On the canonical ideals of one-dimensional Cohen-Macaulay local rings

In this paper we consider the problem of finding explicitly canonical ideals of one-dimensional Cohen-Macaulay local rings. We show that Gorenstein ideals contained in a high power of the maximal ideal are canonical ideals. In the codimension two case, from a Hilbert-Burch resolution, we show how to construct canonical ideals of curve singularities. Finally, we translate the problem of the analytic classification of curve singularities to the classification of local Artin Gorenstein rings with suitable length.


Introduction
Let (R, m) be a one-dimensional Cohen-Macaulay local ring with maximal ideal m for which there exists a canonical module ω R ; this is the case, for instance, if R is the quotient of a local Gorenstein ring. Recall that R possesses a canonical ideal (more precisely, the canonical module ω R of R exists and is contained in R) if and only if the total ring of fractions of the m-adic completion of R is Gorenstein [13,Satz 6.21]. See [4,Chapter 3] for the basic properties of canonical modules and canonical ideals.
The question that motivates this paper is, can we explicitly describe canonical ideals? Recall that Boij [3] addressed this problem for projective zero-dimensional schemes. A second question that we consider is, can we use canonical ideals for the analytic classification of singularities? In this paper we study these questions in the one-dimensional case.
The contents of this paper are as follows. In § 2 we show that Gorenstein ideals contained in a high power of the maximal ideal are canonical ideals (see Proposition 2.3). In the codimension 2 case, and following [3], from a Hilbert-Burch resolution we show how to construct canonical ideals for curve singularities (see Proposition 2.8). In this section we recall how to 'explicitly' describe the canonical module by using Rosenlicht's regular differential forms [20,Chapter IV,§ 3.9]. This strategy is very useful in the case of branches and especially in the case of monomial curve singularities. In § 3 we address the problem of the analytic classification of curve singularities. We show canonical ideals I ⊂ O (X,0) , where X is a curve singularity, for which we compute the multiplicity and the socle degree of the Artin Gorenstein quotient O (X,0) /I (see Proposition 3.4). In Theorem 3.5, we translate the problem of the analytic classification of curve singularities X to the classification of local Artin Gorenstein rings O (X,0) /I of suitable length.
Notation. Let (R, m) be a one-dimensional Noetherian local ring with maximal ideal m and residue field k = R/m. If I is a m-primary ideal of R, we denote by HF 1 Next we recall some basic facts regarding curve singularities. Let (X, 0) be a reduced curve singularity of (C n , . We assume that n is the embedding dimension of (X, 0), which is equivalent to saying that m X /m 2 X is isomorphic as a C-vector space to the homogeneous linear forms of P = C[X 1 , . . . , X n ]. Hence, all elements x ∈ m X /m 2 X define an element in the quotient O (X,0) that we will denote again by x. Let ν :X = Spec(O (X,0) ) → (X, 0) be the normalization of (X, 0), where O (X,0) is the integral closure of O (X,0) on its full ring tot(O (X,0) ) of fractions. The singularity order of (X, 0) is δ(X) = dim C (OX /O (X,0) ). We denote by C the conductor of the finite extension ν * : O (X,0) → O (X,0) and by c(X) the dimension of O (X,0) /C. Let be the dualizing module of (X, 0). We can consider the composition morphism of O (X,0)modules Let d : O (X,0) → Ω (X,0) be the universal derivation. We then have a C-map γ X d that we also denote by d : O (X,0) → ω (X,0) . The Milnor number of (X, 0) is μ(X) = dim C (ω (X,0) /dO (X,0) ) [6]. Notice that (X, 0) is non-singular if and only if μ(X) = 0 if and only if δ(X) = 0 if and only if c(X) = 0. In the following result we collect some basic results on μ and other numerical invariants that we will use later on. Proposition 1.1. Let (X, 0) be a reduced curve singularity of embedding dimension n. Then the following statements hold.
and e 1 (X)

Canonical ideals
The first aim of this section is to find conditions on an m-primary ideal I to be a canonical ideal.  (ii) Assume that I ⊂ xR. If R/I is a Gorenstein ring, then R/x n I is a Gorenstein ring for all n 1.

Proof .
(i) The short exact sequence of socles, which shows that because Hom R (R/m, R/I) = 0 and Length R (Hom R (R/m, R/x n I)) = 1. Hence, R/I is a Gorenstein ring.
(ii) Let α ∈ (x n I : R m). Then, since xα ∈ x n I ⊂ x n+1 R, we get α ∈ x n R, which shows that π((x n I : R m)/x n I) = 0, where π : R/x n I → R/x n R denotes the canonical epimorphism. Hence, we get the isomorphism in the exact sequence of socles. Thus, Length R (Hom R (R/m, R/x n I)) = 1, so that R/x n I is a Gorenstein ring for all n 1.

Proposition 2.2. Let I be an m-primary ideal of R. Then I is a canonical ideal of R if and only if there exists a parameter x ∈ m of R such that R/xI is a Gorenstein ring.
Proof . We have only to prove the if part. Let Q be the total ring of fractions of R. In the next result we prove that an m-primary Gorenstein ideal contained in a high power of the maximal ideal is a canonical ideal. Notice that this result cannot be extended to any m-primary Gorenstein ideal. Let R be a one-dimensional Cohen-Macaulay local ring of Cohen-Macaulay type 2 and embedding dimension b 3, for example, R = k[[t 5 , t 6 , t 7 ]]. Then the maximal ideal is a Gorenstein ideal minimally generated by b 3 elements. Since the minimal number of generators of a canonical ideal is the Cohen-Macaulay type of R, m is not a canonical ideal.
Proof . Since we have that I ⊂ m r+1 ⊂ xR, the assertion follows from Lemma 2.1 and Proposition 2.2.

Remark 2.4. Recall that pn(R)
e 0 (R) − 1 [16,Proposition 12.14]. Hence, if I ⊂ m e0(R) is an m-primary ideal such that R/I is a Gorenstein ring, then I is a canonical ideal.
The last result points out that a basic problem in commutative algebra is to find methods to construct Gorenstein ideals. We know that complete intersection ideals are Gorenstein; by a result of Serre, in codimension 2, being Gorenstein is equivalent to being a complete intersection; in codimension 3, Gorenstein ideals are the ideals generated by the Pfaffians of skew-symmetric matrixes [5]. In Proposition 2.8 we show how to construct canonical ideals, which are Gorenstein, from a Hilbert-Burch resolution. On the other hand, notice that if I is a canonical ideal and y ∈ m is a non-zero divisor of R, then y t I, t 1, is a canonical ideal as well, but the length of R/yI is not under control. In fact, for all t 1 we have [16,Theorem 12.5], In the next result we find canonical ideals for which we compute the multiplicity or the socle degree; in the second part we take t 4μ(X) + 1, where (X, 0) is a reduced curve singularity, because we have to consider a large t in Theorem 3.5. See Example 2.7 for an explicit application of the next result. Proposition 2.5. Let (X, 0) be a reduced curve singularity.
(iii) If (X, 0) is Gorenstein, then O (X,0) is a canonical ideal and for every superficial element z of degree t 1, zO (X,0) is a canonical ideal.
and notice that for all λ ∈ R it holds that Hence, since m c(X) annihilates the quotient ν * OX /O (X,0) , we get that m c(X) annihi- 0) . On the other hand, the epimor- Let us consider the sequence From the perfect pairing from the beginning of the proof, we get dim C (ω (X,0) /O (X,0) ) = 2δ(X). Since z is a degree t 2μ(X) superficial element of O (X,0) , dim C (ω (X,0) /zω (X,0) ) = te 0 (X) (see [16,Theorem 12.5]). From this identity and Proposition 1.1, we get the first part of the claim.
(ii) We have the following inequality for t 4μ(X) + 1: From Proposition 1.1 and the first part of this result, we obtain From Proposition 1.1 we obtain that the socle degree is bounded from above by δ(X)(4e 0 (X) + 3).
(iii) Since any superficial element is a non-zero divisor, we get the claim.
Recall that it is possible to give an 'explicit' description of ω (X,0) by using Rosenlicht's regular differential forms (see [20, Chapter IV, § 3.9] and [6, § 1]). This strategy is very useful in the case of branches, especially in the case of monomial curve singularities.
We denote by ΩX (p · ) the set of meromorphic forms inX with at most a single pole in the set {p 1 , . . . , p r }. Then Rosenlicht's differential forms are defined as follows: ω R Notice that we have a mapping that we also denote by 0) .
. From now on we assume that (X, 0) is a branch, i.e. r = 1. Let t ∈ tot(O (X,0) ) be a uniformizing parameter of (X, 0); this means that O (X,0) ∼ = C[[t]]. We can consider O (X,0) as a sub-C-algebra of C[[t]] and then we may assume that there exists a parametrization of (X, 0), For any subset N of tot(O (X,0) ) we denote by Γ N the set of rational numbers val t (a) for all a ∈ N \ {0}. We assume that Γ N contains the zero element.
Let us assume now that (X, 0) is a monomial curve singularity and let n be an integer of Z \ (−Γ X − 1). Consider the differential α = t n dt: we only have to prove that α ∈ ω (X,0) . Let us consider F = i 0 a i t i , a power series with coefficients a i ∈ C, i 0. Since (X, 0) is monomial, we get that F ∈ O (X,0) if and only if for all a i = 0 it holds that i ∈ Γ X . If α ∈ ω (X,0) , then there exists F ∈ O (X,0) such that res 0 (αF ) = a −n−1 = 0. This implies that −n − 1 ∈ Γ X , which is in contradiction to the hypothesis n / ∈ −Γ X − 1.
The next step is to find explicit canonical ideals from the resolution of O (X,0) when X is a reduced curve singularity of (C 3 , 0). By the Hilbert-Burch theorem, we know that there exists a minimal free resolution of O (X,0) as an O (C 3 ,0) -module [7], where M is a v × (v − 1) matrix with entries belonging to the maximal ideal O (C 3 ,0) and I X is minimally generated by the maximal minors of M . The canonical module of O (X,0) is minimally generated by v−1 elements. Following [3,15], we consider a (2v−1)×(2v−1) block-matrix Notice that M A is also a skew-symmetric matrix and, by the main result of [5], we have a complex

Proof . (i) This is a consequence of [5] and Corollary 2.3.
(ii) The ideal I A is generated by the Pfaffians of M A and these elements take the following form (see [15]). Notice that F 1 , . . . , F v is a system of generators of I X . Hence, I A /I X is generated by the 0) be the natural projection. Let J X be the ideal generated by the 2 × 2 minors of the v × 3 matrix Jac X whose ith row is the gradient vector ∇F i = (∂F 1 /∂x 1 , ∂F 1 /∂x 3 , ∂F 1 /∂x 3 ), i = 1, . . . , v. The image π(J X ) is the Jacobian ideal of X. Since the (X, 0) is an isolated curve singularity, we have that π(J X ) is m X -primary, On the other hand, it is easy to prove that Let p 1 , . . . , p m be the set of minimal primes of O (X,0) . Then p i = m X , i = 1, . . . , m, and K/I X ⊂ p 1 ∪ · · · ∪ p m . Since K/I X is an m-primary ideal of O (X,0) , there is an integer w such that Hence, every element of (x 1 , x 2 , x 3 ) w is a zero divisor of O (X,0) , but this is not possible because O (X,0) is a Cohen-Macaulay ring. We have proved that there exist integers 0) . Let x ∈ m \ m 2 be a non-zero divisor of O (X,0) . Let us consider the skew-symmetric matrix such that a i,j = x pn(O (X,0) ) , a α,β = 0, α < β, (α, β) = (i, j).
Then F k = ±x pn(O (X,0) ) det(M i,j,k ) is a non-zero divisor of O (X,0) . Hence, I A /I X is an m-primary ideal of O (X,0) contained in m pn(O (X,0) )+1 . By (i), we obtain that I A /I X is a canonical ideal of O (X,0) . Example 2.9. Let us consider a monomial curve singularity X with parametrization (t n1 , t n2 , t n3 ) such that n 1 < n 2 < n 3 and gcd(n 1 , n 2 , n 3 ) = 1. Then, e 0 (X) = n 1 and The ideal I X is generated by where r i,j 0 and c i > 0 is the least integer such that c i n i = i =j r i,j n j , i = 1, 2, 3 [12]. We assume that X is not a complete intersection, so r i,j > 0. We then have that c 1 = r 2,1 + r 3,1 , c 2 = r 1,2 + r 3,2 , c 3 = r 1,3 + r 2,3 and F 1 , F 2 , F 3 are the maximal minors of a matrix 3 3 x r1, 3 3 x r3,1 1 ⎞ ⎟ ⎠ (see [18]). Then the matrix M A takes the form 2 ) is a canonical ideal of X.

Canonical ideals and the classification of curve singularities
In this section we translate the classification of curve singularities to the classification of local Artin Gorenstein rings by means of the quotients with canonical ideals.
First we have to define what generic means in our context. We denote by S t the C-vector space of forms of degree t of C[[x 1 , . . . , x n ]].  (1) For all z ∈ U t (X),z ∈ O (X,0) is a degree t superficial element and non-zero divisor.

Proof . (i) From
For each n = 1, . . . , , let W n be a non-empty Zariski open set W n ⊂ W 0 such that σ n (z) = min{σ n } for all z ∈ W n . We set U t (X) = W 0 ∩ · · · ∩ W . From the definition of U t (X), it is easy to get (2) and (3).
We write (X) = e 0 (X)(4μ(X) + 1) − 2δ(X). Notice that this is the length of the quotients O (X,0) /zω (X,0) forz ∈ U 4μ(X)+1 (X).  Given a non-negative integer t, we denote by Hilb t (C n ,0) the Hilbert scheme of length t subschemes Z of (C n , 0). We denote by [Z] the closed point of Hilb t (C n ,0) defined by Z. From the universal property of Hilb t (C n ,0) we deduce that any analytic isomorphism φ : (C n , 0) → (C n , 0) induces a C-scheme isomorphismφ : Hilb t (C n ,0) → Hilb t (C n ,0) such thatφ([Z]) = [φ(Z)]. Given a canonical ideal I of a reduced curve singularity (X, 0), we denote by (X, 0) I the zero-dimensional scheme Spec(O (X,0) /I). We know that (X, 0) I is an Artin Gorenstein scheme [4,Proposition 3.3.18]. It is well known that two canonical ideals are isomorphic [4,Theorem 3.3.4]. In the one-dimensional case one can prove more: if I 1 and I 2 are canonical ideals, there exist non-zero divisors y 1 , y 2 ∈ O (X,0) such that y 1 I 1 = y 2 I 2 [16,Theorem 15.8]. Notice that from the proof of this result we obtain that for all z ∈ U t (X) there exists integer α and a regular element y such that z α I = yzω (X,0) . The ideals I 1 and I 2 are isomorphic as O (X,0) -modules, but, in general, they are not analytic isomorphic. Since K t = x t ω (X,0) is a canonical ideal for all t 1 with x a degree 1 superficial element, the Hilbert function of O (X,0) /K t varies with t.