J/J^2$ is called normal module. We study properties of conormal and normal modules of almost complete intersection perfect ideals of codimension $3$.
Assume $R$ is a polynomial ring over a field and $I$ is a homogeneous Gorenstein ideal of codimension $g\ge3$ and initial degree $p\ge2$. We prove that the number of minimal generators $ν(I_p)$ of $I$ that are in degree $p$ is bounded above by $ν_0={p+g-1\choose g-1}-{p+g-3\choose g-1}$, which is the number of minimal generators of the defining ideal of the extremal Gorenstein algebra of codimension $g$ and initial degree $p$. Further, $I$ is itself extremal if $ν(I_p)=ν_0$.
We focus on the structure of a homogeneous Gorenstein ideal $I$ of codimension three in a standard polynomial ring $R=\kk[x_1,\ldots,x_n]$ over a field $\kk$, assuming that $I$ is generated in a fixed degree $d$. For such an ideal $I$ this degree comes along with the minimal number of generators of $I$ and the degree of the entries of the associated skew-symmetric matrix in a simple formula. 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 filling them. We conjecture that, f…
Codimension two Artinian algebras $A$ have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three AG algebras - the most promising results so far have concerned the weak Lefschetz property for such algebras. We here show that every standard-graded codimension three Artinian Gorenstein algebra $A$ having low maximum value of the Hilbert function - at most six - has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is grea…
We establish the upper bound in the multiplicity conjecture of Herzog, Huneke and Srinivasan for the codimension three almost complete intersections. We also give some partial results in the case where I is the aci linked to a complete intersection in one step.
I describe the projective resolution of a codimension 4 Gorenstein ideal, aiming to extend Buchsbaum and Eisenbud's famous result in codimension 3. The main result is a structure theorem stating that the ideal is determined by its (k+1) x 2k matrix of first syzygies, viewed as a morphism from the ambient regular space to the Spin-Hom variety SpH_k in Mat(k+1,2k). This is a general result encapsulating some theoretical aspects of the problem, but, as it stands, is still some way from tractable applications.
A Gorenstein A-algebra R of codimension 2 is a perfect finite A-algebra such that R=Ext^2(R,A) holds as R-modules, A being a Cohen-Macaulay local ring with dim(A)-dim_A(R)=2. I prove a structure theorem for these algebras improving on an old theorem of M. Grassi. Special attention is paid to the question how the ring structure of R is encoded in its Hilbert resolution. It is shown that R is automatically a ring once one imposes a weak depth condition on a determinantal ideal derived from a presentation matrix of R over A. The interplay of Gorenstein algebras and Koszul modules as introduced by…
We deal with the Weak Lefschetz property (WLP) for Artinian standard graded Gorenstein algebras of codimension $3.$ We prove that many Gorenstein sequences force the WLP for such algebras. Moreover for every Gorenstein sequence $H$ of codimension 3 we found several Gorenstein Betti sequences compatible with $H$ which again force the WLP. Finally we show that for every Gorenstein Betti sequence the general Artinian standard graded Gorenstein algebra with such Betti sequence has the WLP.
In this paper I investigate minimal surfaces of general type with p_g=5, q=0 for which the 1-canonical map is a birational morphism onto a surface in P^4 (so called canonical surfaces in P^4) via a structure theorem for the Hilbert resolutions of the canonical rings of the afore-mentioned surfaces, viewed as Gorenstein algebras of codimension 2 over the homogeneous coordinate ring of P^4. I discuss how the ring structure of such an algebra is encoded in its resolution. Among other things I show how this method can be applied to analyze the moduli space of canonical surfaces with p_g=5, q=0, K^…
We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle degree. In the first open case, namely Hilbert function (1,3,6,6,3,1), we give a complete answer in every characteristic by translating the problem to one of studying geometric aspects of certain morphisms from $\mathbb P^2$ to $\mathbb P^3$, and Hesse configurations in $\mathbb P^2$.
We study the Rees algebra of a perfect Gorenstein ideal of codimension 3 in a hypersurface ring. We provide a minimal generating set of the defining ideal of these rings by introducing a modified Jacobian dual and applying a recursive algorithm. Once the defining equations are known, we explore properties of these Rees algebras such as Cohen-Macaulayness and Castelnuovo-Mumford regularity.
Let $R$ be a polynomial ring over a field and $I \subset R$ 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 arXiv:1805.05180, 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 $I$.
We study Gorenstein liaison of codimension two subschemes of an arithmetically Gorenstein scheme X. Our main result is a criterion for two such subschemes to be in the same Gorenstein liaison class, in terms of the category of ACM sheaves on X. As a consequence we obtain a criterion for X to have the property that every codimension 2 arithmetically Cohen-Macaulay subscheme is in the Gorenstein liaison class of a complete intersection. Using these tools we prove that every arithmetically Gorenstein subscheme of $\mathbb{P}^n$ is in the Gorenstein liaison class of a complete intesection and we a…
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 foll…
A projectively normal Calabi-Yau threefold $X \subseteq \mathbb{P}^n$ has an ideal $I_X$ which is arithmetically Gorenstein, of Castelnuovo-Mumford regularity four. Such ideals have been intensively studied when $I_X$ is a complete intersection, as well as in the case where $X$ is codimension three. In the latter case, the Buchsbaum-Eisenbud theorem shows that $I_X$ is given by the Pfaffians of a skew-symmetric matrix. A number of recent papers study the situation when $I_X$ has codimension four. We prove there are 16 possible betti tables for an arithmetically Gorenstein ideal $I$ with $\math…
In this paper we investigate the question of when the determinantal ring $R$ over a field $k$ is an almost Gorenstein local/graded ring in the sense of Goto, Takahashi, and the author. As a consequence of the main result, we see that if $R$ is a non-Gorenstein almost Gorenstein local/graded ring, then the ring $R$ has a minimal multiplicity.
We study the concept of liaison addition for codimension two subschemes of an arithmetically Gorenstein projective scheme. We show how it relates to liaison and biliaison classes of subschemes and use it to investigate the structure of Gorenstein liaison equivalence classes, extending the known theory for complete intersection liaison of codimension two subschemes. In particular, we show that on the non-singular quadric threefold in projective 4-space, every non-licci ACM curve can be obtained from a single line by successive liaison additions with lines and CI-biliaisons.
It has been conjectured that {\it all} graded Artinian Gorenstein algebras of codimension three have the weak Lefschetz property over a field of characteristic zero. In this paper, we study the weak Lefschetz property of associated graded algebras $A$ of the Apéry set of $M$-pure symmetric numerical semigroups generated by four natural numbers. In 2010, Bryant proved that these algebras are graded Artinian Gorenstein algebras of codimension three. In a recent article, Guerrieri showed that if $A$ is not a complete intersection, then $A$ is of form $A=R/I$ with $R=K[x,y,z]$ and \begin{align*} I…
The Jordan type of an Artinian algebra is the Jordan block partition associated to multiplication by a generic element of the maximal ideal. We study the Jordan type for Artinian Gorenstein (AG) local algebras A, and the interaction of Jordan type with the symmetric decomposition of the Hilbert function H(A). We give examples of Gorenstein sequences H for which the family Gor(H) of AG algebras having Hilbert function H has several irreducible components, each corresponding to a symmetric decomposition of H. The component structure results from the intersection of two opposing filtrations of th…
We classify all possible $h$-vectors of graded artinian Gorenstein algebras in socle degree 4 and codimension $\leq 17$, and in socle degree 5 and codimension $\leq 25$. We obtain as a consequence that the least number of variables allowing the existence of a nonunimodal Gorenstein $h$-vector is 13 for socle degree 4, and 17 for socle degree 5. In particular, the smallest nonunimodal Gorenstein $h$-vector is $(1,13,12,13,1)$, which was constructed by Stanley in his 1978 seminal paper on level algebras. This solves a long-standing open question in this area. All of our results are characteristi…
We establish the multiplicity conjecture of Herzog, Huneke, and Srinivasan about the multiplicity of graded Cohen-Macaulay algebras over a field, for codimension two algebras and for Gorenstein algebras of codimension three. In fact, we prove stronger bounds than the conjectured ones allowing us to characterize the extremal cases. This may be seen as a converse to the multiplicity formula of Huneke and Miller that inspired the conjectural bounds.
This is the second paper in a series of three papers. In the first paper of the series, "Artinian Gorenstein algebras with linear resolutions", (arXiv:1306.2523, J. of Algebra, to appear) we prove that it is possible to give the minimal resolution of the rings from the title in terms of the coefficients of the corresponding Macaulay inverse system. In this context, the word "give" means, "give in a polynomial manner". The first paper in the series proves, essentially, an existence theorem. The second and third papers in the series construct the explicit formulas for the resolution. The present…
We prove upper and lower bounds for all the coefficients in the Hilbert Polynomial of a graded Gorenstein algebra $S=R/I$ with a quasi-pure resolution over $R$. The bounds are in terms of the minimal and the maximal shifts in the resolution of $R$ . These bounds are analogous to the bounds for the multiplicity found in \cite{S} and are stronger than the bounds for the Cohen Macaulay algebras found in \cite{HZ}.
Let $X$ be a normal arithmetically Gorenstein scheme in ${\mathbb P}^n$. We give a criterion for all codimension two ACM subschemes of $X$ to be in the same Gorenstein biliaison class on $X$, in terms of the category of ACM sheaves on $X$. These are sheaves that correspond to the graded maximal Cohen--Macaulay modules on the homogeneous coordinate ring of $X$. Using known results on MCM modules, we are able to determine the Gorenstein biliaison classes of codimension two subschemes of certain varieties, including the nonsingular quadric surface in ${\mathbb P}^3$, and the cone over it in ${\ma…
This paper gives a necessary and sufficient condition for Gorensteinness in Rees algebras of the $d$-th power of parameter ideals in certain Noetherian local rings of dimension $d\ge 2$. The main result of this paper produces many Gorenstein Rees algebras over non-Cohen-Macaulay local rings. For example, the Rees algebra $\mathcal{R}(\mathfrak{q}^d)=\oplus_{i\ge 0}\mathfrak{q}^{di}$ is Gorenstein for every parameter ideal $\mathfrak{q}$ that is a reduction of the maximal ideal in a $d$-dimensional Buchsbaum local ring of depth 1 and multiplicity 2.
We study Stanley decompositions and show that Stanley's conjecture on Stanley decompositions implies his conjecture on partitionable Cohen-Macaulay simplicial complexes. We also prove these conjectures for all Cohen-Macaulay monomial ideals of codimension 2 and all Gorenstein monomial ideals of codimension 3.
We propose a concept of module liaison that extends Gorenstein liaison of ideals and provides an equivalence relation among unmixed modules over a commutative Gorenstein ring. Analyzing the resulting equivalence classes we show that several results known for Gorenstein liaison are still true in the more general case of module liaison. In particular, we construct two maps from the set of even liaison classes of modules of fixed codimension into stable equivalence classes of certain reflexive modules. As a consequence, we show that the intermediate cohomology modules and properties like being pe…
In this paper we show a partial answer the a question of C. Huneke and G. Leuschke (2003): Let R be a standard graded Cohen-Macaulay ring of graded countable Cohen-Macaulay representation type, and assume that R has an isolated singularity. Is R then necessarily of graded finite Cohen-Macaulay representation type? In particular, this question has an affirmative answer for standard graded non-Gorenstein rings as well as for standard graded Gorenstein rings of minimal multiplicity. Along the way, we obtain a partial classification of graded Cohen-Macaulay rings of graded countable Cohen-Macaulay…
We study the weak Lefschetz property of artinian Gorenstein algebras and in particular of artinian complete intersections. In codimension four and higher, it is an open problem whether all complete intersections have the weak Lefschetz property. For a given artinian Gorenstein algebra $A$ we ask what linear forms are Lefschetz elements for this particular algebra, i.e., which linear forms $\ell$ give maximal rank for all the multiplication maps $\times \ell: [A]_i \longrightarrow [A]_{i+1}$. This is a Zariski open set and its complement is the \emph{non-Lefschetz locus}. For monomial complete …
The Multiplicity Conjecture (MC) of Huneke and Srinivasan provides upper and lower bounds for the multiplicity of a Cohen-Macaulay algebra $A$ in terms of the shifts appearing in the modules of the minimal free resolution (MFR) of $A$. All the examples studied so far have lead to conjecture (see $[HZ]$ and $[MNR2]$) that, moreover, the bounds of the MC are sharp if and only if $A$ has a pure MFR. Therefore, it seems a reasonable - and useful - idea to seek better, if possibly {\it ad hoc}, bounds for particular classes of Cohen-Macaulay algebras. In this work we will only consider the codimens…
We first determine all height four Gorenstein sequences beginning H=(1,4,7,...), and we show that their first differences satisfy $ΔH_{\le j/2}$ is an O-sequence. We then study the family PGor(H) parametrizing all graded Artinian Gorenstein [AG] quotients A=R/I of the polynomial ring R=K[w,x,y,z] having a Hilbert function H as above. We give a structure theorem for such AG quotients with $I_2\cong < wx,wy,wz>$. For most H this subfamily forms an irreducible component of PGor(H), and for a slightly more restrictive set, PGor(H) has several irreducible components. M. Boij and others had already …
The study of the $h$-vectors of graded Gorenstein algebras is an important topic in combinatorial commutative algebra, which despite the large amount of literature produced during the last several years, still presents many interesting open questions. In this note, we commence a study of those unimodal Gorenstein $h$-vectors that do \emph{not} satisfy the Stanley-Iarrobino property. Our main results, which are characteristic free, show that such $h$-vectors exist: 1) In socle degree $e$ if and only if $e\ge 6$; and 2) In every codimension five or greater. The main case that remains open is tha…
Let $X$ be a codimension 1 subvariety of dimension $>1$ of a variety of minimal degree $Y$. If $X$ is subcanonical with Gorenstein canonical singularities admitting a crepant resolution, then $X$ is Arithmetically Gorenstein and we characterise such subvarieties $X$ of $Y$ via apolarity as those whose apolar hypersurfaces are Fermat.
We give conjectures on the possible graded Betti numbers of Cohen-Macaulay modules up to multiplication by positive rational numbers. The idea is that the Betti diagrams should be non-negative linear combinations of pure diagrams. The conjectures are verified in the cases where the structure of resolutions are known, i.e., for modules of codimension two, for Gorenstein algebras of codimension three and for complete intersections. The motivation for the conjectures comes from the Multiplicity conjecture of Herzog, Huneke and Srinivasan.
The Nakayama conjecture is one of the most important conjectures in ring theory. The Auslander-Reiten conjecture is closely related to it. The purpose of this note is to show that if the Auslander-Reiten conjecture holds in codimension one for a commutative Gorenstein ring $R$, then it holds for $R$.
Macaulay's Inverse System gives an effective method to construct Artinian Gorenstein k-algebras. To date a general structure for Gorenstein k-algebras of any dimension (and codimension) is not understood. In this paper we extend Macaulay's correspondence characterizing the submodules of the divided power ring in one-to-one correspondence with Gorenstein d-dimensional k-algebras. We discuss effective methods for constructing Gorenstein graded rings. Several examples illustrating our results are given.
