---
title: Gorenstein Sally Semigroups
url: https://www.emergentmind.com/topics/gorenstein-sally-semigroups
type: topic
---

# Gorenstein Sally Semigroups

Gorenstein Sally semigroups are numerical semigroups of multiplicity \(e\) and width \(e-1\) whose semigroup rings realize the Gorenstein side of Sally’s multiplicity–embedding-dimension constraints. In the narrow historical sense, the term refers to the Herzog–Stamate family
\[
\langle e,e+1,e+4,\dots,2e-1\rangle,
\]
which is the two-deletion semigroup obtained from \(\{e,e+1,\dots,2e-1\}\) by removing \(e+2\) and \(e+3\). In the broader recent literature, this example is placed inside the class of Sally type numerical semigroups, namely semigroups minimally generated by a subset of \([e,2e-1]\); in this one-dimensional setting, Gorensteinness is equivalent to symmetry of the numerical semigroup [2507.11738], [2512.18136].

## 1. Origin of the notion and basic definitions

The subject is rooted in Judith Sally’s 1980 theorem that the associated graded ring of one-dimensional Gorenstein local rings of multiplicity \(e\) and embedding dimension \(e-2\) is Cohen–Macaulay, with defining ideal generated by \({e-2\choose 2}\) elements. Numerical semigroup rings provide a large class of one-dimensional Cohen–Macaulay rings, and Herzog and Stamate isolated the semigroup
\[
\langle e,e+1,e+4,\ldots,2e-1\rangle
\]
as a Gorenstein example satisfying Sally’s conditions; this is the source of the designation “Gorenstein Sally semigroup” [2507.11738].

A Sally type numerical semigroup has multiplicity \(e\) and width \(e-1\). Concretely, its minimal generators lie inside the interval \([e,2e-1]\), the smallest generator is \(e\), and the largest is at most \(2e-1\). A central family is obtained by deleting generators from the full interval. In the two-deletion case,
\[
S(e,m,n)=\left\langle \{e,e+1,\dots,2e-1\}\setminus\{e+m,e+n\}\right\rangle,
\qquad 1\le m<n\le e-2,
\]
and the Herzog–Stamate semigroup is exactly the case \((m,n)=(2,3)\). In the more general consecutive-deletion family,
\[
S_k^e(j)=\langle e,\dots,e+j-1,\; e+j+k,\dots,2e-1\rangle,
\]
one removes the block \(j,j+1,\dots,j+k-1\) from \([e,2e-1]\), with
\[
1\le k,\qquad 1\le j,\qquad j+k\le e-1,
\]
and the embedding dimension is \(\nu=e-k\) [2512.18136].

## 2. Symmetry and exact Gorenstein classifications

For numerical semigroups, symmetry is characterized by
\[
2g=F+1,
\]
where \(g\) is the genus and \(F\) is the Frobenius number. For one-dimensional semigroup rings, symmetric is equivalent to Gorenstein. Accordingly, the classification of Gorenstein Sally semigroups is a classification of symmetric members in these Sally-type families [2512.18136].

| Family | Parameters | Symmetric/Gorenstein members |
|---|---|---|
| \(S(e,m,n)\) | \(2\le m<n\le e-2\) | only \((m,n)=(2,3)\) |
| \(S_k^e(j)\) | \(1\le j,\ 1\le k,\ j+k\le e-1\) | for \(1\le k<e/2\), iff \(j=k\) |
| \(\mathcal S_m\) | \(1\le m\le e-1\) | \(m\in\{1,e-1\}\) |
| \(\mathcal S_{m,n}\) | \(1\le m<n\le e-1\) | only \((m,n)=(2,3)\) |

In the two-deletion Sally type family,
\[
S_{e,m,n}\text{ is Gorenstein } \iff (m,n)=(2,3).
\]
Thus the Herzog–Stamate semigroup
\[
S_{e,2,3}=\langle e,e+1,e+4,\dots,2e-1\rangle
\]
is the unique Gorenstein member of that family. Its Frobenius number is exceptional:
\[
F_{e,m,n}=
\begin{cases}
2e+3,&(m,n)=(2,3),\\
e+n,&\text{otherwise.}
\end{cases}
\]
The uniqueness statement is one of the defining rigidity results of the subject [2507.11738].

The consecutive-deletion theory gives a second rigidity theorem. For any \(1\le k<e/2\),
\[
S_k^e(j)\text{ is Gorenstein } \iff j=k.
\]
More sharply, if \(k\le e/2\) and \((j,k)\neq(1,e/2)\), then
\[
S_k^e(j)\text{ is symmetric } \Longleftrightarrow j=k,
\]
while the boundary case \((j,k)=(1,e/2)\) is always symmetric. In the symmetric case \(j=k\),
\[
F=2e+2k-1,\qquad g=e+k.
\]
Hence, in the small-\(k\) regime, deleting a block of \(k\) consecutive generators produces exactly one Gorenstein Sally type semigroup, namely
\[
S_k^e(k)=\langle e,e+1,\dots,e+k-1,\; e+2k,\dots,2e-1\rangle
\]
[2512.18136].

A parallel “type II” classification studies one- and two-deletion semigroups
\[
\mathcal S_m=\langle e,e+1,\dots,2e-1\rangle\setminus\{e+m\},
\qquad
\mathcal S_{m,n}=\langle e,e+1,\dots,2e-1\rangle\setminus\{e+m,e+n\}.
\]
Here
\[
\mathcal S_m \text{ is symmetric } \iff m\in\{1,e-1\},
\qquad
\mathcal S_{m,n} \text{ is symmetric } \iff (m,n)=(2,3).
\]
Thus the Gorenstein Sally semigroup \(\mathcal S_{2,3}\) remains the unique symmetric two-deletion example, while \(\mathcal S_1\) and \(\mathcal S_{e-1}\) are the one-deletion Gorenstein cases [2512.10812].

## 3. Defining ideals and determinantal presentations

A notable structural feature of Gorenstein Sally semigroup rings is that their defining ideals admit determinantal descriptions. This realizes the semigroup ring as a deformation of structured \(2\times n\) minor ideals rather than as an arbitrary toric ideal.

For the symmetric consecutive-deletion semigroup
\[
S_k=S_k^e(k)=\langle e,\dots,e+k-1,\; e+2k,\dots,2e-1\rangle,
\]
the semigroup ring has the form
\[
k[S_k]\cong \frac{R_k}{I_k},
\]
and the defining ideal satisfies
\[
I_k=I_2(A_k^e)+I_2(B_k^e),
\]
where \(I_2(M)\) denotes the ideal of \(2\times2\) minors of the matrix \(M\). The matrix \(A_k^e\) supplies the main scroll-type relations, while \(B_k^e\) provides the extra relations needed to cut out the Sally semigroup ring [2512.18136].

The same pattern appears in Sally type II. For the classical Gorenstein two-deletion case,
\[
I_{\mathcal S_{2,3}}=I_2(A_{2,3})+I_2(B_{2,3}),
\]
with a minimal generating set consisting of the \(\binom{e-3}{2}\) minors of \(A_{2,3}\) together with the \(e-4\) minors of \(B_{2,3}\) involving the first column. For the nearby non-Gorenstein example \((m,n)=(3,4)\),
\[
I_{\mathcal S_{3,4}}=I_2(A_{3,4})+I_2(B_{3,4}),
\]
with \(\binom{e-3}{2}\) minors from \(A_{3,4}\) and \(e-3\) relevant minors from \(B_{3,4}\) [2512.10812].

The broader two-deletion family \(S_{e,m,n}\) also has explicitly described minimal generating sets. These are organized into families of binomials \(f_1,\dots,f_{15}\), with representative formulas
\[
f_1(j,k)=X_{k+1}X_j-X_kX_{j+1},\qquad
f_2(j)=X_jX_0^2-X_{j+1}X_{e-1},
\]
and additional relations depending on the relative position of \(m\) and \(n\). The exact list varies across the cases \(n=m+1\), \(n=m+2\), \(3\le m<n-2\), \(m=2\) with \(n=4,5\), \(m=2\) with \(n\ge6\), and the special cases \((2,3)\) and \((3,4)\) [2507.11738].

## 4. Minimal free resolutions and Betti numbers

The homological theory of Gorenstein Sally semigroup rings is unusually explicit. In the symmetric cases, the minimal free resolutions are built by mapping-cone or mapping-cylinder constructions combining an Eagon–Northcott complex with a dual copy, reflecting Gorenstein self-duality.

For the general Gorenstein consecutive-deletion semigroup \(S_k^e(k)\), the minimal free resolution of \(k[S_k]\) is obtained from the Eagon–Northcott resolution \(\mathbf E(k)\) of \(R_k/I_2(A_k^e)\), its dual \(\mathbf E^*(k)\), and a map
\[
\psi:\mathbf E^*(k)\to \mathbf E(k).
\]
The resulting Betti numbers are
\[
\beta_t=t\binom{e-k-1}{t+1}+(e-k-1-t)\binom{e-k-1}{t-1},
\qquad t\le e-k-2,
\]
and
\[
\beta_{e-k-1}=1.
\]
These are the Betti numbers of the minimal free resolution of the symmetric/Gorenstein Sally semigroup ring [2512.18136].

For the classical Gorenstein Sally semigroup \(\mathcal S_{2,3}\), the minimal free resolution of \(k[\mathcal S_{2,3}]\) is constructed as a mapping cone
\[
\mathbf E^\ast \to \mathbf E,
\]
where \(\mathbf E\) is the Eagon–Northcott complex for \(A_{2,3}\). Its Betti numbers are
\[
\beta_t=\frac{t(e-1)}{e-t-2}\binom{e-3}{t+1},
\qquad 1\le t\le e-4,
\]
with
\[
\beta_0=\beta_{e-3}=1.
\]
The symmetry of the Betti numbers reflects the Gorenstein property. In the one-deletion Gorenstein cases \(\mathcal S_1\) and \(\mathcal S_{e-1}\), the analogous formula is
\[
\beta_t=\frac{et}{e-t-1}\binom{e-2}{t+1},
\qquad 1\le t\le e-3,
\]
with
\[
\beta_{e-2}=1.
\]
By contrast, the non-Gorenstein case \(\mathcal S_2\) has
\[
\beta_t=t\binom{e-1}{t+1},
\qquad 1\le t\le e-2,
\]
and the Betti table is not symmetric [2512.10812].

The first Betti number in the broader two-deletion family \(S_{e,m,n}\) is also completely determined. Writing \(\mu(I_{e,m,n})\) for the minimal number of generators of the defining ideal, one has
\[
\mu(e,m,n)=
\begin{cases}
{e-2\choose 2},&(m,n)\in\{(2,4),(2,5),(3,4)\},\\[2mm]
{e-2\choose 2}-1,&m=2 \text{ and }(n=3\text{ or }n\ge 6),\\[2mm]
{e-2\choose 2}-2,&\text{otherwise.}
\end{cases}
\]
This is derived through Hochster’s combinatorial formula, together with a GAP algorithm `GradedBetti` that computes the graded contributions \(\beta_{1,\lambda}\) degree by degree [2507.11738].

## 5. Projective monomial curves and the rarity of projective Gorensteinness

The affine Gorenstein theory does not transfer unchanged to projective closure. For the projective monomial curves associated to
\[
S(e,m,n)=\left\langle \{e,e+1,\dots,2e-1\}\setminus\{e+m,e+n\}\right\rangle,
\]
the homogeneous coordinate ring is
\[
\mathbb K[\overline{S(e,m,n)}]\cong \frac{R(e,m,n)[y]}{I_{\overline{S(e,m,n)}}},
\]
and the analysis proceeds by computing a Gröbner basis of the toric ideal \(I_{S(e,m,n)}\), homogenizing it, applying Herzog–Stamate’s Cohen–Macaulay criterion, and then testing symmetry of the Hilbert function of an Artinian reduction [2511.06482].

In this projective setting, the Cohen–Macaulay locus is large but not total:
\[
\mathbb K[\overline{S(e,m,n)}]\text{ is Cohen--Macaulay}
\quad\Longleftrightarrow\quad
(m,n)\neq (e-4,e-3).
\]
The Gorenstein locus is much smaller:
\[
\mathbb K[\overline{S(e,m,n)}]\text{ is Gorenstein}
\quad\Longleftrightarrow\quad
(e,m,n)\in\{(4,1,2),(5,2,3)\}.
\]
Thus the classical Sally semigroup survives projectively only in the smallest nontrivial case \(S(5,2,3)\), and there is one additional exceptional projective Gorenstein curve, \(S(4,1,2)\) [2511.06482].

This difference is a recurrent source of misunderstanding. The affine semigroup ring of \(S_{e,2,3}\) is Gorenstein for all \(e\) in the two-deletion Sally family, but the associated projective monomial curve is Gorenstein only for \(e=5\). The same paper also computes the Castelnuovo–Mumford regularity in every Cohen–Macaulay case; the resulting values are \(6\), \(4\), \(3\), or \(2\), depending on the parameters [2511.06482].

## 6. Sally modules, near-Gorenstein hierarchies, and adjacent theories

The expression “Sally” in this area has a second meaning, through the Sally modules of canonical ideals. This line of work does not define Sally semigroups directly, but it provides a one-dimensional commutative-algebraic framework for measuring how far a Cohen–Macaulay ring is from being Gorenstein.

In dimension one, almost Gorenstein local rings correspond to canonical ideals whose Sally module has rank \(1\), and 2-almost Gorenstein local rings are defined by
\[
\operatorname{rank} S_Q(I)=2.
\]
For such rings, the following are equivalent:
\[
R\text{ is 2-AGL}
\iff
K^2=K^3\text{ and }\ell_R(K^2/K)=2
\iff
\ell_R(S/K)=2
\iff
\ell_R(R/c)=2.
\]
The theory is developed explicitly for numerical semigroup rings, where the canonical fractional ideal and pseudo-Frobenius numbers can be written in semigroup-theoretic terms [1704.00997].

The Goto-ring formalism extends this hierarchy. An \(n\)-Goto ring is defined by an extended canonical ideal whose Sally module has rank \(n\) and is generated in degree \(1\). In this stratification, \(0\)-Goto rings are Gorenstein, \(1\)-Goto rings are non-Gorenstein almost Gorenstein, and in dimension \(1\), \(2\)-Goto rings are 2-almost Gorenstein. For numerical semigroup rings this becomes a condition on the fractional canonical ideal \(K\) and the conductor \(c\):
\[
K^2=K^3,\qquad \ell_R(R/c)=n.
\]
The same paper gives explicit \(n\)-Goto semigroup families such as
\[
k[[t^3,t^{3n+1},t^{3n+2}]]
\]
for every \(n\ge1\) [2312.14379].

A complementary stability result comes from dilatations \(T=S+a=\{0\}\cup(M+a)\). Under the condition \(a\in M-2M\), the properties “almost symmetric,” “2-AGL,” and “nearly Gorenstein” are preserved in both directions:
\[
S\text{ has }\mathcal P \iff S+a\text{ has }\mathcal P,
\]
for each of these three properties. This shows that several Gorenstein-adjacent classes are stable under a natural semigroup construction, even though genuine Gorensteinness in Sally-type families is much more rigid [1710.07586].

Within the Sally type II program, this rigidity remains the organizing theme. The Gorenstein semigroups
\[
\mathcal S_1,\qquad \mathcal S_{e-1},\qquad \mathcal S_{2,3}
\]
have self-dual resolutions and symmetric Betti tables, while the non-symmetric families are compared to these Gorenstein models through explicit conjectures on Betti numbers for \(\mathcal S_m\), \(\mathcal S_{2,n}\), \(\mathcal S_{2,4}\), \(\mathcal S_{2,5}\), \(\mathcal S_{m,m+1}\), and \(\mathcal S_{1,e-1}\) [2512.10812].

The resulting picture is highly constrained. In the classical two-deletion setting, the Gorenstein Sally semigroup is unique; in the consecutive-deletion setting with \(k<e/2\), there is again exactly one Gorenstein member; and in projective closure even these affine Gorenstein examples almost always cease to be Gorenstein. The term therefore denotes not a broad generic phenomenon, but a sharply delimited locus inside the combinatorics and homological algebra of Sally type numerical semigroups.

Source: https://www.emergentmind.com/topics/gorenstein-sally-semigroups