Socle Degree Research Articles

Socle degrees, resolutions, and Frobenius powers

We first describe a situation in which every graded Betti number in the tail of the resolution of R J may be read from the socle degrees of R J . Then we apply the above result to the ideals J and J [ q ] ; and thereby describe a situation in which the graded Betti numbers in the tail of the resolution of R / J [ q ] are equal to the graded Betti numbers in the tail of a shift of the resolution of R / J .

Interval conjectures for level Hilbert functions

We conjecture that the set of all Hilbert functions of (artinian) level algebras enjoys a very natural form of regularity, which we call the Interval Conjecture (IC): If, for some positive integer α, ( 1 , h 1 , … , h i , … , h e ) and ( 1 , h 1 , … , h i + α , … , h e ) are both level h-vectors, then ( 1 , h 1 , … , h i + β , … , h e ) is also level for each integer β = 0 , 1 , … , α . In the Gorenstein case, i.e. when h e = 1 , we also supply the Gorenstein Interval Conjecture (GIC), which naturally generalizes the IC, and basically states that the same property simultaneously holds for any two symmetric entries, say h i and h e − i , of a Gorenstein h-vector. These conjectures are inspired by the research performed in this area over the last few years. A series of recent results seems to indicate that it will be nearly impossible to characterize explicitly the sets of all Gorenstein or of all level Hilbert functions. Therefore, our conjectures would at least provide the existence of a very strong — and natural — form of order in the structure of such important and complicated sets. We are still far from proving the conjectures at this point. However, we will already solve a few interesting cases, especially when it comes to the IC, in this paper. Among them, that of Gorenstein h-vectors of socle degree 4, that of level h-vectors of socle degree 2, and that of non-unimodal level h-vectors of socle degree 3 and any given codimension.

Hilbert function of local Artinian level rings in codimension two

In this paper we characterize the possible Hilbert functions of a local Artinian level ring of given type and socle degree in the codimension two case. In particular, given an “admissible” numerical function h, an effective method is given to produce a zero-dimensional ideal I of R = k 〚 x , y 〛 such that R / I is a local Artinian level ring having h as Hilbert function. As a consequence we recover the well-known result proved by Briançon and Iarrobino concerning the Hilbert functions of a complete intersection of height two. We extend notions of standard or Gröbner bases from the graded to the local algebra context. The principal tools concern the concepts of enhanced standard basis and generic initial ideals which are defined for an ideal in the formal power series ring R = k 〚 x 1 , … , x n 〛 .

Bounds and asymptotic minimal growth for Gorenstein Hilbert functions

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 ⩾ 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 Stanley that all Gorenstein h-vectors in codimension three are unimodal. Our second main theorem is an asymptotic formula for the least value that the ith entry of a Gorenstein h-vector may assume, in terms of codimension, r, and socle degree, e. This theorem broadly generalizes a recent result of ours, where we proved a conjecture of Stanley predicting that asymptotic value in the specific case e = 4 and i = 2 , as well as a result of Kleinschmidt which concerned the logarithmic asymptotic behavior in degree i = ⌊ e 2 ⌋ .

The strength of the Weak Lefschetz Property

We study a number of conditions on the Hilbert function of a level Artinian algebra which imply the Weak Lefschetz Property (WLP). Possibly the most important open case is whether a codimension 3 SI-sequence forces the WLP for level algebras. In other words, does every codimension 3 Gorenstein algebra have the WLP? We give some new partial answers to this old question: we prove an affirmative answer when the initial degree is 2, or when the Hilbert function is relatively small. Then we give a complete answer to the question of what is the largest socle degree forcing the WLP.

Free resolutions over short local rings

Let R be a local ring with maximal ideal m admitting a non-zero element a\in\fm for which the ideal (0:a) is isomorphic to R/aR. We study minimal free resolutions of finitely generated R-modules M, with particular attention to the case when m^4=0. Let e denote the minimal number of generators of m. If R is Gorenstein with m^4=0 and e\ge 3, we show that \Poi MRt is rational with denominator \HH R{-t} =1-et+et^2-t^3, for each finitely generated R-module M. In particular, this conclusion applies to generic Gorenstein algebras of socle degree 3.

On the degree two entry of a Gorenstein $h$-vector and a conjecture of Stanley

In this short paper we establish a (non-trivial) lower bound on the degree two entry h 2 of a Gorenstein h-vector of any given socle degree e and any codimension r. In particular, when e = 4, that is, for Gorenstein h-vectors of the form h = (1, r, h 2 , r, 1), our lower bound allows us to prove a conjecture of Stanley on the order of magnitude of the minimum value, say f(r), that h 2 may assume. In fact, we show that limr→∞ f(r) r 2/3 =6 2/3 . In general, we wonder whether our lower bound is sharp for all integers e > 4 and r > 2.

The Geometry of the Weak Lefschetz Property and Level Sets of Points

AbstractIn a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property (WLP), and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its Artinian reduction from having WLP, which in itself is a very algebraic notion. More precisely, we produce level sets of points whose Artinian reductions have socle types 3 and 4 and arbitrary socle degree ≥ 12 (in the worst case), but fail to have WLP. We also produce a level set of points whose Artinian reduction fails to have unimodal Hilbert function; our example is based on Zanello's example. Finally, we show that a level set of points can have Artinian reduction that has WLP but fails to have the Strong Lefschetz Property. While our constructions are all based on basic double G-linkage, the implementations use very different methods.

Socle degrees of Frobenius powers

Let $k$ be a field of positive characteristic $p$, $R$ be a Gorenstein graded $k$-algebra, and $S=R/J$ be an artinian quotient of $R$ by a homogeneous ideal. We ask how the socle degrees of $S$ are related to the socle degrees of $F_R^e(S)=R/J^{[q]}$. If $S$ has finite projective dimension as an $R$-module, then the socles of $S$ and $F_R^e(S)$ have the same dimension and the socle degrees are related by the formula $D_i=qd_i-(q-1)a(R)$, where $d_1\le \dots\le d_{\ell}$ and $D_1\le \dots \le D_{\ell}$ are the socle degrees of $S$ and $F_R^e(S)$, respectively, and $a(R)$ is the $a$-invariant of the graded ring $R$, as introduced by Goto and Watanabe. We prove the converse when $R$ is a complete intersection.

New approaches to bounding the multiplicity of an ideal

We use the theory of resolutions for a given Hilbert function to investigate the multiplicity conjectures of Huneke and Srinivasan, and Herzog and Srinivasan. To prove the conjectures for all modules with a particular Hilbert function, we show that it is enough to prove the statements only for elements at the bottom of the partially ordered set of resolutions with that Hilbert function. This enables us to test the conjectured upper bound for the multiplicity efficiently with the computer algebra system Macaulay 2, and we verify the upper bound for many Artinian modules in three variables with small socle degree. Moreover, with this approach, we show that though numerical techniques have been sufficient in several of the known special cases, they are insufficient to prove the conjectures in general. Finally, we apply a result of Herzog and Srinivasan on ideals with a quasipure resolution to prove the upper bound for Cohen–Macaulay quotients by ideals with generators in high degrees relative to the regularity.

CANCELLATION IN RESOLUTIONS AND LEVEL ALGEBRAS

ABSTRACT A difficult open problem (even in codimension 3) asks: what can be the Hilbert function of a standard graded Artinian level algebra? In this paper we solve the related problem (in codimension 3): What are the Hilbert functions H for which the minimal resolution of the lex-segment ideal with Hilbert function H permit (at least theoretically) enough cancellation to give a possible resolution for a level algebra? This gives a necessary (but not sufficient, as we show by example) response to the original open problem. The answer is given in terms of type vectors (which are equivalent to, but different from, Hilbert functions of Artinian algebras). We also give an algorithm (implemented in C.C.A) which describes how to construct all the type vectors (in codimension 3) with fixed socle degree and fixed value in that degree.

On the Resolution of Certain Level Algebras#

In this paper, we investigate the minimal free resolution of certain level algebras which have minimal Hilbert Function among Artinian graded algebras of given socle degree and type. As a consequence of this results, we discover a new class of Hilbert Functions which do not support a unique smallest set of graded Betti numbers and another class which do not support all the possible resolutions between its maximum and minimum elements.

Minimal resolution of relatively compressed level algebras

A relatively compressed algebra with given socle degrees is an Artinian quotient A of a given graded algebra R / c , whose Hilbert function is maximal among such quotients with the given socle degrees. For us c is usually a “general” complete intersection and we usually require that A be level. The precise value of the Hilbert function of a relatively compressed algebra is open, and we show that finding this value is equivalent to the Fröberg conjecture. We then turn to the minimal free resolution of a level algebra relatively compressed with respect to a general complete intersection. When the algebra is Gorenstein of even socle degree we give the precise graded Betti numbers. When it is of odd socle degree we give good bounds on the graded Betti numbers. We also relate this case to the Minimal Resolution Conjecture of Mustaţǎ for points on a projective variety. Finding the graded Betti numbers is essentially equivalent to determining to what extent there can be redundant summands (i.e., “ghost terms”) in the minimal free resolution, i.e., when copies of the same R ( − t ) can occur in two consecutive free modules. This is easy to arrange using Koszul syzygies; we show that it can also occur in more surprising situations that are not Koszul. Using the equivalence to the Fröberg conjecture, we show that in a polynomial ring where that conjecture holds (e.g., in three variables), the possible non-Koszul ghost terms are extremely limited. Finally, we use the connection to the Fröberg conjecture, as well as the calculation of the minimal free resolution for relatively compressed Gorenstein algebras, to find the minimal free resolution of general Artinian almost complete intersections in many new cases. This greatly extends previous work of the first two authors.

On parameter spaces for artin level algebras

The set of all artin level quotients of a polynomial algebra in n variables having specified socle degree and type admits a parameter space. It is in fact a quasiprojective variety, naturally embedded in a Grassmannian. We give a geometric description of this variety in the case of two variables. Then we give some sufficient conditions for the parameter variety to be projectively normal in the Plucker embedding.

Level Algebras, Lex Segments, and Minimal Hilbert Functions

In this paper we prove the existence of minimal level artinian graded algebras having socle degree r and type t and describe their h-vector in terms of the r-binomial expansion of t. We also investigate the graded Betti numbers of such algebras and completely describe their extremal resolutions. We also show that any set of points in ℙ n whose Hilbert function has first difference as described above, must satisfy the Cayley-Bacharach property.

On the minimal free resolution of $n+1$ general forms

O-ROIG ⁄⁄ Abstract. Let R = k(x1;:::;xn) and let I be the ideal of n + 1 generically chosen forms of degrees d1 • ¢¢¢ • dn+1. We give the precise graded Betti numbers of R=I in the following cases: † n = 3. † n = 4 and P5=1 di is even. † n = 4, P 5=1 di is odd and d2 + d3 + d4 < d1 + d5 + 4. † n is even and all generators have the same degree, a, which is even. † ( Pn+1 i=1 di) i n is even and d2 + ¢¢¢ + dn < d1 + dn+1 + n. † ( P n+1 i=1 di) i n is odd, n ‚ 6 is even, d2 + ¢¢¢ + dn < d1 + dn+1 + n and d1 + ¢¢¢ + dn i dn+1 i n ? 0. We give very good bounds on the graded Betti numbers in many other cases. We also extend a result of M. Boij by giving the graded Betti numbers for a generic compressed Gorenstein algebra (i.e. one for which the Hilbert function is maximal given n and the socle degree) when n is even and the socle degree is large. A recurring theme is to examine when and why the minimal free resolution may be forced to have redundant summands. We conjecture that if the forms all have the same degree then there are no redundant summands, and we present some evidence for this conjecture.

Hilbert Functions and Level Algebras

The authors consider certain quotients A=R/I of the polynomial ring R=K[x1,…,xr] over an arbitrary field K. They first determine upper and lower bounds on the Hilbert functions of any algebra having the form A=R/V, where V̄ is the largest ideal of R agreeing in degrees at least j with the ideal (V) generated by a vector subspace V⊂Rj of degree j forms: these bounds extend the Macaulay bounds (Theorem 1.4). A level Artinian quotient of A=R/I has socle Soc(A)=(0:M),M=(x1,…,xr) in a single degree j. The authors next determine the extremal Hilbert functions Hmax(t,j,r) and Hmin(t,j,r) that occur for level graded Artin algebra quotients of R having socle degree j and “type” t=dimKSoc(A), and they describe the extremal strata (Theorem 1.8).They next give a natural upper bound for the Hilbert function of level algebra quotients of the coordinate ring OZ of a punctual subscheme Z⊂Pn,n=r−1, in terms of the Hilbert function HZ and j,t. This bound was known and known to be sharp in the case Z is locally Gorenstein and t=1. Finally, they show that there are no level algebras of Hilbert function T=(1,3,4,5,6,…,2) (Proposition 2.7). Given that there are smooth punctual subschemes Z of P2 with Hilbert function HZ=(1,3,4,5,6,6,…), this result shows that the sharpness of the natural upper bound in the case where Z is Gorenstein and t=1 does not extend to level algebra quotients of type 2, even when Z is smooth.The extremality results for the Hilbert functions of level algebras, and some counterexamples, use the extremality theorem of F. H. S. Macaulay [1927, Proc. London Math. Soc.26, 531–555] and G. Gotzmann's [1978, Math. Z.158, 61–70] results on the Hilbert scheme.

Gröbner Flags and Gorenstein Algebras

The goal of this paper is to study the Koszul property and the property of having a Gröbner basis of quadrics for classical varieties and algebras as canonical curves, finite sets of points and Artinian Gorenstein algebras with socle in low degree. Our approach is based on the notion of Gröbner flags and Koszul filtrations. The main results are the existence of a Gröbner basis of quadrics for the ideal of the canonical curve whenever it is defined by quadrics, the existence of a Gröbner basis of quadrics for the defining ideal of s [les ] 2n points in general linear position in Pn, and the Koszul property of the ‘generic’ Artinian Gorenstein algebra of socle degree 3.

High-order vanishing ideals at n+3 points of Pn

The authors conjecture that the ideal IZ[t−1] of functions vanishing to order t−1 at the subscheme Z of Pn, n=2t−1 comprised of 2t+2 generic smooth points, satisfies dimk(IZ[t−1])t=2t, in its initial degree, t. They show that this dimension is at least t+1, by a direct construction of suitable vanishing forms. This result is complementary to those of M.V. Catalisano, P. Ellia, and A. Gimigliano in [5]. The authors also consider related problems, including the Macaulay dual problem, of determining the Hilbert function H(A), A=R/(x12,…,xr2,(x1+⋯+xr)2,L2) — where R=k[x1,…,xr], r=2t and L is a generic linear form — in the socle degree t of A.

Gorenstein Witt Rings II

AbstractThe abstract Witt rings which are Gorenstein have been classified when the dimension is one and the classification problem for those of dimension zero has been reduced to the case of socle degree three. Here we classify the Gorenstein Witt rings of fields with dimension zero and socle degree three. They are of elementary type.