Strongly Gorenstein C-Homological Modules Under Change of Rings

Home , Completion of a ring

Bull. Korean Math. Soc. 58 (2021), No. 4, pp. 939–958 https://doi.org/10.4134/BKMS.b200673 pISSN: 1015-8634 / eISSN: 2234-3016

STRONGLY GORENSTEIN C-HOMOLOGICAL MODULES UNDER CHANGE OF RINGS

Yajuan Liu and Cuiping Zhang

Abstract. Some properties of strongly Gorenstein C-projective, C-injective and C-ﬂat modules are studied, mainly considering these properties under change of rings. Speciﬁcally, the completions of rings, the localizations and the polynomial rings are considered.

1. Introduction Unless stated otherwise, throughout this paper R is a commutative and noetherian ring with identity and C is a semidualizing R-module. Let R be a ring and we denote the category of left R-module by R-Mod. By PC (R), FC (R) and IC (R) denote the classes of all C-projective, C-flat and C-injective R-modules, respectively. For any R-module M, pdR(M) denotes the projective dimension of M. When R is two-sided noetherian, Auslander and Bridger [1] introduced the G-dimension, G-dimR(M) for every finitely generated R-module M. They proved the inequality G-dimR(M) ≤ pdR(M), with equality G-dimR(M) = pdR(M) when pdR(M) is finite. Several decades later, Enochs and Jenda [4,5] extended the ideas of Auslander and Bridger and introduced the Gorenstein projective, injective and flat dimensions. Bennis and Mahdou [3] studied a particular case of Gorenstein projective, injective and flat modules, which they called strongly Gorenstein projective, injective and flat modules. Yang and Liu [16] discussed some connections between strongly Gorenstein projective, injective and flat modules, and they considered these properties under change of rings. Specifically, they considered completions of rings and localizations. The nation of a “semidualizing module” is one of the most central notions in the relative homological algebra. This notion was first introduced by Foxby [7]. This notion has been investigated by many authors from different points

Received August 9, 2020; Revised March 5, 2021; Accepted March 10, 2021. 2010 Mathematics Subject Classiﬁcation. Primary 13D07, 16E65. Key words and phrases. Semidualizing module, strongly Gorenstein C-projective (C- injective, C-ﬂat) module, completion, localization, polynomial. This work was supported by National Natural Science Foundation of China (No. 11761061).

c 2021 Korean Mathematical Society 939 940 Y. J. LIU AND C. P. ZHANG of view. A semidualizing R-module C gives rise to several distinguished classes of modules. For instance, one has the classes PC (R), IC (R) and FC (R) of C- projective, C-injective and C-flat R-modules. Over a commutative noetherian ring, Holm and Jørgensen [8] introduced GC -projective, GC -injective and GC - flat modules. For Gorenstein homological theory with respect to semidualizing modules, Wang [14] gave another definition. An R-module M is called Goren- stein C-projective if there exists an exact sequence of R-modules

0 1 f1 f0 0 f 1 f P = ··· −→ C ⊗R P0 −→ C ⊗R P −→ C ⊗R P −→· · · , i ∼ 0 where each Pi and P is projective with M = Kerf , such that the complex HomR(P,Q) is exact for each C-projective R-module Q. Dually, Gorenstein C-injective modules are defined. An R-module M is said to be Gorenstein C-flat, if there exists an exact sequence of C-flat R-module

0 1 f1 f0 0 f 1 f F = ··· −→ C ⊗R F0 −→ C ⊗R F −→ C ⊗R F −→· · · , ∼ 0 with M = Kerf , such that complex E ⊗R F is exact for each C-injective R- modules E. Recently, Zhang, Liu, and Yang [17] introduced the concepts of strongly WP -Gorenstein, WI -Gorenstein and WF -Gorenstein modules and discussed the basic properties of these modules. Some results related to strongly Goren- stein projective, injective and flat modules were extended to strongly WP - Gorenstein, WI -Gorenstein and WF -Gorenstein modules. In this paper, we in- troduce the concepts of strongly Gorenstein projective, injective, flat modules with respect to a semidualizing module, which are different from the definition above and those in [17] and mainly consider these properties under change of rings. Specifically, the completions of rings, the localizations and the polynomial rings are considered.

2. Preliminaries In this section, we recall some definitions and known facts needed in the sequel. Definition ([17]). Let R be a ring. A finitely generated R-module C is called semidualizing if the following conditions are satisfied: (1) The natural homothety morphism R → HomR(C,C) is an isomorphism, i≥1 (2) ExtR (C,C) = 0. Definition ([17]). An R-module is C-flat (resp., C-projective) if it has the form C ⊗R F for some flat (resp., projective) R-module F . An R-module is C-injective if it has the form HomR(C,I) for some injective R-module I. Set

FC (R) = {C ⊗R F | F is a ﬂat R-module},

PC (R) = {C ⊗R P | P is a projective R-module}, STRONGLY GORENSTEIN C-HOMOLOGICAL MODULES 941

IC (R) = {HomR(C,I) | I is an injective R-module}.

Then PC (R) ⊆ FC (R). Clearly, if C = R, then PC (R), FC (R) and IC (R) are just the classes of ordinary projective, flat and injective R-modules. Definition. An R-module M is said to be strongly Gorenstein C-projective, if there exists an exact sequence of C-projective R-modules f f f f P = ··· −→ C ⊗R P −→ C ⊗R P −→ C ⊗R P −→· · · , ∼ with M = Kerf, such that the complex HomR(P,Q) is exact for each C- projective R-module Q. An R-module M is said to be strongly Gorenstein C-injective, if there exists an exact sequence of C-injective R-module f f f f I = ··· −→ HomR(C,I) −→ HomR(C,I) −→ HomR(C,I) −→· · · , ∼ with M = Kerf, such that the complex HomR(E, I) is exact for each C- injective R-module E. An R-module M is said to be strongly Gorenstein C-flat, if there exists an exact sequence of C-flat R-module f f f f F = ··· −→ C ⊗R F −→ C ⊗R F −→ C ⊗R F −→· · · , ∼ with M = Kerf, such that complex E ⊗R F is exact for each C-flat R- modules E. By SGPC (R), SGIC (R), SGF C (R) we denote the classes of all strongly Gorenstein C-projective, C-injective and C-flat R-modules, respectively. It is easy to see that, when C = R, the definitions correspond to those of strongly Gorenstein projective, injective and flat modules. In particular, for more similarly details about strongly Gorenstein C-projective, C-injective and C-flat R-modules, the reader may consult [3, 16, 17]. We have the following simple facts. Proposition 2.1. Let C be a semidualizing R-module. For any module M, the following are equivalent: (1) M is a strongly Gorenstein C-projective R-module; (2) there exists a short exact sequence 0 → M → C ⊗R P → M → 0, i≥1 where P is a projective R-module, and ExtR (M,Q) = 0 for any C-projective R-module Q; (3) there exists a short exact sequence 0 → M → C ⊗R P → M → 0, i≥1 where P is a projective R-module, and ExtR (M,L) = 0 for any module L with finite C-projective dimension; (4) there exists a short exact sequence 0 → M → C ⊗R P → M → 0, where P is a projective R-module, such that the sequence 0 → HomR(M,Q) → HomR(C ⊗R P,Q) → HomR(M,Q) → 0 is exact for any C-projective R- module Q. 942 Y. J. LIU AND C. P. ZHANG

Proof. (1) ⇔ (2) and (2) ⇔ (4) are obvious. (2) ⇒ (3) holds by dimension shifting. (3) ⇒ (2) is clear. 3. Main results Let (R, m) be a commutative local noetherian ring with the maximal ideal m and its residue field k and let E(k) be the injective envelope of k. R,b Mc will denote the m-adic completion of a ring R and an R-module M, respectively. In [16], Yang and Liu discussed some properties of strongly Gorenstein projective (injective, flat) R-modules in the completions of the rings. In the following, we give some properties of strongly Gorenstein C-projective (C-injective, C- flat) R-modules in the completion of the ring. Lemma 3.1. Let (R,m) be a local ring. If C be a semidualizing R-module, then Cb is a semidualizing module of Rb. Proof. This follows from [6, Theorem 2.5.14] and [6, Theorem 3.2.5]. Next are key lemmas, which play an important part. Lemma 3.2. Let (R, m) be a local ring. (1) If P is a projective R-module, then P ⊗R Rb is a projective Rb-module. (2) Assume that Rb is a projective R-module. If P is a projective Rb-module, then P is a projective R-module. i≥1 i≥1 Proof. (1) Since Ext (Rb ⊗R P, −) = HomR(P, Ext (R,b −)) = 0 by [12, Rb Rb i≥1 p. 258, 9.20], Ext (Rb ⊗R P, −) = 0. Hence Rb ⊗R P is a projective Rb-module. Rb (2) Since P is isomorphic to a summand of Rb(X) for some set X and Rb(X) is a projective R-module, it follows that P is a projective R-module. Lemma 3.3. Let (R, m) be a local ring. (1) If E is an injective R-module, then E ⊗R Rb is an injective Rb-module. (2) If E is an injective Rb-module, then E is an injective R-module. Proof. (1) Let E be any injective R-module. Then E is isomorphic to a sum- X mand of E(k) for some set X, and hence E ⊗R Rb is isomorphic to a summand of E(k)X ⊗ R = E (R/m)X ⊗ R by [6, Theorem 3.4.1]. Since E (R/m)X ⊗ R b Rb b b R b Rb b b R Rb is an injective Rb-module, E ⊗R Rb is an injective Rb-module. (2) Since Rb is a faithfully flat R-module by [6, Theorem 2.5.18]. Let M be a i≥1 i≥1 finitely generated R-module. Then Ext (M, E)⊗R Rb = Ext (M ⊗R R,b E⊗R R Rb Rb) = 0 by [6, Theorem 3.2.5]. Since E ⊗R Rb is an injective Rb-module by i≥1 [6, Theorem 3.2.16], ExtR (M, E) = 0. Hence E is an injective R-module. Lemma 3.4. Let (R, m) be a local ring. (1) If F is a flat R-module, then F ⊗R Rb is a flat Rb-module. (2) If F is a flat Rb-module, then F is a flat R-module. STRONGLY GORENSTEIN C-HOMOLOGICAL MODULES 943

Proof. (1) It is clear by [6, p. 43 Exercise 9]. R ∼ Rb (2) Let M be any R-module. Then Tori≥1(M, F ) ⊗R Rb = Tori≥1(M ⊗R R,b F ⊗R Rb) by [6, Theorem 2.1.11]. Since Rb is a faithfully ﬂat R-module by [6, Rb R Theorem 2.5.18], Tori≥1(M ⊗R R,b F ⊗R Rb) = 0. Hence Tori≥1(M, F ) = 0. It follows that F is a ﬂat R-module.

Proposition 3.5. Let (R, m) be a local ring. (1) If Rb is a projective R-module and M ∈ SGPC (R), then M ⊗R Rb ∈ SGP (R). Cb b (2) Let M be a ﬁnitely generated R-module. If M ∈ SGP (R), then M ∈ c Cb b c SGPC (R).

Proof. (1) There is an exact sequence 0 → M → C ⊗R P → M → 0 in R-Mod i≥1 with P projective and ExtR (M,Q) = 0 for any C-projective R-module Q. Then

0 → M ⊗R Rb → (C ⊗R P ) ⊗R Rb → M ⊗R Rb → 0 is exact. Since C⊗ P ⊗ R ∼ (C⊗ R)⊗ (R⊗ P ) ∼ C⊗ (R⊗ P ), R⊗ P is R R b = R b Rb b R = b Rb b R b R a projective Rb-module by Lemma 3.2(1). Then

0 → M ⊗ R → C ⊗ (R ⊗ P ) → M ⊗ R → 0 R b b Rb b R R b is exact. Let P be any projective Rb-module. Then P is a projective R-module by Lemma 3.2(2). So

i≥1 ∼ i≥1 Ext (M ⊗R R,b Cb ⊗ P ) = Ext (M ⊗R R,Cb ⊗R Rb ⊗ P ) Rb Rb Rb Rb ∼ i≥1 = Ext (M ⊗R R,Cb ⊗R P ) Rb ∼ Exti≥1(M, Hom (R,C ⊗ P )) = R Rb b R ∼ i≥1 = ExtR (M,C ⊗R P ) = 0 by [12, p. 258, 9.20]. Hence M ∈ SGP (R). c Cb b (2) There is an exact sequence 0 → M → C ⊗ P → M → 0 in R-Mod c b Rb c b with P projective. Since C ⊗ P ∼ C ⊗ R ⊗ P ∼ C ⊗ (R ⊗ P ) ∼ b Rb = R b Rb = R b Rb = C ⊗R P with P projective in R-Mod by Lemma 3.2(2). The following sequence

0 → Mc → C ⊗R P → Mc → 0 is exact in R-Mod. Let P be any projective R-module. Then i≥1 ∼ i≥1 Ext (M,C ⊗R P ) ⊗R Rb = Ext (M ⊗R R,Cb ⊗R P ⊗R Rb) R Rb ∼ i≥1 = Ext (M,c Cb ⊗ (Rb ⊗R P )) = 0 Rb Rb 944 Y. J. LIU AND C. P. ZHANG by [6, Theorem 3.2.5] and Lemma 3.2(1). Since Rb is a faithfully ﬂat R- i≥1 module, ExtR (M,C ⊗R P ) = 0. Since i≥1 ∼ i≥1 ExtR (M,Cc ⊗R P ) = ExtR (M ⊗R R,Cb ⊗R P ) ∼ i≥1 = HomR(R,b ExtR (M,C ⊗R P )) = 0

i≥1 by [12, p. 258, 9.20], we have ExtR (M,Cc ⊗R P ) = 0. Hence Mc ∈ SGPC (R). Dually, we discuss the properties of strongly Gorenstein C-injective R-modules. Proposition 3.6. Let (R, m) be a local ring. If Rb is a projective R-module, then the following statements hold. (1) If M ∈ SGI (R), then Hom (R,M) ∈ SGI (R). C R b Cb b (2) If Hom (R,M) ∈ SGI (R), then Hom (R,M) ∈ SGI (R). R b Cb b R b C

Proof. (1) There is an exact sequence 0 → M → HomR(C,E) → M → 0 in R- i≥1 Mod with E injective and ExtR (I,M) = 0 for any C-injective R-module I. Then

0 → HomR(R,Mb ) → HomR(R,b HomR(C,E)) → HomR(R,Mb ) → 0 is exact. Since Hom (R, Hom (C,E)) ∼ Hom (C ⊗ R,E) ∼ Hom (C ⊗ R ⊗ R,E) R b R = R R b = R R b Rb b ∼ Hom (C ⊗ R,E) = R b Rb b ∼ Hom (C, Hom (R,E)) = Rb b R b by [12, p. 258, 9.20] and [12, p. 258, 9.21], the sequence 0 → Hom (R,M) → Hom (C, Hom (R,E)) → Hom (R,M) → 0 R b Rb b R b R b is exact in Rb-Mod with HomR(R,Eb ) injective in Rb-module. Let E be any injective Rb-module. Then E is an injective R-module by Lemma 3.3(2). Since Hom (C, E) ∼ Hom (C ⊗ R, E) ∼ Hom (C, Hom (R, E)) Rb b = Rb R b = R Rb b ∼ = HomR(C, E) by [12, p. 258, 9.21], i≥1 ∼ i≥1 Ext (Hom (C,b E), HomR(R,Mb )) = Ext (Hom (C,b E) ⊗ R,Mb ) Rb Rb R Rb Rb ∼ Exti≥1(Hom (C, E),M) = R Rb b ∼ i≥1 = ExtR (HomR(C, E),M) = 0 by [12, p. 258, 9.21] and [6, Lemma 3.2.4]. Therefore Hom (R,M) ∈ SGI (R). R b Cb b STRONGLY GORENSTEIN C-HOMOLOGICAL MODULES 945

(2) There is an exact sequence 0 → Hom (R,M) → Hom (C, E) → Hom (R,M) → 0 R b Rb b R b in R-Mod with E injective in Rb-Mod. Note that Hom (C, E) ∼ Hom (C ⊗ R, E) ∼ Hom (C, Hom (R, E)) ∼ Hom (C, E) Rb b = Rb R b = R Rb b = R by [12, p. 258, 9.21]. Let I be any injective R-module. Then I ⊗R Rb is an injective Rb-module by Lemma 3.3(1), and hence i≥1 ExtR (HomR(C,I), HomR(R,Mb )) ∼ Exti≥1(Hom (C,I), Hom (R, Hom (R,M))) = R R Rb b R b ∼ i≥1 = Ext (Rb ⊗R HomR(C,I), HomR(R,Mb )) Rb ∼ i≥1 = Ext (Hom (C,Ib ⊗R Rb), HomR(R,Mb )) = 0. Rb Rb So HomR(R,Mb ) ∈ SGIC (R). Theorem 3.7. Let (R, m) be a local ring and M a finitely generated R-module. Consider the following statements: (1) M ∈ SGF C (R); (2) M ∈ SGF (R); c Cb b (3) Mc ∈ SGF C (R). Then (3) ⇒ (2) ⇔ (1). If Rb is a finitely generated projective R-module, then (2) ⇒ (3). Proof. TorRb (Hom (C,E(k)), M) ∼ TorRb (Hom (C,E(k)) ⊗ R, M) i≥1 Rb b c = i≥1 R R b c ∼ R = Tori≥1(HomR(C,E(k)),M) ⊗R R.b Since Rb is a faithfully flat R-module, TorRb (Hom (C,E(k)), M) = 0 ⇔ TorR (Hom (C,E(k)),M) = 0. i≥1 Rb b c i≥1 R

(1) ⇒ (2) There is an exact sequence 0 → M → C ⊗R F → M → 0 in R-Mod with F ﬂat. Then

0 → M → C ⊗ (F ⊗R R) → M → 0 c b Rb b c is exact with (C ⊗ F ) ⊗ R ∼ C ⊗ (F ⊗ R). Then F ⊗ R is a ﬂat R- R Rb b = b Rb R b R b b module by Lemma 3.4(1). Let I be any injective Rb-module. Then I is an injective R-module by Lemma 3.3(2). Hence I is isomorphic to a summand of E(k)X for some set X. Since Rb is a faithfully ﬂat R-module by [6, Theo- rem 3.2.26], TorRb (Hom (C,E(k)X ), M) ∼ TorRb (Hom (C,E(k))X , M) i≥1 Rb b c = i≥1 Rb b c ∼ TorRb (Hom (C,E(k)), M)X = 0. = i≥1 Rb b c 946 Y. J. LIU AND C. P. ZHANG

Thus TorRb (Hom (C, I), M) = 0. So M ∈ SGF (R). i≥1 Rb b c c Cb b (2) ⇒ (1) There is an exact sequence 0 → M → C ⊗ F → M → 0 in R-Mod c b Rb c b with F flat. Then F is a flat R-module by Lemma 3.4(2). Since ∼ ∼ HomR(Rb ⊗R R,Eb (k)) = HomR(R,b HomR(R,Eb (k))) = HomR(R,Eb (k)) ∼ ∼ ∼ by the proof of [13, Corollary 2.5], we have R ⊗R R = R. Since F = F ⊗ R = b b b Rb b ∼ ∼ ∼ F ⊗ (R ⊗R R) = (F ⊗ R) ⊗R R = F ⊗R R, we have C ⊗ F = (C ⊗R F ) ⊗R Rb b b Rb b b b b Rb b Rb. Since Rb is a faithfully flat R-module, it follows that

0 → M → C ⊗R F → M → 0 is exact in R-Mod. Let I be any injective R-module. Since I is isomorphic to a summand of E(k)X for some set X, then R ∼ R X Tori≥1(HomR(C,I),M) = Tori≥1(HomR(C,E(k)),M) = 0 R by [6, Theorem 3.2.26]. So Tori≥1(HomR(C,I),M) = 0. It follows that M ∈ SGF C (R). (3)⇒ (2) There is an exact sequence 0 → Mc → C ⊗R F → Mc → 0 in R-Mod with F ﬂat. Since Rb is a ﬂat R-module,

0 → Mc ⊗R Rb → (C ⊗R F ) ⊗R Rb → Mc ⊗R Rb → 0 is exact with C ⊗ F ⊗ R ∼ (C ⊗ R) ⊗ (R ⊗ F ) ∼ C ⊗ (R ⊗ F ). Since R R b = R b Rb b R = b Rb b R Rb ⊗R F is a ﬂat Rb-module by Lemma 3.4(1), 0 → M → C ⊗ (R ⊗ F ) → M → 0 c b Rb b R c ∼ is exact. Since Rb ⊗R Rb = Rb, TorRb (Hom (C,E(k)), M) ∼ TorRb (Hom (C,E(k)) ⊗ R,M ⊗ R ⊗ R) i≥1 Rb b c = i≥1 R R b R b R b ∼ R = Tori≥1(HomR(C,E(k)),M ⊗R Rb) ⊗R Rb ∼ R = Tori≥1(HomR(C,E(k)), Mc) ⊗R R.b

R Rb Since Tori≥1(HomR(C,E(k)), Mc) = 0, Tori≥1(HomR(C,E(k)), Mc) = 0. It follows that M ∈ SGF (R). c Cb b (2)⇒ (3) There is an exact sequence 0 → M → C ⊗ F → M → 0 in R-Mod c b Rb c b with F ﬂat. Since C ⊗ F ∼ C ⊗ R ⊗ F ∼ C ⊗ (R ⊗ F ) ∼ C ⊗ F, b Rb = R b Rb = R b Rb = R

0 → Mc → C ⊗R F → Mc → 0 ∼ is exact in R-Mod. Since Rb ⊗R Rb = Rb, 0 = TorRb (Hom (C,E(k)), M) i≥1 Rb b c ∼ Rb = Tori≥1(HomR(C,E(k)) ⊗R R,Mb ⊗R Rb ⊗R Rb) STRONGLY GORENSTEIN C-HOMOLOGICAL MODULES 947

∼ R = Tori≥1(HomR(C,E(k)),M ⊗R Rb) ⊗R Rb ∼ R = Tori≥1(HomR(C,E(k)), Mc) ⊗R R.b

R Hence Tori≥1(HomR(C,E(k)), Mc) = 0, and thus Mc ∈ SGF C (R).

Proposition 3.8. Let (R, m) be a local ring. Assume that Rb is a projective R-module. If M is a strongly Gorenstein C-flat R-module, then Rb ⊗R M is a strongly Gorenstein Cb-flat Rb-module. Proof. There exists an exact sequence of C-flat R-module

f f f f F = ··· −→ C ⊗R F −→ C ⊗R F −→ C ⊗R F −→· · · with M =∼ Kerf. Then

f f f f LF = ··· −→ Rb ⊗R C ⊗R F −→ Rb ⊗R C ⊗R F −→ Rb ⊗R C ⊗R F −→· · · is exact. So

f f f F = ··· −→ (R ⊗ C) ⊗ (R ⊗ F ) −→ (R ⊗ C) ⊗ (R ⊗ F ) −→ b R Rb b R b R Rb b R f (R ⊗ C) ⊗ (R ⊗ F ) −→· · · b R Rb b R ∼ is exact with Rb ⊗R M = Kerf. Then

f f f f ··· −→ C ⊗ (R ⊗ F ) −→ C ⊗ (R ⊗ F ) −→ C ⊗ (R ⊗ F ) −→· · · b Rb b R b Rb b R b Rb b R is exact in Rb-Mod with Rb ⊗R F ﬂat in Rb-Mod. Let E be any injective Rb- module. Then E is an injective R-module by Lemma 3.3(2). Hom (C, E) ⊗ (C ⊗ (R ⊗ F )) ∼ Hom (C, E) ⊗ (C ⊗ R ⊗ F ) Rb b Rb b Rb b R = Rb b Rb R b R ∼ Hom (C, E) ⊗ R ⊗ (C ⊗ F ) = Rb b Rb b R R ∼ Hom (C, E) ⊗ (C ⊗ F ) = Rb b R R ∼ Hom (C, Hom (R, E)) ⊗ (C ⊗ F ) = R Rb b R R ∼ = HomR(C, E) ⊗R (C ⊗R F ).

Sine HomR(C, E)⊗R F is exact, then HomR(C, E)⊗R LF is exact. So Rb⊗R M is a strongly Gorenstein Cb-ﬂat Rb-module. In this part, we consider these properties under localizations of rings. Let R be a commutative ring and S a multiplicatively closed set of R. Then S−1R = (R × S)/ ∼= {a/s | a ∈ R, s ∈ S} is a ring and S−1M = M × S/ ∼= {x/s | x ∈ M, s ∈ S} is an S−1R-module. If p is a prime ideal of R and −1 −1 S = R − p, then we will denote S M,S R by Mp,Rp, respectively. 948 Y. J. LIU AND C. P. ZHANG

Lemma 3.9. Let R be a commutative ring and S a multiplicatively closed set of R. If C is a semidualizing R-module, then S−1C is a semidualizing module of S−1R. −1 −1 ∼ −1 −1 ∼ Proof. Since HomS−1R(S C,S C) = HomS−1R(C ⊗R S R,C ⊗R S R) = −1 −1 −1 HomR(C,C) ⊗R S R = R ⊗R S R = S R by [6, Proposition 2.2.4] and −1 −1 [6, Theorem 3.2.5], S C is a semidualizing module of S R. Lemma 3.10 ([16, Lemma 3.16]). Let R be a commutative ring and S a multiplicatively closed set of R and A ∈ S−1R-Mod. If S−1R is a projective R-module, then A is a projective R-module if and only if A is a projective S−1R-module. Theorem 3.11. Let R be a commutative ring and S a multiplicatively closed set of R, and S−1R a projective R-module. Then the following statements hold. (1) If A is a strongly Gorenstein C-projective R-module, then S−1A is a strongly Gorenstein S−1C-projective S−1R-module. (2) If S−1R is a ﬁnitely generated R-module, then B is a strongly Goren- stein C-projective R-module if and only if B is a strongly Gorenstein S−1C- projective S−1R-module for any B ∈ S−1R-Mod.

Proof. (1) There is an exact sequence 0 → A → C ⊗R P → A → 0 in R-Mod with P projective. Then −1 −1 −1 0 → S A → S (C ⊗R P ) → S A → 0 −1 −1 ∼ −1 −1 is exact in S R-Mod. Since S (C ⊗R P ) = S C ⊗S−1R S P by [10, Proposition 5.17], S−1P is a projective S−1R-module by Lemma 3.10. Let Q be any projective S−1R-module. Since Q is a projective R-module by Lemma 3.10, i≥1 −1 −1 ExtS−1R(S A, S C ⊗S−1R Q) ∼ i≥1 −1 −1 = ExtS−1R(A ⊗R S R,S C ⊗S−1R Q) ∼ i≥1 −1 −1 = ExtR (A, HomS−1R(S R,S C ⊗S−1R Q)) ∼ i≥1 −1 = ExtR (A, S C ⊗S−1R Q) ∼ i≥1 −1 = ExtR (A, C ⊗R S R ⊗S−1R Q) ∼ i≥1 = ExtR (A, C ⊗R Q) = 0. So S−1A is a strongly Gorenstein S−1C-projective S−1R-module. (2) ⇒) Since B =∼ S−1B by [10, Proposition 5.17], the result is true by (1). ⇐) There is an exact sequence 0 → B → S−1C ⊗ P → B → 0 in S−1R- S−1R module with P projective. Then P is a projective R -module by Lemma 3.10 and S−1C ⊗ P ∼ C ⊗ S−1R ⊗ P ∼ C ⊗ P. S−1R = R S−1R = R Then

0 → B → C ⊗R P → B → 0 STRONGLY GORENSTEIN C-HOMOLOGICAL MODULES 949 is exact. Let Q be any projective R-module. Then S−1Q is a projective S−1R- module. Since S−1R is a finitely generated projective R-module, it follows that i≥1 ∼ i≥1 −1 ExtR (B,C ⊗R Q) = ExtR (B ⊗S−1R S R,C ⊗R Q) ∼ i≥1 −1 = ExtS−1R(B, HomR(S R,C ⊗R Q)) ∼ i≥1 −1 −1 = ExtS−1R(B,S HomR(S R,C ⊗R Q)) ∼ i≥1 −1 −1 −1 = ExtS−1R(B, HomS−1R(S R,S C ⊗S−1R S Q)) ∼ i≥1 −1 −1 = ExtS−1R(B,S C ⊗S−1R S Q) = 0 by [10, Proposition 5.17], [12, p. 258, 9.21] and [12, Theorem 3.84]. Thus B is a strongly Gorenstein C-projective R-module. Proposition 3.12. Let R be a commutative ring and S a multiplicatively closed set of R. Assume that S−1R is a faithfully flat R-module. If B is a finitely generated strongly Gorenstein S−1C-projective S−1R-module, then B is a strongly Gorenstein C-flat R-module.

−1 −1 Proof. There is an exact sequence 0 → B → S C ⊗S−1R P → B → 0 in S R- Mod with P projective. Then P is a flat R-module by [10, Theorem 5.18] and −1 ∼ −1 ∼ S C ⊗S−1R P = C ⊗R S R ⊗S−1R P = C ⊗R P. Let I be any injective R-module. Then S−1I is an injective S−1R-module. Since S−1R is a noetherian ring by [9, Theorem 85], i≥1 −1 −1 0 = HomS−1R(ExtS−1R(B,S C),S I) ∼ S−1R −1 −1 = Tori≥1 (HomS−1R(S C,S I), B) ∼ S−1R −1 = Tori≥1 (S HomR(C,I), B) ∼ −1 R = S Tori≥1(HomR(C,I), B) ∼ −1 R = S R ⊗R Tori≥1(HomR(C,I), B) by [6, Theorem 3.2.13] and [12, Theorem 9.49]. Since S−1R is a faithfully flat R R-module, Tori≥1(HomR(C,I), B) = 0, and hence B is a strongly Gorenstein C-flat R-module. Lemma 3.13. Let R be a commutative ring and S a multiplicatively closed set of R. Assume that S−1R is a finitely generated R-module. If I is an injective R- −1 −1 −1 module, then HomR(S R, HomR(C,I)) is an S C-injective S R-module. Proof. Since S−1R is a finitely generated R-module, −1 ∼ −1 −1 HomR(S R, HomR(C,I)) = S HomR(S R, HomR(C,I)) ∼ −1 −1 = HomS−1R(S R,S HomR(C,I)) ∼ −1 −1 −1 = HomS−1R(S R, HomS−1R(S C,S I)) 950 Y. J. LIU AND C. P. ZHANG

∼ −1 −1 = HomS−1R(S C,S I) by [10, Proposition 5.17] and [12, Theorem 3.84]. Since I is injective in R- −1 −1 −1 Mod, S I is injective in S R-Mod by [2, Lemma 1.2]. Hence HomR(S R, −1 −1 HomR(C,I)) is an S C-injective S R-module. Proposition 3.14. Let R be a commutative ring and S a multiplicatively closed set of R. If S−1R is a ﬁnitely generated projective R-module, then the following statements hold. −1 (1) If A is a strongly Gorenstein C-injective R-module, then HomR(S R,A) is a strongly Gorenstein S−1C-injective S−1R-module. −1 (2) For any B ∈ R-Mod, HomR(S R,B) is a strongly Gorenstein C- −1 injective R-module if and only if HomR(S R,B) is a strongly Gorenstein S−1C-injective S−1R-module.

Proof. (1) There is an exact sequence 0 → A → HomR(C,E) → A → 0 in R-Mod with E injective. So −1 −1 −1 0 → HomR(S R,A) → HomR(S R, HomR(C,E)) → HomR(S R,A) → 0 −1 −1 −1 −1 is exact in S R-Mod, HomR(S R, HomR(C,E)) is an S C-injective S R- module by Lemma 3.13. Let E be any injective S−1R-module. Then E is an injective R-module by [2, Lemma 1.2]. i≥1 −1 −1 ExtS−1R(HomS−1R(S C, E), HomR(S R,A)) ∼ i≥1 −1 −1 = ExtR (S R ⊗S−1R HomS−1R(S C, E),A) ∼ i≥1 −1 = ExtR (HomS−1R(S C, E),A) ∼ i≥1 −1 = ExtR (HomR(C, HomS−1R(S R, E)),A) ∼ i≥1 = ExtR (HomR(C, E),A) = 0 −1 −1 by [12, p. 258, 9.21]. So HomR(S R,A) is a strongly Gorenstein S C- injective S−1R-module. (2) ⇒) There is an exact sequence −1 −1 0 → HomR(S R,B) → HomR(C,E) → HomR(S R,B) → 0 in R-Mod with E injective. Then −1 −1 −1 −1 −1 0 → S HomR(S R,B) → S HomR(C,E) → S HomR(S R,B) → 0 −1 −1 −1 ∼ −1 is exact in S R-Mod and S HomR(S R,B) = HomR(S R,B) by [10, −1 ∼ −1 −1 Proposition 5.17]. Since S HomR(C,E) = HomS−1R(S C,S E) by [12, Theorem 3.84], −1 −1 −1 −1 0 → HomR(S R,B) → HomS−1R(S C,S E) → HomR(S R,B) → 0 is exact in S−1R-Mod and S−1E is an injective S−1R-module. Let E be any injective S−1R-module. Then E is an injective R-module by [2, Lemma 1.2]. So i≥1 −1 −1 −1 ExtS−1R(HomS−1R(S C,S E), HomR(S R,B)) STRONGLY GORENSTEIN C-HOMOLOGICAL MODULES 951

∼ i≥1 −1 −1 = ExtS−1R(S HomR(C, E), HomR(S R,B)) ∼ i≥1 −1 −1 = ExtR (HomR(C, E), HomS−1R(S R, HomR(S R,B))) ∼ i≥1 −1 = ExtR (HomR(C, E), HomR(S R,B)) = 0 −1 by [12, p. 258, 9.21] and [12, Theorem 3.84]. It follows that HomR(S R,B) is a strongly Gorenstein S−1C-injective S−1R-module. ⇐) There is an exact sequence −1 −1 −1 0 → HomR(S R,B) → HomS−1R(S C, E) → HomR(S R,B) → 0 in S−1R-Mod and E is an injective S−1R-module. Then E is an injective R- module by [2, Lemma 1.2]. Since −1 ∼ −1 HomS−1R(S C, E) = HomS−1R(S R ⊗R C, E) ∼ −1 = HomR(C, HomS−1R(S R, E)) ∼ = HomR(C, E),

−1 HomS−1R(S C, E) is an C-injective R-module. Let E be any injective R- module. Then S−1E is an injective S−1R-module. i≥1 −1 ExtR (HomR(C,E), HomR(S R,B)) ∼ i≥1 −1 −1 = ExtR (HomR(C,E), HomS−1R(S R, HomR(S R,B))) ∼ i≥1 −1 −1 = ExtS−1R(S R ⊗R HomR(C,E), HomR(S R,B)) ∼ i≥1 −1 −1 −1 = ExtS−1R(HomS−1R(S C,S E), HomR(S R,B)) = 0

−1 by [12, p. 258, 9.21] and [12, Theorem 9.50]. Hence HomR(S R,B) is a strongly Gorenstein C-injective R-module. Lemma 3.15. Let F be an S−1R-module. If F is a flat S−1R-module, then F is a flat R-module. Proof. There is an exact sequence 0 → M → N → Q → 0 in R-Mod. Then 0 → S−1M → S−1N → S−1Q → 0 is exact in S−1R-Mod. Since F is a flat S−1R-module, the sequence −1 −1 −1 0 → S M ⊗S−1R F → S N ⊗S−1R F → S Q ⊗S−1R F → 0 is exact. So −1 −1 −1 0 → M ⊗R S F → N ⊗R S F → Q ⊗R S F → 0 is exact by [10, Proposition 5.17]. Since S−1F =∼ F , the sequence

0 → M ⊗R F → N ⊗R F → Q ⊗R F → 0 is exact in R-Mod. Hence F is a ﬂat R-module. 952 Y. J. LIU AND C. P. ZHANG

Proposition 3.16. Let R be a commutative ring and S a multiplicatively closed set of R. (1) If A is a strongly Gorenstein C-flat R-module, then S−1A is a strongly Gorenstein C-flat R-module. (2) If A is a strongly Gorenstein C-flat R-module, then S−1A is a strongly Gorenstein S−1C-flat S−1R-module. (3) Let B be an S−1R-module. Then B is a strongly Gorenstein C-flat R- module if and only if B is a Gorenstein S−1C-flat S−1R-module. Proof. (1) There is a complete C-flat R-module

f f f f F = ··· −→ C ⊗R F −→ C ⊗R F −→ C ⊗R F −→· · · in R-Mod with A =∼ Kerf. Then

−1 −1 −1 −1 f −1 S f −1 S f −1 S f S F = ··· −→ S (C⊗RF ) −−−→ S (C⊗RF ) −−−→ S (C⊗RF ) −−−→· · · −1 −1 ∼ −1 −1 ∼ −1 is exact in S R-Mod with S A = KerS f. Since S (C⊗RF ) = C⊗RS F by [10, Proposition 5.17],

−1 −1 −1 −1 f −1 S f −1 S f −1 S f C ⊗ S F = ··· −→ C ⊗R S F −−−→ C ⊗R S F −−−→ C ⊗R S F −−−→· · · is exact in R-Mod. Since S−1F is a flat S−1R-module, S−1F is a flat R- module by Lemma 3.15. Let I be any injective R-module. Then HomR(C,I)⊗R −1 F is exact, and so S (HomR(C,I) ⊗R F) is exact. −1 ∼ −1 S (HomR(C,I) ⊗R F) = HomR(C,E) ⊗R S F ∼ −1 = HomR(C,E) ⊗R (C ⊗ S F) −1 by [10, Proposition 5.17], and hence HomR(C,E) ⊗R (C ⊗ S F) is exact. It follows that S−1A is a strongly Gorenstein C-flat R-module.

(2) There is an exact sequence 0 → A → C ⊗R F → A → 0 in R-Mod with F flat. Then −1 −1 −1 0 → S A → S (C ⊗R F ) → S A → 0 −1 −1 ∼ −1 −1 is exact in S R-Mod. S (C ⊗R F ) = S C ⊗S−1R S F by [10, Proposi- tion 5.17], and so −1 −1 −1 −1 0 → S A → S C ⊗S−1R S F → S A → 0 is exact in S−1R-Mod with S−1F flat in S−1R-Mod. Let E be any injective S−1R-module. Then E is an injective R-module by [2, Lemma 1.2]. So S−1R −1 −1 ∼ S−1R −1 −1 Tori≥1 (HomS−1R(S C,E),S A) = Tori≥1 (S HomR(C,E),S A) ∼ −1 R = S Tori≥1(HomR(C,E),A) = 0 by [12, Theorem 9.49]. Hence S−1A is a strongly Gorenstein S−1C-flat S−1R- module. STRONGLY GORENSTEIN C-HOMOLOGICAL MODULES 953

(3) ⇒) It is clear by (2). ⇐) There is an exact sequence of S−1R-modules

f −1 f −1 f −1 f F = ··· −→ S C ⊗S−1R F −→ S C ⊗S−1R F −→ S C ⊗S−1R F −→· · · , where F is a ﬂat S−1R-module with B =∼ Kerf. Since −1 ∼ −1 ∼ S C ⊗S−1R F = C ⊗R S F = C ⊗R F by [10, Proposition 5.17]. Then

f f f f LF = ··· −→ C ⊗R F −→ C ⊗R F −→ C ⊗R F −→· · · is exact. Since F is a ﬂat S−1R-module, F is a ﬂat R-module by Lemma 3.15. Let E be any injective R-module. Then S−1E is an injective S−1R-module by [2, Lemma 1.2].

−1 −1 −1 HomS−1R(S C,S E) ⊗S−1R (S C ⊗S−1R F ) ∼ −1 −1 = S HomR(C,E) ⊗S−1R (S C ⊗S−1R F ) ∼ −1 = HomR(C,E) ⊗R S (C ⊗R F ) ∼ −1 = HomR(C,E) ⊗R (C ⊗R S F ) ∼ = HomR(C,E) ⊗R (C ⊗R F )

−1 −1 by [10, Lemma 5.17]. Since HomS−1R(S C,S E) ⊗S−1R F is exact,

HomR(C,E) ⊗R LF is exact. So B is a strongly Gorenstein C-ﬂat R-module. If R is a ring, then R[x] is the polynomial ring. If M is a left R-module, write M[x] = R[x]⊗M. Since R[x] is a free R-module and since tensor product i commutes with sums, we may regard the elements of M[x] as ‘Vectors’ (x ⊗R −1 mi), i ≥ 0,Mi ∈ M with almost all mi = 0. M[[x ]] is the R[x]-module such −1 −1 −1 that x(m0 + m1x + ··· ) = m1 + m2x + ··· and r(m0 + m1 + ··· ) = −1 rm0 + rm1 + ··· , where r ∈ R. Lemma 3.17. If C is a semidualizing R-module, then C[x] is a semidualizing R-module of R[x].

Proof. This follows from [6, Theorem 3.2.15]. Proposition 3.18. Let m be an R-module. Then the following statements hold. (1) If M is a strongly Gorenstein C-projective R-module, then M[x] is a strongly Gorenstein C[x]-projective R[x]-module. (2) If M[x] is a strongly Gorenstein C[x]-projective R[x]-module, then M[x] is a strongly Gorenstein C-projective R-module. 954 Y. J. LIU AND C. P. ZHANG

Proof. (1) There is an exact sequence 0 → M → C ⊗R P → M → 0 in R-Mod i≥1 with P projective and ExtR (M,Q) = 0 for any C-projective R-module Q.

0 → M ⊗R R[x] → (C ⊗R P ) ⊗R R[x] → M ⊗R R[x] → 0 ∼ ∼ is exact. Since (C ⊗R P ) ⊗R R[x] = (C ⊗R R[x]) ⊗R[x] (R[x] ⊗R P ) = C[x] ⊗R[x] (P [x]). P [x] is a projective R[x]-module by [10, Proposition 5.11], then

0 → M[x] → C[x] ⊗R[x] P [x] → M[x] → 0 ∼ ∼ (N) is exact. Let Q be any projective R[x]-module, Q[x] = R[x] ⊗R Q = Q , (N) then Q is a projective R[x]-module, and so Q is a projective R-module by [10, Proposition 5.11]. Thus i≥1 ∼ i≥1 ∼ i≥1 ExtR[x](M[x], Q) = ExtR[x](R[x] ⊗R M, Q) = ExtR (M, HomR[x](R[x], Q)) ∼ i≥1 = ExtR (M, Q) = 0 by [12, p. 258, 9.21], and hence M[x] is a strongly Gorenstein C[x]-projective R[x]-module. (2) There is an exact sequence 0 → M[x] → C[x] ⊗R[x] P → M[x] → 0 ∼ ∼ in R[x]-Mod with P projective. Since C[x] ⊗R[x] P = C ⊗R R[x] ⊗R[x] P = ∼ C ⊗R (R[x] ⊗R[x] P ) = C ⊗R P with P projective in R-Mod by [10, Proposition 5.11], we have the following exact sequence

0 → M[x] → C ⊗R P → M[x] → 0. Let P be any projective R-module. Then P [x] is a projective R[x]-module by [10, Proposition 5.11]. i≥1 ∼ i≥1 0 = ExtR[x](M[x],C[x] ⊗R[x] P [x]) = ExtR[x](M[x],C ⊗R P [x]) by [12, p. 258, 9.20], Since P is isomorphic to a summand of P [x], it follows i≥1 that ExtR (M,C ⊗R P ) = 0. i≥1 ∼ i≥1 ExtR (M[x],C ⊗R P ) = ExtR (M ⊗R R[x],C ⊗R P ) ∼ i≥1 = HomR(R[x], ExtR (M,C ⊗R P ))

i≥1 i≥1 by [12, p. 258, 9.20]. Since ExtR (M,C ⊗R P ) = 0, ExtR (M[x],C ⊗R P ) = 0. Hence M[x] is a strongly Gorenstein C-projective R-module. Proposition 3.19. Let m be an R-module. Then the following statements hold. (1) If M is a strongly Gorenstein C-injective R-module, then M[[x−1]] is a strongly Gorenstein C[x]-injective R[x]-module. (2) If M[[x−1]] is a strongly Gorenstein C[x]-injective R[x]-module, then M[[x−1]] is a strongly Gorenstein C-injective R-module. STRONGLY GORENSTEIN C-HOMOLOGICAL MODULES 955

Proof. (1) There is an exact sequence 0 → M → HomR(C,E) → M → 0 in i≥1 R-Mod with E injective and ExtR (I,M) = 0 for any C-injective R-module I. So

0 → HomR(R[x],M) → HomR(R[x], HomR(C,E)) → HomR(R[x],M) → 0 is exact in R[x]-module. Since ∼ HomR(R[x], HomR(C,E)) = HomR(C ⊗R R[x],E) ∼ = HomR(C ⊗R R[x] ⊗R[x] R[x],E) ∼ = HomR(C[x] ⊗R[x] R[x],E) ∼ = HomR[x](C[x], HomR(R[x],E)), the sequence

0 → HomR(R[x],M) → HomR[x](C[x], HomR(R[x],E)) → HomR(R[x],M) → 0 is exact with HomR(R[x],E) injective in R[x]-Mod by [11, Lemma 1.2]. Let E be any injective R[x]-module. Then E is an injective R-module. i≥1 ExtR[x](HomR[x](C[x], E), HomR(R[x],M)) ∼ i≥1 = ExtR (HomR[x](C[x], E) ⊗R[x] R[x],M) ∼ i≥1 = ExtR (HomR[x](C[x], E),M) ∼ i≥1 = ExtR (HomR(C, E),M) = 0 −1 ∼ by [12, p. 258, 9.21] and [6, Lemma 3.2.4]. Since M[[x ]] = HomR(R[x],M) by [11, Lemma 1.2]. Thus M[[x−1]] is a strongly Gorenstein C[x]-injective R[x]- module. −1 (2) There is an exact sequence 0 → M[[x ]] → HomR[x](C[x], E) → M[[x−1]] → 0 in R[x]-Mod with E injective. Since ∼ ∼ HomR[x](C[x], E) = HomR[x](C ⊗R R[x], E) = HomR(C, HomR[x](R[x], E)) ∼ = HomR(C, E), the sequence −1 −1 0 → M[[x ]] → HomR(C, E) → M[[x ]] → 0 is exact. Let E be any injective R-module. Then HomR(R[x],E) is an injective R[x]-module. i≥1 −1 ExtR (HomR(C, HomR(R[x],E)),M[[x ]]) ∼ i≥1 −1 = ExtR[x](HomR[x](C[x], HomR(R[x],E)),M[[x ]]) = 0 by [12, p. 258, 9.21]. Since E is isomorphic to a summand of HomR(R[x],E), i≥1 −1 ExtR (HomR(C,E),M[[x ]]) = 0. −1 Thus M[[x ]] is a strongly Gorenstein C-injective R-module. 956 Y. J. LIU AND C. P. ZHANG

Proposition 3.20. Let m be an R-module. Then the following statements hold. (1) If M is a strongly Gorenstein C-flat R-module, then M[x] is a strongly Gorenstein C[x]-flat R[x]-module. (2) If M[x] is a strongly Gorenstein C[x]-flat R[x]-module, then M[x] is a strongly Gorenstein C-flat R-module.

Proof. (1) There is an exact sequence 0 → M → C ⊗R F → M → 0 in R-Mod with F ﬂat. Then

0 → M ⊗R R[x] → C ⊗R F ⊗R R[x]) → M ⊗R R[x] → 0 ∼ ∼ is exact with (C ⊗R F ) ⊗R[x] R[x] = C ⊗R R[x] ⊗R[x] (F ⊗R R[x]) = C[x] ⊗R[x] F [x]. F [x] is a ﬂat R[x]-module by [15, Theorem, 3.8.21]. Let I be any injective R[x]-module. Then I is an injective R-module.

R[x] + ∼ R[x] + Tori≥1 (HomR[x](C[x], I),M[x]) = Tori≥1 (HomR[x](C ⊗R R[x], I),M[x]) ∼ R[x] + = Tori≥1 (HomR(C, I),M[x]) ∼ i≥1 + = ExtR[x](M[x], HomR(C, I) ) ∼ i≥1 + = ExtR[x](M ⊗R R[x], HomR(C, I) ) ∼ i≥1 + = ExtR (M, HomR(C, I) ) ∼ R + = Tori≥1(HomR(C, I),M) = 0 by [12, p. 258, 9.21] and [6, Lemma 3.2.1]. Thus M[x] is a strongly Goren- stein C[x]-ﬂat R[x]-module.

(2) There is an exact sequence 0 → M[x] → C[x] ⊗R[x] F → M[x] → 0 ∼ ∼ in R[x]-Mod with F ﬂat. C[x]⊗R[x] F = C ⊗R (R[x]⊗R[x] F = C ⊗R (R[x]⊗R[x] ∼ F ) = C ⊗R F . Then

0 → M[x] → C ⊗R F → M[x] → 0 is exact. Let E be any injective R-module. Then HomR(R[x],E) is an injective R[x]-module.

R[x] + Tori≥1 (M[x], HomR[x](C[x], HomR(R[x],E))) ∼ R[x] + = Tori≥1 (M[x], HomR[x](C ⊗R R[x], HomR(R[x],E))) ∼ R[x] + = Tori≥1 (M[x], HomR(C, HomR[x](R[x], HomR(R[x],E)))) ∼ R[x] + = Tori≥1 (M[x], HomR(C, HomR(R[x],E))) ∼ i≥1 + = ExtR (M[x], HomR(C, HomR(R[x],E)) ) ∼ R + = Tori≥1(M[x], HomR(C, HomR(R[x],E))) STRONGLY GORENSTEIN C-HOMOLOGICAL MODULES 957 by [12, p. 258, 9.21] and [6, Lemma 3.2.1]. Thus M[x] is a strongly Gorenstein C[x]-ﬂat R[x]-module. Since E is isomorphic to a summand of HomR(R[x],E),

R Tori≥1(M[x], HomR(C,E)) = 0.

Thus M[x] is a strongly Gorenstein C-ﬂat R-module.

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Yajuan Liu Department of Applied Mathematics Northwest Normal University Lanzhou 730070, P. R. China Email address: [email protected]

Cuiping Zhang Department of Mathematics Northwest Normal University Lanzhou 730070, P. R. China Email address: [email protected]