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-injec- tive and C-flat modules are studied, mainly considering these properties under change of rings. Specifically, the completions of rings, the localiza- tions 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 com- plex 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 polyno- mial 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) = fC ⊗R F j F is a flat R-moduleg; PC (R) = fC ⊗R P j P is a projective R-moduleg; STRONGLY GORENSTEIN C-HOMOLOGICAL MODULES 941 IC (R) = fHomR(C; I) j I is an injective R-moduleg: 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.