Artinian level algebras of socle degree 4

TL;DR

This paper characterizes the O-sequences of Artinian level and Gorenstein algebras with socle degree 4, providing necessary and sufficient conditions and constructive methods for their realization.

Contribution

It establishes exact criteria for O-sequences of level and Gorenstein algebras of socle degree 4 and offers explicit construction methods.

Findings

Characterization of O-sequences for level algebras with socle degree 4.

Criteria for Gorenstein algebra O-sequences with socle degree 4.

Refinement of conditions for Gorenstein algebras to be canonically graded.

Abstract

In this paper we study the O-sequences of the local (or graded) $K$-algebras of socle degree $4.$ More precisely, we prove that an O-sequence $h=(1, 3, h_2, h_3, h_4)$, where $h_4 \geq 2,$ is the $h$-vector of a local level $K$-algebra if and only if $h_3\leq 3 h_4.$ We also prove that $h=(1, 3, h_2, h_3, 1)$ is the $h$-vector of a local Gorenstein $K$-algebra if and only if $h_3 \leq 3$ and $h_2 \leq \binom{h_3+1}{2}+(3-h_3).$ In each of these cases we give an effective method to construct a local level $K$-algebra with a given $h$-vector. Moreover we refine a result by Elias and Rossi by showing that if $h=(1,h_1, h_2, h_3, 1)$ is an unimodal Gorenstein O-sequence, then $h$ forces the corresponding Gorenstein $K$-algebra to be canonically graded if and only if $h_1=h_3 $ and $h_2=\binom{h_1+1}{2}, $ that is the $h$-vector is maximal.

Tables

Table 1. Table 1.

$(1,3,2,2,2)$	$(1,3,3,2,2)$	$(1,3,4,2,2)$	$(1,3,5,2,2)$	$(1,3,6,2,2)$	$(1,3,5,3,2)$
$(1,3,6,3,2)$	$(1,3,3,4,2)$	$(1,3,4,3,3)$	$(1,3,5,3,3)$	$(1,3,6,3,3)$	$(1,3,3,4,3)$
$(1,3,6,4,3)$	$(1,3,3,4,4)$	$(1,3,5,4,4)$	$(1,3,6,4,4)$	$(1,3,3,4,5)$	$(1,3,4,4,5)$
$(1,3,5,4,5)$	$(1,3,6,4,5)$	$(1,3,6,5,5)$	$(1,3,5,5,6)$	$(1,3,6,5,6)$	$(1,3,6,6,7)$
$(1,3,6,7,9)$

Table 2. Table 2.

$(1,3,1,1,1)$

$(1,3,2,1,1)$

$(1,3,3,1,1)$

$(1,3,2,2,1)$

$(1,3,3,2,1)$

$(1,3,4,2,1)$

Equations

$$e_i:={\dim }_K((0:\mathfrak{m})\cap {\mathfrak{m}}^i)/(0:\mathfrak{m})\cap {\mathfrak{m}}^{i+1}).$$

$$h_i\le \min \{{\dim }_K{R}_i,e_i{\dim }_K{R}_0+e_{i+1}{\dim }_K{R}_1+\cdots +e_s{\dim }_K{R}_{s-i}\}$$

$$h_{s-1}\le h_1\text{mbox}and{h}_{s-2}\le \left(\genfrac{}{}{0ex}{}{h_{s-1}+1}{2}\right)+(h_1-h_{s-1}).$$

$$\begin{array}[]{ cccc}\circ:&R\times P&\longrightarrow&P\\ &(f,g)&\longrightarrow&f\circ g=f(\partial_{X_{1}},\dots,\partial_{X_{r}})(g)\end{array}$$

$$\begin{array}[]{ cccc}\langle\ ,\ \rangle:&R\times P&\longrightarrow&K\\ &(f,g)&\longrightarrow&(f\circ g)(0).\end{array}$$

$$I^{\perp }:=\{g\in P|⟨f,g⟩=0\forall f\in I\}.$$

$${\mathrm{Ann}}_R(M):=\{g\in R|⟨g,f⟩=0\forall f\in M\}$$

$$\displaystyle\left\{\begin{array}[]{cc}I\subseteq R\mbox{ such that }R/I\\ \mbox{ is a level local ring of }\\ \mbox{ socle degree $s$ and type }\tau\end{array}\right\}\$$

$$\displaystyle I\ \ \ \ \$$

$$\displaystyle\operatorname{Ann}_{R}(M)\ \ \ \ \$$

$$(R/I{)}^{\ast }\simeq I^{\perp },$$

$$(I^{\perp }{)}_i:=\frac{I^{\perp }\cap P_{\le i}+P_{<i}}{P_{<i}}.$$

$$h_i(R/I)={\dim }_K(I^{\perp }{)}_i.$$

$$G=C(0)\supseteq C(1)\supseteq \cdots \supseteq C(s)=0,$$

$$C(a{)}_i:=\frac{(0:{\mathfrak{m}}^{s+1-a-i})\cap {\mathfrak{m}}^i}{(0:{\mathfrak{m}}^{s+1-a-i})\cap {\mathfrak{m}}^{i+1}}.$$

$$Q(a)=C(a)/C(a+1).$$

$$\{Q(a):a=0,\dots ,s-1\}$$

$$h_i(A)={\dim }_K{G}_i=\sum _{a=0}^{s-1}{\dim }_{K}Q(a{)}_i.$$

$$h_{s-1}\le h_1\text{mbox}and{h}_{s-2}\le \left(\left(\genfrac{}{}{0ex}{}{h_{s-1}+1}{2}\right)+(h_1-h_{s-1})\right).$$

$$h_{s-2}$$

$$f=x_1^4+\cdots +x_{h_3}^4+x_{h_3+1}^3+\cdots +x_{h_2}^3+x_{h_2+1}^2+\cdots +x_{h_1}^2.$$

$$g_i=\{\begin{array}{ll}{x}_i^2& \text{mbox}if1\le i\le n\\ x_{i-n}{x}_{i-n+1}& \text{mbox}ifn+1\le i\le 2n-1\\ x_n{x}_1& \text{mbox}ifi=2n.\end{array}$$

$$x^{\underline{i}}\notin \{g_i:1\le i\le 2n\}.$$

$$f=\{\begin{array}{ll}\sum _{i=1}^n{x}_i^2{g}_i+\sum _{i=1}^{h_2-h_1}{x}_i^2{g}_{n+i}+x_{n+1}^3+\cdots +x_{h_1}^3& \text{mbox}if{h}_2-h_1\le n\\ \sum _{i=1}^n{x}_i^2{g}_i+\sum _{i=1}^n{x}_i^2{g}_{n+i}+\sum _{2n+1}^{h_2-h_1+n}{g}_i^2+x_{n+1}^3+\cdots +x_{h_1}^3& \text{mbox}if{h}_2-h_1>n.\end{array}$$

$$\displaystyle h_{2}=\dim_{K}\left(\{g_{i}:1\leq i\leq h_{2}-h_{1}+n\}\ooalign{\Large$\bigcup$\cr\hss\raisebox{-0.21529pt}{\scriptsize$\cdot$}\hss}\{x_{n+1}^{2},\ldots,x_{h_{1}}^{2}\}\right).$$

$$f_1=x_1^4+g_1+g_2\text{mbox}and{f}_2=x_2^4+g_3+g_4.$$

$$h_3$$

$$h_2$$

$$f_4=\{\begin{array}{ll}{x}_2^3{x}_3& \text{mbox}if\{m\ge n\text{mbox}andn\le 6\}\text{mbox}OR\{n\ge 7\text{mbox}andm=6\}\\ x_1^3{x}_2& \text{mbox}if\{m<n\le 6\}\text{mbox}OR\{(m,n)=(5,7)\}.\end{array}$$

$$f_5$$

$$f_6$$

$$f_7$$

$$f_8=x_1^2{x}_2^2,f_9=x_1^2{x}_3^2,f_{10}=x_2{x}_3^3,f_{11}=x_1{x}_2^3,f_{12}=x_1^3{x}_3,f_{13}=x_2^3{x}_3,f_{14}=x_1{x}_2^2{x}_3,f_{15}=x_1{x}_2{x}_3^2.$$

Full text

The first author was supported by INdAM COFOUND Fellowships cofounded by Marie Curie actions, Italy. The second author was partially supported by PRIN 2014 “Geometria delle varieta’ algebriche”

1. Introduction

Let $(A,{\mathfrak{m}})$ be an Artinian local or graded $K$-algebra where $K$ is an algebraically closed field of characteristic zero. Let $\operatorname{Soc}(A)=(0:{\mathfrak{m}})$ be the socle of $A.$ We denote by $s$ the socle degree of $A,$ that is the maximum integer $j$ such that ${\mathfrak{m}}^{j}\neq 0.$ The type of $A$ is $\tau:=\dim_{K}\operatorname{Soc}(A).$ Recall that $A$ is said to be level of type $\tau$ if $\operatorname{Soc}(A)={\mathfrak{m}}^{s}$ and $\dim_{K}{\mathfrak{m}}^{s}=\tau.$ If $A$ has type $1,$ equivalently $\dim_{K}\operatorname{Soc}(A)=1,$ then $A$ is Gorenstein. In the literature local rings with low socle degree, also called short local rings, have emerged as a testing ground for properties of infinite free resolutions (see [AIS08], [AHS13], [HS11], [Sjo79], [Roos05], [CRV01]). They have been also extensively studied in problems related to the irreducibility and the smoothness of the punctual Hilbert scheme $\operatorname{Hilb}_{d}(\mathbb{P}^{n}_{K})$ parameterizing zero-dimensional subschemes $\mathbb{P}^{n}_{K}$ of degree $d,$ see among others [Poo08], [EV08], [CEVV09], [CENR13]. In this paper we study the structure of level $K$-algebras of socle degree $4,$ hence ${\mathfrak{m}}^{5}=0.$ One of the most significant information on the structure is given by the Hilbert function.

By definition, the Hilbert function of $A,$ $h_{i}=h_{i}(A)=\dim_{K}{\mathfrak{m}}^{i}/{\mathfrak{m}}^{i+1},$ is the Hilbert function of the associated graded ring $gr_{\mathfrak{m}}(A)=\oplus_{i\geq 0}{\mathfrak{m}}^{i}/{\mathfrak{m}}^{i+1}.$ We also say that $h=(h_{0},h_{1},\ldots,h_{s})$ is the $h$-vector of $A.$ In [Mac27] Macaulay characterized the possible sequences of positive integers $h_{i}$ that can occur as the Hilbert function of $A.$ Since then there has been a great interest in commutative algebra in determining the $h$-vectors that can occur as the Hilbert function of $A$ with additional properties (for example, complete intersection, Gorenstein, level, etc). A sequence of positive integers $h=(h_{0},h_{1},\ldots,h_{s})$ satisfying Macaulay’s criterion, that is $h_{0}=1$ and $h_{i+1}\leq h_{i}^{\langle i\rangle}$ for $i=1,\ldots,s-1,$ is called an O-sequence. A sequence $h=(1,h_{1},\ldots,h_{s})$ is said to be a level (resp. Gorenstein) O-sequence if $h$ is the Hilbert function of some Artinian level (resp. Gorenstein) $K$-algebra $A.$ Remark that $h_{1}$ is the embedding dimension and, if $A$ is level, $h_{s}$ is the type of $A.$ Notice that a level O-sequence is not necessarily the Hilbert function of an Artinian level graded $K$-algebra. This is because the Hilbert function of the level ring $(A,{\mathfrak{m}})$ is the Hilbert funtion of $gr_{{\mathfrak{m}}}(A)$ which is not necessarily level. From now on we say that $h$ is a graded level O-sequence if $h$ is the Hilbert function of a level graded standard $K$-algebra. For instance it is well known that the $h$-vector of a Gorenstein graded $K$-algebra is symmetric, but this is no longer true for a Gorenstein local ring. Characterize level O-sequences is a wide open problem in commutative algebra. The problem is difficult and very few is known even in the graded case as evidenced by [GHMY07]. In the following table we give a summary of known results:

[TABLE]

In this paper we fill the above table by characterizing the Gorenstein and level O-sequences with a particular attention to socle degree $4$ and embedding dimension $h_{1}=3.$

In our setting we can write $A=R/I$ where $R=K[[x_{1},\dots,x_{r}]]$ (resp. $K[x_{1},\ldots,x_{r}]$) is the formal power series ring (resp. the polynomial ring with standard grading) and $I$ an ideal of $R.$ We say that $A$ is graded when it can be presented as $R/I$ where $I$ is an homogeneous ideal in $R=K[x_{1},\ldots,x_{r}].$ Without loss of generality we assume that $h_{1}=\dim_{K}{\mathfrak{m}}/{\mathfrak{m}}^{2}=r.$

Recall that the socle type of $A=R/I$ is the sequence $E=(0,e_{1},\ldots,e_{s}),$ where

[TABLE]

It is known that for all $i\geq 0,$

[TABLE]

(see [Iar84]). Hence a necessary condition for $h$ to be a level O-sequence is that $h_{s-1}\leq h_{1}h_{s}$ being $e_{s}=h_{s}$ and $e_{i}=0$ otherwise. In the following theorem we prove that this condition is also sufficient for $h=(1,3,h_{2},h_{3},h_{4})$ to be a level O-sequence, provided $h_{4}\geq 2.$ However, if $h_{4}=1,$ we need an additional assumption for $h$ to be a Gorenstein O-sequence. We remark that the result can not be extended to $h_{1}>3$ (see Example 3.8).

Theorem 1**.**

Let $h=(1,3,h_{2},h_{3},h_{4})$ be an O-sequence.

1. (a)

Let $h_{4}=1.$ Then $h$ is a Gorenstein O-sequence if and only if $h_{3}\leq 3$ and $h_{2}\leq\binom{h_{3}+1}{2}+(3-h_{3}).$ 2. (b)

Let $h_{4}\geq 2$. Then $h$ is a level O-sequence if and only if $h_{3}\leq 3h_{4}.$

The proof of the above result is effective in the sense that in each case we construct a local level $K$-algebra with a given $h$-vector verifying the necessary conditions (see Theorem 3.7).

Combining Theorem 1(b) and results in [GHMY07] we show that there are level O-sequences which are not admissible in the graded case (see Section 3, Table 1). A similar behavior was observed in [Ste14] for socle degree $3.$ Theorem 1(a) is a consequence of the following more general result which holds for any embedding dimension:

Theorem 2**.**

1. (a)

If $(1,h_{1},\ldots,h_{s-2},h_{s-1},1)$ is a Gorenstein O-sequence, then

[TABLE] 2. (b)

If $h=(1,h_{1},h_{2},h_{3},1)$ is an unimodal O-sequence satisfying (1.2), then $h$ is a Gorenstein O-sequence.

The bound (1.2) improves the bound for $h_{s-2}$ given in (1.1).

If $A$ is a Gorenstein local $K$-algebra with symmetric $h$-vector, then $gr_{{\mathfrak{m}}}(A)$ is Gorenstein, see [Iar94]. It is a natural question to ask, in this case, whether $A$ is analytically isomorphic to $gr_{{\mathfrak{m}}}(A).$ Accordingly with the definition given in [Ems78, Page 408] and in [ER12], recall that an Artinian local $K$-algebra $(A,{\mathfrak{m}})$ is said to be canonically graded if there exists a $K$-algebra isomorphism between $A$ and its associated graded ring $gr_{{\mathfrak{m}}}(A).$

For instance J. Elias and M. E. Rossi in [ER12] proved that every Gorenstein $K$-algebra with symmetric $h$-vector and ${\mathfrak{m}}^{4}=0$ ($s\leq 3$) is canonically graded. A local $K$-algebra $A$ of socle type $E$ is said to be compressed if equality holds in (1.1) for all $1\leq i\leq s,$ equivalently the $h$-vector is maximal (see [Iar84, Definition 2.3]). In [ER15, Theorem 3.1] Elias and Rossi proved that if $A$ is any compressed Gorenstein local $K$-algebra of socle degree $s\leq 4,$ then $A$ is canonically graded. In Section 4 we prove that if the socle degree is $4,$ then the assumption can not be relaxed. More precisely only the maximal $h$-vector forces every corresponding Gorenstein $K$-algebra to be canonically graded. We prove that if $h$ is unimodal and not maximal, then there exists a Gorenstein $K$-algebra with Hilbert function $h$ which is not canonically graded. To prove that a local $K$-algebra is not canonically graded is in general a very difficult task. See also [Jel16] for interesting discussions.

Theorem 3**.**

Let $h=(1,h_{1},h_{2},h_{3},1)$ be a Gorenstein O-sequence with $h_{2}\geq h_{1}.$ Then every local Gorenstein $K$-algebra with Hilbert function $h$ is necessarily canonically graded if and only if $h_{1}=h_{3}$ and $h_{2}=\binom{h_{1}+1}{2}.$

The main tool of the paper is Macaulay’s inverse system [Mac1916] which gives a one-to-one correspondence between ideals $I\subseteq R$ such that $R/I$ is an Artinian local ring and finitely generated $R$-submodules of a polynomial ring. In Section 2 we gather preliminary results needed for this purpose. We prove Theorems 1 and 2 in Section 3, and Theorem 3 in Section 4.

We have used Singular [Singular], [Eli15] and CoCoA [CoCoA] for various computations and examples.

Acknowledgements

We thank Juan Elias for providing us the updated version of Inverse-syst.lib for computations.

2. preliminaries

2.1. Macaulay’s Inverse System

In this subsection we recall some results on Macaulay’s inverse system which we will use in the subsequent sections. This theory is well-known in the literature, especially in the graded setting (see for example [Mac1916, Chapter IV] and [IK99]). However, the local case is not so well explored. We refer the reader to [Ems78], [Iar94] for an extended treatment.

It is known that the injective hull of $K$ as $R$-module is isomorphic to a divided power ring $P=K[X_{1},\ldots,X_{r}]$ which has a structure of $R$-module by means of the following action

[TABLE]

where $\partial_{X_{i}}$ denotes the partial derivative with respect to $X_{i}.$ For the sake of simplicity from now on we will use $x_{i}$ instead of the capital letters $X_{i}.$ If $\{f_{1},\ldots,f_{t}\}\subseteq P$ is a set of polynomials, we will denote by $\langle f_{1},\ldots,f_{t}\rangle_{R}$ the $R$-submodule of $P$ generated by $f_{1},\ldots,f_{t}$, i.e., the $K$-vector space generated by $f_{1},\ldots,f_{t}$ and by the corresponding derivatives of all orders. We consider the exact pairing of $K$-vector spaces:

[TABLE]

For any ideal $I\subset R$ we define the following $R$-submodule of $P$ called Macaulay’s inverse system:

[TABLE]

Conversely, for every $R$-submodule $M$ of $P$ we define

[TABLE]

which is an ideal of $R.$ If $M$ is generated by polynomials $\underline{f}:=f_{1},\ldots,f_{t},$ with $f_{i}\in P,$ then we will write $\operatorname{Ann}_{R}(M)=\operatorname{Ann}_{R}(\underline{f})$ and $A_{\underline{f}}=R/\operatorname{Ann}_{R}(\underline{f}).$

By using Matlis duality one proves that there exists a one-to-one correspondence between ideals $I\subseteq R$ such that $R/I$ is an Artinian local ring and $R$-submodules $M$ of $P$ which are finitely generated. More precisely, Emsalem in [Ems78, Proposition 2] and Iarrobino in [Iar94, Lemma 1.2] proved that:

Proposition 2.1**.**

There is a one-to-one correspondence between ideals $I$ such that $R/I$ is a level local ring of socle degree $s$ and type $\tau$ and $R$-submodules of $P$ generated by $\tau$ polynomials of degree $s$ having linearly independent forms of degree $s.$ The correspondence is defined as follows:

[TABLE]

By [Ems78, Proposition 2(a)], the action $\langle~{},~{}\rangle$ induces the following isomorphism of $K$-vector spaces:

[TABLE]

(where $(R/I)^{*}$ denotes the dual with respect to the pairing $\langle~{},~{}\rangle$ induced on $R/I$). Hence $\dim_{K}R/I=\dim_{K}I^{\perp}.$ As in the graded case, it is possible to compute the Hilbert function of $A=R/I$ via the inverse system. We define the following $K$-vector space:

[TABLE]

Then, by (2.3), it is known that

[TABLE]

2.2. Q-decomposition

It is well-known that the Hilbert function of an Artinian graded Gorenstein $K$-algebra is symmetric, which is not true in the local case. The problem comes from the fact that, in general, the associated algebra $G:=gr_{{\mathfrak{m}}}(A)$ of a Gorenstein local algebra $A$ is no longer Gorenstein. However, in [Iar94] Iarrobino proved that the Hilbert function of a Gorenstein local $K$-algebra $A$ admits a “symmetric” decomposition. To be more precise, consider a filtration of $G$ by a descending sequence of ideals:

[TABLE]

where

[TABLE]

Let

[TABLE]

Then

[TABLE]

is called Q-decomposition of the associated graded ring $G.$ We have

[TABLE]

Iarrobino [Iar94, Theorem 1.5] proved that if $A=R/I$ is a Gorenstein local ring then for all $a=0,\ldots,s-1,$ $Q(a)$ is a reflexive graded $G$-module, up to a shift in degree: $\operatorname{Hom}_{K}(Q(a)_{i},K)\cong Q(a)_{s-a-i}.$ Hence the Hilbert function of $Q(a)$ is symmetric about $\frac{s-a}{2}.$ Moreover, he showed that $Q(0)=G/C(1)$ is the unique (up isomorphism) socle degree $s$ graded Gorenstein quotient of $G.$ Let $f=f[s]+{\text{ lower degree terms...}}$ be a polynomial in $P$ of degree $s$ where $f[s]$ is the homogeneous part of degree $s$ and consider $A_{f}$ the corresponding Gorenstein local $K$-algebra. Then, $Q(0)\cong R/\operatorname{Ann}_{R}(f[s])$ (see [Ems78, Proposition 7] and [Iar94, Lemma 1.10]).

3. Characterization of level O-sequences

In this section we characterize Gorenstein and level O-sequences of socle degree $4$ and embedding dimension $3.$ First if $h=(1,h_{1},\ldots,h_{s})$ is a Gorenstein O-sequence (in any embedding dimension), we give an upper bound on $h_{s-2}$ in terms of $h_{s-1}$ which improves the already known inequality given in (1.1).

Theorem 3.1**.**

1. (a)

If $(1,h_{1},\ldots,h_{s-2},h_{s-1},1)$ is a Gorenstein O-sequence, then

[TABLE] 2. (b)

If $h=(1,h_{1},h_{2},h_{3},1)$ is an O-sequence such that $h_{2}\geq h_{3}$ and it satisfies (3.2), then $h$ is a Gorenstein O-sequence.

Proof.

(a): From (1.1) it follows that $h_{s-1}\leq h_{1}.$ Let $A$ be a Gorenstein local $K$-algebra with the Hilbert function $h$ and $\{Q(a):a=0,\ldots,s-1\}$ be a $Q$-decomposition of $gr_{\mathfrak{m}}(A).$ Since $Q(i)_{s-1}=0$ for $i>0,$ $\dim_{K}Q(0)_{s-1}=h_{s-1}.$ Hence $\dim_{K}Q(0)_{1}=\dim_{K}Q(0)_{s-1}=h_{s-1}$ and $\dim_{K}Q(0)_{s-2}=\dim_{K}Q(0)_{2}\leq h_{s-1}^{\langle 1\rangle}.$ This in turn implies that $\dim_{K}Q(1)_{s-2}=\dim_{K}Q(1)_{1}\leq h_{1}-h_{s-1}.$ Therefore

[TABLE]

(b): Suppose $h_{2}\leq h_{1}.$ Set

[TABLE]

Then $A_{f}$ has the Hilbert function $h.$ Now assume $h_{2}>h_{1}.$ Denote $h_{3}:=n$ and define monomials $g_{i}\in K[x_{1},\ldots,x_{n}]$ as follows:

[TABLE]

For $\underline{i}=(i_{1},\ldots,i_{n})\in\mathbb{N}^{n},$ let $x^{\underline{i}}:=x_{1}^{i_{1}}\ldots x_{n}^{i_{n}}.$ Let $T$ be the set of monomials $x^{\underline{i}}$ of degree $2$ in $K[x_{1},\ldots,x_{n}]$ such that

[TABLE]

Then $|T|=\binom{n+1}{2}-2n.$ We write $T=\{g_{i}:2n<i\leq\binom{n+1}{2}\}.$ Define

[TABLE]

Then $h_{3}=\dim_{K}\left(\frac{\partial f}{\partial x_{1}},\ldots,\frac{\partial f}{\partial x_{n}}\right)=n$ and

[TABLE]

Thus $A_{f}$ has the Hilbert function $(1,h_{1},h_{2},h_{3},1).$ ∎

Remark 3.4**.**

If $h_{3}=h_{1},$ then $f$ in the proof of Theorem 3.1(b) is homogeneous and hence $A_{f}$ is a graded Gorenstein $K$-algebra.

If the socle degree is $4$ and $h_{1}\leq 12,$ then the converse holds in (a).

Corollary 3.5**.**

An O-sequence $h=(1,h_{1},h_{2},h_{3},1)$ with $h_{1}\leq 12$ is a Gorenstein O-sequence if and only if $h$ satisfies (3.2).

Proof.

By Theorem 3.1, it suffices to show that $h_{2}\geq h_{3}$ if $h_{1}\leq 12.$ Let $A$ be a local Gorenstein $K$-algebra with the Hilbert function $h.$ Considering the symmetric $Q$-decomposition of $gr_{{\mathfrak{m}}}(A),$ we observe that in this case $Q(0)$ has the Hilbert function $(1,h_{3},h_{2}-m,h_{3},1),$ for some non-negative integer $m.$ Since $h_{1}\leq 12,$ by (3.2) $h_{3}\leq 12.$ As $Q(0)$ is a graded Gorenstein $K$-algebra, by [MZ16, Theorem 3.2] we conclude that $h_{2}-m\geq h_{3}$ since $Q(0)$ has unimodal Hilbert function, hence $h_{2}\geq h_{3}.$ ∎

Remark 3.6**.**

A Gorenstein O-sequence $(1,h_{1},h_{2},h_{3},1)$ does not necessarily satisfy $h_{2}\geq h_{3}.$ For example, consider the the sequence $h=(1,13,12,13,1).$ By [Sta78, Example 4.3] there exists a graded Gorenstein $K$-algebra with $h$ as $h$-vector.

In the following theorem we characterize the $h$-vector of local level algebras of socle degree $4$ and embedding dimension $3.$

Theorem 3.7**.**

Let $h=(1,3,h_{2},h_{3},h_{4})$ be an O-sequence.

1. (a)

Let $h_{4}=1.$ Then $h$ is a Gorenstein O-sequence if and only if $h_{3}\leq 3$ and $h_{2}\leq\binom{h_{3}+1}{2}+(3-h_{3}).$ 2. (b)

Let $h_{4}\geq 2$. Then $h$ is a level O-sequence if and only if $h_{3}\leq 3h_{4}.$

Proof.

(a): Follows from Corollary 3.5.

(b): The “only if” part follows from (1.1). The converse is constructive and we prove it by inductive steps on $h_{4}.$ First we consider the cases $h_{4}=2,3,4$ and then $h_{4}\geq 5.$ In each case we define the polynomials $\underline{f}:=f_{1},\ldots,f_{h_{4}}\in P=K[x_{1},x_{2},x_{3}]$ of degree $4$ such that $A_{\underline{f}}$ has the Hilbert function $h.$ We set $g_{1}^{\prime}=x_{3}^{3},g_{2}^{\prime}=x_{2}^{2}x_{3},g_{3}^{\prime}=x_{1}^{2}x_{2},g_{4}^{\prime}=x_{1}x_{3}^{2}.$ For short, in this proof we use the following notation: $m:=h_{2}$ and $n:=h_{3}.$

Case 1: $h_{4}=2.$

In this case $n\leq 6$ as $h_{3}\leq 3h_{4}$ by assumption. Suppose $m=2.$ Then $h$ is an O-sequence implies that $n=2.$ In this case, let $f_{1}=x_{1}^{4}+x_{3}^{2}$ and $f_{2}=x_{2}^{4}.$ Then $A_{\underline{f}}$ has the Hilbert function $(1,3,2,2,2).$ Now assume that $m\geq 3.$

Subase 1: $m\geq n.$ We set $g_{i}=\begin{cases}x_{4-i}g_{i}^{\prime}&\mbox{ if }1\leq i\leq n-2\\ g_{i}^{\prime}&\mbox{ if }n-1\leq i\leq m-2\\ 0&\mbox{ if }m-1\leq i\leq 4.\end{cases}$

(Here $x_{0}=x_{3}$). Define

[TABLE]

Then

[TABLE]

and hence $A_{\underline{f}}$ has the required Hilbert function $h.$

Subcase 2: $m<n.$ The only possible ordered tuples $(m,n)$ with $m<n\leq 6$ such that $h$ is an O-sequence are $\{(3,4),(4,5),(5,6)\}.$ For each $2$-tuple $(m,n)$ we define $f_{1},f_{2}$ as:

$a.(m,n)=(3,4):$ $f_{1}=x_{1}^{4}+x_{1}^{2}x_{2}^{2}+x_{3}^{2};f_{2}=x_{2}^{4}+x_{1}^{2}x_{2}^{2}.$

$b.(m,n)=(4,5):$ $f_{1}=x_{1}^{4}+x_{1}^{2}x_{2}^{2}+x_{3}^{4};f_{2}=x_{2}^{4}+x_{1}^{2}x_{2}^{2}.$

$c.(m,n)=(5,6):$ $f_{1}=x_{1}^{4}+x_{1}^{2}x_{2}^{2}+x_{3}^{4};f_{2}=x_{2}^{4}+x_{1}^{2}x_{2}^{2}+x_{2}^{3}x_{3}.$

Case 2: $h_{4}=3.$

In this case $n\leq 9.$ We consider the following subcases:

Subcase 1: $n\leq 6.$ Let $\underline{f}^{\prime}=f_{1},f_{2}$ be polynomials defined as in Case 1 such that $A_{\underline{f}^{\prime}}$ has the Hilbert function $(1,3,m,n,2).$ Now define $f_{3}=\begin{cases}x_{3}^{4}&\mbox{ if }m\geq n\\ x_{1}^{2}x_{2}^{2}&\mbox{ if }m<n.\end{cases}$

Then $A_{\underline{f}}$ has the required Hilbert function $h.$

Subcase 2: $7\leq n\leq 9.$ Let $\underline{f}^{\prime}=f_{1},f_{2}$ be polynomials defined as in Case 1 such that $A_{\underline{f}^{\prime}}$ has the Hilbert function $(1,3,m,6,2).$ We set $p_{1}=x_{2}^{2}x_{3}^{2},p_{2}=x_{1}^{2}x_{2}^{2}$ and $p_{3}=x_{1}^{2}x_{3}^{2}.$ Since $h$ is an O-sequence and $n\geq 7$, we get $m\geq 5.$ Now define $f_{3}=\begin{cases}\sum_{i=1}^{n-6}p_{i}&\mbox{ if }m=6\\ x_{2}^{2}x_{3}^{2}&\mbox{ if }m=5.\end{cases}$

Then $A_{\underline{f}}$ has the required Hilbert function $h.$

Case 3: $h_{4}=4.$

Since $h$ is an O-sequence, $n\leq 10.$ We consider the following subcases:

Subcase 1: $n\leq 9.$ Let $\underline{f}^{\prime}=f_{1},f_{2},f_{3}$ be polynomials defined as in Case 2 such that $A_{\underline{f}^{\prime}}$ has the Hilbert function $(1,3,m,n,3).$ Define

[TABLE]

Then $A_{\underline{f}}$ has the Hilbert function $(1,3,m,n,4).$

Subcase 2: $n=10.$ As $h$ is an O-sequence, we conclude that $m=6.$ Let $\underline{f}^{\prime}=f_{1},f_{2},f_{3}$ be polynomials defined as in Case 2 such that $A_{\underline{f}^{\prime}}$ has the Hilbert function $(1,3,6,9,3).$ Define $f_{4}=x_{1}^{2}x_{2}x_{3}.$ Then $A_{\underline{f}}$ has the Hilbert function $(1,3,6,10,4).$

Case 4: $h_{4}\geq 5.$

Since $h$ is an O-sequence, $n\leq 10$ and $h_{4}\leq 15$

**Subcase 1: $n\geq h_{4}$ OR $h_{4}\geq 11.$ ** Let $\underline{f}^{\prime}=f_{1},f_{2},f_{3},f_{4}$ be defined as in Case 3 such that $A_{\underline{f}^{\prime}}$ has the Hilbert function $(1,3,m,n,4).$ For $5\leq i\leq 15,$ define $f_{i}$ as follows:

[TABLE]

(Note that in the last case, $h_{4}\geq 7$ implies that $n\geq 7$). If $h_{4}\geq 8,$ then $n\geq 8$ which implies that $m=6.$ We set

[TABLE]

Now $A_{\underline{f}}$ has the Hilbert function $(1,3,m,n,h_{4}).$

**Subcase 2: $n<h_{4}\leq 10.$ ** The smallest ordered tuple $(n,h_{4})$ such that $h$ is an O-sequence and $n<h_{4}$ is $(4,5).$ (Here smallest ordered tuple means smallest with respect to the order $\leq$ defined as: $(n_{1},n_{2})\leq(m_{1},m_{2})$ if and only if $n_{1}\leq m_{1}$ and $n_{2}\leq m_{2}$). Let

[TABLE]

We define

[TABLE]

Then $A_{\underline{f}}$ has the Hilbert function $(1,3,m,4,5).$

Let $h_{4}\geq 6.$ We set

[TABLE]

Then $A_{\underline{f}}$ has the Hilbert function $(1,3,m,n,h_{4}).$ ∎

Using [GHMY07, Appendix D] and Theorem 3.7(b) we list in the following table all the O-sequences which are realizable for local level $K$-algebras with $h_{4}\geq 2$, but not for graded level $K$-algebras.

Analogously, by Theorem 3.7(a), the following O-sequences are Gorenstein sequences, but they are not graded Gorenstein sequences since they are not symmetric. If an O-sequence $h$ is symmetric with $h_{1}=3$ then $h$ is also a graded Gorenstein O-sequence by Remark 3.4.

The following example shows that Theorem 3.7(b) can not be extended to $h_{1}\geq 4$ because the necessary condition $h_{3}\leq h_{1}h_{s}$ is not longer sufficient for characterizing level O-sequences of socle degree $4.$

Example 3.8**.**

The O-sequence $h=(1,4,9,2,2)$ is not a level O-sequence.

Proof.

Let $A=R/I$ be a local level $K$-algebra with the Hilbert function $h.$ The lex-ideal $L\in P=K[x_{1},\dots,x_{4}]$ with the Hilbert function $h$ is

[TABLE]

A minimal graded $P$-free resolution of $P/L$ is:

[TABLE]

By [RS10, Theorem 4.1] the Betti numbers of $A$ can be obtained from the Betti numbers of $P/L$ by a sequence of negative and zero consecutive cancellations. This implies that $\beta_{4}(A)\geq 3$ and hence $A$ has type at least $3,$ which leads to a contradiction. ∎

4. Canonically graded algebras with ${\mathfrak{m}}^{5}=0.$

It is clear that a necessary condition for a Gorenstein local $K$-algebra $A$ being canonically graded is that the Hilbert function of $A$ must be symmetric. Hence we investigate whether a Gorenstein $K$-algebra $A$ with the Hilbert function $(1,h_{1},h_{2},h_{1},1)$ is necessarily canonically graded. If $h_{2}=\binom{h_{1}+1}{2}$ (equiv. $A$ is compressed), then by [ER15, Theorem 3.1] $A$ is canonically graded. In this section we prove that if $h=(1,h_{1},h_{2},h_{1},1)$ is an O-sequence with $h_{1}\leq h_{2}<\binom{h_{1}+1}{2},$ then there exists a polynomial $F$ of degree $4$ such that $A_{F}$ has the Hilbert function $h$ and it is not canonically graded.

Theorem 4.1**.**

Let $h=(1,h_{1},h_{2},h_{3},1)$ be a Gorenstein O-sequence with $h_{2}\geq h_{1}$. Then every local Gorenstein $K$-algebra with Hilbert function $h$ is necessarily canonically graded if and only if $h_{1}=h_{3}$ and $h_{2}=\binom{h_{1}+1}{2}.$

Proof.

The assertion is clear for $h_{1}=1.$ Hence we assume $h_{1}>1.$ The “if” part of the theorem follows from [ER15, Theorem 3.1]. We prove the converse, that is we show that if $h_{2}<\binom{h_{1}+1}{2},$ then there exists a polynomial $G$ of degree $4$ such that $A_{G}$ has the Hilbert function $h$ and it is not canonically graded. We may assume $h_{3}=h_{1},$ otherwise the result is clear. For simplicity in the notation we put $h_{1}:=n$ and $h_{2}:=m.$

First we prove the assertion for $n\leq 3.$ We define

[TABLE]

Let $G=F+x_{n}^{3}.$ It is easy to check that $A_{G}$ has the Hilbert function $h.$ We claim that $A_{G}$ is not canonically graded. Suppose that $A_{G}$ is canonically graded. Then $A_{F}\cong gr_{{\mathfrak{m}}}(A)\cong A_{G}.$ Let $\varphi:A_{F}\longrightarrow A_{G}$ be a $K$-algebra automorphism. Since $x_{n}^{2}\circ F=0,$ $x_{n}^{2}\in\operatorname{Ann}_{R}(F).$ This implies that $\varphi(x_{n})^{2}\in\operatorname{Ann}_{R}(G)$ and hence $\varphi(x_{n})^{2}\circ G=0.$ For $\underline{i}=(i_{1},\ldots,i_{n})\in\mathbb{N}^{n},$ let $|\underline{i}|=i_{1}+\cdots+i_{n}.$ Suppose

[TABLE]

Comparing the coefficients of the monomials of degree $\leq 2$ in $\varphi(x_{n})^{2}\circ G=0,$ it is easy to verify that $u_{1}=\cdots=u_{n}=0.$ This implies that $\varphi(x_{n})$ has no linear terms and thus $\varphi$ is not an automorphisms, a contradiction.

Suppose $n>3.$ First we define a homogeneous polynomial $F\in P$ of degree $4$ such that $A_{F}$ has the Hilbert function $h$ and $x_{n}^{2}$ does not divide any monomial in $F$ (in other words, if $x^{\underline{i}}$ is a monomial that occurs in $F$ with nonzero coefficient, then $i_{n}\leq 1$).

Let $T$ be a monomials basis of $P_{2}.$ We split the set $T\setminus\{x_{n}^{2}\}$ into a disjoint union of monomials as follows. We set

[TABLE]

Let $E=\{p_{i}:1\leq i\leq n\},B=\{p_{i}:n+1\leq i\leq 2n\},C=\{x_{i}x_{j}:1\leq i<j<n\mbox{ such that }j-i>1\}$ and $D:=\{x_{i}x_{n}:3\leq i\leq n-2\}.$ Then

[TABLE]

Denote by $|\cdot|$ the cardinality, then $|C|=\binom{n+1}{2}-2n-(n-4)-1$ and $|D|=n-4.$ Hence we write $C=\{p_{i}:2n<i\leq\binom{n+1}{2}-(n-4)-1\}$ and $D=\{p_{i}:\binom{n+1}{2}-(n-4)-1<i\leq\binom{n+1}{2}-1\}.$ We set

[TABLE]

Define

[TABLE]

Since $m\geq n,$ $\dim_{K}(\langle F\rangle_{R})_{i}=n$ for $i=1,3.$ Also, $\dim_{K}(\langle F\rangle_{R})_{2}=\dim_{K}\{p_{i}:1\leq i\leq m\}=m.$ Hence $A_{F}$ has the Hilbert function $h.$

Let $G=F+x_{n}^{3}.$ We prove that $A_{G}$ is not canonically graded. Suppose that $A_{G}$ is canonically graded. Then, as before, $A_{G}\cong A_{F}.$ Let $\varphi:A_{F}\longrightarrow A_{G}$ be a $K$-algebra automorphism. Since $F$ does not contain a monomial multiple of $x_{n}^{2},$ $x_{n}^{2}\circ F=0$ and hence $x_{n}^{2}\in\operatorname{Ann}_{R}(F)$ which implies that $\varphi(x_{n})^{2}\in\operatorname{Ann}_{R}(G).$ Let

[TABLE]

We claim that $u_{1}=\cdots=u_{n-1}=0.$

Case 1: $m=n.$ Comparing the coefficients of $x_{1}^{2},x_{2}x_{n},x_{3}^{2},\ldots,x_{n-1}^{2}$ in $\varphi(x_{n})^{2}\circ G=0,$ we get $u_{1}=\cdots=u_{n-1}=0.$

Case 2: $m=n+1$ OR $m=n+2.$ Compare the coefficients of $x_{1}x_{2},x_{2}x_{n},x_{3}^{2},\ldots,x_{n-1}^{2}$ in $\varphi(x_{n})^{2}\circ G=0,$ to get $u_{1}=\cdots=u_{n-1}=0.$

Case 3: $n+2<m<2n.$ Comparing the coefficients of $x_{1}x_{2},x_{2}x_{n},x_{3}x_{4},\ldots,x_{m-n}x_{m-n+1},x_{m-n+1}^{2},$ $x_{m-n+2}^{2}\ldots,x_{n-1}^{2}$ in $\varphi(x_{n})^{2}\circ G=0,$ we get $u_{1}=\cdots=u_{n-1}=0.$

Case 4: $m\geq 2n.$ Comparing the coefficients of $x_{1}x_{2},x_{1}x_{n},x_{3}x_{4},\ldots,x_{n-1}x_{n}$ in $\varphi(x_{n})^{2}\circ G=0,$ we get $u_{1}=\cdots=u_{n-1}=0.$

This proves the claim. Now, comparing the coefficients of $x_{n}$ in $\varphi(x_{n})^{2}\circ G=0,$ we get $u_{n}=0$ (since $F$ does not contain a monomial divisible by $x_{n}^{2}$). This implies that $\varphi(x_{n})$ has no linear terms and hence $\varphi$ is not an automorphism, a contradiction. ∎

We expect that the Theorem 4.1 holds true without the assumption $h_{2}\geq h_{1}$. The problem is that, as far as we know, the admissible Gorenstein not unimodal $h$-vectors are not classified even if $s=4.$ However, starting from an example by Stanley, we are able to construct a not canonically graded Gorenstein $K$-algebra with (not unimodal) $h$-vector $(1,13,12,13,1).$

Corollary 4.2**.**

Let $h=(1,h_{1},h_{2},h_{1},1)$ where $h_{1}\leq 13$ be a Gorenstein O-sequence. Then every Gorenstein $K$-algebra with the Hilbert function $h$ is necessarily canonically graded if and only if $h_{2}=\binom{h_{1}+1}{2}.$

Proof.

If a local Gorenstein $K$-algebra $A$ has Hilbert function $h=(1,h_{1},h_{2},h_{1},1),$ then by considering Q-decomposition of $gr_{\mathfrak{m}}(A)$ we conclude that $gr_{\mathfrak{m}}(A)\cong Q(0).$ This implies that $h$ is also the Hilbert function of a graded Gorenstein $K$-algebra. By [MZ16, Theorem 3.2] if $h_{1}\leq 12,$ then the Hilbert function of a graded Gorenstein $K$-algebra is unimodal. Hence by Theorem 4.1 the result follows.

If $h_{1}=13$ and $h$ is unimodal, then the assertion follows from Theorem 4.1. Now, by [MZ16, Theorem 3.2] the only nonunimodal graded Gorenstein O-sequence with $h_{1}=13$ is $h=(1,13,12,13,1).$ In this case we write $P=[x_{1},\ldots,x_{10},x,y,z].$ Let

[TABLE]

where $\mu=\{x^{3},x^{2}y,x^{2}z,xy^{2},xyz,xz^{2},y^{3},y^{2}z,yz^{2},z^{3}\}=\{\mu_{1},\ldots,\mu_{10}\}.$ Let $G=F+x_{1}^{3}+\cdots+x_{10}^{3}.$ Then $A_{G}$ has the Hilbert function $h.$ We claim that $A_{G}$ is not canonically graded. Suppose $A_{G}$ is canonically graded. Then $A_{G}\cong A_{F}.$ Let $\varphi:A_{F}\longrightarrow A_{G}$ be a $K$-algebra automorphism. Since $x_{1}^{2}\in\operatorname{Ann}_{R}(F),$ $\varphi(x_{1})^{2}\in\operatorname{Ann}_{R}(G).$ Let

[TABLE]

Comparing the coefficients of $x_{1}x,x_{7}y,x_{10}z$ in $\varphi(x_{1})^{2}\circ G=0,$ we get $u_{11}=u_{12}=u_{13}=0.$ Now, comparing the coefficients of $x_{1},\ldots,x_{10}$ in $\varphi(x_{1})^{2}\circ G=0,$ we get $u_{1}=\cdots=u_{10}=0.$ This implies that $\varphi(x_{1})$ has no linear terms and thus $\varphi$ is not an automorphism, a contradiction. ∎