OPENNESS OF FID-LOCI

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1 OPENNESS OF FID-LOCI YO TAKAHASHI Abstract. Let be a commutative Noetherian ring and M a finite -module. In this paper, we consider Zariski-openness of the FID-locus of M, namely, the subset of Spec consisting of all prime ideals p such that M p has finite injective dimension as an p -module. We prove that the FID-locus of M is an open subset of Spec whenever is excellent. 1. Introduction Throughout the present paper, we assume that all rings are commutative and Noetherian. Let P be a property of local rings. The P-locus of a ring is the set of prime ideals p of such that the local ring p satisfies the property P. It is a natural question to ask whether the P-locus of is an open subset of Spec in the Zariski topology, and it has been considered for a long time. For example, it is known that the P-locus of an excellent ring is open if P is any of the regular property, the complete intersection property, the Gorenstein property, and the Cohen-Macaulay property. As to the details of openness of loci for properties of local rings, see [3], [4, 6 7], [6], [7, 24], [8], and [9]. On the other hand, let P be a property of modules over a local ring. The P-locus of a module M over a ring is defined to be the subset of Spec consisting of all prime ideals p such that the p -module M p satisfies P. The locus of a finite module for the property of finite projective dimension is known to be an open subset [1, Corollary 9.4.7], and so is the locus of a finite module for the Gorenstein property if the base ring is acceptable, and therefore if it is excellent [5, Corollaries 4.6 and 4.7]. In this paper, we will consider openness of the locus of a finite module for the property of finite injective dimension, which we call the FID-locus. We shall prove that the FID-locus of a finite module satisfying certain conditions is an open subset. Using this result, we will show the following: Theorem. Let be an excellent ring and M a finite -module. Then the FIDlocus FID (M) = { p Spec idp M p < } of M is an open subset of Spec in the Zariski topology. Of course, this theorem implies the result of Greco and Marinari [3, Corollary 1.5] asserting that the Gorenstein locus of an excellent ring is open. Key words and phrases: injective dimension, excellent ring Mathematics Subject Classification: 13D05, 13F40. 1

2 2 YO TAKAHASHI 2. The results Throughout this section, let be a commutative Noetherian ring. ecall that a subset U of Spec is called stable under generalization provided that if p U and q Spec with q p then q U. We begin by stating two lemmas. The former is called the topological Nagata criterion ; it is a criterion for Zariski-openness which is due to Nagata. Lemma 2.1. [7, Theorem 24.2] The following are equivalent for a subset U of Spec : (1) U is an open subset of Spec ; (2) U is stable under generalization, and contains a nonempty open subset of V (p) for any p U. Lemma 2.2. [3, Lemma 1.1] Let p be a minimal prime of a finite -module M. Then there exist an element f \ p and a chain 0 = N 0 N 1 N n = M f of f -submodules of M f such that N i /N i 1 = f /p f for 1 i n. Next, we study an easy lemma. Lemma 2.3. Let p be a prime ideal of and M a finite -module. If M p = 0, then M f = 0 for some f \ p. Proof. If M p = 0, then p is not in the support of the -module M, hence p does not contain the annihilator ideal Ann M. Therefore there is an element f Ann M \ p. We easily obtain M f = 0. We define the FID-locus of an -module M to be the set of prime ideals p of such that the p -module M p has finite injective dimension, and denote it by FID (M). Now, we can prove the following proposition, which will play a key role in the proof of our main result. Proposition 2.4. Let M be a finite -module, and let p FID (M). Suppose that the FID-locus FID /p (Ext j (/p, M)) contains a nonempty open subset of Spec /p for each integer j with 0 j ht p. Then there exists an element f \ p such that the FID-locus FID (M) contains V (p) D(f). Proof. First of all, we note that to prove the proposition we can freely replace our ring with its localization g for an element g \ p. In fact, we have p g FID g (M g ) and ht p g = ht p. Let U j be a nonempty open subset of Spec /p which is contained in FID /p (Ext j (/p, M)) for 0 j ht p. Write U j = D(I j /p) for some ideal I j of containing p, and we see that D(I j g /p g ) is a nonempty open subset of Spec g /p g which is contained in FID g /p g (Ext j g ( g /p g, M g )). If there exists an element h g n g \ p g with h and n 0 such that V (p g ) D( h g ) is contained in FID n g (M g ), then h is an element of \ p and V (p) D(gh) is contained in FID (M). Suppose that M p = 0. Then we have M f = 0 for some f \ p by Lemma 2.3. Hence the set D(f) is itself contained in the locus FID (M), and there is nothing more to prove. Therefore in what follows we consider the case where M p 0. Since M p is a finite p -module of finite injective dimension, p is a Cohen-Macaulay local

3 OPENNESS OF FID-LOCI 3 ring by virtue of [2, Corollary 9.6.2, emark 9.6.4(a)]. Put n = dim p, and take a sequence x = x 1, x 2,..., x n of elements in p which forms an p -regular sequence. Then, putting H i = (0 : /(x1,x 2,...,x i 1 ) x i ), we have (H i ) p = 0 for 1 i n. Hence Lemma 2.3 implies that (H i ) fi = 0 for some f i \ p. Setting f = f 1 f 2 f n, we see that f is in \ p and that x is an f -regular sequence. eplacing with f, we may assume that x is an -regular sequence. Set = /(x) and p = p/(x). Then p is a minimal prime of, hence is an associated prime of. Let P 1 = p, P 2,..., P s be the associated primes of. Taking an element of the set s i=2 P i \ P 1, we easily see that there is an element f \ p such that Ass f = { } p f, where f denotes the residue class of f in. eplacing with f, we may assume that Ass = { p }. On the other hand, since id p M p <, we have id p M p = n by [2, Theorem ] and hence Ext n+1 p (κ(p), M p ) = 0, where κ(p) denotes the residue field of p. Therefore it follows from Lemma 2.3 that Ext n+1 f ( f /p f, M f ) = 0 for some f \ p. eplacing with f, we may assume that (2.4.1) Ext n+1 (/p, M) = 0. Here, we establish a claim. Claim. One may assume that Ext j (/p, M) = 0 for all integers j > n. Proof of Claim. If p = 0, then p = (x) and the -module /p has projective dimension n since x is an -regular sequence of length n. Hence Ext j (/p, M) = 0 for j > n, as desired. Assume p 0. Then we have = Min (p) Ass (p) Ass = { p }, and therefore Min (p) = { p }. According to Lemma 2.2, for some element f \ p there is a chain 0 = N 0 N 1 N l = p f of f -modules such that N i /N i 1 = f /p f = f /p f for any 1 i l. eplacing with f, we may assume that there is a chain 0 = N 0 N 1 N l = p of -modules such that each N i /N i 1 is isomorphic to /p. We have obtained a series of exact sequences of -modules (2.4.2) 0 N i 1 N i /p 0 (1 i l). Using these sequences and (2.4.1), we can get Ext n+1 (p, M) = 0. The natural exact sequence 0 p /p 0 induces an exact sequence of Ext modules: 0 = Ext n+1 (p, M) Extn+2 (/p, M) Extn+2 (, M). Noting that has projective dimension n as an -module, we have Ext i (, M) = 0 for every i > n, and Ext n+2 (/p, M) = 0. Using the sequences (2.4.2) again, we get Extn+2 (p, M) = 0. Iterating this procedure shows the claim. The assumption of the proposition yields a nonempty open subset U j of Spec /p contained in the locus FID /p (Ext j (/p, M)) for 0 j n. We can write U j = D(I j /p) for some ideal I j of which strictly contains p. Hence there exists an element f j I j \p, and setting f = f 0 f 1 f n, we see that the set D(f) is contained in D(I j ) for any 0 j n. Fix a prime ideal q V (p) D(f). Then q/p belongs to D(I j /p) = U j, which is contained in FID /p (Ext j (/p, M)) for 0 j n. Hence Extj q ( q /p q, M q ) has finite injective dimension as an q /p q -module for any integer j with 0 j n.

4 4 YO TAKAHASHI Put m = max{ id q/p q (Ext j q ( q /p q, M q )) 0 j n }. Consider the following spectral sequence: E i,j 2 = Ext i q/p q (κ(q), Ext j q ( q /p q, M q )) = Ext i+j q (κ(q), M q ). We have E i,j 2 = 0 if i > m, and the above claim shows that E i,j 2 = 0 if j > n. From this spectral sequence, we see that Ext i q (κ(q), M q ) = 0 for i > m+n. This implies that the q -module M q has finite injective dimension (cf. [2, Proposition ]), that is q FID (M). It follows that V (p) D(f) is contained in FID (M), which completes the proof of the proposition. Now we state and prove our main result of this paper. Theorem 2.5. Let M be a finite -module. Suppose that FID /p (Ext j (/p, M)) contains a nonempty open subset of Spec /p for any prime ideal p FID (M) and any integer 0 j ht p. Then FID (M) is an open subset of Spec. Proof. Proposition 2.4 shows that for any p FID (M) there exists f \ p such that FID (M) contains V (p) D(f). Note that V (p) D(f) is not an empty set since p belongs to it. On the other hand, it is easy to see from [2, Proposition 3.1.9] that FID (M) is stable under generalization. Thus the theorem follows from Lemma 2.1. We denote by eg() the regular locus of, namely, the set of prime ideals p of such that the local ring p is regular. The following result can be obtained from the above theorem. Corollary 2.6. Let be an excellent ring. Then FID (M) is an open subset of Spec for any finite -module M. Proof. Fix a prime ideal p FID (M) and an integer j with 0 j ht p. By the definition of an excellent ring, the regular locus eg(/p) is an open subset of Spec /p. The zero ideal of /p belongs to eg(/p), hence it is nonempty. Noting that any module over a regular local ring has finite injective dimension, we see that eg(/p) is contained in the locus FID /p (Ext j (/p, M)). Thus all the assumptions of Theorem 2.5 are satisfied, and it follows that FID (M) is open in Spec. We denote by Gor() the Gorenstein locus of, that is, the subset of Spec consisting of all prime ideals p of such that p is a Gorenstein local ring. Since Gor() coincides with FID (), the above corollary yields a result of Greco and Marinari [3, Corollary 1.5]: Corollary 2.7 (Greco-Marinari). Let be an excellent ring. Then the Gorenstein locus Gor() is open in Spec. eferences [1] Brodmann, M. P.; Sharp,. Y. Local cohomology: an algebraic introduction with geometric applications. Cambridge Studies in Advanced Mathematics, 60. Cambridge University Press, Cambridge, [2] Bruns, W.; Herzog, J. Cohen-Macaulay rings. revised edition. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, [3] Greco, S.; Marinari, M. G. Nagata s criterion and openness of loci for Gorenstein and complete intersection. Math. Z. 160 (1978), no. 3,

5 OPENNESS OF FID-LOCI 5 [4] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II. (French) Inst. Hautes Études Sci. Publ. Math. No. 24, [5] Leuschke, G. J. Gorenstein modules, finite index, and finite Cohen-Macaulay type. Comm. Algebra 30 (2002), no. 4, [6] Massaza, C.; Valabrega, P. Sull apertura di luoghi in uno schema localmente noetheriano. (Italian) Boll. Un. Mat. Ital. A (5) 14 (1977), no. 3, [7] Matsumura, H. Commutative ring theory. Translated from the Japanese by M. eid. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, [8] Nagata, M. On the closedness of singular loci. Inst. Hautes Études Sci. Publ. Math [9] Sharp,. Y. Acceptable rings and homomorphic images of Gorenstein rings. J. Algebra 44 (1977), no. 1, Department of Mathematics, School of Science and Technology, Meiji University, Higashimita, Tama-ku, Kawasaki , Japan address: takahasi@math.meiji.ac.jp