CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Journal of Algebra 306 (2006) 507–519 www.elsevier.com/locate/jalgebra A construction of Gorenstein rings Marco D’Anna Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria, 6, I-95125 Catania, Italy Received 8 November 2005 Available online 23 January 2006 Communicated by Luchezar L. Avramov Abstract In this paper, starting with a commutative ring R and a proper ideal I ⊂ R, we construct and study a new ring denoted by R 1 I. In particular, we prove that if R is a CM local ring, then R 1 I is Gorenstein if and only if I is a canonical ideal of R and we apply this construction to algebroid curves. © 2006 Elsevier Inc. All rights reserved. Keywords: Gorenstein ring; Canonical ideal; Algebroid curve 1. Introduction If R is a commutative ring with unity and M is an R-module, the idealization R M (also called trivial extension), introduced by Nagata in 1956 (cf. Nagata’s book [12, p. 2]), is a new ring where the module M can be viewed as an ideal such that its square is (0).In [13] it is proved that, if R is a local Cohen–Macaulay ring, then R M is Gorenstein if and only if M is a canonical module of R; the “if” direction of this result was generalized by Fossum in [7], for any “commutative extension.” In this paper we consider a different type of construction, obtained involving a ring R and an ideal I ⊂ R, that will be denoted by R 1 I and it is defined as the following subring E-mail address: [email protected]. URL: http://www.dmi.unict.it/~mdanna/. 0021-8693/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2005.12.023 508 M. D’Anna / Journal of Algebra 306 (2006) 507–519 of R × R: R 1 I = (r, r + i) | r ∈ R, i ∈ I . More generally this construction can be given starting with a ring R and an ideal E of an overring S of R (such that S ⊆ Q(R), where Q(R) is the total ring of fractions of R); this extension has been studied, in the general case and from the different point of view of pullbacks, by Fontana and the author in [5]. One main difference of this construction, with respect to the idealization (or with respect to any commutative extension, in the sense of Fossum) is that the ring R 1 I is reduced whenever R is reduced. One main result of this paper is the following (see Theorem 11): if R is a Cohen– Macaulay local ring, then the ring R 1 I is Gorenstein if and only if R has a canonical ∼ ideal ωR and I = ωR. In order to get this result it is important to prove that I and HomR1I (R, R 1 I) are isomorphic as R-modules (see Proposition 3) and to understand when the ring R 1 I is Cohen–Macaulay (see Section 3). We also discuss the general (nonlocal) situation using a localization statement (see Proposition 7) to reduce to the lo- cal case. One direction of this result can be obtained as a corollary of a more general (but unpublished) result, due to Eisenbud (see Theorem 12). The construction studied in this paper has an interesting application to curve singulari- ties. We show that, if we start with an algebroid curve R with h branches, the ring R 1 I is again an algebroid curve with 2h branches; moreover for every branch of R there are ex- actly two corresponding branches of R 1 I both isomorphic to the branch of R we started with (see Theorem 14). This result gives also an explicit presentation of the algebroid curve R 1 I as a quotient of a power series ring; in particular, we get an explicit construction of Gorenstein algebroid curves (see Corollary 17). The paper is organized as follows: Section 2 contains general results on the ring R 1 I . In Section 3 we study when R 1 I is CM and when it is Gorenstein. Section 4 contains the application to curve singularities. 2. The ring R 1 I Let R be a commutative ring with unit element 1 and let I be a proper ideal of R.We define R 1 I ={(r, s) | r, s ∈ R, s − r ∈ I}. It is easy to check that R 1 I is a subring, with unit element (1, 1),ofR × R (with the usual componentwise operations) and that R 1 I ={(r, r + i) | r ∈ R, i ∈ I}. Some results in this section are proved in a more general context in [5] and we omit the proofs. We recall that the idealization R M, introduced by Nagata for every R-module M (cf. [12, p. 2]), is defined as the R-module R ⊕ M endowed with the multiplication (r, m)(s, n) = (rs, rn + sm). We also recall that a commutative extension of R by an R- module M is defined as an exact sequence of abelian groups i π 0 M T R 0 M. D’Anna / Journal of Algebra 306 (2006) 507–519 509 where T is a commutative ring, π is a ring homomorphism and the R-module structure of M is related to (T,i,π)by the equation t · i(m) = i(π(t) · m), for every t ∈ T and m ∈ M (cf. [7]). Notice that R M is a commutative extension and that, for any commutative extension, i(M) is a nilpotent ideal of T , of index 2. Therefore a commutative extension is never a reduced ring. On the other hand, R 1 I is reduced if and only if R is reduced (see Proposition 2); hence, in general, the ring R 1 I is not a commutative extension (see also the exact se- quence (2) below); more precisely, it is isomorphic to the idealization if and only if I is a nilpotent ideal of index 2 in R. If in the R-module direct sum R ⊕ I we introduce a multiplicative structure by setting (r, i)(s, j) = (rs, rj + si + ij ), it is not difficult to check that the map f : R ⊕ I → R 1 I defined by f ((r, i)) = (r, r + i) is a ring isomorphism. Moreover, the diagonal embedding ϕ : R → R 1 I , defined by ϕ(r) = (r, r), is an injective ring homomorphism. Hence we have the following short exact sequence of R-modules: ϕ ψ 0 R R 1 I I 0 (1) where ψ((r, s)) = s − r, for every (r, s) ∈ R 1 I . Notice that this sequence splits; hence we also have the short exact sequence of R-modules: ψ ϕ 0 I R 1 I R 0 (2) where ψ(i) = (0,i)and ϕ((r, s)) = r, for every i ∈ I and (r, s) ∈ R 1 I .Wewillseelater (cf. Remark 4) that the exact sequence (2) is also a sequence of R 1 I -modules, while the other one is not. Remark 1. (a) If we set R = ϕ(R), then R ⊆ R 1 I ⊆ R × R.NowR→ R × R is a finite homomorphism, since R × R is generated by (1, 0) and (0, 1) as R-module. It follows that R ⊆ R × R is an integral extension; hence both R ⊆ R 1 I and R 1 I ⊆ R × R are integral extensions. It follows that R and R 1 I have the same Krull dimension (cf. [10, Theorem 48]). (b) It is not difficult to see that, if πi (i = 1, 2) are the projections of R × R on R, ∼ then πi(R 1 I)= R; hence, if Oi = Ker(πi|R1I ), then (R 1 I)/Oi = R; moreover O1 = {(0,i)| i ∈ I}, O2 ={(i, 0) | i ∈ I} and O1 ∩ O2 = (0). (c) R is Noetherian if and only if R 1 I is Noetherian. In fact, if R is Noetherian, then also R × R is Noetherian, hence R 1 I is finitely generated as R-module and so is ∼ Noetherian; conversely, if R 1 I is Noetherian, then R = (R 1 I)/Oi is Noetherian. Proposition 2. [5] R 1 I is a reduced ring if and only if R is reduced. In particular, if R is an integral domain, R 1 I is reduced and its only minimal primes are O1 and O2. 510 M. D’Anna / Journal of Algebra 306 (2006) 507–519 The next result will be very important for one direction of Theorem 11. Proposition 3. The following isomorphism of R-modules holds: ∼ I = HomR1I (R, R 1 I). ∼ Proof. Recall that R = (R 1 I)/O1, with O1 ={(0,i)| i ∈ I}, hence R is a cyclic R 1 I - module generated by 1 and (r, r + i)· s = π1((r, r + i))s = rs (where s ∈ R and (r, r + i) ∈ R 1 I ). It follows that, if we fix an element i ∈ I ,themapgi : R → R 1 I , defined by gi(r) = (ri, 0),isanR 1 I -homomorphism. Hence the following map is induced: f : I → HomR1I (R, R 1 I), i → gi. It is easy to see that f is an injective homomorphism of R-modules. It remains to check that f is surjective: if h : R → R 1 I is a R 1 I -homomorphism, then it is determined by h(1) = (x, y) (where x,y ∈ R and y − x ∈ I ).