arXiv:1612.02723v2 [] 23 Dec 2016

THE TRACE OF THE CANONICAL MODULE
JÜRGEN HERZOG, TAKAYUKI HIBI AND DUMITRU I. STAMATE
Abstract. The trace of the canonical module (the canonical trace) determines
the non-Gorenstein locus of a local Cohen–Macaulay ring. We call a local Cohen–
Macaulay ring nearly Gorenstein, if its canonical trace contains the maximal ideal.
Similar definitions can be made for positively graded Cohen–Macaulay K-algebras.
We study the canonical trace for tensor products and Segre products of algebras,
as well as of (squarefree) Veronese subalgebras. The results are used to classify
the nearly Gorenstein Hibi rings. We also consider the canonical trace of onedimensional rings with a focus on numerical semigroup rings.

Contents
Introduction
1. Basic properties of the trace
2. Nearly Gorenstein rings
3. Computing the trace of the canonical module
4. The canonical trace ideal for special constructions of algebras
5. Nearly Gorenstein Hibi rings
6. The one-dimensional case
7. The trace of the canonical ideal of a numerical semigroup ring
References

1
5
7
8
10
20
22
25
36

Introduction
In this paper we study the trace of the canonical module of a Cohen-Macaulay
ring R. The Cohen-Macaulay ring may either be a local ring admitting a canonical
module or else a finitely generated positively graded K-algebra, where K is a field.
All of the definitions and results which are phrased for local rings have their analogous correspondence for graded rings. Thus for the general facts about traces we
will restrict ourselves to local rings, unless otherwise stated.
Recall that for an R-module M one defines the trace of M, denoted tr(M), as the
sum of the ideals ϕ(M), where the sum is taken over all R-module homomorphisms
2010 Mathematics Subject Classification. Primary 13H10, 13D02, 05E40, 16S36; Secondary
14M25, 13A02, 13F20, 13A18, 06A11.
Key words and phrases. canonical module, trace, nearly Gorenstein, almost Gorenstein,
Veronese algebras, Segre product, Hibi ring, conductor, numerical semigroup, residue, shifting.
1

ϕ : M → R. Traces of modules have been considered in various contexts, in particular to better understand the center of the ring of endomorphisms of a module, cf.
[25]. Here we are only interested in the trace of the canonical module.
The significance of the trace of the canonical module ωR arises from the fact
that it describes the non-Gorenstein locus of R, see Lemma 2.1. In particular, R
is Gorenstein if tr(ωR ) = R, and it is Gorenstein on the punctured spectrum of R
if tr(ωR ) is m-primary, where m is the maximal ideal of R. Ding [9] has already
considered tr(ωR ). He showed that tr(ωR ) is m-primary if and only if the Auslander
index of R is finite, see [9, Theorem 1.1].
We call R nearly Gorenstein, if m ⊆ tr(ωR ). Thus R is nearly Gorenstein but
not Gorenstein, if and only if tr(ωR ) = m. We would like to understand how much
this class differs from the Gorenstein one, and compare it with related concepts
like almost Gorenstein, in the sense of [2] and [14]. Using a result of Teter, in [22,
Corollary 2.2] Huneke and Vraciu showed that quotients R of Gorenstein artinian
local rings by their socle satisfy the condition m ⊆ tr(ωR ). They also classified the
m-primary monomial ideals I in a polynomial ring S such that S/I is of small type
and it is nearly Gorenstein, see [22, Example 4.3, Theorem 4.5].
In this paper, after proving general statements about tr(ωR ) and how it can be
computed, we apply these results to study the nearly Gorenstein property for several
classes of algebras, including the ones of Veronese type, Segre products, Hibi rings,
or one dimensional rings, with a focus on the toric case.
It is easily seen that if I ⊂ R is an ideal with grade I > 0, then tr(I) = I · I −1 .
Here I −1 denotes the inverse ideal of I, namely I −1 = {x ∈ Q(R) : xI ⊆ R} with
Q(R) the total ring of fractions of R. In particular, if R is generically Gorenstein, ωR
may be identified with an ideal of grade 1, and hence in this case tr(ωR ) = ωR ωR−1,
where ωR−1 is the anti-canonical ideal of R. While the canonical ideal ωR is only
determined up to isomorphism, its trace is uniquely determined.
For the convenience of the reader we collect and record in Section 1 a few general
and basic properties on the trace of modules, some of them well-known. In Proposition 1.4 the trace of a product of ideals is considered in relation to the traces of its
factors, while Lemma 1.5 describes the behavior of the trace under change of rings.
In Section 2 nearly Gorenstein rings are introduced and it is shown in Proposition
2.3 that if a ring is nearly Gorenstein, then any reduction of the ring modulo a
regular sequence is again nearly Gorenstein. It is also observed that the converse
does not hold in general.
In the case that R is a residue class ring of a regular local ring S, tr(ωR ) can
be computed in terms of the free S-resolution of R, more precisely in terms of the
last step of the resolution, as shown in Corollary 3.2. This is the consequence of a
result of Vasconcelos [37] which we recall in Proposition 3.1. In the rest of Section 3
other, more special, situations are considered in which tr(ωR ) can be computed more
explicitly. Notable is the situation when R is a domain and the Cohen–Macaulay
type of R is 2. It is shown in Corollary 3.4 that in this case the entries of the last
map in the free S-resolution of R = S/I generate tr(ωR ) modulo I.
2

Section 4 is devoted to the study of the canonical trace ideal for special algebra constructions. There we have in mind tensor products of algebras, Veronese
algebras, squarefree Veronese algebras and Segre products. The result for tensor
algebras (Theorem 4.2) asserts the following: let K be a field and R1 and R2 be
positively graded Cohen-Macaulay K-algebras, and let ωR1 , ωR2 be their respective
canonical modules. Then trR (ωR1 ⊗K R2 ) = trR1 (ωR1 )R·trR2 (ωR2 )R. As an immediate
consequence one obtains that the tensor product R1 ⊗K R2 is nearly Gorenstein if
and only if it is is Gorenstein, which in turn is the case if and only if both R1 and
R2 are Gorenstein, see Corollary 4.3. In particular, a polynomial ring extension of
a positively graded K-algebra R is nearly Gorenstein if and only if R is Gorenstein,
see Corollary 4.4. A similar statement holds for power series over a local ring, as
stated in Proposition 4.5
Next we consider Veronese subalgebras
of a standard graded K-algebra R. Let
L
(d)
d > 0 be an integer and
R
=
R
be
the d-th Veronese subalgebra of R. The
i id
L
(d)
R -modules Mj =
i∈N Rdi+j for j = 0, . . . , d − 1 are called the d-th Veronese
submodules of R. In Theorem 4.6 we show that m(d) ⊆ trR(d) (Mj ) for any d-th
Veronese submodule Mj of R. Here m(d) denotes the graded maximal ideal of R(d) .
By using a result of Goto and Watanabe [15] one deduces then from Theorem 4.6
that all Veronese subalgebras of a standard graded K-algebra R over an infinite field
K are nearly Gorenstein, if R is Gorenstein.
The situation for squarefree Veronese subalgebras is more complicated. Given
integers n ≥ d > 0, the d-th squarefree Veronese subalgebra Rn,d of the polynomial
ring S in n variables over a field K is the K-algebra generated by the squarefree
monomials in S of degree d. Based on a theorem of Bruns, Vasconcelos and Villarreal
[5] we give in Theorem 4.12 an explicit description of the anti-canonical ideal of Rn,d ,
and we use this result to show in Theorem 4.14 that the following conditions are
equivalent: (i) Rn,d is nearly Gorenstein, (ii) Rn,d is Gorenstein, (iii) d = 1 or
d = n − 1 or n = 2d.
By another result of Goto and Watanabe [15], the canonical module for the Segre
product T = R#S of positively graded Cohen-Macaulay K-algebras R and S of
Krull dimension at least 2 is just the Segre product of the respective canonical
modules, assuming that T is Cohen–Macaulay. We use this result in Theorem 4.15
to compute tr(ωT ) in the case that R and S are standard graded Gorenstein K|r−s|
algebras. It is shown in Theorem 4.15 that mT
⊆ tr(ωT ), where mT is the graded
maximal ideal of T and where r and s are the respective a-invariants of R and S.
Equality holds, if T is a domain. If follows that under these conditions T is nearly
Gorenstein if and only if |r − s| ≤ 1, see Corollary 4.16. The section ends with
Proposition 4.18 in which the anti-canonical ideal of the Segre product is computed
in the case that R and S are polynomial rings.
These results on Segre products are used in Section 5 to give a complete classification of all nearly Gorenstein Hibi rings. Given a finite distributive lattice L and a
field K, the Hibi ring of L defined over K is the toric ring RK [L] whose relations are
the meet-join relations of L. By a fundamental theorem of Birkhoff, L is the ideal
lattice J (P ) of its poset of join irreducible elements. It is shown in Theorem 5.4
3

that RK [L] is nearly Gorenstein if and only if P is the disjoint union of pure connected posets P1 , . . . , Pq such that | rank(Pi ) − rank(Pj )| ≤ 1 for 1 ≤ i < j ≤ q.
This naturally complements the result of the second author in [19] that RK [L] is
Gorenstein if and only if P is pure.
The last two sections deal with one-dimensional rings, and especially with numerical semigroup rings. In Section 6 we consider one-dimensional local Cohen–
Macaulay rings admitting a canonical module. Our first observation is that any
one-dimensional almost Gorenstein ring as defined by Barucci and Fröberg [2] is
nearly Gorenstein, see Proposition 6.1. For this proof we use the description of
almost Gorenstein rings as given by Goto et al. [14]. This description has also the
advantage that the concept of almost Gorenstein rings can be naturally extended to
higher dimensions. Unfortunately, in higher dimensions almost Gorenstein rings and
nearly Gorenstein rings are not related to each other, as it is shown by examples. If
the one-dimensional Cohen–Macaulay domain R is of embedding dimension 3 and it
has the presentation S/I with S a regular local ring of dimension 3, then R is nearly
Gorenstein if and only if the entries of the relation matrix of I generate the maximal
ideal of S, see Proposition 6.3. Here the difference between almost Gorenstein and
nearly Gorenstein becomes tangible, especially when we consider the semigroup ring
of a 3-generated numerical semigroup. By a result of Nari, Numata and K.-i Watanabe [29] (see also [28]) the ring is almost Gorenstein if the entries of a row of the
relation matrix (and not necessarily all entries of the relation matrix) generate the
maximal ideal of S.
In the case that R is a one-dimensional domain which is a subring of a discrete
valuation ring R̃, and for which R̃ is a finite R-module, it can be seen (Proposition
6.5) that the trace of an ideal in R, and hence also the trace of the canonical
e As a consequence, one obtains
module, contains the conductor CR̃/R = R :R R.
(Corollary 6.6) that R is nearly Gorenstein if m = CR̃/R . For any ideal I ⊂ R,
Proposition 6.8 provides several conditions on I which are equivalent to tr(I) = CR̃/R .
Finally, it is shown in Corollary 6.9 that the equality tr(ωR ) = CR̃/R , forces R to
have minimal multiplicity, if R is an almost complete intersection of embedding
dimension 3.
In Section 7 we focus on toric rings coming from numerical semigroups. This is
a rich source of examples and we can use specific techniques to study the nearly
Gorenstein property. A numerical semigroup H is a subsemigroup of N containing 0
such that the number of gaps g(H) = |N \ H| is finite. The largest gap (i.e. positive
integer not in H) is the Frobenius number F(H). In Proposition 7.1 we show that if
H is generated by an arithmetic sequence, then K[H] is nearly Gorenstein. For such
semigroups, only in some special situations, the semigroup ring K[H] is (almost)
Gorenstein. However, if H has minimal multiplicity (i.e. its smallest positive element
equals the size of its minimal generating set), then we prove in Theorem 7.4 that
K[H] is nearly Gorenstein if and only if it is almost Gorenstein.
As a measure of how far is K[H] from being Gorenstein (equivalently, that H is
symmetric, cf. [24]), we introduce the residue of H defined as
res(H) = dimK K[H]/ tr(ωK[H]).
4

Clearly, res(H) = 0 when H is symmetric, and res(H) ≤ 1 precisely when K[H] is
nearly Gorenstein. The exponents of the monomials in tr(ωK[H]) form a semigroup
ideal tr(H) ⊆ H. By applying the results from Section 6, we see in Proposition 7.5
that if H is not symmetric, then CH ⊆ tr(H) ⊆ H \ {0}, where CH is the semigroup
ideal generated by the elements of H larger than F(H).
This observation gives a first estimate res(H) ≤ n(H) := |{x ∈ H : x < F(H)}|
in Corollary 7.6. However, examples computed with the NumericalSgps package [8]
in GAP [12] indicate (Question 7.7) that a tighter bound might hold:
(1)

res(H) ≤ n(H) − g(H).

This bound is correct if K[H] is nearly Gorenstein, and also if H is 3-generated, cf.
Proposition 7.10.
When H is 3-generated and not symmetric, the relation ideal IH ⊂ K[x1 , x2 , x3 ]
of K[H] is given by the maximal minors of the structure matrix of H, which is of
the form
 a1 a2 a3 
x1 x2 x3
A=
(2)
.
xb22 xb33 xb11
Q
With this notation we derive in Proposition 7.9 that res(H) = 3i=1 min{ai , bi }.
Working with the structure matrix of H allows us to parametrize explicitly the
non-symmetric 3-generated semigroups H whose trace is at either end of the interval
[CH , H \ {0}], see Theorem 7.11 and Proposition 7.13.
Example 7.8 shows that res(H) may take any nonnegative integer value, even if
we fix the number of generators of H. Still, once we fix n1 < · · · < ne , the residue
of the semigroups in the shifted family {hn1 + j, . . . , ne + ji}j≥0 seem to change
periodically with j, for j ≫ 0. This goes in the same direction as a recent number
of other results about eventually periodic properties in this shifted family, see [23],
[39], [18]. Using [34], we prove in Theorem 7.16 that given n1 < n2 < n3 and letting
Hj = hn1 + j, n2 + j, n3 + ji we have res(Hj ) = res(Hj+(n3 −n1 ) ) for all j ≫ 0. In this
setup, in Corollary 7.18 we obtain another upper bound for res(Hj ) when j ≫ 0,
depending on n3 − n1 .
1. Basic properties of the trace
For an R-module M, its trace, denoted trR (M), is the sum of the ideals ϕ(M)
with ϕ ∈ HomR (M, R). Thus,
X
trR (M) =
ϕ(M).
ϕ∈HomR (M,R)

When there is no risk of confusion about the ring we simply write tr(M).
Note that if M is finitely generated, the trace localizes. In other words, tr(M)RP =
tr(MP ) for all P ∈ Spec(R).
If M1 and M2 are isomorphic R-modules, then trR (M1 ) = trR (M2 ).
Given any ideal I ⊂ R of positive grade, we set

I −1 = {x ∈ Q(R) : xI ⊆ R},
5

where Q(R) is the total ring of fractions of R.
Lemma 1.1. Let I ⊂ R be an ideal of positive grade. Then tr(I) = I · I −1 .
Proof. Pick b ∈ I and which is regular on R. For any a ∈ I and any ϕ ∈ HomR (I, R)
one has ϕ(ab) = bϕ(a) = aϕ(b), hence ϕ(a) = a · (ϕ(b)/b). We claim that ϕ(b)/b ∈
I −1 . Indeed, for any c ∈ I one has (ϕ(b)/b)c = (ϕ(b)c)/b = ϕ(bc)/b = (bϕ(c))/b =
ϕ(c) ∈ R. This shows that tr(I) ⊆ I · I −1 .
P
For the reverse inclusion, note that I · I −1 = x∈I −1 xI. Since xI is the image of
the R-linear map θx : I → R with θx (a) = xa, the assertion follows.


Remark 1.2. According to [4, Exercise 1.2.24], for any ideal I of the noetherian
ring R, grade I ≥ 2 if and only if the canonical homomorphism R → HomR (I, R)
is an isomorphism. Equivalently, any R-module homomorphism from I into R is
just the multiplication by some element in R. Therefore, if grade I ≥ 2 one has
tr(I) = I.
We record some properties of the trace ideals that we need for later.
Proposition 1.3. Let M and N be two R-modules. Then
tr(M) tr(N) ⊆ tr(M ⊗R N) ⊆ tr(M) ∩ tr(N).

Proof. Let a ∈ M, b ∈ N, ϕ ∈ HomR (M, R) and ψ ∈ HomR (N, R). If we denote
ε : R ⊗R R → R the canonical map letting ε(u ⊗ v) = uv for any u, v in R, then
ϕ(a)ψ(b) = (ε ◦ (ϕ ⊗ ψ))(a ⊗ b) ∈ tr(M ⊗R N). This shows that tr(M) tr(N) ⊆
tr(M ⊗R N).
Let a ∈ M, b ∈ N and ϕ ∈ HomR (M ⊗R N, R). The map ψ : N → R given by
ψ(n) = ϕ(a ⊗ n) is R-linear and ϕ(a ⊗ b) = ψ(b) ∈ tr(N). From here we get that
tr(M ⊗R N) ⊆ tr(N), and by symmetry, also that tr(M ⊗R N) ⊆ tr(M) ∩ tr(N).

Proposition 1.4. Let I and J be ideals of positive grade in the ring R. Then
tr(I) tr(J) ⊆ tr(IJ) ⊆ tr(I) ∩ tr(J).
Proof. Notice first that IJ has positive grade and Lemma 1.1 applies to it, as well.
Let a ∈ I, b ∈ I −1 , a′ ∈ J and b′ ∈ J −1 . Clearly (ab)(a′ b′ ) = (aa′ )(bb′ ) and
aa′ ∈ IJ. We claim that bb′ ∈ (IJ)−1 . For that, it is enough to consider u ∈ I, v ∈ J
and show that (bb′ )(uv) ∈ (IJ)−1 . This is the case, since (bb′ )(uv) = (bu)(b′ v) ∈ R.
We conclude that tr(I) tr(J) ⊆ tr(IJ).
For the other
z ∈ (IJ)−1 .
P inclusion in the text we consider x = az with a ∈ IJ and−1
One has a = λ uλ vλ with uλ ∈ I and vλ ∈ J for all λ. Clearly, vλ z ∈ I , since for
any b ∈ I one has b(vλ z) = (bvλ )z ∈ (IJ)(IJ)−1 ⊆ R. Therefore, uλ (vλ z) ∈ I · I −1 ,
and also az ∈ I · I −1 . Conclusion follows by Lemma 1.1.

Next we study how the trace behaves under a base change.

Lemma 1.5. Let ϕ : R1 → R be a ring homomorphism, M an R-module, and M1
an R1 -module. Then
(i) if ϕ is surjective, one has (trR1 M)R ⊆ trR M;
6

(ii) (trR1 M1 )R ⊆ trR (M1 ⊗R1 R);
(iii) if ϕ is a flat morphism and M1 has a finite presentation, one has
(trR1 M1 )R = trR (M1 ⊗R1 R).

Proof. For (i) let us consider an R1 -linear map f : M → R1 . We claim that the map
g = ϕ ◦ f : M → R is R-linear. It is clearly additive. For r ∈ R we pick r1 ∈ R1 with
ϕ(r1 ) = r. Hence g(rm) = g(r1 · m) = ϕ(f (r1 · m)) = ϕ(r1 f (x)) = ϕ(r1 )ϕ(f (m)) =
rg(m), for all m in M. This shows that ϕ(Im ϕ) ⊆ Im g, hence (trR1 M)R ⊆ trR M.
For (ii), let f : M1 → R1 be an R1 -linear map. Denoting g = f ⊗R1 1R :
M1 ⊗R1 R → R1 ⊗R1 R ∼
= R the R-linear map given by g(m ⊗ r) = rf (m), we have
that Im(f )R ⊆ Im(g), hence (trR1 M1 )R ⊆ trR (M1 ⊗R1 R).
Under the assumptions of (iii), by [26, 3.E] there is an isomorphism
HomR (M1 ⊗R R, R) ∼
= HomR (M1 , R1 ) ⊗R R.
1

1

1

This implies that the image of any g ∈ HomR (M1 ⊗R1 R, R) is obtained by extending
into R the image of a suitable f ∈ HomR1 (M1 , R1 ), by the method described in the
proof of part (ii). This gives that trR (M1 ⊗R1 R) ⊆ (trR1 M1 )R, and using part (ii),
the conclusion follows.


Remark 1.6. The inclusion in part (i) of Lemma 1.5 may be strict. Let R be any
ring and I an ideal of positive grade. Then trR/I (R/I) = R/I and trR/I (R) = 0.
Indeed, if x ∈ I is a regular on R and ϕ ∈ HomR (R/I, R), then 0 = ϕ(x̂) = ϕ(x· 1̂) =
xϕ(1̂) in R, hence ϕ(1̂) = 0. This implies ϕ(r̂) = rϕ(1̂) = 0 for any r in R.
2. Nearly Gorenstein rings
Let (R, m) be a Cohen–Macaulay local ring which admits a canonical module ωR .
The trace of ωR describes the non-Gorenstein locus of R. Indeed, one has
Lemma 2.1. Let P ∈ Spec(R). Then RP is not a Gorenstein ring if and only if
tr(ωR ) ⊆ P.

Proof. Let tr(ωR ) ⊆ P . Suppose that RP is Gorenstein. Then ωRP ∼
= RP , and hence
there exists ϕ ∈ HomRP (ωRP , RP ) with ϕ(ωRP ) = RP . It follows that tr(ωR )P =
tr(ωRP ) = RP . Thus tr(ωR ) 6⊆ P .
Conversely, suppose that tr(ωR ) 6⊆ P . Then tr(ωRP ) = RP . Therefore, there
exists a surjective RP -module homomorphism ϕ : ωRP → RP . Since RP is free, the
map ϕ splits, and we get ωRP ∼
= RP ⊕ U. Since ωRP is a maximal Cohen–Macaulay
module of type 1, it follows that ωRP ∼
= RP , and this implies that RP is Gorenstein.

If R is Gorenstein on the punctured spectrum Spec(R) \ {m} we define the residue
of R as the numerical invariant
res(R) = dimR/m (R/ tr(ωR )).
Ananthnarayan [1, Corollary 3.8] shows that if moreover R is artinian, then res(R)
is bounded above by the Gorenstein colength of R, introduced by him in [1, Definition 1.2].
7

Definition 2.2. A Cohen-Macaulay local ring (or positively graded K-algebra) R
admitting a canonical module ωR is called nearly Gorenstein, if tr(ωR ) contains the
(graded) maximal ideal m of R.
It follows from this definition that Gorenstein rings are nearly Gorentein, and
that R is nearly Gorenstein but not Gorenstein, if and only if tr(ωR ) = m.
We recall that the ring R is called generically Gorenstein if RP is Gorenstein for
all minimal prime ideals P of R. Since any field is Gorenstein, we get that domains
are generically Gorenstein. If we assume that R is generically Gorenstein, then by
[4, Proposition 3.3.18], ωR is either isomorphic to the whole ring R if the latter is
Gorenstein, or otherwise it can be identified with an ideal of R of grade 1. It follows
from Lemma 1.1 that tr(ωR ) = ωR · ωR−1 , where ωR−1 is also called the anti-canonical
ideal of R.
Proposition 2.3. (a) Let R be nearly Gorenstein. Then R is Gorenstein on the
punctured spectrum of R.
(b) Let x = x1 , . . . , xr be a regular sequence on R, and set R̄ = R/(x). If R is
nearly Gorenstein, then so is R̄. The converse does not hold in general.
Proof. (a) is an immediate consequence of Lemma 2.1
(b) Observing that ωR̄ = ωR ⊗ R̄ by [4, Theorem 3.3.5], Lemma 1.5 implies that
trR (ωR )R̄ ⊆ trR̄ (ωR̄ ). This shows that if R is nearly Gorenstein, then so is R̄.
On the other hand, assume that R̄ is nearly Gorenstein but not Gorenstein, and
let R = R̄[|x|] be the formal power series over R̄. Then x is a non-zerodivisor of R,
and R/(x)R = R̄ is nearly Gorenstein. But R itself is not nearly Gorenstein, see
Proposition 4.5.

3. Computing the trace of the canonical module
Let α : Rp → Rq be an R-linear map, and let A be the matrix representing α with
respect to some bases of Rp and Rq . We denote by It (A) the ideal of t-minors of A.
This ideal depends only on α and not on the chosen bases. Therefore, we also write
It (α) for It (A). Following Vasconcelos [37, Remark 3.3], the trace of a module can
be computed as follows.
ϕ

Proposition 3.1. Let F1 → F0 → M be a minimal free presentation of the Rmodule M. Let ϕ∗ denote the dual of ϕ, and consider the beginning of a minimal
free resolution
α

ϕ∗

G −−−→ F0∗ −−−→ F1∗ −−−→ D(M) −−−→ 0.

of the Auslander dual D(M) of M (which is defined to be the cokernel of ϕ∗ ). Then
tr(M) = I1 (α).
Proof. The R-module homomorphisms γ : M → R are induced
P by R-module homomorphisms β : F0 → R with β ◦ ϕ = 0. Thus tr(M) =
β(F0 ), where the sum
is taken over all R-module homomorphisms β : F0 → R with β ◦ ϕ = 0. Let β be
such an R-module homomorphism, and let B : e1 , . . . , em be a basis of F0 . Then
8

P
∗
β(F0 ) is the ideal (β(e1 ), . . . , β(em )) ⊆ R. Notice that β ∗ (1) = m
i=1 β(ei )ei , where
∗
∗
∗
∗
∗
e1 , . . . , em is the dual basis of B. Since ϕ ◦ β = (β ◦ ϕ) = 0, it follows that
P
m
∗
I1 (α).
i=1 β(ei )ei is in the image of α, and this shows that tr(M) ⊆ P
m
∗
Conversely, let g1 , . . . , gr be a basis of G and let α(gk ) =
j=1 ajk ej for k =
1, . . . , r. Then I1 (α) is the sum of the ideals Jk with Jk = (a1k , . . . , amk ) for k =
1, . . . , r. For such k, let βk : F0 → P
R be the R-module homomorphism defined by
∗
∗
βk (ej ) = ajk for j = 1, . . . , m. Since m
j=1 ajk ej is in the kernel of ϕ , it follows that
βk ◦ ϕ = 0. Therefore, Jk ⊆ tr(M) for all k. This shows that I1 (α) ⊆ tr(M).

Corollary 3.2. Let S be a regular local ring and let
ϕp

F : 0 −→ Fp −−−→ Fp−1 −→ · · · −→ F1 −→ F0 −→ R −→ 0

be a minimal free S-resolution of the Cohen-Macaulay ring R
S/J. Let e1 , . . . , et
P=
t
be a basis of Fp . Suppose that for i = 1, . . . , s the elements j=1 rij ej generate the
kernel of
ψp

Fp ⊗ R −−−→ Fp−1 ⊗ R,
where ψp = ϕp ⊗ R. Then tr(ωR ) is generated by the elements rij with i = 1, . . . , s
and j = 1, . . . , t.
Proof. The canonical module ωR of R can be computed as the cokernel of the map
∗
ϕ∗p : Fp−1
→ Fp∗ , see [4, Corollary 3.3.9]. Hence, as an R-module its presentation is
given as follows:
ϕ∗p ⊗R

∗
Fp−1
⊗ R −−−→ Fp∗ ⊗ R −→ ωR −→ 0.
Thus the desired conclusion follows from Proposition 3.1.



Recall that the (Cohen–Macaulay) type of a Cohen–Macaulay local ring (R, m)
of dimension d is the number dimR/m ExtdR (R/m, R).
In the following corollaries we refer to the notation of Corollary 3.2.
Corollary 3.3. Let R be Cohen–Macaulay of type t, and assume that R is generically Gorenstein. Then It−1 (ψp ) ⊆ tr(ωR ).

Proof. Since ϕ∗p ⊗ R = ψp∗ , it follows that It−1 (ψp ) = It−1 (ψp∗ ) = It−1 (ϕ∗p ⊗ R).
Let U = Im(ϕ∗p ⊗ R). Then we obtain an exact sequence 0 → U → Rt → ωR → 0.
Here we identified Fp∗ ⊗ R with Rt , since Fp is free of a rank t.
Since R is generically Gorenstein, ωR is an ideal of rank 1, so that U is a module
of rank t − 1. Let A be the t × r-matrix which describes U as a submodule of Rt .
Then A is just the relation matrix of ωR and rank A = t − 1. This implies ([4,
Proposition 1.4.11]) that all t-minors of A vanish.
We need to prove that It−1 (A) ⊆ tr(ωR ).
Let ∆ be any (t − 1)-minor of A, and let B be the t × (t − 1) submatrix of A
whose columns are involved in computing ∆. For any j = 1, . . . , r, adding the
column (a1j , . . . , atj )T of A to B, we obtain a t × t-matrix B ′ whose determinant
is zero, since It (A) = 0. P
Expanding B ′ with respect to the new column we have
added to B, we see that ti=1 (−1)i aij ∆i = 0, where ∆i is the determinant of the
(t − 1) × (t − 1)-matrix which is obtained from B by dropping the ith row.
9

In conclusion, we see that U is in the kernel of the map α : Rt → R which
assigns to the ith canonical basis vector of Rt the element (−1)i ∆i . This shows that
(∆1 , . . . , ∆t )R ⊆ Im(ᾱ), where ᾱ : Rt /U → R is the R-linear map induced by α. In
particular, ∆ ∈ tr(ωR ). This finishes the proof.

Corollary 3.4. Suppose R is a domain of type 2. Then
tr(ωR ) = I1 (ψp ) and ωR−1 ∼
= Im(ψp∗ ).
In particular, µ(ωR−1) = rank Fp−1 .
Proof. Regarding the first equation, we only need to show that tr(ωR ) ⊆ I1 (A), since
I1 (ψp ) = I1 (A) and since the other inclusion is already shown in Corollary 3.3.
As R is a domain, it follows that tr(ωR ) = ωR ·ωR−1. Let f1 and f2 be the generators
of ωR . We may assume that A is the relation matrix of ωR with respect to these
generators. Then it suffices to show that the elements xf1 and xf2 belong to I1 (A)
for any x ∈ ωR−1 . To see this, let a = xf1 and b = xf2 . Then x(bf1 − af2 ) = 0. Since
R is a domain, we conclude that bf1 − af2 = 0. This implies that a, b ∈ I1 (ψp ).
Letting r = rank Fp−1 ,
ψp∗

Rr −−−→ R2 −→ ωR −→ 0

is the beginning of the minimal free resolution for the ideal ωR = (f1 , f2 ). Then
U = Im(ψp∗ ) = {(a, b) ∈ R2 : af1 + bf2 = 0}. It is an easy exercise to check that
the map ωR−1 → U with g 7→ (gf2, −gf1 ) establishes an isomorphism, hence also
µ(ωR−1) = µ(U) = r = rank Fp−1 .

By using Corollary 3.4 we obtain the following characterization of nearly Gorenstein rings of type 2.
Corollary 3.5. Let R be a domain of type 2. Then R is nearly Gorenstein if and
only if I1 (ϕp ) = n, where n is the maximal ideal of S.
Proof. Corollary 3.4 implies that I1 (ψp ) = m if and only if R is nearly Gorenstein.
Let R = S/J. Then we see that R is nearly Gorenstein if and only if I1 (ϕp ) + J =
n + J. The desired result follows from Nakayama’s lemma.


4. The canonical trace ideal for special constructions of algebras
4.1. Tensor products. The next result describes the trace of a tensor product of
two K-algebras as the product of the trace ideals of the factors.
Proposition 4.1. Let R1 and R2 be K-algebras over the field K, and set R =
R1 ⊗K R2 . For any finitely presented R1 -module M1 and R2 -module M2 one has that
(i) TorR
i (M1 ⊗R1 R, M2 ⊗R2 R) = 0 for all i > 0;
(ii) trR1 (M1 )R · trR2 (M2 )R = trR (M1 ⊗K M2 ) = trR1 (M1 )R ∩ trR2 (M2 )R.
10

Proof. Let F′ → 0 and G′ → 0 be free resolutions of M1 over R1 , and of M2 over
R2 , respectively. General homological algebra facts (see [40, Theorem 2.7.2]) imply
that for all i > 0 one has Hi (F′ ⊗K G′ ) ∼
= TorK
i (M1 , M2 ) = 0, since M1 is free over
K.
Since the canonical maps R1 → R and R2 → R are flat, it follows that the chain
complexes F = F′ ⊗R1 R and G = G′ ⊗R2 R are free resolutions over R of M1 ⊗R1 R,
∼
and of M2 ⊗R2 R, respectively. As before, we get that TorR
i (M1 ⊗R1 R, M2 ⊗R2 R) =
Hi (F ⊗R G) for all i.
For any R1 -module N1 and any R2 -module N2 there exists a canonical isomorphism N1 ⊗K N2 ∼
= (N1 ⊗R1 R) ⊗R (N2 ⊗R2 R), which can be used to construct an
isomorphism between the chain complexes F′ ⊗K G′ and F ⊗R G. This implies that
F ⊗R G is also acyclic, which proves (i).
For (ii) we apply the identity in part (i) to the modules R1 / trR1 (M1 ) and R2 / trR2 (M2 ).
Combined with [40, Exercise 3.1.3], that gives


R
R
(trR1 (M1 ))R ∩ (trR1 (M2 ))R
R
∼
0 = Tor1
,
.
=
(trR1 (M1 ))R (trR1 (M2 ))R
(trR1 (M1 ))R · (trR1 (M2 ))R
These equations together with Proposition 1.3, Lemma 1.5(iii) and the isomorphism
of R-modules M1 ⊗K M2 ∼
= (M1 ⊗R1 R) ⊗R (M2 ⊗R2 R) yield the desired statement.


Theorem 4.2. Let R1 and R2 be positively graded Cohen-Macaulay K-algebras over
a field K, and ωR1 , ωR2 their respective canonical modules. Denote R = R1 ⊗K R2 .
Then
ωR ∼
= ωR1 ⊗K ωR2 and
trR (ωR ) = trR1 (ωR1 )R · trR2 (ωR2 )R.

Proof. We consider minimal presentations R1 ∼
= S2 /I2 as quotients of
= S1 /I1 , R2 ∼
the polynomial rings S1 and S2 , and we let S = S1 ⊗K S2 . Clearly R ∼
= S/(I1 , I2 )S.
We note that the fibers of the flat extensions S1 ⊂ S and S2 ⊂ S are Gorenstein,
hence using [4, Theorem 3.3.14] we obtain that ω1 = ωR1 ⊗S1 S and ω2 = ωR2 ⊗S2 S
are the canonical modules for S1 /I1 ⊗S1 S ∼
= S/I1 S, and for S/I2 S, respectively.
Let
′
F′ : 0 → Fp′ → Fp−1
→ · · · → F0′ → S1 /I1 → 0

be a minimal free S1 -resolution of S1 /I1 . Since the extension S1 ⊂ S is flat, it follows
that the complex F = F′ ⊗S1 S is a minimal free resolution of S/I1 S over S. Then
by [4, Corollary 3.3.9], the dual complex F∗ = HomS (F, S) ∼
= HomS1 (F′ , S1 ) ⊗S1 S is
a minimal free resolution of ω1 .
Similarly, if we start with
G′ :

0 → G′q → G′q−1 → · · · → G′0 → S2 /I2 → 0

a minimal free S2 -resolution of S2 /I2 , the complex G = G′ ⊗S2 S is a minimal free
resolution of S/I2 S over S, and G∗ minimally resolves ω2 over S.
By [40, Theorem 2.7.2], Hi (F⊗S G) = TorSi (S/I1 S, S/I2 S) for all i. We may apply
Proposition 4.1 to the S1 -module S1 /I1 and to the S2 -module S2 /I2 and we obtain
11

that the complex F ⊗S G is acyclic and that it minimally resolves S/I1 S ⊗S S/I2 S ∼
=
R
over
S.
Thus
proj
dim
R
=
proj
dim
R
+
proj
dim
R
.
S/(I1 , I2 )S ∼
=
S
S1 1
S2 2
On the other hand, by the formula of Hoa and Tam in [21, Theorem 1.3] we have
that dim R = dim R1 + dim R2 , hence R is also Cohen-Macaulay. This implies that
(F ⊗S G)∗ is a minimal S-free resolution of ωR .
From the isomorphism (F ⊗S G)∗ ∼
= F∗ ⊗S G∗ we derive that F∗ ⊗S G∗ is acyclic,
hence it resolves (minimally) the S-module ω1 ⊗S ω2 . Since

ω1 ⊗S ω2 ∼
= ωR1 ⊗S1 S ⊗S ω2 ∼
= ωR1 ⊗S1 (S1 ⊗K S2 ) ⊗S2 ωR2 ∼
= ωR1 ⊗K ωR2 ,
we get that ωR ∼
= ωR1 ⊗K ωR2 . Now using Proposition 4.1 we obtain that trR (ωR ) =
trR1 (ωR1 )R · trR2 (ωR2 )R.


Corollary 4.3. Let R1 and R2 be positively graded Cohen-Macaulay K-algebras
over a field K with emb dim Ri > 0 for i = 1, 2, and let R = R1 ⊗K R2 . Then the
following conditions are equivalent:
(i) R is nearly Gorenstein;
(ii) R is Gorenstein;
(iii) R1 and R2 are Gorenstein.
Proof. (i) ⇒ (iii): Since tr(ωR ) = tr(ωR1 )R · tr(ωR2 )R, our assumption implies that
m ⊆ tr(ωR1 )R · tr(ωR2 )R. Suppose tr(ωRi ) ⊆ mRi for some i, say i = 1. Then
tr(ωR ) ⊆ mR1 R ⊆ mR . This is a contradiction since the last inclusion is proper
if emb dim R2 > 0. Thus tr(ωRi ) = Ri for i = 1, 2, and this implies that Ri is
Gorenstein for i = 1, 2, see Lemma 2.1.
(iii) ⇒ (ii): The condition implies that ωRi ∼
= Ri (up to a shift in the grading).
∼
Therefore, ωR = R1 ⊗K R2 (up to a shift in the grading). Hence R is Gorenstein.
(ii) ⇒ (i) is obvious.

Corollary 4.4. Let R be a positively graded K-algebra and S = R[x1 , . . . , xn ] with
n ≥ 1 be the polynomial ring over R. Then S is nearly Gorenstein if and only if R
is Gorenstein.
There is a local analogue to this corollary.
Proposition 4.5. Let R be a local Cohen-Macaulay ring admitting a canonical
module, and let S = R[|x1 , . . . , xn |] with n ≥ 1 be the formal power series over R.
Then S is nearly Gorenstein if and only if R is Gorenstein.
Proof. If R is Gorenstein, then S is Gorenstein and hence nearly Gorenstein. For the
converse implication, observe that ωS = ωR ⊗R S. Thus we may apply Lemma 1.5
and see that tr(ωS ) = tr(ωR )S. If R is not Gorenstein, then tr(ωR ) ⊆ mR , and hence
tr(ωS ) = mR S ( mS . Therefore, S is not nearly Gorenstein.

L
4.2. Veronese subalgebras. Let R =
i∈Z Ri be a standard graded K-algebra
over the field K and d a positive integer. The ring R(d) = ⊕i∈Z Rid is called the d-th
Veronese subring of R. We will consider R(d) as a standard graded K-algebra by
letting the elements (generators) in Rd have degree 1.
12

L
L
Let Mj = i∈N Rdi+j for j = 0, . . . , d − 1. Then R = d−1
j=0 Mj , and Mj is a finite
(d)
R -module. This explains why the Mj ’s are called the d-th Veronese submodules
of R. We refer to [4, Exercise 3.6.21] and [15] for more background on Veronese
algebras and their properties.
We denote m and m(d) the maximal graded ideal of R, and of R(d) , respectively.
In the sequel we show that the trace of any d-th Veronese module of R contains
(d)
m .
Theorem 4.6. Let R be a standard graded K-algebra over an infinite field K and
d a positive integer. Then m(d) ⊆ trR(d) (Mj ) for any d-th Veronese submodule Mj of
R.
Proof. If d = 1, there is only one Veronese submodule M0 = R(d) and its trace equals
R(d) = R.
Assume d > 1. Since the field K is infinite, we may choose a K-basis x1 , . . . , xn
for R1 consisting of nonzero divisors. These generate the algebra R, as well. Fix
0 ≤ j < d. As an R(d) -module, Mj is generated by the set G = {xa11 · · · xann :
Pn
d−j
(d)
i=1 ai = j}. Since x1 is regular on R, the R -module I = x1 Mj is isomorphic
to Mj , hence they have the same trace ideal. Note that I is an ideal of R(d) which
contains the regular element xd1 , and therefore the formula in Lemma 1.1 applies.
P
For a vector of nonnegative integers α = (a1 , . . . , an ) we denote |α| = ni=1 ai
and xα = xa11 · · · xann . Given a product xα with |α| = d, there exist vectors β, γ with
nonnegative integer entries such that α = β + γ and |β| = j. We may write
xα = (xd−j
· xβ ) · (xγ /xd−j
1
1 ).
· xβ is in I and it is regular on R (and on R(d) ). Hence
Clearly, xd−j
1
xγ /xd−j
= xα /(xd−j
· xβ ) ∈ Q(R(d) ).
1
1
(d)
α
−1
Since xxd−j (xd−j
and m(d) ⊆ trR(d) (Mj ), which
1 G) ⊆ R , we conclude that x ∈ I ·I
1
finishes the proof.

γ

Corollary 4.7. If R is a Gorenstein standard graded K-algebra over an infinite
field K, then all its Veronese subalgebras are nearly Gorenstein.
Proof. Goto and Watanabe in [15] showed that if R is Gorenstein, then up to a shift
ωR(d) ∼
= Mj for some j ∈ 1, . . . , d − 1. This fact together with Theorem 4.6 proves
the statement.

Note that in the proof of Theorem 4.6 the assumption on the field K to be infinite
was only required to insure the existence of a system of algebra generators of degree
1. This observation proves the next corollary.
Corollary 4.8. The Veronese subalgebras of a polynomial ring over any field are
nearly Gorenstein.
13

4.3. Squarefree Veronese subalgebras. For 1 ≤ d ≤ n, the d-th squarefree
Veronese subalgebra of S = K[x1 , . . . , xn ] is the subalgebra Rn,d generated by the
set Vn,d of squarefree monomials in S of degree d. As with the regular Veronese,
Rn,d is standard graded by letting deg m = 1 for all m ∈ Vn,d . Moreover, Rn,d is
normal (cf. [38], and [35]), hence Cohen-Macaulay.
In contrast to the regular Veronese subalgebras, we show that Rn,d is not nearly
Gorenstein unless it is already Gorenstein. By work of De Negri and Hibi in [6] (see
also [5, Corollary 2.14]) the latter property holds if and only if
(n, d) ∈ {(n, 1), (n, n − 1), (2d, d)}.
The monomials in Rn,d correspond to lattice points in some rational cone C ⊂
Rn , and by Danilov’s Theorem the canonical module ωRn,d is the K-span of the
monomials in the relative interior of C. We refer to [4, Chapter 6] for the necessary
background. The next results of Bruns, Vasconcelos and Villarreal in [5] describe
the lattice points of the cone C and a generating system for ωRn,d .
Lemma 4.9. ([5, Proposition 2.4]) A monomial m = xa11 · · · xann is in Rn,d if and
only if
n
X
i=1

dai ≤

n
X

ai ≡ 0 mod d, and

aj , for all i = 1, . . . , n.

j=1

Theorem 4.10. ([5, Theorem 2.6]) If n ≥ 2d ≥ 4, the ideal ωRn,d is generated by
the monomials m = xa11 · · · xann which satisfy the conditions:
P
(i) P
ai ≥ 1 and dai ≤ −1 + ni=1 ai , for all i = 1, . . . , n;
(ii) ni=1 ai ≡ 0 mod d;
(iii) |{i : ai ≥ 2}| ≤ d − 1.
The restrictions on n and d in Theorem 4.10 do not leave out any interesting
cases. If d = 1, then Rn,d = S. As noted in [5, Remark 2.13], the bijective map
ρ : Vn,d → Vn,n−d given by ρ(m) = x1 · · · xn /m for all m in Vn,d induces a graded
isomorphism between the K-algebras Rn,d and Rn,n−d , so we may restrict our study
to the case n ≥ 2d ≥ 4.
Our aim is to describe a generating set for the anti-canonical ideal of a squarefree
Veronese algebra Rn,d . The following lemma is a partial result in this direction.
Lemma 4.11. If n > 2d ≥ 4, then

(1/xi1 · · · xid : 1 ≤ i1 < · · · < id ≤ n)Rn,d ⊆ ωR−1n,d .

Equality holds if and only if n = 2d + 1.
Proof. Let f = 1/xi1 · · · xid with 1 ≤ i1 < · · · < id ≤ n, and mP= xa11 · · · xann a
generator for ωRn,d as given by Theorem 4.10. Then dai ≤ −1 + ni=1 ai for all i,
14

and since d divides
(3)

Pn

j=1 aj

we get
n
X

dai ≤ −d +

aj , for i = 1, . . . , n.

j=1

Since ai > 0 for all i, it follows that
mf =

d
Y
j=1

ai −1

xij j

!


Y

i∈{i
/ 1 ,...,id }



xai i 

is a monomial in S of degree divisible by d. The inequalities (3) assure that the
exponents of mf satify the conditions in Lemma 4.9, therefore mf ∈ Rn,d . This
confirms that 1/xi1 · · · xid ∈ ωR−1n,d .
We defer the discussion of the equality case after the proof of Theorem 4.12.

For any monomial u is S we denote its vector of exponents by log u. Even if u
is in Rn,d , its degree will be considered with respect to the standard grading on S
assigning to each variable the degree 1.
Theorem 4.12. Let n > 2d ≥ 4 . Then ωR−1n,d is minimally generated by the fractions
u/x1 · · · xn , where u is a monomial in S such that either it is a product of n − d
distinct variables, or else, up to a permutation of the entries, log u belongs to the
set Pn,d , where Pn,d is the set of vectors of the form
(c, . . . , c, bd+2 , . . . , bn−d−c+1 , 0, . . . , 0) ∈ Nn
| {z }
| {z }
d+1

d+c−1

with 2 ≤ c ≤ n − 2d, c ≥ bd+2 ≥ · · · ≥ bn−d−c+1 ≥ 0 and

Pn−d−c+1
i=d+2

bi = n − 2d − c.

Proof. Let R = Rn,d and A denote the set of generators of ωR as given in Theorem 4.10.
Step 1. Note that by permuting the exponents of the variables of any monomial
in A we get another monomial in A. This observation together with (i) and (iii) in
Theorem 4.10 assures that x1 · · · xn is the greatest common divisor (computed in S)
of the monomials in A.
Step 2. Since ωR is generated by monomials in R, it follows that ωR−1 is also
−1
±1
generated by monomials in K[x±1
1 , . . . , xn ]. If f = u/v ∈ ωR with u and v coprime
polynomials in S, then v divides the greatest common divisor of the monomials in
A, which is x1 · · · xn . This shows that for the purpose of determining a generating
set for ωR−1 , it is enough to consider fractions f = u/(x1 · · · xn ) where u is in the set
B = {u monomial in S : deg u ≡ n mod d, u · ωR ⊆ x1 · · · xn R}.
A priori, for u ∈ B the fraction u/(x1 · · · xn ) is not necessarily in Q(R).
Step 3. We identify the elements of B as follows. A monomial u = xb11 · · · xbnn is
in B if and only if
xa11 +b1 −1 · · · xnan +bn −1 ∈ R for all xa11 · · · xann ∈ A.
15

This is equivalent, by Lemma 4.9, to the fact that
(4)

deg u ≡ n mod d, and
n
X
dbi + (dai + n − d) ≤ deg u +
aj , for 1 ≤ i ≤ n, and xa11 · · · xann ∈ A.
j=1

Letting

E = max{dai + n − d −

n
X
j=1

aj : 1 ≤ i ≤ n, xa11 · · · xann ∈ A},

the condition (4) is equivalent to
(5)

deg u ≡ n mod d, and dbi + E ≤ deg u, for 1 ≤ i ≤ n.

It immediately follows that if u′ is obtained from u ∈ B by permuting its exponents, then also u′ ∈ B.
Step 4. We claim that
E = n − 2d.
It is easy to check with Theorem 4.10 that

(n − 2d + 1, . . . , n − 2d + 1, 1, . . . , 1)
|
{z
} | {z }
d−1

n−d+1

is the exponent of a monomial in A, hence E ≥ n − 2d.
On the other hand, 1/(x1 · · · xd ) ∈ ωR−1 by Lemma 4.11, hence xd+1 · · · xn ∈ B.
From (5) we get that d + E ≤ n − d, which shows that E = n − 2d. Therefore (5)
becomes
(6)

deg u ≡ n mod d, and dbi + (n − 2d) ≤ deg u, for 1 ≤ i ≤ n.

Step 5. A consequence of (5) is that there are at least d + 1 distinct variables
dividing u. Indeed, arguing by contradiction, if there are at most d distinct variables
dividing u and say, bd+1 = · · · = bn = 0, we get by summing the inequality in (5)
for i = 1, . . . , d that
d
d
X
X
bi ,
(dbi + E) ≤ d deg u = d
i=1

i=1

equivalently dE ≤ 0, which is false.
Step 6. We define a partial order on B by letting u ≤ v if v/u ∈ R. Our next goal
is to identify the elements of B minimal with respect to this partial order.
For u = xb11 · · · xbnn in B we set
I(u) = {xi : dbi + (n − 2d) = deg u}.

It follows from (6), arguing modulo d, that if for some i we have dbi + E < deg u,
then dbi + E ≤ deg u − d.
We claim that
(7)

u is minimal in B ⇐⇒ |I(u)| ≥ d + 1.
16

Indeed, if |I(u)| ≤ d, then there exists m a squarefree monomial of degree d which
contains the variables in I(u). Then u′ = u/m verifies (6), hence u′ ∈ B and u is
not minimal in B.
On the other hand, if |I(u)| ≥ d + 1, then no matter by which product of d
distinct variables we divide u to obtain a monomial u′, for at least some xi0 ∈ I(u)
the inequality in (6) applied to u′ and i0 does not hold. This confirms (7).
Based on (7), we describe the minimal elements in B as follows. From (6) we infer
that deg u ≥ n − d, and equality holds if and only if u is a product of n − d distinct
variables. These monomials are clearly minimal in B.
Assume u is minimal in B and deg u > n − d. Since deg u ≡ n mod d we have
deg u ≥ n. We may pick i1 < · · · < id+1 in I(u) and we let c = bi1 = · · · = bid+1 .
Clearly, c = max{bi : 1 ≤ i ≤ n} is greater than 1, otherwise u = x1 · · · xn for which
I(u) = ∅, a contradiction. Then
X
n − 2d = c +
bj .
j ∈{i
/ 1 ,...,id+1 }

It follows that 2 ≤ c ≤ n−2d and that for at least (n−d−1)−(n−2d−c) = d+c−1
indices j ∈
/ {i1 , . . . , id+1 } we have bj = 0. Equivalently, up to a permutation of the
entries, log u is in Pn,d .
Step 7. We claim that for any u a minimal element in B we have that f =
u/(x1 · · · xn ) is in Q(R). That would imply that these f ’s generate ωR−1 minimally.
We write n = ℓd + r with 1 ≤ r ≤ d and we let m be the product of some distinct
r variables dividing u. Since n > 2d, we get n ≥ 2d + r, hence by (6) we obtain
dbi ≤ deg u − (n − 2d) ≤ deg u − r, for 1 ≤ i ≤ n.
This implies that u/m is in R, using Lemma 4.9. Clearly (x1 · · · xn )/m is also in R as
a product of ℓd distinct variables, hence f ∈ Q(R). This concludes the proof.

We can now finish the proof of Lemma 4.11.
Proof. (of Lemma 4.11, continued) According to Theorem 4.12, the equality holds
in Lemma 4.11 if and only if the set Pn,d is empty.
As noted in the proof of Theorem 4.12, any vector in Pn,d has at least d + 1 non
zero entries, and at least d + 1 zero entries, hence n ≥ 2d + 2. This shows that
Pn,d = ∅ for n = 2d + 1. Conversely, if n ≥ 2d + 2, it is easy to check that
(n − 2d, . . . , n − 2d, 0, . . . , 0) ∈ Pn,d ,
|
{z
} | {z }
d+1

n−d−1

hence Pn,d is not empty in this case.



Example 4.13. We computed the sets Pn,2 for small values of n. For brevity, we
record only the non zero entries of the vectors in Pn,2 , as is it clear how many zeroes
one should add to reconstruct log u.
P5,2 = ∅, P6,2 = {(2, 2, 2)}, P7,2 = {(2, 2, 2, 1), (3, 3, 3)},
P8,2 = {(2, 2, 2, 2), (2, 2, 2, 1, 1), (3, 3, 3, 1), (4, 4, 4)},
P9,2 = {(2, 2, 2, 1, 1, 1), (2, 2, 2, 2, 1), (3, 3, 3, 1, 1), (3, 3, 3, 2), (4, 4, 4, 1), (5, 5, 5)},
17

P10,2 = {(2, 2, 2, 2, 2), (2, 2, 2, 2, 1, 1), (2, 2, 2, 1, 1, 1, 1), (3, 3, 3, 3), (3, 3, 3, 2, 1),
(3, 3, 3, 1, 1, 1), (4, 4, 4, 2), (4, 4, 4, 1, 1), (5, 5, 5, 1), (6, 6, 6)}.
The next result shows that a squarefree Veronese subalgebras of S is not nearly
Gorenstein, unless it is already Gorenstein.
Theorem 4.14. For 1 ≤ d ≤ n let R = Rn,d . The following are equivalent:
(i) R is nearly Gorenstein;
(ii) R is Gorenstein;
(iii) d = 1 or d = n − 1 or n = 2d.

Proof. It is clear that (ii) implies (i). The equivalence between (ii) and (iii) was
showed by De Negri and Hibi in [6], and also by Bruns, Vasconcelos and Villarreal
in [5]. So, it is enough to prove that if 2 ≤ 2d < n, then tr(ωR ) ( mR . Indeed, if m
is a monomial generator for ωR as given by Theorem 4.10, and f = u/(x1 · · · xn ) is
a generator for ωR−1 as given by Theorem 4.12, then deg m ≥ n and deg u ≥ n − d.
Thus, mf is a monomial of degree at least n − d ≥ d + 1 and tr(ωR ) ( mR .

4.4. Segre products. If R and S are N-graded K-algebras, their Segre product
T = R♯S is the graded K-algebra whose components are Ti = Ri ⊗K Si for all i.
In this part we assume that R and S are standard graded Gorenstein rings.
We further need that T is Cohen–Macaulay. For example, if one assumes that
dim R, dim S ≥ 2, then T is Cohen–Macaulay if and only if a(R), a(S) < 0, see [15,
Theorem 4.4.4]. Here a(R′ ) := − min{i : (ωR′ )i 6= 0} denotes the a-invariant of a
graded Cohen–Macaulay K-algebra R′ .
Theorem 4.15. Let R and S be standard graded Gorenstein K-algebras of Krull
dimension at least 2 with a-invariants a(R) = r and a(S) = s, and assume that
T = R♯S is Cohen–Macaulay. Then the following hold:
|r−s|
(i) tr(ωT ) ⊇ mT ;
|r−s|
(ii) if T is a domain, then tr(ωT ) = mT .
Proof. (i) By [15, Theorem 4.3.1] of Goto and Watanabe one has that ωT ∼
= ωR ♯ωS .
∼
∼
Since ωR = R(r) and ωS = S(s), it follows that we may identify ωT with R(r)#S(s).
We may assume s ≥ r. Then it can be seen that ωT is generated as a T -module by
(ωT )−r = R0 ⊗K Ss−r . Each f ∈ Rs−r induces for each i a K-linear map
(ωT )i = Ri+r ⊗ Si+s → Ri+s ⊗ Si+s = Ti+s ,

a ⊗ b 7→ f a ⊗ b.

These K-linear maps compose to a T -linear map f : ωT → T . The image of this
map is generated by f ⊗ Ss−r . Thus, as f varies over all f ∈ Rs−r we see that
tr(ωT ) ⊇ Rs−r ⊗ Ss−r . This yields the desired conclusion, since mTs−r is generated by
the elements of Rs−r ⊗ Ss−r .
(ii) Since T is a domain, R is a domain, as well. We choose a non-zero element
f ∈ Rs−r . Since T is a domain, the T -module homomorphism f : ωT → T , defined
in (i) is injective. Therefore, ωT is isomorphic to its image in T , and hence due to
(i) it suffices to show that I −1 · I ⊆ mTs−r for the ideal I in T generated by f ⊗ Ss−r .
Since I is a graded ideal, its inverse I −1 is graded, as well. Let x ∈ I −1 , where
x = (g1 ⊗ g2 )/(h1 ⊗ h2 ) = (g1 /h1 ) ⊗ (g2 /h2 ) with g1 ⊗ g2 ∈ Ta and h1 ⊗ h2 ∈ Tb .
18

Then f g1 /h1 ⊗ g2 /h2 Ss−r ⊆ T . In particular, g2 /h2 ∈ J −1 where J = mSs−r . Since
dim S ≥ 2, it follows that grade J ≥ 2, so that J −1 = S. Therefore, a − b ≥ 0, which
implies that xI ∈ mTs−r .

Corollary 4.16. Assume the hypotheses of Theorem 4.15.
If |r − s| ≤ 1 then the Segre product T = R#S is nearly Gorenstein. Moreover, if
T is a nearly Gorenstein domain, then |r − s| ≤ 1.
Remark 4.17. If R⊗K S is a domain, then R♯S is also a domain being a subalgebra
of R ⊗K S. It is known from [3, Chapter 5, §17] that if the field K is algebraically
closed and R and S are domains, then so is R ⊗K S.
In the special case that R = K[x1 , . . . , xr ] and S = K[y1 , . . . , ys ] are polynomial
rings and r ≥ s ≥ 2, the anti-canonical ideal of T can be easily computed. We
remark that T may be identified with
K[xi yj : 1 ≤ i ≤ r, 1 ≤ j ≤ s] ⊂ K[x, y].
Any monomial u ∈ K[x1 , . . . , xr , y1, . . . , ys ] may be uniquely written u = ux uy , where
ux and uy are monomials in R, and respectively in S. After this identification, a Kbasis of T is given by the monomials u = ux uy with deg ux = deg uy . Furthermore,
by Theorem 4.15, since a(R) = −r and a(S) = −s, the canonical module of T is
isomorphic to the ideal
I = (x1r−s y β : |β| = r − s) ⊂ T.
Proposition 4.18. With notation introduced, I −1 is generated as a T -module by
the fractions xα /x1r−s with 0 ≤ α and |α| = r − s.
Proof. Let f = xα /x1r−s with 0 ≤ α and |α| = r − s. Then f = (xα y1r−s )/(x1 y1 )r−s is
in Q(T ) and for any generator of I, g = x1r−s y β with |β| = r − s, clearly f g = xα y β is
in T . Therefore, the given set of fractions is part of I −1 . We show that it generates
it, too.
Let u, v be coprime polynomials in K[x, y] with u/v ∈ I −1 . Then v divides x1r−s y β
for all β ≥ 0. Since s ≥ 2 we get that v divides x1r−s . As T is a monomial subalgebra
of K[x, y], it follows that for determining a generating system of I −1 it is enough to
consider fractions of the form f = u/x1r−s with u a monomial in K[x, y]. Such an
f belongs to Q(T ) if and only if deg ux = (r − s) + deg uy If that is the case, we
′
may decompose ux = xα · xα with α, α′ vectors with nonnegative integer entries and
′
|α| = r − s. Letting w = xα uy , since |α′ | = deg uy we get that w ∈ T . Therefore
f = (xα /x1r−s )w, which shows that the given set of fractions generates I −1 . This
completes the proof of the proposition.

Remark 4.19. Proposition 4.18 implies that ωT−1 is isomorphic to the ideal
J = (xα y1r−s : |α| = r − s) ⊂ T.
19

5. Nearly Gorenstein Hibi rings
As an application of the results of the previous section on Segre products we will
classify in this section the nearly Gorenstein Hibi rings.
Let P = {p1 , . . . , pn } be a finite partially ordered set (“poset” for short). A chain
of P is a totally ordered subset of P . The length of a chain C is |C| − 1. The rank
of P is the maximal length of chains of P and it is denoted by rank(P ). A poset P
is called pure if the length of each maximal chain of P is equal to rank(P ).
A poset ideal of P is a subset α ⊂ P with the property that if a ∈ α and b ∈ P
with b ≤ a, then b ∈ α. In particular, the empty set, as well as P itself, is a poset
ideal. Let J (P ) denote the set of poset ideals of P . If α and β are poset ideals of
P , then each of α ∩ β and α ∪ β is again a poset ideal of P . Hence J (P ) is a finite
lattice ([17, p. 157]) ordered by inclusion.
A finite lattice L is called distributive if, for all a, b, c belonging to L, one has
(a ∨ b) ∧ c = (a ∧ c) ∨ (b ∧ c),
(a ∧ b) ∨ c = (a ∨ c) ∧ (b ∨ c).

For example, for an arbitrary finite poset P , the finite lattice J (P ) is distributive.
Furthermore, Birkhoff’s fundamental structure theorem for finite distributive lattices ([17, Theorem 9.1.7]) guarantees the converse. More precisely, given a finite
distributive lattice L, there is a unique finite poset P with L = J (P ).
Let S = K[x1 , . . . , xn , s] denote the polynomial ring in n + 1 variables over a field
K. Given a poset ideal α ∈ J (P ), we introduce the squarefree monomial
!
Y
uα =
xi s
pi ∈α

which belongs to S. In particular, u∅ = s and uP = x1 · · · xn s. In [19] the toric ring
RK [L] = K[{ uα : α ∈ J (P ) }] ⊂ S

is introduced and it is nowadays called the Hibi ring of L defined over K. The
toric ring RK [L] is normal and Cohen–Macaulay of dimension |P | + 1, see [19]. It
follows that the quotient field of RK [L] is the rational function field K(x1 , . . . , xn , s).
Furthermore, the a-invariant a(RK [L]) of RK [L] coincides with −(rank(P ) + 2).
Let Pb = P ∪ {−∞, +∞} with −∞ < a < +∞ for each a ∈ P . Write Ω(P ) for the
set of those order-reversing maps δ : Pb → Z≥0 with δ(+∞) = 0 and Ω∗ (P ) for the
set of those strictly order-reversing maps δ : Pb → Z≥0 with δ(+∞) = 0. Let M(P )
denote the set of those maps γ : Pb → Z with γ(+∞) = 0 for which δ + γ ∈ Ω(P )
for each δ ∈ Ω∗ (P ).
Given a map ξ : Pb → Z≥0 with ξ(+∞) = 0, we introduce the monomial
!
n
Y
ξ(p )
uξ =
xi i sξ(−∞)
i=1

which belongs to S. It is shown in [19] that RK [L] with L = J (P ) is spanned
by those monomials uδ with δ ∈ Ω(P ). Let ωRK [L] denote the ideal of RK [L]
20

which is generated by those monomials uδ with δ ∈ Ω∗ (P ). Then the ideal ωRK [L] is
isomorphic to the canonical ideal of RK [L]. It then follows that RK [L] is Gorenstein
if and only if P is pure.
We now turn to the problem of finding a characterization on P for which the toric
ring RK [L] with L = J (P ) is nearly Gorenstein. Given a ∈ P , we introduce the
intervals [−∞, a] = { b ∈ Pb : −∞ ≤ b ≤ a } and [a, +∞] = { b ∈ Pb : a ≤ b ≤ +∞ }
of Pb.
Lemma 5.1. Let P be a finite poset and suppose that the toric ring RK [L] of the
distributive lattice L = J (P ) is nearly Gorenstein. Then, for each element a ∈ P ,
the intervals [−∞, a] and [a, +∞] of Pb are pure.

Proof. Let a ∈ P for which [−∞, a] is nonpure and define ρ ∈ Ω(P ) by setting ρ(z) =
1 if z ∈ [−∞, a] and ρ(z) = 0 if z 6∈ [−∞, a]. Since RK [L] is nearly Gorenstein,
there exist δ ∈ Ω∗ (P ) and γ ∈ M(P ) with ρ = δ + γ. Since [−∞, a] is nonpure,
there exist y and y ′ belonging to [−∞, a] for which y ′ covers y with δ(y) > δ(y ′ ) + 1.
Since y ′ covers y, it follows from [20, Corollary (2.8)] that there is δ ′ ∈ Ω∗ (P ) with
δ ′ (y) = δ ′ (y ′ ) + 1. Furthermore, since γ ∈ M(P ), one has δ ′ + γ ∈ Ω(P ). Now,
since (δ + γ)(y) = (δ + γ)(y ′) = 1, it follows that (δ ′ + γ)(y) < (δ ′ + γ)(y ′), which
contradicts δ ′ + γ ∈ Ω(P ).
Let a ∈ P for which [a, +∞] is nonpure and define ρ ∈ Ω(P ) by setting ρ(z) = 1
if z < a and ρ(z) = 0 if z 6< a. Again, since RK [L] is nearly Gorenstein, there
exist δ ∈ Ω∗ (P ) and γ ∈ M(P ) with ρ = δ + γ. Since [a, +∞] is nonpure, there
exist y and y ′ belonging to [a, +∞] for which y ′ covers y with δ(y) > δ(y ′) + 1.
Since y ′ covers y, again by using [20, Corollary (2.8)], one has δ ′ ∈ Ω∗ (P ) with
δ ′ (y) = δ ′ (y ′ ) + 1. Furthermore, since γ ∈ M(P ), one has δ ′ + γ ∈ Ω(P ). Now,
since (δ + γ)(y) = (δ + γ)(y ′) = 0, it follows that (δ ′ + γ)(y) < (δ ′ + γ)(y ′), which
contradicts δ ′ + γ ∈ Ω(P ).

Lemma 5.2. Let P be a finite connected poset and suppose that, for each element
a ∈ P , the intervals [−∞, a] and [a, +∞] of Pb are pure. Then P is pure.

Proof. Let a1 , . . . , ar denote the maximal elements of P . Then (−∞, a1 ], . . . , (−∞, ar ]
are pure, where (−∞, ai ] = { b ∈ P : b ≤ ai }. If (−∞, ai ] ∩ (−∞, aj ] 6= ∅, then
rank((−∞, ai ]) = rank((−∞, aj ]). In fact, if a ∈ (−∞, ai ] ∩ (−∞, aj ], then the fact
that [−∞, a] and [a, +∞] are pure guarantees rank((−∞, ai ]) = rank((−∞, aj ]).
Since P is connected, after rearranging a1 , . . . , as , it follows that (∪ji=1 (−∞, ai ]) ∩
(−∞, aj+1 ] 6= ∅ for 1 ≤ j < r. Hence rank((−∞, a1 ]) = · · · = rank((−∞, ar ]). Since
every maximal chain of P belongs to one of (−∞, a1 ], . . . , (−∞, ar ], it follows that
P is pure, as desired.

Combined, Lemmata 5.1 and 5.2 guarantee the following.
Corollary 5.3. Let P be a finite poset and suppose that the toric ring RK [L] of the
distributive lattice L = J (P ) is nearly Gorenstein. Then every connected components of P is pure.
21

In general, if a finite P is the disjoint union of finite posets P1 , . . . , Pq , then RK [L]
is the Segre product RK [L1 ]♯RK [L2 ]♯ · · · ♯RK [Lq ], where L = J (P ) and Li = J (Pi )
for 1 ≤ i ≤ q.
Theorem 5.4. Let P be a finite poset. Then the toric ring RK [L] of the distributive
lattice L = J (P ) is nearly Gorenstein if and only if P is the disjoint union of pure
connected posets P1 , . . . , Pq such that | rank(Pi ) − rank(Pj )| ≤ 1 for 1 ≤ i < j ≤ q.
Proof. Let P be the disjoint union of connected posets P1 , . . . , Pq and Li = J (Pi )
for 1 ≤ i ≤ q. Then RK [L] = RK [L1 ]♯ · · · ♯RK [Lq ]. Recall that RK [Li ] is Gorenstein
if and only if Pi is pure. Furthermore, a(RK [Li ]) = −(rank(Pi ) + 2). Hence the “if”
part follows from Theorem 4.15. Now, Corollary 5.3 says that if RK [L] is nearly
Gorenstein, then each of P1 , . . . , Pq is pure. Thus the “only if” part follows again
from Theorem 4.15.

6. The one-dimensional case
Let (R, m, K) be a 1-dimensional local Cohen–Macaulay ring with canonical module ωR . The results and proofs of this section are equally valid for 1-dimensional
positively graded K-algebras.
Barucci and Fröberg [2] introduced almost Gorenstein rings for one-dimensional
local rings which are analytically unramified. We use here the description of almost
Gorenstein rings due to Goto, Takahashi and Taniguchi [14] which allows an extension of this concept to higher dimensions. Goto et al. [14] call R almost Gorenstein
if there exists an exact sequence
(8)

0 → R → ωR → C → 0

with mC = 0.
We have the following implication.

Proposition 6.1. Let R be almost Gorenstein. Then R is nearly Gorenstein.
Proof. The exact sequence (8) yields the exact sequence
0 → HomR (ωR , R) → R → Ext1R (C, R)

Let f ∈ ωR be the image of 1 ∈ R under the map R → ωR . Then the map
HomR (ωR , R) → R is defined by assigning to each ϕ ∈ HomR (ωR , R) the element
ϕ(f ) ∈ R. Thus tr(ωR ) contains the image of HomR (ωR , R) in R. Since m annihilates
Ext1R (C, R), this image is equal to m, and the desired result follows.

Remark 6.2. (a) In [14] almost Gorenstein rings are defined also for higher dimensional local rings. The definition is similar to that in the 1-dimensional case. The
only difference is that the module C in the exact sequence 0 → R → ωR → C → 0
should be an Ulrich module. In the case that R is 1-dimensional, the Krull dimension of C is equal to 0. Since an Ulrich module of dimension zero is annihilated by
the maximal ideal, one recovers the definition of 1-dimensional almost Gorenstein
rings.
22

In contrast to the 1-dimensional case, it does not follow in higher dimensions that
an almost Gorenstein ring is nearly Gorenstein. Indeed, if R is almost Gorenstein,
then the formal power series ring R[|x|] is again almost Gorenstein, see [14, Theorem
3.7]. On the other hand by Proposition 4.5, if R is not Gorenstein, then R[|x|] is
not nearly Gorenstein.
(b) In general, the class of 1-dimensional nearly Gorenstein rings is much larger
than that of 1-dimensional almost Gorenstein rings. Examples of 1-dimensional
non-Gorenstein rings rings which are nearly Gorenstein, but not almost Gorenstein
will be discussed in the next section. A simple example of this kind is the subring
R = K[|t5 , t6 , t7 |] of the formal power series ring K[|t|]. That R is indeed nearly
Gorenstein can be seen from the next result.
Proposition 6.3. Let (S, n) be a 3-dimensional regular local ring, I ⊂ S an ideal
generated by 3 elements such that R = S/I is a 1-dimensional Cohen-Macaulay
domain. Let A be the relation matrix of I. Then R is nearly Gorenstein if and only
if I1 (A) = n.
Proof. The assumptions imply that I has the resolution
(9)

0 → S 2 → S 3 → I → 0.

In particular, the type of R is 2. Now we apply Corollary 3.5.



Coming back to the example in Remark 6.2(b) we write R = K[|t5 , t6 , t7 |] as
a factor ring of S = K[|x1 , x2 , x3 |] by considering the K-algebra homomorphism
ε : S → R with ε(x1 ) = t5 , ε(x2 ) = t6 , ε(x3 ) = t6 . The kernel I of ε is generated by
x41 − x2 x23 , x22 − x1 x3 , x33 − x31 x2 .

The relation matrix A of this ideal is


x1 x2 x23
,
A=
x2 x3 x31

Thus I1 (A) = (x1 , x2 , x3 ), and hence R is nearly Gorenstein.

Proposition 6.4. Under the assumptions of Proposition 6.3, ωR is an ideal in R
and µ(ωR−1 ) = 3.
Proof. In our situation R is a domain of type 2, and we apply Corollary 3.4.



e n) is an extension of local rings, where R
e is a finite
Assume now that (R, m) ⊂ (R,
e ⊆ Q(R). We also assume that
R-module and a discrete valuation ring such that R
e induces an isomorphism R/m → R/n.
e
the inclusion map R → R
The set
e ⊆ R}
C e = {x ∈ R : xR
R/R

e With
is called the conductor of this extension and it is an ideal of both R and R.
the notation introduced we have

Proposition 6.5. For any ideal I ⊂ R one has CR/R
⊆ tr(I). In particular, CR/R
⊆
e
e
tr(ωR ).
23

e = (f )R.
e Now let g ∈ C e . Then (g/f )I ⊆
Proof. There exists f ∈ I such that I R
R/R
−1
e
e
(g/f )I R = g R ⊆ R. Thus g/f ∈ I . Since g = (g/f )f with f ∈ I, it follows that
g ∈ I −1 · I = tr(I).

Corollary 6.6. R is nearly Gorenstein, if CR/R
= m.
e

e
For the rest of this section, the ideal quotients are computed in Q(R) = Q(R).
The following lemma will be used in the proof of Proposition 6.8.
e = R : Ce
Lemma 6.7. R
R/R

e ⊆ R : Ce .
Proof. It is clear from the definition of CR/R
that R
e
R/R
e
e
For the reverse inclusion, let f ∈ R : CR/R
/ R, then since Q(R) = Q(R)
e . If f ∈
e a positive integer and t a generator
we may write f = εt−a with ε invertible in R,
e
e there exists b > 0 such that
of the maximal ideal of R. Since C e is an ideal in R,
R/R

e
CR/R
= (t )R.
e
The property f ∈ R : CR/R
now reads as εt−a CR/R
⊆ R, or equivalently, that
e
e
a
CR/R
⊆ t R. This implies that b ≥ a. Since we may write tb−1 = f · g where
e
b−1
g = ε−1 ta+b−1 is clearly in CR/R
∈ R.
e , it follows that t
b−1 e
We claim that t R ⊆ R. This will then lead to a contradiction, since tb−1 6∈ C e .
b

R/R

e is a unit,
It is clear that t n = CR/R
⊆ R. Thus is suffices to show that if ν ∈ R
e
e is an
then νtb−1 ∈ R. To prove this, we use our assumption that R/m → R/n
e
isomorphism. In order to simplify notation we may assume that R/m = R/n.
Then
this implies that there exists h ∈ R such that ν − h ∈ n. Thus, ν = h + h1 with
h ∈ R and h1 ∈ n, and therefore νtb−1 = htb−1 + h1 tb−1 . Since both summands on
the right hand side of this equation belong to R, the claim follows.
e
This concludes the proof of the inclusion R : CR/R
⊆ R.

e
b−1

Our next result gives several characterizations of the situation when tr(I) = CR/R
e ,
−1
in terms of the fractionary ideal I . We first recall that for any fractionary ideal J
of R, the ideal quotient J : J may be identified with the endomorphism ring End(J)
of J, see [25, Lemma 3.14].
Proposition 6.8. Let I ⊂ R be an ideal. The following conditions are equivalent:
(i) tr(I) = CR/R
e .
e
(ii) End(I −1 ) = R.
e
(iii) I −1 ∼
= R.

Proof. (i) ⇒ (ii): If tr(I) = CR/R
e , then using Lemma 6.7 and the remark following
it, we get
e = R : C e = R : (I · I −1 ) = (R : I) : I −1 = End(I −1 ).
R
R/R

(ii) ⇒ (i): Suppose that tr(I) 6= CR/R
is properly contained in tr(I).
e . Then CR/R
e
e Let t be generator of the
It follows that CR/R
is properly contained in tr(I)R.
e
24

e Then there exist integers a < b such that tr(I)R
e = ta R
e and
maximal ideal of R.
be
b
e
e
CR/R
= t R. Clearly, t ∈ CR/R
e
e R = CR/R
e . Since tr(I)R is a principal ideal, there
e = (f )R.
e Therefore, there exists u invertible in R
e
exists f ∈ tr(I) such that tr(I)R
a
such that t = u · f .
e = End(I −1 ) (which is R : tr(I)), we obtain that ta ∈ R.
Since by assumption R
e and ta ∈ tr(I). Using again that
We may write tb−1 = tb−a−1 · ta , where tb−a−1 ∈ R
e = R : tr(I), it follows that tb−1 ∈ R. Arguing as in the proof of Lemma 6.7 we
R
obtain tb−1 ∈ CR/R
e , which is a contradiction.
e then I −1 R
e = I −1 . Since any nonzero R-ideal
e
(ii) ⇒ (iii): If End(I −1 ) = R,
is
e
isomorphic to R, the assertion follows.
(iii) ⇒ (ii) is obvious.

For an R-module M we let e(M) denote its multiplicity.

Corollary 6.9. Let R be as in Proposition 6.8, and assume in addition that R is
of the form R = S/J with (S, n) regular local ring of dimenson 3 and J ⊆ n2 .
If tr(ωR ) = CR/R
then e(R) = µ(J). In particular, if R is an almost complete
e
intersection, then R has minimal multiplicity.
Proof. Since R is a 1-dimensional domain, it it Cohen-Macaulay and proj dimS (R) =
1. Thus J has a minimal presentation 0 → S g−1 → S g → J, where g = µ(J).
e are
Now Proposition 6.8(iii) together with tr(ωR ) = CR/R
imply that ωR−1 and R
e
isomorphic R-modules, and they must have the same number of minimal generators
e R)
e = g, since µ(ω −1) = g, by Corollary 3.4.
over R. Hence dimR/m (R/m
R
e is a discrete valuation ring, there exists f ∈ m such that mR
e = f R.
e Since
As R
e is a finitely generated R-module of rank 1, it follows that e(R) = e(R),
e see [4,
R
e = dimR/m mk R/m
e k+1 R
e for k ≫ 0. Since mk R/m
e k+1 R
e=
Corollary 4.7.9]. Now e(R)
e R,
e we conclude from the above considerations that
e R
e ∼
e k+1R
e ∼
f k R/f
= R/m
= R/f
e(R) = g, as desired.
If R is an almost complete intersection, then µ(J) = height(J) + 1 = dim S, hence
e(R) = emb dim R.

7. The trace of the canonical ideal of a numerical semigroup ring
Let H be a numerical semigroup minimally generated by n1 < n2 < . . . < ne with
e > 1. We write H = hn1 , . . . , ne i. The number e is called the embedding dimension
of H and the number n1 the multiplicity of H. One always has n1 ≤ e. We say that
H has minimal multiplicity if n1 = e. In this case, one also says that H has maximal
embedding dimension, cf. [33].
The elements in the set G(H) = N \ H are called the gaps of H. Unless otherwise
stated we will assume that gcd(n1 , . . . , ne ) = 1. Then |G(H)| < ∞, and there exists
a largest integer F(H), called the Frobenius number of H, such that F(H) 6∈ H.
We denote by M the semigroup ideal H \ {0}. The elements f ∈ G(H) with
f + M ∈ H are called pseudo-Frobenius numbers. The set of pseudo-Frobenius
25

numbers will be denoted by PF(H). The cardinality of PF(H) is called the type of
H, denoted type(H).
We fix a field K. The positively graded K-subalgebra K[H] = K[tn1 , . . . , tne ] of
K[t] is the semigroup ring of H. Its graded maximal ideal is m = (tn1 , . . . , tne ).
The embedding dimension (resp. multiplicity) of H is also the embedding dimension
(resp. multiplicity) of K[H] in the algebraic sense. The polynomial ring K[t] is a
finite module over K[H] and is the integral closure of K[H] in its quotient field.
K[t]/K[H] is a module of finite length with K-basis {ta : a ∈ G(H)}.
We may write K[H] ∼
= S/IH , where S = K[x1 , . . . , xe ] is the polynomial ring
and where IH is the kernel of the K-algebra homomorphism ε : S → K[H] with
xi 7→ tni for i = 1, . . . , e. If we assign to each xi the degree ni , then ε becomes a
homomorphism of graded K-algebras and IH a graded ideal.
The canonical module ωK[H] of K[H] is the fractionary K[H]-ideal generated by
the elements t−f with f ∈ PF(H), see [10, Exercise 21.11]. Therefore, the CohenMacaulay type of K[H] is equal to type(H). In particular, K[H] is Gorenstein if
and only if PF(H) = {F(H)}. Kunz [24] showed that K[H] is Gorenstein if and
only if H is symmetric, i.e. for all x ∈ Z either x ∈ H, or F(H) − x ∈ H.
Let PF(H) = {f1 , . . . , fτ −1 , F(H)}, with fi < fi+1 for 1 ≤ i < τ − 2. It is known
by Nari [27] that K[H] is almost Gorenstein, if and only if
(10)

fi + fτ −i = F(H) for i = 1, . . . , ⌊τ /2⌋.

The semigroup H is called almost symmetric if K[H] is almost Gorenstein, and H
is called nearly Gorenstein, if K[H] is nearly Gorenstein.
A subset I ⊂ Z is called a relative ideal of H if I + H ⊆ I and h + I ⊆ H for
some h ∈ H. If moreover I ⊆ H, then I is called an ideal of H.
Let ΩH and Ω−1
H be the set of exponents of the monomials in ωK[H] , and in
−1
ωK[H] respectively. Then ΩH and Ω−1
H are relative ideals of H called the canonical,
respectively the anti-canonical ideal of H. We define the trace of H as tr(H) =
ΩH + Ω−1
H . It is clear that tr(H) is an ideal in H consisting of the exponents of the
monomials in tr(K[H]).
In this notation, H is nearly Gorenstein if and only if M ⊆ tr(H).
The following result shows that a numerical semigroup generated by an arithmetic
sequence is nearly Gorenstein. We also characterize when such semigroups are almost symmetric, taking into account that the symmetric case was known from work
of Gimenez, Sengupta and Srinivasan in [13].
Proposition 7.1. Let e > 2, and H = ha, a + d, . . . , a + (e − 1)di with a, d coprime
nonnegative integers and e ≤ a. Then
(a) H is nearly Gorenstein;
(b) H is symmetric if and only if a ≡ 2 mod(e − 1);
(c) H is almost symmetric if and only if a = e or a ≡ 2 mod(e − 1).

Proof. It is known from [13, Theorem 4.7] that τ = type(H) is the unique integer
1 ≤ τ ≤ e − 1 such that a = k(e − 1) + τ + 1 with k integer. Equivalently, k = ⌊ a−2
⌋.
e−1
26

Tripathi [36, Theorem on page 3] shows that
 


x−1
+ dx : a − τ ≤ x ≤ a − 1 .
PF(H) = a
e−1

For a − τ ≤ x ≤ a − 1 we get k(e − 1) ≤ x − 1 ≤ k(e − 1) + (τ − 1), hence ⌊ x−1
⌋ = k.
e−1
This implies that F(H) = ak + d(a − 1) and
(11)

PF(H) = {F(H) − (τ − 1)d, . . . , F(H) − d, F(H)},

hence the canonical ideal ΩH is generated by

W = {− F(H), − F(H) + d, . . . , − F(H) + (τ − 1)d}.

For part (a) we consider the set

W ′ = {F(H) + a, F(H) + a + d, . . . , F(H) + a + (e − τ )d} ⊂ H.

An element in W+W ′ is of the form a+(i+j)d with 0 ≤ i ≤ τ −1 and 1 ≤ j ≤ e−τ .
This way we obtain the generators of H: a, a + d, . . . , a + (e − 1)d, which shows that
−1
W ′ ⊂ Ω−1
H and ΩH + ΩH ⊇ M. Equivalently, H is nearly Gorenstein.
Part (b) is known and may be traced back to [13, Theorem 2.2] or (less explicitly
in) [30]. The statement is an immediate consequence of the fact that K[H] is
Gorenstein if and only if τ = 1.
For part (c), using (b), it is enough to treat the case of H being almost symmetric,
but not symmetric. This is equivalent (using (10) and (11)) to
(F(H) − (τ − 1)d) + (F(H) − d) = F(H), which is equivalent to
F(H) = τ d.
After we substitute the values of F(H) and τ in the previous equation, we get
ak + d(a − 1) = (a − 1 − k(e − 1))d,
k(a + d(e − 1)) = 0,
k = 0.
Note that e ≤ a and by the way k was defined, we may express k =
Therefore k = 0 if and only if a = e.


 a−2 
.
e−1

In the following we present a class of numerical semigroups which are nearly
Gorenstein if and only if they are almost symmetric. Unless otherwise stated, we
assume that H is minimally generated by n1 < · · · < ne .

Lemma 7.2. Let H = hn1 , . . . , ne i be a numerical semigroup with minimal multiplicity. Then PF(H) = {n2 − n1 , . . . , ne − n1 }. In particular, F(H) = ne − n1 .
Proof. The Apéry set of H with respect to an element a ∈ H is the set
Ap(a, H) = {h ∈ H : h − a 6∈ H}.

The number of elements of Ap(H, a) is equal to a. Thus, Ap(n1 , H) = {0, n2 , . . . , ne }
since H has minimal multiplicity, which means that n1 = e. For any f ∈ PF(H)
one has f + n1 ∈ Ap(n1 , H), and conversely, if h ∈ Ap(n1 , H), then h − n1 ∈ PF(H)
if and only h + ni 6∈ Ap(n1 , H) for all i. The elements in Ap(n1 , H) having this
property are exactly the elements n2 , . . . , ne . The desired result follows.

27

Corollary 7.3. Let H = hn1 , . . . , ne i be a numerical semigroup with minimal multiplicity. Then H is almost symmetric if and only if
for i = 2, . . . , ⌊e/2⌋.

ni + ne−i+1 = ne + n1

Proof. The result is an immediate consequence of Lemma 7.2 and of Nari’s characterization (10) of almost symmetric numerical semigroups.

Theorem 7.4. Let H be a numerical semigroup with minimal multiplicity. Then
the following conditions are equivalent:
(i) H is almost symmetric.
(ii) H is nearly Gorenstein.
Proof. Since (i) ⇒ (ii) is always valid, as shown in Proposition 6.1, it suffices to show
that (ii) ⇒ (i). It follows from Lemma 7.2 that ΩH is generated as a relative ideal
of H by −ne + n1 , . . . , −n2 + n1 . Since H is nearly Gorenstein, we get n1 ∈ tr(H),
and hence there exists a ∈ H such that
(α) −ni + n1 + a ∈ H for i = 2, . . . , e;
(β) −nj + n1 + a = n1 for some j.
Condition (β) implies that a = nj , and then (α) implies that −ni +n1 +nj ∈ H for
i = 2, . . . , e. Suppose that j < e. Then −ne +n1 +nj < n1 . Since −ne +n1 +nj ∈ H,
we must have −ne + n1 + nj = 0. But then ne = n1 + nj , a contradiction. Thus we
see that j = e, and hence (α) implies that
n1 + ne = ni + hi

for i = 2, . . . , e

with ne > h2 > · · · > he = n1 > 0 and hi ∈ H. Since hi − n1 = ne − ni 6∈ H for
i < e, we see that hi ∈ Ap(n1 , H) = {0, n2 , n3 , . . . , ne } for 2 ≤ i < e. This implies
that hi = ne−i+1 for i = 2, . . . , e, and hence
n1 + ne = ni + ne−i+1

for i = 2, . . . , ⌊e/2⌋.

It follows from Corollary 7.3 that H is almost symmetric, as desired.



The semigroup ring K[H] is 1-dimensional, so its canonical trace ideal is either
the whole ring, or it is an m-primary ideal. Equivalently, K[H]/ tr(ωK[H]) is a finite
dimensional vector space with a K-basis given by {th : h ∈ H \ tr(H)}. We define
the residue of H as the residue of K[H], namely
(12)

res(H) = dimK K[H]/ tr(ωK[H] ) = |H \ tr(H)|.

Thus res(H) = 0 means that H is symmetric, and res(H) ≤ 1 if and only if H is
nearly Gorenstein.
The conductor of the extension K[H] ⊆ K[t] is the ideal
CK[t]/K[H] = (th : h > F(H))K[H],

which explains why the quantity c(H) := F(H) + 1 is named the conductor of H.
We denote CH = {h : th ∈ CK[t]/K[H]}, which is an ideal in H minimally generated
by c(H), c(H) + 1, . . . , c(H) + n1 − 1. Proposition 6.5 applied to K[H] gives the
following result.
28

Proposition 7.5. For any numerical semigroup H one has
CH ⊆ tr(H) ⊆ H.
If H is not symmetric then CH ⊆ tr(H) ⊆ M.
As a corollary we obtain an upper bound for res(H).
We define the set of non-gaps of H to be NG(H) = {x ∈ H : x < F(H)} and we
denote n(H) = |NG(H)|.
Corollary 7.6. For any numerical semigroup H one has res(H) ≤ n(H), with
equality if and only if tr(H) = CH .
Proof. The desired inequality follows from the observation that
n(H) = |NG(H)| = |H \ CH | ≥ |H \ tr(H)| = res(H). 
The map ρ : NG(H) → G(H) given by ρ(x) = F(H) − x for all x in NG(H) is
well defined and injective. Also, |NG(H)| + |G(H)| = F(H) + 1, hence denoting
g(H) = |G(H)| we have n(H) ≤ g(H).
Numerical experiments with GAP ([12]) indicate that the bound in Corollary 7.6
might be improved. We formulate the following question.
Question 7.7. Given a numerical semigroup H, is it true that
res(H) ≤ g(H) − n(H)?
This question has a positive answer for symmetric semigroups: by [11, Lemma
1(f)] H is symmetric if and only if n(H) = g(H). In Proposition 7.10 we also confirm
Question 7.7, when H is 3-generated.
Next, we present a family of numerical semigroups H such that CH = tr(H).
Example 7.8. For the integers m > 1 and q > 0 we let
H = hm, qm + 1, qm + 2, . . . , qm + m − 1i.
This is a semigroup with minimal multiplicity, hence by Lemma 7.2 we get that
PF(H) = {(q −1)m+1, (q −1)m+2, . . . , qm−1}, a list of m−1 consecutive integers.
Let x ∈ Ω−1
H , i.e. − PF(H)+x ⊂ H. This can happen only if x−F(H) = x−qm+1 ≥
qm, equivalently x ≥ 2qm − 1. Consequently, tr(H) = {x : x ≥ qm} = CH , and
res(H) = |{0, m, . . . , (q − 1)m}| = q.
When H is 3-generated, the results in Section 3 can be applied to obtain a simple
formula of res(H).
Assume H is minimally generated by n1 , n2 , n3 , not necessarily listed increasingly.
Let ϕ : S = K[x1 , x2 , x3 ] → K[H] the algebra map given by ϕ(xi ) = tni for i =
1, . . . , 3. Then ker(ϕ) = IH , the defining ideal of K[H].
It is proven in [16] that H is symmetric, equivalently K[H] is a complete intersection, if and only if, up to a permutation, d = gcd(n1 , n2 ) > 1 and n3 ∈ hn1 /d, n2 /di.
29

Assume H is not symmetric. We recall from [16] how to compute the ideal IH
in this case. We find the positive integers c1 , c2 , c3 minimal with the property that
there exist nonnegative integers ai , bi , i = 1, . . . , 3 such that
(13)

c1 n1 = b2 n2 + a3 n3 ,
c2 n2 = a1 n1 + b3 n3 ,
c3 n3 = b1 n1 + a2 n2 .

Such ai , bi are positive, unique, and ci = ai + bi for i = 1, . . . , 3. In this notation,
the ideal IH is the ideal of maximal minors of the matrix
 a1 a2 a3 
x1 x2 x3
(14)
A=
,
xb22 xb33 xb11

that we call the structure matrix of the semigroup H.
It is noticed in [29, page 69] that one can recover n1 , n2 , n3 from the matrix A by
computing the K-vector space dimension for the isomorphic rings
K[H]/(tn1 ) ∼
= K[x2 , x3 ]/(xa2 +b2 , xb2 xa3 , xa3 +b3 ),
= S/(x1 , IH ) ∼
2

2

3

3

and the other two cases, see [33, Lemma 10.23] for a different approach. Namely,
we get
(15)

n1 = a2 a3 + b2 a3 + b2 b3 ,
n2 = a1 a3 + a1 b3 + b1 b3 ,
n3 = a1 a2 + b1 a2 + b1 b2 .

It follows from the Hilbert-Burch theorem ([4, Theorem 1.4.17]) that the transpose
AT is the relation matrix of IH , i.e. the sequence
AT

0 → S 2 −→ S 3 → IH → 0
is exact. The type of R = K[H] is 2, hence by Corollary 3.4 we get tr(ωR ) =
I1 (A¯T ) = (tni ai , tni bi : i = 1, . . . , 3), where A¯T is obtained by applying ϕ on the
entries of AT . We may formulate the following result.
Proposition 7.9. Assume H is a non-symmetric 3-generated numerical semigroup
and let R = K[H]. With notation as in (13), we set di = min{ai , bi } for 1 ≤ i ≤ 3.
Then
tr(ωR ) = (td1 n1 , td2 n2 , td3 n3 )R, and res(H) = d1 d2 d3 .
Proof. The first part is clear from the discussion above. Since
R/ tr(ωR ) ∼
= S/(IH , xd1 , xd2 , xd3 ) ∼
= S/(xd1 , xd2 , xd3 )
1

2

3

we obtain that res(H) = dimK R/ tr(ωR ) = d1 d2 d3 .

1

2

3



We may now give a positive answer to Question 7.7, in embedding dimension 3.
Proposition 7.10. For any 3-generated numerical semigroup H one has
res(H) ≤ g(H) − n(H).
30

Proof. If H is symmetric we actually have equality 0 = res(H) = g(H) − n(H), as
noted in [11, Lemma 1(f)]. Assume H is not symmetric and that it has a structure
matrix A denoted as in (14). Nari et al. prove in [29, Theorem 3.2] that
2g(H) − (F(H) + 1) ∈ {a1 b1 c1 , a2 b2 c2 }.
Using Proposition 7.9, we obtain
res(H) ≤ min{a1 b1 c1 , a2 b2 c2 } ≤ 2g(H) − (F(H) + 1) = g(H) − n(H). 
We will characterize the 3-generated numerical semigroups such that their trace
is at either end of the interval [CH , M].
Theorem 7.11. Let H be a 3-generated numerical semigroup. Then tr(H) = M if
and only if one of the following cases occurs:
(i) H = hab + b + 1, b + c + 1, ac + a + ci where a, b, c are positive integers with
gcd(b + c − 1, ab − c) = 1, or
(ii) H = hbc + b + 1, ca + c + 1, ab + a + 1i, where a, b, c are positive integers with
gcd(bc + b + 1, ca + c + 1) = 1.
In case (i), F(H) = abc+bc−b−1+max{0, ab−c}, and in case (ii), F(H) = 2abc−2.
Proof. Assume H = hn1 , n2 , n3 i such that tr(H) = M. By Corollary 3.5, that is
equivalent to I1 (A) = (x1 , x2 , x3 ), where A is the matrix attached to H as in (14).
Clearly, H is not symmetric, hence up to a permutation of the variables, there are
essentially two (overlapping) cases to consider.
Case 1:


x1 xa2 xb3
A=
, with a, b, c > 0.
x2 x3 xc1
Using (15) we get n1 = ab+b+1, n2 = b+c+1, n3 = ac+a+c, as desired. It is easy
to check that gcd(n1 , n2 ) = gcd(n2 , n3 ) = gcd(n1 , n3 ), hence 1 = gcd(n1 , n2 , n3 ) =
gcd(n2 , n1 − n2 ) = gcd(b + c − 1, ab − c).
Conversely, let n1 = ab + b + 1, n2 = b + c + 1, n3 = ac + a + c for some positive
integers a, b, c such that gcd(b + c − 1, ab − c) = 1. Arguing as above we see that
the generators of H are pairwise coprime, hence H is a numerical semigroup which
is not symmetric. It is easy to verify the following equations:
(16)

(1 + c)n1 = n2 + bn3 ,
(1 + a)n2 = n1 + n3 ,
(1 + b)n3 = cn1 + an2 .

We claim that these are the minimal relations (13) among n1 , n2 , n3 .
Since a1 , b3 in (13) are positive, unique and (1 + a)n2 = n1 + n3 , we may identify
c2 = 1 + a and a1 = b3 = 1.
After substituting n1 = (1 + a)n2 − n3 into c1 n1 = b2 n2 + a3 n3 , we get c1 ((1 +
a)n2 − n3 ) = b2 n2 + a3 n3 , hence
(c1 (1 + a) − b2 )n2 = (c1 + a3 )n3 .
31

Since n2 and n3 are coprime, there exists a positive integer ℓ so that c1 + a3 = ℓn2 .
Thus, c1 + a3 ≥ b + c + 1.
On the other hand, comparing (16) and (13) we obtain that c1 ≤ 1 + c and
a3 = c3 − b3 ≤ (b + 1) − 1 = b, hence c1 + a3 ≤ 1 + c + b. This implies that
c1 + a3 = b + c + 1, and moreover c1 = c + 1 and a3 = b. We can now identify the rest
of the coefficients in (13): c1 = 1 + b, b1 = c, a2 = a, which shows that the matrix A
has the desired entries.
Case 2:


x1 x2 x3
A=
, with a, b, c > 0.
xb2 xc3 xa1
Using (15) we get n1 = bc + b + 1, n2 = ca + c + 1, n3 = ab + a + 1. It is easy to see
that gcd(n1 , n2 ) = gcd(n2 , n3 ) = gcd(n1 , n3 ), hence the desired description for H.
Conversely, let a, b, c be positive integers with gcd(bc+b+1, ca+c+1) = 1. It now
follows from [32, Theorem 14] (and its proof) that H = hbc+b+1, ca+c+1, ab+a+1i
is a pseudo-symmetric numerical semigroup whose matrix A is the one we started
with this case.
It is shown in [31, Theorem 2.2.3] that for any non-symmetric numerical semigroup
H = hn1 , n2 , n3 i one has
F(H) = max{c1 n1 + b3 n3 , c2 n2 + a3 n3 },
where c1 , c2 , a3 , b3 are as in (13). It is now an easy exercise to derive the announced
formulas for F(H), when H belongs to either one of the two families considered
above.

Remark 7.12. As noticed by Nari, Numata and Keiichi Watanabe in [29, Corollary 3.3] (see also [28, Corollary 2.9]), the format of the matrix A in case (ii) of
Theorem 7.11 corresponds to H being pseudo-symmetric, which is equivalent in
embedding dimension 3 to H being almost symmetric and not symmetric, see [28,
Proposition 2.3]. The complete parametrization of 3-generated pseudo-symmetric
numerical semigroups was obtained by Rosales and Garcı́a-Sánchez in [32].
Proposition 7.13. Assume H is a non-symmetric 3-generated numerical semigroup. Then tr(H) = CH if and only if H = h3, 3a + 1, 3a + 2i for some positive
integer a.
Proof. Assume tr(H) = CH .
It follows from Proposition 7.9 that µ(tr(H)) = 3, hence H has multiplicity 3. (We
can get the same thing by applying directly Corollary 6.9.) Listing its generators
increasingly we have that either H = h3, 3a + 1, 3b + 2i with 0 < a ≤ b, or H =
h3, 3b + 2, 3a + 1i with a > b > 0.
Assume a ≤ b. Then 3b + 2 ∈
/ h3, 3a + 1i, hence 3b + 2 ≤ F(h3, 3a + 1i) = 6a − 1,
by [31]. Thus b < 2a. It is easy to check that the structure matrix (14) is

 2a−b
x1
x2
x3
.
A=
x2
x3 x2b−a+1
1
32

Hence res(H) = 2a−b by Proposition 7.9. Note that 0, 3, 6, . . . , 3(a−1) are not in CH ,
hence 2a − b = res(H) = |H \ CH | ≥ a. This gives a = b and H = h3, 3a + 1, 3a + 2i.
If a > b then arguing as in the previous case we obtain 3a+1 ≤ F(h3, 3b+2i) = 6b+
1, and a < 2b. Clearly 0, 3, 6, . . . , 3b are not in CH , hence res(H) = |H \ CH | ≥ b + 1.
On the other hand, the structure matrix of H is
 2b−a+1

x1
x2 x3
A=
,
x2
x3 x2a−b
1

and Proposition 7.9 gives res(H) = 2b − a + 1. Thus 2b − a + 1 ≥ b + 1, and b ≥ a,
a contradiction.
Example 7.8 confirms that for any a > 0 the semigroup H = h3, 3a + 1, 3a + 2i
satisfies tr(H) = CH .

Remark 7.14. From the proof of Proposition 7.13 we see that for any a > 0 we
have that res(h3, 3a + 1, 3a + 2i) = a.
In particular, the residue of a 3-generated numerical semigroup H may be as large
as possible. However, this is not the case in a shifted family of semigroups, as we
verify below.
Firstly, we extend the definition of residue from (12) to arbitrary affine subsemigroups of N. In this sense, for any semigroup H ⊂ N containing 0 we let


1
res(H) = res
H , where d = gcd(h : h ∈ H).
d

Given the sequence of integers a : a1 < · · · < ae , for any j we denote a + j :
a1 + j, . . . , ae + j. The shifted family of a is the family {a+ j}j≥0 . It has been proved
that for large enough shifts several properties occur periodically in the shifted family
of semigroups {ha + ji}j≥0 and their semigroup rings {K[ha + ji]}j≥0, see [23], [39],
[18], [34]. For instance, Jayanthan and Srinivasan [23] showed that for j ≫ 0
K[ha + ji] is complete intersection (CI) ⇐⇒ K[ha + j + (ae − a1 )i] is CI.

More generally, Vu ([39, Theorem 1.1]) showed that
for j ≫ 0,

βi (K[ha + ji]) = βi (K[ha + j + (ae − a1 )i]) for all i.

In particular, for j ≫ 0 the algebras K[ha + ji] and K[ha + j + (ae − a1 )i] are
Gorenstein at the same time. This implies that the semigroups ha + ji and ha + j +
(ae − a1 )i are symmetric at the same time. Equivalently,
for j ≫ 0,

res(ha + ji) = 0 ⇐⇒ res(ha + j + (ae − a1 )i) = 0.

It is natural to ask the following.

Question 7.15. Given the list of integers a : a1 < · · · < ae , is it true that
for j ≫ 0,

res(ha + ji) = res(ha + j + (ae − a1 )i)?

Numerical experiments with GAP ([12], [8]) indicate that Question 7.15 might
have a positive answer. Next we confirm it in case e ≤ 3. If e = 2, then ha1 +j, a2 +ji
is symmetric for all j, and we are done. The case e = 3 is proved in the following
33

theorem. We first note that when studying asymptotic properties in a shifted family
{a + j}j , we may assume a1 = 0.
Theorem 7.16. Given the integers 0 < a < b, let D = gcd(a, b) and ka,b =
max{b( b−a
− 1), ba
}. For any integer j we denote Hj = hj, j + a, j + bi.
D
D
Then res(Hj ) = res(Hj+b) for all j > 2ka,b .
Before giving the proof of Theorem 7.16 we recall a result from [34] (extending
Jayanthan and Srinivasan’s [23, Theorem 1.4]) about the occurence of symmetric
semigroups in a shifted family {hj, a + j, b + ji}j≥0.
Lemma 7.17. ([34, Theorem 3.1]) With notation as in Theorem 7.16, let
Y
(17)
T =
pνp (b) ,
p prime, νp (a)<νp (b)

where for any integer n we denote νp (n) = max{i : pi divides n}. Then for j > ka,b
the semigroup Hj is symmetric if and only if j is a multiple of T . In particular, in
the family of semigroups {Hj }j>ka,b the symmetric property occurs periodically with
principal period T .

Proof. (of Theorem 7.16).
We start with j > ka,b . By Lemma 7.17, if Hj is symmetric then Hj+b is symmetric,
too, hence res(Hj ) = res(Hj+b) = 0.
Assume Hj is not symmetric. By Lemma 7.17, Hj+ℓb is not symmetric for all
ℓ ≥ 0. Denote Aj+ℓb the structure matrix (14) of the non-symmetric semigroup
Hj+ℓb.
For (n1 , n2 , n3 ) = (0, a, b) + j + ℓb, it is proved in [34, Theorem 2.2] that for any
ℓ ≥ 0 the middle equation in (13) is
b
b−a
a
(18)
n2 =
n1 + n3 .
D
D
D
This implies that
!
(b−a)/D
x1
xa22 xa33
,
Aj =
a/D
xb22
x3
xb11
where a2 , a3 , b1 , b2 are positive integers (depending on j) such that a2 + b2 = b/D,
by (18).
Let e = gcd(a, b)/ gcd(j, a, b). Proposition 4.2 in [34] explains how the equations
(13) change when we shift up by b. According to this result, only the last column
of Aj changes and we obtain
!
(b−a)/D
x1
xa22 x3a3 +e
.
Aj+b =
a/D
xb22
x3
xb11 +e
Iterating this, we have that
(b−a)/D

Aj+ℓb =

x1

xb22

xa22 xa33 +ℓe
a/D
x3
xb11 +ℓe
34

!

, for ℓ ≥ 0.

Proposition 7.9 gives
(19)

res(Hj+ℓb) = min{(b − a)/D, b1 + ℓe} · min{a/D, a3 + ℓe} · min{a2 , b2 }.

For ℓ ≥ max{ b−a
− 1, Da } = 1b ka,b it is easy to see that b1 + ℓe ≥ (b − a)/D and
D
a3 + ℓe ≥ a/D. Hence, (19) becomes
(20)

res(Hj+ℓb) = min{a2 , b2 } · a(b − a)/D 2 ,

which is a formula not involving ℓ.
The argument above shows that for any j > 2ka,b we have that res(Hj ) =
res(Hj+b). This concludes the proof.

Corollary 7.18. With notation as in Theorem 7.16, for j > 2ka,b the residue of Hj
is an integer divisible by (b − a)a/D 2 and
res(Hj ) < 8b3 /27D 3 .

Proof. If Hj is symmetric, the inequality to prove is clear. Assume Hj is not symmetric and j > 2ka,b . By (20), we have res(Hj ) = min{a2 , b2 } · a(b − a)/D 2 , with
a2 , b2 positive integers such that a2 + b2 = b/D. This shows the first part of the
claim. The second part is obtained from the following chain of inequalities


 3
a(b − a) b
ab(b − a)
8b3
1
2b
res(Hj ) ≤
−
1
<
≤
=
,
·
D2
D
D3
3
D3
27D 3
√
where for the last inequality we used the known fact that 3 xyz ≤ (x + y + z)/3 for
x, y, z > 0.

Corollary 7.19. With notation as in Theorem 7.16, for j > 2ka,b the semigroup Hj
is nearly Gorenstein if and only if Hj+b is nearly Gorenstein.
We make a comment about the frequency of occurences of the symmetric, almost
symmetric and nearly Gorenstein property in a shifted family.
Remark 7.20. Let 0 < a < b. For j ≥ 0 we denote Hj = hj, j + a, j + bi. We use
the constant ka,b introduced in Theorem 7.16.
Lemma 7.17 shows that we find symmetric semigroups for arbitrarily large shifts
j. From the formula (17) for the principal period T we infer that T > 1, otherwise
a = b, which is false. This means that there is no j0 such that Hj is symmetric for
all j > j0 .
When b = 2a, the semigroup Hj is generated by an arithmetic sequence. Using
Proposition 7.1 we get that Hj is nearly Gorenstein for all j > 0. On the other
hand, by Lemma 7.17, when j > b we have that Hj is symmetric if and only if j is
divisible by 2ν2 (b) .
It is however possible that on the shifted family {Hj }j≥0, for large j the only
nearly Gorenstein semigroups are the symmetric ones. Indeed, if a, b are coprime
and a > 1, by Corollary 7.18 we have that for j > 2ka,b , either res(H) = 0, or
res(H) ≥ a(b − a) > 1.
The first author was informed by Kei-ichi Watanabe that there are only finitely
many almost symmetric semigroups in the shifted family {Hj }j≥0. This can also be
seen as follows. According to Nari et al. [29], the structure matrix Aj for an almost
35

symmetric semigroup Hj must have one row consisting of linear forms. However, it
is proven in [34, Proposition 4.2] that for j > ka,b the matrix Aj+b is obtained from
Aj by increasing the exponents of the last column by gcd(a, b)/ gcd(a, b, j). This
shows that for j > ka,b + b the semigroup Hj is not almost symmetric.
Acknowledgement. We gratefully acknowledge the use of SINGULAR ([7]) and
of the numericalsgps package ([8]) in GAP ([12]) for our computations.
Dumitru Stamate was supported by a fellowship at the Research Institute of the
University of Bucharest (ICUB).
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Jürgen Herzog, Fachbereich Mathematik, Universität Duisburg-Essen, Campus
Essen, 45117 Essen, Germany
E-mail address: juergen.herzog@uni-essen.de
Takayuki Hibi, Department of Pure and Applied Mathematics, Graduate School
of Information Science and Technology, Osaka University, Toyonaka, Osaka 5600043, Japan
E-mail address: hibi@math.sci.osaka-u.ac.jp
Dumitru I. Stamate, ICUB/Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, Bucharest – 010014, Romania
E-mail address: dumitru.stamate@fmi.unibuc.ro
37

