WHEN IS R ⋉ I AN ALMOST GORENSTEIN LOCAL RING?

arXiv:1704.05961v1 [] 20 Apr 2017

SHIRO GOTO AND SHINYA KUMASHIRO

Abstract. Let (R, m) be a Gorenstein local ring of dimension d > 0 and let I be an
ideal of R such that (0) 6= I ( R and R/I is a Cohen-Macaulay ring of dimension d.
There is given a complete answer to the question of when the idealization A = R ⋉ I of
I over R is an almost Gorenstein local ring.

1. Introduction
Let (R, m) be a Gorenstein local ring of dimension d > 0 with infinite residue class
field. Assume that R is a homomorphic image of a regular local ring. With this notation
the purpose of this paper is to prove the following theorem.
Theorem 1.1. Let I be a non-zero ideal of R and suppose that R/I is a Cohen-Macaulay
ring of dimension d. Let A = R⋉I denote the idealization of I over R. Then the following
conditions are equivalent.
(1) A = R ⋉ I is an almost Gorenstein local ring.
(2) R has the presentation R = S/[(X) ∩ (Y )] where S is a regular local ring of
dimension d + 1 and X, Y is a part of a regular system of parameters of S such
that I = XR.
The notion of almost Gorenstein local ring (AGL ring for short) is one of the generalization of Gorenstein rings, which originated in the paper [1] of V. Barucci and R. Fröberg
in 1997. They introduced the notion for one-dimensional analytically unramified local
rings and developed a beautiful theory, investigating the semigroup rings of numerical
semigroups. In 2013 the first author, N. Matsuoka, and T. T. Phuong [5] extended the
notion to arbitrary Cohen-Macaulay local rings but still of dimension one. The research of
[5] has been succeeded by two works [11] and [3] in 2015 and 2017, respectively. In [3] one
can find the notion of 2-almost Gorenstein local ring (2-AGL ring for short) of dimension
one, which is a generalization of AGL rings. Using the Sally modules of canonical ideals,
the authors show that 2-AGL rings behave well as if they were twins of AGL rings. The
purpose of the research [11] of the first author, R. Takahashi, and N, Taniguchi started in
a different direction. They have extended the notion of AGL ring to higher dimensional
Cohen-Macaulay local/graded rings, using the notion of Ulrich modules ([2]). Here let us
briefly recall their definition for the local case.
The first author was partially supported by JSPS Grant-in-Aid for Scientific Research (C) 25400051.
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SHIRO GOTO AND SHINYA KUMASHIRO

Definition 1.2. Let (R, m) be a Cohen-Macaulay local ring of dimension d, possessing
the canonical module KR . Then we say that R is an AGL ring, if there exists an exact
sequence
0 → R → KR → C → 0
of R-modules such that either C = (0) or C 6= (0) and µR (C) = e0m (C), where µR (C)
denotes the number of elements in a minimal system of generators of C and
e0m (C) = lim (d − 1)!·
n→∞

ℓR (C/mn+1 C)
nd−1

denotes the multiplicity of C with respect to the maximal ideal m (here ℓR (∗) stands for
the length).
We explain a little about Definition 1.2. Let (R, m) be a Cohen-Macaulay local ring
of dimension d and assume that R possesses the canonical module KR . The condition
of Definition 1.2 requires that R is embedded into KR and even though R 6= KR , the
difference C = KR /R between KR and R is an Ulrich R-module ([2]) and behaves well.
In particular, the condition is equivalent to saying that mC = (0), when dim R = 1 ([11,
Proposition 3.4]). In general, if R is an AGL ring of dimension d > 0, then Rp is a
Gorenstein ring for every p ∈ Ass R, because dimR C ≤ d − 1 ([11, Lemma 3.1]).
The research on almost Gorenstein local/graded rings is still in progress, exploring, e.g.,
the problem of when the Rees algebras of ideals/modules are almost Gorenstein rings (see
[6, 7, 8, 9, 10, 15]) and the reader can consult [11] for several basic results on almost
Gorenstein local/graded rings. For instance, non-Gorenstein AGL rings are G-regular in
the sense of [14] and all the known Cohen-Macaulay local rings of finite Cohen-Macaulay
representation type are AGL rings. Besides, the authors explored the question of when
the idealization A = R ⋉ M is an AGL ring, where (R, m) is a Cohen-Macaulay local ring
and M is a maximal Cohen-Macaulay R-module. Because A = R ⋉ M is a Gorenstein
ring if and only if M ∼
= KR as an R-module ([13]), this question seems quite natural and
in [11, Section 6] the authors actually gave a complete answer to the question in the case
where M is a faithful R-module, that is the case (0) :R M = (0). However, the case
where M is not faithful has been left open, which our Theorem 1.1 settles in the special
case where R is a Gorenstein local ring and M = I is an ideal of R such that R/I is
a Cohen-Macaulay ring with dim R/I = dim R. For the case where dim R/I = d but
depth R/I = d − 1 the question remains open (see Remark 2.6).
2. Proof of Theorem 1.1
The purpose of this section is to prove Theorem 1.1. To begin with, let us fix our
notation. Unless otherwise specified, throughout this paper let (R, m) be a Gorenstein
local ring with d = dim R > 0. Let I be a non-zero ideal of R such that R/I is a

WHEN IS R ⋉ I AN ALMOST GORENSTEIN LOCAL RING?

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Cohen-Macaulay ring with dim R/I = d. Let A = R ⋉ I be the idealization of I over R.
Therefore, A = R ⊕ I as an R-module and the multiplication in A is given by
(a, x)(b, y) = (ab, bx + ay)
where a, b ∈ R and x, y ∈ I. Hence A is a Cohen-Macaulay local ring with dim A = d,
because I is a maximal Cohen-Macaulay R-module.
For each R-module N let N ∨ = HomR (N, R). We set L = I ∨ ⊕ R and consider L to
be an A-module under the following action of A
(a, x) ◦ (f, y) = (af, f (x) + ay),
where (a, x) ∈ A and (f, y) ∈ L. Then it is standard to check that the map
A∨ → L, α 7→ (α ◦ j, α(1))
is an isomorphism of A-modules, where j : I → A, x 7→ (0, x) and 1 = (1, 0) denotes the
identity of the ring A. Hence by [12, Satz 5.12] we get the following.
Fact 2.1. KA = L, where KA denotes the canonical module of A.
We set J = (0) :R I. Let ι : I → R denote the embedding. Then taking the R-dual of
the exact sequence
ι
0→I−
→ R → R/I → 0,
we get the exact sequence
ι∨

0 → (R/I)∨ → R∨ −
→ I ∨ → 0 = Ext1R (R/I, R) → · · ·
of R-modules, which shows I ∨ = R·ι. Hence J = (0) :R I ∨ because I = I ∨∨ ([12, Korollar
6.8]), so that I ∨ = R·ι ∼
= (R/J)∨ = KR/J ([12, Satz
= R/J as an R-module. Hence I ∼
5.12]). Therefore, taking again the R-dual of the exact sequence
ι∨

0 → J → R∨ −
→ I ∨ → 0,
ι
we get the exact sequence 0 → I −
→ R → J ∨ → 0 of R-modules, whence J ∨ ∼
= R/I, so
∨
∼
that J = (R/I) = KR/I . Summarizing the arguments, we get the following.

Fact 2.2. I ∼
= (R/I)∨ = KR/I .
= (R/J)∨ = KR/J and J ∼
Notice that r(A) = 2 by [12, Satz 6.10] where r(A) denotes the Cohen-Macaulay type
of A, because A is not a Gorenstein ring (as I ∼
6= R; see [13]) but KA is generated by two
elements; KA = R·(ι, 0) + R·(0, 1).
We denote by M = m × I the maximal ideal of A. Let us begin with the following.
Lemma 2.3. Let d = 1. Then the following conditions are equivalent.
(1) A is an AGL ring.
(2) I + J = m.
When this is the case, I ∩ J = (0).

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SHIRO GOTO AND SHINYA KUMASHIRO

Proof. (2) ⇒ (1) We set f = (ι, 1) ∈ KA and C = KA /Af . Let α ∈ m and β ∈ I. Let us
write α = a + b with a ∈ I and b ∈ J. Then because
(α, 0)(0, 1) = (0, α) = (bι, a + b) = (b, a)(ι, 1), (0, β)(0, 1) = (0, 0),
we get MC = (0), whence A is an AGL ring.
(1) ⇒ (2) We have I ∩ J = (0). In fact, let p ∈ Ass R and set P = p × I. Hence
P ∈ Min A. Assume that IRp 6= (0). Then since AP = Rp ⋉ IRp and AP is a Gorenstein
local ring, IRp ∼
= Rp ([13]), so that JRp = (0). Therefore, (I ∩ J)Rp = (0) for every
p ∈ Ass R, whence I ∩ J = (0).
Now consider the exact sequence
ϕ

0→A−
→ KA → C → 0
of A-modules such that MC = (0). We set f = ϕ(1). Then f 6∈ MKA by [11, Corollary
3.10], because A is not a discrete valuation ring (DVR for short). We identify KA =
I ∨ × R (Fact 2.1) and write f = (aι, b) with a, b ∈ R. Then a 6∈ m or b 6∈ m, since
f = (a, 0)(ι, 0) + (b, 0)(0, 1) 6∈ MKA .
Firstly, assume that a 6∈ m. Without loss of generality, we may assume a = 1, whence
f = (ι, b). Let α ∈ m. Then since (α, 0)(0, 1) ∈ Af , we can write (α, 0)(0, 1) = (r, x)(ι, b)
with some r ∈ R and x ∈ I. Because
(0, α) = (α, 0)(0, 1) = (r, x)(ι, b) = (rι, x + rb),
we get
r ∈ (0) :R ι = J,

α = x + rb ∈ I + J.

Therefore, m = I + J.
Now assume that a ∈ m. Then since b 6∈ m, we may assume b = 1, whence f = (aι, 1).
Let α ∈ m and write (α, 0)(ι, 0) = (r, x)(aι, 1) with r ∈ R and x ∈ I. Then since
(αι, 0) = ((ra)ι, ax + r), we get
α − ra ∈ J,

r = −xa ∈ (a),

so that α ∈ J + (a2 ) ⊆ J + m2 , whence m = J. Because I ∩ J = (0), this implies I = (0),
which is absurd. Therefore, a 6∈ m, whence I + J = m.

Corollary 2.4. Let d = 1. Assume that A = R ⋉ I is an AGL ring. Then both R/I
and R/J are discrete valuation rings and µR (I) = µR (J) = 1. Consequently, if R is a
homomorphic image of a regular local ring, then R has the presentation
R = S/[(X) ∩ (Y )]
for some two-dimensional regular local ring (S, n) with n = (X, Y ), so that I = (x) and
J = (y), where x, y respectively denote the images of X, Y in R.

WHEN IS R ⋉ I AN ALMOST GORENSTEIN LOCAL RING?

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Proof of Corollary 2.4. Since I + J = m and I ∩ J = (0), KR/I ∼
= m/I by Fact
= J ∼
2.2. Hence R/I is a DVR by Burch’s Theorem (see, e.g., [4, Theorem 1.1 (1)]), because
idR/I m/I = idR/I KR/I = 1 < ∞, where idR/I (∗) denotes the injective dimension. We
similarly get that R/J is a DVR, since KR/J ∼
= m/J . Consequently, µR (I) = µR (J) =
=I∼
1. We write I = (x) and J = (y). Hence m = I + J = (x, y). Since xy = 0, we have
m2 = (x2 , y 2) = (x + y)m. Therefore, v(R) = e(R) = 2 because R is not a DVR, where
v(R) (resp. e(R)) denotes the embedding dimension of R (resp. the multiplicity e0m (R)
of R with respect to m). Suppose now that R is a homomorphic image of a regular local
ring. Let us write R = S/a where a is an ideal in a two-dimensional regular local ring
(S, n) and choose X, Y ∈ n so that x, y are the images of X, Y in R, respectively. Then
n = (X, Y ), since a ⊆ n2 . We consider the canonical epimorphism
ϕ : S/[(X) ∩ (Y )] → R
and get that ϕ is an isomorphism, because
ℓS (S/(XY, X + Y )) = 2 = ℓR (R/(x + y)R) .
Thus a = (X) ∩ (Y ) and R = S/[(X) ∩ (Y )].



We note the following.
Proposition 2.5. Let S be a regular local ring of dimension d + 1 (d > 0) and let X, Y
be a part of a regular system of parameters of S. We set R = S/[(X) ∩ (Y )] and I = (x),
where x denotes the image of X in R. Then I 6= (0), R/I is a Cohen-Macaulay ring with
dim R/I = d, and the idealization A = R ⋉ I is an AGL ring.
Proof. Let y be the image of Y in R. Then (y) = (0) :R x and we have the presentation
0 → (y) → R → (x) → 0
of the R-module I = (x), whence A = R[T ]/(yT, T 2), where T is an indeterminate.
Therefore
A = S[T ]/(XY, Y T, T 2 ).
Notice that (XY, Y T, T 2 ) is equal to the ideal generated by the 2 × 2 minors of the matrix
Y T
M = (X
T Y 0 ) and we readily get by [11, Theorem 7.8] that A = R ⋉ I is an AGL ring,
because X, Y, T is a part of a regular system of parameters of the regular local ring S[T ]P ,
where P = nS[T ] + (T ).

We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1. By Proposition 2.5 we have only to show the implication (1) ⇒ (2).
Consider the exact sequence
0 → A → KA → C → 0

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SHIRO GOTO AND SHINYA KUMASHIRO

of A-modules such that C is an Ulrich A-module. Let M = m × I stand for the maximal
ideal of A. Then since mA ⊆ M ⊆ mA (here mA denotes the integral closure of mA)
and the field R/m is infinite, we can choose a superficial sequence f1 , f2 , . . . , fd−1 ∈ m for
C with respect to M so that f1 , f2 , . . . , fd−1 is also a part of a system of parameters for
both R and R/I. We set q = (f1 , f2 , . . . , fd−1 ) and R = R/q. Let I = (I + q)/q and
J = (J + q)/q. Then since f1 , f2 , . . . , fd−1 is a regular sequence for R/I, by the exact
sequence
0 → I → R → R/I → 0
we get the exact sequence
0 → I/qI → R → R/(I + q) → 0,
so that I/qI ∼
= I as an R-module. Hence
A/qA = R ⋉ (I/qI) ∼
= R ⋉ I.
Remember that A/qA is an AGL ring by [11, Theorem 3.7], because f1 , f2 , . . . , fd−1 is a
superficial sequence of C with respect M and f1 , f2 , . . . , fd−1 is an A-regular sequence.
Consequently, thanks to Corollary 2.4, R is a DVR and µR (I) = 1. Hence R/I is a
regular local ring and µR (I) = 1, because I/qI ∼
= I. Let I = (x). Then R/J ∼
= I = (x),
since J = (0) :R I. Because f1 , f2 , . . . , fd−1 is a regular sequence for the R-module I,
f1 , f2 , . . . , fd−1 is a regular sequence for R/J, so that we get the exact sequence
0 → J/qJ → R → R/(J + q) → 0.
Therefore, J ∼
= I, we have J = (0) :R I. Hence R/J
= I/qI ∼
= J/qJ and since R/(J + q) ∼
is a regular local ring and µR (J) = 1, because R/J is a DVR and µR (J ) = 1 by Corollary
2.4.
Let J = (y) and let m = m/q. Then by Lemma 2.3 we have m = I + J, whence
m = (x, y, f1 , f2 , . . . , fd−1 ). Therefore µR (m) = d + 1, since R is not a regular local
ring. On the other hand, since both R/I and R/J are regular local rings, considering the
canonical exact sequence
0 → R → R/I ⊕ R/J → R/(I + J) → 0
(notice that I ∩ J = (0) for the same reason as in the proof of Lemma 2.3), we readily
get e(R) = 2. We now choose a regular local ring (S, n) of dimension d + 1 and an ideal
a of S so that R = S/a. Let X, Y, Z1 , Z2 , . . . , Zd−1 be the elements of n whose images in
R are equal to x, y, f1, f2 , . . . , fd−1 , respectively. Then n = (X, Y, Z1 , Z2 , . . . , Zd−1 ), since
a ⊆ n2 . Because (X) ∩ (Y ) ⊆ a as xy = 0, we get a surjective homomorphism
S/[(X) ∩ (Y )] → R

WHEN IS R ⋉ I AN ALMOST GORENSTEIN LOCAL RING?

7

of rings, which has to be an isomorphism, because both the Cohen-Macaulay local rings
S/[(X) ∩ (Y )] and R have the same multiplicity 2. This completes the proof of Theorem
1.1.

Remark 2.6. Let (S, n) be a two-dimensional regular local ring and let X, Y be a regular
system of parameters of S. We set R = S/[(X) ∩(Y )]. Let x, y denote the images of X, Y
in R, respectively. Let n ≥ 2 be an integer. Then dim R/(xn ) = 1 but depth R/(xn ) = 0.
We have xn = xn−1 (x + y), whence (xn ) ∼
= (x) as an R-module because x + y is a nonn
zerodivisor of R, so that R ⋉ (x ) is an AGL ring (Proposition 2.5). This example shows
that there are certain ideals I in Gorenstein local rings R of dimension d > 0 such that
dim R/I = d and depth R/I = d − 1, for which the idealizations R ⋉ I are AGL rings.
However, we have no idea to control them.
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Department of Mathematics, School of Science and Technology, Meiji University,
1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan
E-mail address: shirogoto@gmail.com
Department of Mathematics and Informatics, Graduate School of Science and Technology, Chiba University, Chiba-shi 263, Japan
E-mail address: polar1412@gmail.com

