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Openness of various loci over Noetherian rings

In this paper, we consider the openness of the P-locus of a finitely generated module over a commutative noetherian ring in the case where P is each of the properties FID, Gor, CM, MCM, (S_n), and (T_n). One of the main results asserts that FID-loci over an acceptable ring are open. We give a module version of the Nagata criterion, and prove that it holds for all of the aforementioned properties.


Introduction
Throughout the present paper, all rings are assumed to be commutative and noetherian, and all modules be finitely generated.
Let P be a property of local rings, and R a ring. The set of prime ideals p of R such that the local ring R p satisfies P is called the P-locus of R. For example, the regular, complete intersection, Gorenstein, and Cohen-Macaulay properties and Serre's condition (S n ) can be considered as P. When P is any one of the properties appearing above, the P-locus is stable under generalization. Therefore, it is a natural question to ask when the P-locus is open in the Zariski topology for a given P. This question has been studied for a long time by many people. Nagata [11] produced the following condition, which is called the Nagata criterion: (NC): if the P-locus of R/p contains a nonempty open subset of Spec(R/p) for all prime ideals p of R, then the P-locus of R is an open subset of Spec(R). The Nagata criterion holds for the regular, complete intersection, Gorenstein, and Cohen-Macaulay properties and Serre's condition (S n ); see [6,10,11,16].
Let P be a property of modules over a local ring, and M an R-module. The set of prime ideals p of R such that the module M p over the local ring R p satisfies P is called the P-locus of M (over R). The same question can be asked for the P-locus of M which is stable under generalization, and by the so-called topological Nagata criterion, the P-locus of M is open if and only if the P-locus of M contains a nonempty open subset of V(p) for all p belonging to the P-locus of M . It is well-known fact that the free locus [10] and the finite projective dimension locus [1] are always open. The Cohen-Macaulay locus of a module over an excellent ring is open [7]. Furthermore, the Gorenstein locus of a module over an acceptable ring in the sense of Sharp [15] is open [9], and so is the finite injective dimension locus of a module over an excellent ring [17].
In this paper, we consider the openness of the P-locus of a module in the case where P is each of the finite injective dimension property (FID), the Gorenstein property (Gor), the Cohen-Macaulay property (CM), the maximal Cohen-Macaulay property (MCM), and Serre's conditions (S n ) and (T n ); for the definition of (T n ) see Definition 2.1. First of all, we handle the case P = FID. For a fixed element p of the P-locus of M , we give a (necessary and) sufficient condition for the P-locus of M to contain a nonempty open subset of V(p). Also, we study some rings that satisfy those conditions for all prime ideals. The main result in this direction is the following theorem concerning FID-loci over an acceptable ring. Next, we confirm that some results on the finite injective dimension property hold on other properties. We give a module version of the Nagata criterion for a property P of modules over a local ring: It is natural to ask for which properties (NC) * hold. We prove the following theorem.
The organization of this paper is as follows. Section 2 is denoted to preliminaries for the later sections. In Sections 3 and 4, we study the openness of the FID-locus of a module. We give a sufficient condition for the FID-locus of a module to be open, and obtain Theorem 1.1. In Section 5, we observe that the same results as we gave in the previous sections hold for the Cohen-Macaulay property. In Section 6, we consider when the (S n ) and (T n )-loci of a module are open. It is seen that if the MCM-locus (resp. the CM-locus) is open, then so is the (S n )-locus (resp. the (T n )-locus). In Section 7, we prove Theorem 1.2.

Notation and lemmas
In this section, we state the definitions of notions used in this paper, and give basic lemmas about the Zariski topology. Throughout the present paper, let R be a ring. Let P be a property of local rings. The subset P(R) = {p ∈ Spec(R) | P holds for R p } of Spec(R) is called the P-locus of R. Similarly, if P is a property of modules over a local ring, then P R (M ) = {p ∈ Spec(R) | P holds for M p } is called the P-locus of M (over R) for an R-module M .
Definition 2.1. Let (R, m, k) be a local ring, M an R-module, and n an integer.
• If dim M depth M , then M is called a Cohen-Macaulay module.
• A Gorenstein module of type 1 is called a canonical module. In general, let R be a (not necessarily local) ring, and M an R-module. We denote by Ann R (M ) the annihilator ideal of M . The injective dimension of M is denoted by id R M . We say that M is a Cohen-Macaulay module (resp. a maximal Cohen-Macaulay module, a Gorenstein module, and a canonical module) if so is the module M m over the local ring R m for every maximal ideal m of R. If R is itself a Cohen-Macaulay (resp. Gorenstein) module, then it is called a Cohen-Macaulay (resp. Gorenstein) ring. We say that • M satisfies Serre's condition (S n ) if depth M p inf{n, ht p} for all p ∈ Spec(R), and • M satisfies Serre's condition (T n ) if depth M p inf{n, dim M p } for all p ∈ Spec(R).
In this paper, the following notation is used.
Definition 2.2. Let M be an R-module, I an ideal of R, f an element of R, and n an integer. • From now on, Spec(R) will be considered to be with the Zariski topology, and any subset of Spec(R) will be considered to be with the subspace topology. Below is called the topological Nagata criterion. (1) U is stable under generalization, that is, if p ∈ U and q ∈ Spec(R) with q ⊆ p, then q ∈ U .  Proof. Suppose that there exists an open subset U of Spec(R) such that V(p) ∩ U is nonempty and is contained in S. There exists a prime ideal q of R, which belongs to U and contains p.
Conversely, if there exists a nonempty open subset U of V(pR f ) that is contained in T , then the subset   (3) Suppose that a sequence x = x 1 , . . . , x n of elements in p is an M p -regular sequence. Then there exists f ∈ R \ p such that x is an M f -regular sequence. (4) If p is a minimal prime ideal of an ideal I, then IR f = pR f for some f ∈ R \ p.
Proof. (1): An analogous argument to the former part of the proof of Lemma 2.5 shows the assertion.
(3): We may assume n = 1. Let φ be the multiplication map of M by (4): Let p = p 1 , p 2 , . . . , p n be all the minimal prime ideals of I. Since p i p for any 2 i n, there exists f ∈ R \ p such that f ∈ p i for each 2 i n. We easily obtain IR f = pR f . (5) and (6)

The openness of the FID-locus of a module
In this section, we study the openness of the FID-locus of a module by relating it to the Gor-locus of a ring. More precisely, we examine the relationship between the hypothesis of the Nagata criterion for the Gorenstein property and the openness of the finite injective dimension locus. Recall a few definitions that will be used in the following; see [14,15].
Definition 3.1. Let S be a ring. A ring homomorphism φ : R → S is said to be a Gorenstein homomorphism if φ is flat and all the fiber rings of φ are Gorenstein.

Definition 3.2.
A ring R is said to be well-fibered if the natural ring homomorphism from R p to its completion is Gorenstein for every prime ideal p of R. Definition 3.3. A ring R is said to be acceptable if the following three conditions are satisfied.
(2) For all finitely generated R-algebras S, Gorenstein since all the fiber rings of φ are regular. (3) Any homomorphic image of a Gorenstein ring is acceptable; see [6,15].
The key role is played by the lemma below. The main result of this section is the following theorem, whose proof uses the above lemma. The result below can be obtained from Theorem 3.6. Note that the assumption in Corollary 3.7 (2) is the same as that of (NC) for the Gorensteinness. The above corollary yields the following result; it recovers theorems of Greco and Marinari and of Takahashi. Proof. The assertion (1) follows from Corollary 3.7 (2) because Gor(R) = FID R (R). If R is an acceptable ring, then (3.7.1) holds for R by definition. Hence, Corollary 3.7 (2) gives the assertion (2). Both a homomorphic image of a Gorenstein ring and an excellent ring are acceptable. Thus the assertions (3) and (4) follow from (2).
In the next section, the ring R that satisfies (3.7.1) will further be studied. We close this section by stating a proposition about (NC) for the Gorensteinness. Put Let n 0 be an integer. If nth syzygy of M is totally reflexive, then we say that M has G-dimension at most n, and write G-dim(M ) n. We say that M has infinite G-dimension, and write G-dim(M ) = ∞ if such an integer n does not exist.
Let S be a ring. A ring homomorphism φ : R → S is said to be essentially of finite type if S is the localization of a finitely generated R-algebra. Below is a direct corollary of Lemma 4.1. Suppose that R is an Artinian ring. Then (3.7.1) holds for R since R/p is a field for any prime ideal p of R. More generally, we can prove the result below. Proof. It suffices to verify that (3.7.1) holds for R. Let p be a prime ideal of R, and let m 1 , . . . , m r be all the maximal elements of V(p). Fix an integer 1 i r. By assumption, the local ring R mi is wellfibered. It follows from [15, Corollary 6.5] that Gor(R mi /pR mi ) is an open subset of Spec(R mi /pR mi ).   Proof. Note that we can freely replace our ring R with its localization R f for any element f ∈ R \ p to prove the theorem; we apply Lemma 2.5.
First, we consider the case ht p = 0. Then M p is a nonzero injective R p -module because id Rp M p = depth R p = 0 by [2, Theorem 3.1.17]. Hence, Ext 1 Rp (κ(p), M p ) = 0 and Hom Rp (κ(p), M p ) ≃ κ(p) ⊕n for some integer n > 0; here we set κ(p) = R p /pR p . By (2) and (5) of Lemma 2.7, we may assume that the following conditions (i) and (ii) are satisfied. On the other hand, since p is a minimal prime ideal of R, it follows from (4) and (6) of Lemma 2.7 that we may assume that the following condition (iii) is satisfied.
There exists an integer r > 0 such that p r = 0 and p i /p i+1 is a free R/p-module for each 1 i r.
If Ext j R (R/p, M ) = 0 for an integer j > 0, then it is seen that Ext j+1 R (R/p, M ) ≃ Ext j R (p, M ) = 0 by (iii). Therefore, we have Ext j R (R/p, M ) = 0 for any j > 0 by (i). Let I : 0 → I 0 → I 1 → · · · be an injective resolution of M . It follows from [2, Lemma 3.1.6] and (ii) that the complex Hom R (R/p, I) is an injective resolution of Hom R (R/p, M ) ≃ (R/p) ⊕n as an R/p-module, because Ext j R (R/p, M ) = 0 for any j > 0. Let q ∈ V(p). There is an isomorphism Hom R (R/q, I) ≃ Hom R/p (R/q, Hom R (R/p, I)) of complexes. This says that Ext j R (R/q, M ) ≃ Ext j R/p (R/q, R/p) ⊕n for any j 0. We obtain an equivalence (4.4.1) Ext j Rq (κ(q), M q ) = 0 ⇔ Ext j Rq/pRq (κ(q), R q /pR q ) = 0 for each integer j 0. It follows from (4.4.1) and [2, Proposition 3.1.14] that q is in FID R (M ) if and only if q/p is in Gor(R/p). Applying Lemma 2.6 to I = p and S = FID R (M ), we see that the assertion holds.
Next, we address the general case. Put n = ht p. It follows from [2, Corollary 9.6.2, Remark 9.6.4(a)] that R p is a Cohen-Macaulay local ring since id Rp M p < ∞ and M p = 0. On the other hand, M p is a nonzero maximal Cohen-Macaulay R p -module. By Lemma 2.7 (3), we may assume that there exists a sequence x = x 1 , . . . , x n of elements in p, which is a regular sequence on R and M . Set R = R/xR, is a ring isomorphism. We say that M is with full support if Supp R (M ) = Spec(R).
Remark 4.6. The assumption of Corollary 4.5 says that there is a Gorenstein R-module with full support. Therefore, the above corollary recovers Corollary 3.8 (3) because R is itself a Gorenstein Rmodule with full support if R is Gorenstein. More generally, if R is a Cohen-Macaulay ring and M is a canonical module of R, then the assumption of Corollary 4.5 is also satisfied. However, in this case, note that R ⋉ M is Gorenstein; see the proof of [2, Theorem 3.3.6]. For example, Weston [18] gives a Cohen-Macaulay local ring with a Gorenstein module of rank two (thus with full support), but without a canonical module.

The openness of the CM-locus of a module
In this section, we study the openness of the CM-locus and MCM-locus of a module. More precisely, we observe that the same results as we gave in the previous sections hold for the Cohen-Macaulay property.
Let S be a ring. A ring homomorphism φ : R → S is said to be a Cohen-Macaulay homomorphism if φ is flat and all the fiber rings of φ are Cohen-Macaulay [14]. We give the following condition. Proof. The assertions can be shown in a similar way as in the proof of Lemma 4.1. However, note the following. First, it is seen that the same assertion as [ Corollary 5.2. Let S be a ring, and let φ : R → S be a ring homomorphism. Suppose that (5.0.1) holds for R, and φ is either essentially of finite type or Cohen-Macaulay. Then (5.0.1) also holds for S.
We give a lemma to state the main result of this section.
Lemma 5.3. Let M be a nonzero R-module, I an ideal of R, and x = x 1 , . . . , x n a sequence of elements in R. Suppose that there exists an integer r > 0 such that I r M = 0, and I i−1 M/I i M is a free R/I-module for each 1 i r. Then x is an R/I-regular sequence if and only if it is an M -regular sequence. In particular, depth R p /IR p = depth M p for any p ∈ V(I).
Proof. We prove the former assertion of the lemma by induction on n. Suppose that n = 1, and x = x 1 .
with exact rows by [2, Proposition 1.1.4]. Applying the case n = 1 shows the assertion. Next we prove the latter assertion of the lemma. Let p be a prime ideal of R containing I. Since M/IM is a nonzero free R/I-module, we obtain M p = 0. It is seen that I r M p = 0, and I i−1 M p /I i M p is a free R p /IR p -module for each 1 i r. The assertion follows from the former. Now we can prove the main result of this section by using [10,Theorem 24.5]. It corresponds to Theorems 3.6 and 4.4. Proof. First of all, we can freely replace our ring R with its localization R f for any element f ∈ R \ p to prove the theorem; we apply Lemma 2.5.
(i): We consider the case where p belongs to MCM R (M ). We prove the equivalence (2) ⇔ (3). First, we deal with the case ht p = 0. Since p is a minimal prime ideal of R, we may assume that p r = 0 for some integer r > 0, and p i−1 M/p i M is a free R/p-module for each 1 i r by (4) and (6) of Lemma 2.7. Fix q ∈ V(p). Note that M q = 0. Lemma 5.3 implies that On the other hand, p is the only minimal prime ideal of R since p r = 0. Thus, we obtain It follows from (5.4.1) and (5.4.2) that q/p is in CM(R/p) if and only if q is in MCM R (M ). Applying Lemma 2.6 to I = p and S = MCM R (M ), we see that the equivalence (2) ⇔ (3) holds. Next, we handle the general case. Put n = depth M p = dim R p . We may assume that there exists a sequence x = x 1 , . . . , x n of elements in p such that x is an M -regular sequence and ht xR = ht p = n by Lemma 2.7 (3). Fix q ∈ V(p). We have 6. The openness of the (S n )-locus of a module In this section, we study the openness of the (S n )-locus and (T n )-locus of a module. The main result of this section is the theorem below. Note that the assumptions of the theorem are weaker than the condition (2) in Lemma 2.4. . Let X be the closure of X. It suffices to show that X = X. There exists an ideal I of R such that X = V(I) since X is closed. We may assume I = R.
Claim. If p is a minimal prime ideal of I, then p belongs to X. Proof of Claim. We prove the claim by contradiction. Suppose that there exists a minimal prime ideal p of I, which belongs to S R n (M ).
(i): Suppose either p / ∈ Supp R (M ) or ht p < n. In these cases, S R n (M ) contains a nonempty open subset of V(p); see Remark 2.8 and the assumption of (1). We can choose f ∈ R \ p such that D(f ) ∩ V(p) is contained in S R n (M ) by Lemma 2.7 (1). Furthermore, it follows from Lemma 2.7 (4) that we may assume pR f = IR f . Now p is in V(I) = X, and D(f ) is a neighbourhood of p. Hence D(f ) ∩ X is nonempty. On the other hand, This is a contradiction. (ii): Suppose that p ∈ Supp R (M ) and ht p n. Since p belongs to S R n (M ) and ht p n, we have depth M p inf{n, ht p} = n. Now M p = 0. It follows from (3) and (4) of Lemma 2.7 that there exist a sequence x = x 1 , . . . , x n of elements in p and f ∈ R \ p such that x is an M f -regular sequence and pR f = IR f . Similarly as in the proof of (i), D(f ) ∩ X is nonempty. Let q ∈ D(f ) ∩ X. There exists a prime ideal q ′ of R such that q ′ ⊆ q and depth M q ′ < inf{n, ht q ′ } because q is not in S R n (M ). It is easy to see that q ′ belongs to the subset D(f ) ∩ X of D(f ) ∩ V(I) = D(f ) ∩ V(p). We obtain depth M q ′ grade(q ′ R f , M f ) n since x is contained in q ′ . This is a contradiction.
Let p 1 , . . . , p r be all the minimal prime ideals of I. The locus S R n (M ) is stable under generalization. Therefore, we have X = V(I) = r i=1 V(p i ) ⊆ X.  Proof. First of all, by Remark 2.8, we may assume that p is in Supp R (M ). Also, we can freely replace our ring R with its localization R f for an element f ∈ R \ p to prove the theorem; we apply Lemma 2.5.
(1): Since p belongs to S R n (M ) and ht p n, we have depth M p inf{n, ht p} = ht p. Put d = ht p. Note that M p = 0. It follows from (3), (4), and (6) of Lemma 2.7 that we may assume that the following three conditions are satisfied.
(i) There exist a sequence x = x 1 , . . . , x d in p such that x is an M -regular sequence, and p = √ xR. (ii) Set R = R/xR, p = p/xR, and M = M/xM . Then there exists an integer r > 0 such that p r = 0 and p i M /p i+1 M is a free R/p-module for each 1 i r. (iii) R/p satisfies (S n ). Now we claim that V(p) is contained in S R n (M ). Suppose that p ′ ∈ V(p) and q ∈ Spec(R) with q ⊆ p ′ , then we prove that depth M q inf{n, ht q}.
First, we deal with the case ht(q + p/p) n. Since q + p is contained in p ′ , we can choose q ′ ∈ V(q + p) such that ht(q ′ /p) = ht(q + p/p). It follows from the above three conditions and Lemma 5.3 that depth M q ′ − d = depth M q ′ = depth R q ′ /pR q ′ inf{n, ht(q ′ /p)} = ht(q ′ /p) = ht(q ′ /xR) = ht q ′ − d.
This says that M q ′ is maximal Cohen-Macaulay, and so is M q . We obtain depth M q ht q inf{n, ht q}.
Next, we consider the case ht(q + p/p) > n. It follows from (iii) and [2, Proposition 1.2.10 (a)] that grade(q + p/p, R/p) = inf{depth R q ′ /pR q ′ | q ′ /p ∈ V(q + p/p)} n. Hence, there exists an R/p-regular sequence y = y 1 , . . . , y n in q. It is also an M -regular sequence by (ii) and Lemma 5.3. By (i), the sequence x, y = x 1 , . . . , x d , y 1 , . . . , y n in p ′ is an M p ′ -regular sequence, and so is y. Therefore, y is an M q -regular sequence. We obtain depth M q n inf{n, ht q}.
(2): Put I = Ann R (M ), and R = R/I. We see that dim M q = ht(q/I), and q is in T R n (M ) if and only if q/I is in S R n (M ) for any q ∈ V(I). It follows from the assertion (1) and Lemma 2.6 that T R n (M ) contains a nonempty open subset of V(p).
The same result as Corollary 5.5 holds for Serre's condition (S n ).  Proof. The assertions follow from Theorems 6.1 and 6.4.
Let n 0 be an integer. We give the following condition.
(S n ) † : S n (R/p) contains a nonempty open subset of Spec(R/p) for all prime ideals p of R.
Proposition 6.6. Let m, n 0 be integers. Suppose that (S n ) † holds for R.