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Characterization of generalized Gorenstein rings

The notion of generalized Gorenstein local ring (GGL ring for short) is one of the generalizations of Gorenstein rings. In this article, there is given a characterization of GGL rings in terms of their canonical ideals and related invariants.


Introduction
The notion of a generalized Gorenstein local ring (GGL ring for short) is one of the generalizations of Gorenstein rings. Similarly, for almost Gorenstein local rings (AGL rings for short), the notion is given in terms of a certain specific embedding of the rings into their canonical modules (see [5] by the first and third authors). However, the research on AGL rings developed in [7] by the first author, N. Matsuoka, and T. T. Phuong for arbitrary one-dimensional Cohen-Macaulay local rings is based on an investigation of the relationship between two invariants: the first Hilbert coefficient of the canonical ideals and the Cohen-Macaulay type of the rings. Therefore, it seems natural to ask for a possible characterization of AGL rings of higher dimension and that of GGL rings in terms of their canonical ideals and some related invariants. For AGL rings, this characterization has been accomplished by the first and fourth authors and R. Takahashi. They have already given a satisfactory result [17,Theorem 5.1]. The present purpose is to perform the task for GGL rings of higher dimension.
Originally, the series of studies in [3,4,5,6,7,8,9,10,11,16,17,18,19,20] aimed to find a new class of Cohen-Macaulay local rings, which contains the class of Gorenstein rings. AGL rings are one of the candidates for such a class. Historically, the notion of an AGL ring in our sense has its roots in [1] by V. Barucci and R. Fröberg in 1997, where they dealt with one-dimensional analytically unramified local rings. They also explored numerical semigroup rings, starting a very beautiful theory. In [7], the authors relaxed the notion to arbitrary Cohen-Macaulay local rings of dimension one based on a different point of view. Repairing a gap in the proof of [1, Proposition 25], they provided new possible areas of study for one-dimensional Cohen-Macaulay local rings. Among various results in [7], the most striking achievement seems that their arguments have prepared for a possible definition [17,Definition 3.3] of AGL rings of higher dimension. We now have two more notions: the 2-almost Gorenstein local ring ( [3]) and GGL ring ( [5]), both of which are reasonable candidate generalizations of Gorenstein rings and almost Gorenstein rings as well.
As stated above, the present purpose is to find a characterization of GGL rings in terms of the canonical ideals and related invariants. To state our motivation and the results more precisely, let us review the definition of GGL rings. Throughout this article, let (R, m) be a Cohen-Macaulay local ring with d = dim R > 0, possessing the canonical module K R . For simplicity, let us always assume that the residue class field R/m of R is infinite. Let a be an m-primary ideal of R. With this notation, the definition of a GGL ring is stated as follows. 5]). We say that R is a generalized Gorenstein local ring (GGL for short) if one of the following conditions is satisfied: (1) R is a Gorenstein ring.
(2) R is not a Gorenstein ring, but there exists an exact sequence of R-modules such that (i) C is an Ulrich R-module with respect to a, and (ii) the induced homomorphism R/a ⊗ R ϕ : R/a → K R /aK R is injective. When Case (2) occurs, we especially say that R is a GGL ring with respect to a. ). The notion of an Ulrich R-module with respect to an m-primary ideal is a generalization of a maximally generated maximal Cohen-Macaulay R-module (i.e., a maximal Ulrich R-module with respect to m; see [2]). One can consult [5,14,15,17] for the basic properties of Ulrich modules in our sense. Here, let us note one thing. In the setting of Definition 1.1, suppose that there is an exact  [5], the authors investigate GGL rings, and one can find a report of the basic results for GGL rings, which greatly generalizes several results in [17], clarifying what AGL rings are. The present purpose is to give a characterization of GGL rings. Let r(R) stand for the Cohen-Macaulay type of R. We then have the following, which is the main result of this article.
Theorem 1.2. Let (R, m) be a Cohen-Macaulay local ring with d = dim R > 0 and an infinite residue class field, possessing the canonical module K R . Let I R be an ideal of R such that I ∼ = K R as an R-module. Let a be an m-primary ideal of R. Assume that R is not a Gorenstein ring. Then, the following conditions are equivalent.
(1) R is a GGL ring with respect to a.
(2) There exists a parameter ideal Q = (f 1 , f 2 , . . . , f d ) of R such that f 1 ∈ I and, setting J = I + Q, the following three conditions are satisfied.
. When this is the case, R/a is a Gorenstein ring, and the following assertions hold true.
the Sally module of J with respect to Q (see [21]) (resp. the Rees algebra of Q). (c) f 2 , f 3 , . . . , f d form a super-regular sequence of R with respect to J, and depth gr J (R) = d − 1, where gr J (R) = n≥0 J n /J n+1 denotes the associated graded ring of J. (d) The Hilbert function of R with respect to J is given by The study of GGL rings is still in progress, and Theorem 1.2 now completely generalizes the corresponding assertion [17, Theorem 5.1] of AGL rings to arbitrary GGL rings of higher dimension, certifying that the notion of a GGL ring is a reasonable generalization of AGL rings and that a GGL ring is one of the candidates of broader notion that generalizes Gorenstein rings.
We now briefly explain how this paper is organized. The proof of Theorem 1.2 shall be given in Sections 3 and 4. In Section 2, we summarize some of the known results given by [5], which we use to prove Theorem 1.2. We will explore an example in order to illustrate Theorem 1.2 in Section 5.

Preliminaries
In this section we summarize some of the results in [5,Section 4] about GGL rings , which we use to prove Theorem 1.2.
First, let (R, m) be a Cohen-Macaulay local ring of dimension one, admitting the canonical module K R . Let I R be an ideal of R such that I ∼ = K R as an R-module. We assume that I contains a parameter ideal (a) of R as a reduction. We set is independent of the choice of canonical ideals I and reductions (a) of I ([3, Theorem 2.5]). We set c = R : S. We then have the following, which shows the mprimary ideal a which appears in Definition 1.1 of a GGL ring R is uniquley determined, when dim R = 1. We note a brief proof.
). Suppose R is not a Gorenstein ring but R is a GGL ring with respect to an m-primary ideal a of R. Then a = c.
Proof. We choose an exact sequence of R-modules such that C is an Ulrich R-module with respect to a and the induced homomorphism R/a ⊗ R ϕ : R/a → I/aI is injective. We set f = ϕ(1) and identify C = I/(f ). Then aI ⊆ (f ) since a·(I/(f )) = (0), while (f ) ∩ aI = af since the homomorphism R/a ⊗ R ϕ is injective. Consequently, aI = af , whence (f ) is a reduction of I. We consider L = I f and set S = R[L]. Then aL = a since aI = af , so that aS = a since S = L n for all n ≫ 0. Therefore, a ⊆ c = R : S, so that a = c, because c ⊆ R : L = (f ) : R I = a.
In general we have the following.
The key in the theory of one-dimensional GGL rings is the following, which we shall freely use in the present article. See [5,Section 4] for the proof. (1) R is a GGL ring (necessarily with respect to c).
(2) K/R is a free R/c-module. When this is the case, the following assertions hold true.
(ii) R/c is a Gorenstein ring.
Let us note here the non-zerodivisor characterization given by [5] for GGL rings of higher dimension. To state it, let (R, m) be a Cohen-Macaulay local ring with d = dim R > 0 and infinite residue class field, possessing the canonical module K R . We then have the following.
Theorem 2.4 ( [5]). Suppose that R is not a Gorenstein ring. Then the following assertions hold true.
(1) Let R be a GGL ring with respect to a and d ≥ 2. Consider the exact sequence of R-modules such that C is an Ulrich R-module with respect to a and the induced homomorphism R/a ⊗ R ϕ : R/a → K R /aK R is injective. Choose a superficial element f ∈ a for C with respect to a and assume that f is R-regular. Then R/(f ) is a GGL ring with respect to a/(f ), possessing to be a defining exact sequence.
(2) Let f ∈ m be R-regular and assume that R/(f ) is a GGL ring with respect to [a + (f )]/(f ). Then R is a GGL ring with respect to a + (f ) and f ∈ m[a + (f )].

Proof of the main part of Theorem 1.2
Let (R, m) be a Cohen-Macaulay local ring with d = dim R > 0 and infinite residue class field, possessing the canonical module K R . Let I R be an ideal of R such that I ∼ = K R as an R-module. We choose a parameter ideal Q = (f 1 , f 2 , . . . , f d ) of R so that f 1 ∈ I. We set q = (f 2 , f 3 , . . . , f d ) and J = I + q. Let a be an m-primary ideal of R. In Sections 3 and 4 we throughout assume that R is not a Gorenstein ring. The purpose is to prove the equivalence between Conditions (1) and (2) in Theorem 1.2.
Let us begin with the following. Proof. We get q ∩ I = qI, since q is a parameter ideal of the Gorenstein ring R/I. If J = Q, then I = Q ∩ I = (f 1 ) + (q ∩ I), whence I = (f 1 ) by the first equlaity. Therefore, R is a Gorenstein ring, which is impossible. When this is the case, R/q is a GGL ring with respect to a/q, whence so is the ring R with respect to a.
Proof. We may assume aJ = aQ. Hence, Q is a reduction of J. Because J/Q = (0) by Proposition 3.1, we get a = Q : R J, once J/Q is R/a-free. Consequently, we may also assume that a = Q : R J. First, suppose that d = 1. Hence J = I. We set K = I f 1 in the total ring of fractions of R. Then, since aK = a, we have a = R : R[K] by Fact 2.2. Consequently, by Theorem 2.3, R is a GGL ring (necessarily with respect to c; see Proposition 2.1) if and only if e 1 I (R) = ℓ R (R/a)·r(R). By Theorem 2.3, the former condition is also equivalent to saying that I/Q ( ∼ = K/R) is a free R/a-module, whence the equivalence of Conditions (1) and (2) follows.
Let us consider the case where d ≥ 2. Assume that Condition (2) is satisfied. Let us check that R = R/q is a GGL ring. Set R = R/q, Q = Q/q, J = J/q, and a = a/q.
We then have J = (I + q)/q ∼ = I/qI = K R , since I ∼ = K R and f 2 , f 3 , . . . , f d is a regular sequence for the R-module I. Consequently, because a·J = a·Q and J/Q is R/a-free, from the case of d = 1 it follows that R is a GGL ring (Fact 2.2 and Theorem 2.3), whence so is R with respect a (Theorem 2.4 (2)).
We now assume that the implication (2) ⇒ (1) holds true for d − 1. Since Q is a reduction of J and the field R/m is infinite, there exist elements h 1 , h 2 , . . . , h d ∈ Q such that (i) h 1 ∈ I, (ii) Q = (h 1 , h 2 , . . . , h d ), and (iii) h 2 is superficial for R with respect to J. This time, we consider the ring R = R/(h 2 ) and let * denote the reduction mod (h 2 ). Then I = [I + (h 2 )]/(h 2 ) ∼ = I/h 2 I = K R and h 1 ∈ I. Condition (2) is clearly satisfied for the ring R as for the ideals a, Q, and J. Therefore, by the hypothesis of induction on d we get e 1 J (R) = ℓ R R/a ·r(R). Consequently, e 1 J (R) = ℓ R (R/a)·r(R), because e 1 J (R) = e 1 J (R) (remember that h 2 is superficial for R with respect to J).
The reverse implication (1) ⇒ (2) also follows by induction on d, chasing the above argument in the opposite direction.
Let us suppose, with the same notation as in Theorem 3.2, that the equivalent conditions of Theorem 3.2 are satisfied. Then we get the following.
, and c = R : S. Then since R is a GGL ring with respect a, we get a = c by Proposition 2.1 and c = R : K (see Fact 2.2 and Theorem 2.3). Therefore, I ⊆ c, because c is an ideal of S and f 1 ∈ c (note that I = f 1 K ⊆ R). If d > 1, then passing to R/q, we have J/q ⊆ a/q, whence J ⊆ a. Therefore J 2 ⊆ Q, because a = Q : R J.
We are now ready to prove the equivalence of Conditions (1) and (2) in Theorem 1.2.
Proof of the main part in Theorem 1.2. See Theorem 3.2 for the implication (2) ⇒ (1). To see the implication (1) ⇒ (2), we consider the exact sequence of R-modules such that C is an Ulrich R-module with respect to a and the induced homomorphism R/a ⊗ R ϕ : R/a → I/aI is injective. Let f 1 = ϕ(1) ∈ I. Then f 1 is a non-zerodivisor of R. Choose elements f 2 , f 3 , . . . , f d ∈ a so that in the ring R ′ = R/(f 1 ) these elements generate a reduction of a·R ′ . Then f 1 , f 2 , . . . , f d is a system of parameters of R with f 1 ∈ I. Set q = (f 2 , f 3 , . . . , f d ), Q = (f 1 ) + q, and J = I + q. Then, q is a parameter ideal of R/I, and because a·R ′ (C) = e 0 a (C) = ℓ R (C/aC), we get aC = qC. Therefore, since a n C = q n C for all n ∈ Z, f 2 , f 3 , . . . , f d forms a superregular sequence for C with respect to a, whence it is a superficial sequence for C with respect to a. We set R = R/q, I = I/qI, and C = C/qC. Then, from exact sequence (♯) we get the exact sequence of R-modules. By Theorem 2.4 (1) the ring R is a GGL ring with respect to a/q, possessing sequence (♯) to be a defining exact sequence. Consequently, because J/q = (I + q)/q ∼ = I = K R , by Theorem 2.3 it follows that a·J = a·Q and J/Q is R/a-free, where a = a/q, J = J/q, and Q = Q/q.
Hence J/Q is R/a-free and aJ ⊆ aQ + q. Therefore where the third equality follows from the fact that aJ = aI + aq. Hence aJ = aQ, because I ⊆ a by Proposition 3.3. Thus Theorem 3.2 certifies that Condition (2) in Theorem 1.2 is satisfied for the ideals a, Q, and J. This completes the proof of the equivalence of Conditions (1) and (2)  In what follows, assume that our ideals Q and J satisfy the equivalent conditions in Theorem 3.2. Hence R (resp. R/q) is a GGL ring with respect to a (resp. a/q), and R/a is a Gorenstein ring by Theorem 2.3 (ii). To prove the last assertions of Theorem 1.2, we need some preliminaries. Let us maintain the same notation as in the proof of Theorem 3.2.
We begin with the following.
Proposition 4.2. We set L = Q : R a. Then the following assertions hold true.