Gorenstein Sally Semigroups

• Gorenstein Sally semigroups are numerical semigroups with multiplicity e and width e-1, whose semigroup rings exhibit a unique Gorenstein (symmetric) structure.
• They are constructed via specific deletion operations on the full interval [e, 2e-1], with the classical two-deletion case (Herzog–Stamate semigroup) being uniquely Gorenstein.
• Their algebraic structure features determinantal presentations and explicit Betti numbers, offering deep insights into Cohen–Macaulay and Gorenstein classifications in one-dimensional settings.

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 (Dubey et al., 15 Jul 2025, Singh et al., 19 Dec 2025).

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

$e-1$0

as a Gorenstein example satisfying Sally’s conditions; this is the source of the designation “Gorenstein Sally semigroup” (Dubey et al., 15 Jul 2025).

A Sally type numerical semigroup has multiplicity $e-1$1 and width $e-1$2. Concretely, its minimal generators lie inside the interval $e-1$3, the smallest generator is $e-1$4, and the largest is at most $e-1$5. A central family is obtained by deleting generators from the full interval. In the two-deletion case,

$e-1$6

and the Herzog–Stamate semigroup is exactly the case $e-1$7. In the more general consecutive-deletion family,

$e-1$8

one removes the block $e-1$9 from $\langle e,e+1,e+4,\dots,2e-1\rangle,$0, with

$\langle e,e+1,e+4,\dots,2e-1\rangle,$1

and the embedding dimension is $\langle e,e+1,e+4,\dots,2e-1\rangle,$2 (Singh et al., 19 Dec 2025).

2. Symmetry and exact Gorenstein classifications

For numerical semigroups, symmetry is characterized by

$\langle e,e+1,e+4,\dots,2e-1\rangle,$3

where $\langle e,e+1,e+4,\dots,2e-1\rangle,$4 is the genus and $\langle e,e+1,e+4,\dots,2e-1\rangle,$5 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 (Singh et al., 19 Dec 2025).

Family	Parameters	Symmetric/Gorenstein members
$\langle e,e+1,e+4,\dots,2e-1\rangle,$6	$\langle e,e+1,e+4,\dots,2e-1\rangle,$7	only $\langle e,e+1,e+4,\dots,2e-1\rangle,$8
$\langle e,e+1,e+4,\dots,2e-1\rangle,$9	$\{e,e+1,\dots,2e-1\}$0	for $\{e,e+1,\dots,2e-1\}$1, iff $\{e,e+1,\dots,2e-1\}$2
$\{e,e+1,\dots,2e-1\}$3	$\{e,e+1,\dots,2e-1\}$4	$\{e,e+1,\dots,2e-1\}$5
$\{e,e+1,\dots,2e-1\}$6	$\{e,e+1,\dots,2e-1\}$7	only $\{e,e+1,\dots,2e-1\}$8

In the two-deletion Sally type family,

$\{e,e+1,\dots,2e-1\}$9

Thus the Herzog–Stamate semigroup

$e+2$0

is the unique Gorenstein member of that family. Its Frobenius number is exceptional: $e+2$1 The uniqueness statement is one of the defining rigidity results of the subject (Dubey et al., 15 Jul 2025).

The consecutive-deletion theory gives a second rigidity theorem. For any $e+2$2,

$e+2$3

More sharply, if $e+2$4 and $e+2$5, then

$e+2$6

while the boundary case $e+2$7 is always symmetric. In the symmetric case $e+2$8,

$e+2$9

Hence, in the small-$e+3$0 regime, deleting a block of $e+3$1 consecutive generators produces exactly one Gorenstein Sally type semigroup, namely

$e+3$2

(Singh et al., 19 Dec 2025).

A parallel “type II” classification studies one- and two-deletion semigroups

$e+3$3

Here

$e+3$4

Thus the Gorenstein Sally semigroup $e+3$5 remains the unique symmetric two-deletion example, while $e+3$6 and $e+3$7 are the one-deletion Gorenstein cases (Goel et al., 11 Dec 2025).

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 $e+3$8 minor ideals rather than as an arbitrary toric ideal.

For the symmetric consecutive-deletion semigroup

$e+3$9

the semigroup ring has the form

$[e,2e-1]$0

and the defining ideal satisfies

$[e,2e-1]$1

where $[e,2e-1]$2 denotes the ideal of $[e,2e-1]$3 minors of the matrix $[e,2e-1]$4. The matrix $[e,2e-1]$5 supplies the main scroll-type relations, while $[e,2e-1]$6 provides the extra relations needed to cut out the Sally semigroup ring (Singh et al., 19 Dec 2025).

The same pattern appears in Sally type II. For the classical Gorenstein two-deletion case,

$[e,2e-1]$7

with a minimal generating set consisting of the $[e,2e-1]$8 minors of $[e,2e-1]$9 together with the $e$0 minors of $e$1 involving the first column. For the nearby non-Gorenstein example $e$2,

$e$3

with $e$4 minors from $e$5 and $e$6 relevant minors from $e$7 (Goel et al., 11 Dec 2025).

The broader two-deletion family $e$8 also has explicitly described minimal generating sets. These are organized into families of binomials $e$9, with representative formulas

$e-2$0

and additional relations depending on the relative position of $e-2$1 and $e-2$2. The exact list varies across the cases $e-2$3, $e-2$4, $e-2$5, $e-2$6 with $e-2$7, $e-2$8 with $e-2$9, and the special cases ${e-2\choose 2}$0 and ${e-2\choose 2}$1 (Dubey et al., 15 Jul 2025).

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 ${e-2\choose 2}$2, the minimal free resolution of ${e-2\choose 2}$3 is obtained from the Eagon–Northcott resolution ${e-2\choose 2}$4 of ${e-2\choose 2}$5, its dual ${e-2\choose 2}$6, and a map

${e-2\choose 2}$7

The resulting Betti numbers are

${e-2\choose 2}$8

and

${e-2\choose 2}$9

These are the Betti numbers of the minimal free resolution of the symmetric/Gorenstein Sally semigroup ring (Singh et al., 19 Dec 2025).

For the classical Gorenstein Sally semigroup $e-1$00, the minimal free resolution of $e-1$01 is constructed as a mapping cone

$e-1$02

where $e-1$03 is the Eagon–Northcott complex for $e-1$04. Its Betti numbers are

$e-1$05

with

$e-1$06

The symmetry of the Betti numbers reflects the Gorenstein property. In the one-deletion Gorenstein cases $e-1$07 and $e-1$08, the analogous formula is

$e-1$09

with

$e-1$10

By contrast, the non-Gorenstein case $e-1$11 has

$e-1$12

and the Betti table is not symmetric (Goel et al., 11 Dec 2025).

The first Betti number in the broader two-deletion family $e-1$13 is also completely determined. Writing $e-1$14 for the minimal number of generators of the defining ideal, one has

$e-1$15

This is derived through Hochster’s combinatorial formula, together with a GAP algorithm GradedBetti that computes the graded contributions $e-1$16 degree by degree (Dubey et al., 15 Jul 2025).

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

$e-1$17

the homogeneous coordinate ring is

$e-1$18

and the analysis proceeds by computing a Gröbner basis of the toric ideal $e-1$19, homogenizing it, applying Herzog–Stamate’s Cohen–Macaulay criterion, and then testing symmetry of the Hilbert function of an Artinian reduction (Bhardwaj et al., 9 Nov 2025).

In this projective setting, the Cohen–Macaulay locus is large but not total: $e-1$20 The Gorenstein locus is much smaller: $e-1$21 Thus the classical Sally semigroup survives projectively only in the smallest nontrivial case $e-1$22, and there is one additional exceptional projective Gorenstein curve, $e-1$23 (Bhardwaj et al., 9 Nov 2025).

This difference is a recurrent source of misunderstanding. The affine semigroup ring of $e-1$24 is Gorenstein for all $e-1$25 in the two-deletion Sally family, but the associated projective monomial curve is Gorenstein only for $e-1$26. The same paper also computes the Castelnuovo–Mumford regularity in every Cohen–Macaulay case; the resulting values are $e-1$27, $e-1$28, $e-1$29, or $e-1$30, depending on the parameters (Bhardwaj et al., 9 Nov 2025).

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 $e-1$31, and 2-almost Gorenstein local rings are defined by

$e-1$32

For such rings, the following are equivalent: $e-1$33 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 (Chau et al., 2017).

The Goto-ring formalism extends this hierarchy. An $e-1$34-Goto ring is defined by an extended canonical ideal whose Sally module has rank $e-1$35 and is generated in degree $e-1$36. In this stratification, $e-1$37-Goto rings are Gorenstein, $e-1$38-Goto rings are non-Gorenstein almost Gorenstein, and in dimension $e-1$39, $e-1$40-Goto rings are 2-almost Gorenstein. For numerical semigroup rings this becomes a condition on the fractional canonical ideal $e-1$41 and the conductor $e-1$42: $e-1$43 The same paper gives explicit $e-1$44-Goto semigroup families such as

$e-1$45

for every $e-1$46 (Endo, 2023).

A complementary stability result comes from dilatations $e-1$47. Under the condition $e-1$48, the properties “almost symmetric,” “2-AGL,” and “nearly Gorenstein” are preserved in both directions: $e-1$49 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 (Barucci et al., 2017).

Within the Sally type II program, this rigidity remains the organizing theme. The Gorenstein semigroups

$e-1$50

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 $e-1$51, $e-1$52, $e-1$53, $e-1$54, $e-1$55, and $e-1$56 (Goel et al., 11 Dec 2025).

The resulting picture is highly constrained. In the classical two-deletion setting, the Gorenstein Sally semigroup is unique; in the consecutive-deletion setting with $e-1$57, 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.