On Ding Projective Complexes

Transcription

1 Acta Mathematica Sinica, English Series Nov, 218, Vol 34, No 11, pp Published online: June 6, Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 218 On Ding Projective Complexes Gang YANG Xuan Shang DA Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 737, P R China yanggang@maillzjtucn @qqcom Abstract In the paper, Ding projective modules and Ding projective complexes are considered In particular, it is proven that Ding projective complexes are precisely the complexes X for which each X m is a Ding projective R-module for all m Z Keywords Gorenstein projective modules, Ding projective and Ding injective modules, Ding projective complexes MR(21) Subject Classification 18G25, 18G35, 55U15, 55U35 1 Introduction In 1966, Auslander introduced the notion of G-dimension of a finite R-module over a commutative noetherian local ring In 1969, Auslander and Bridger extended this notion to two sided noetherian rings Calling the modules of G-dimension zero Gorenstein projective modules, in 1995, Enochs and Jenda defined Gorenstein projective (whether finitely generated or not) and Gorenstein injective modules over an arbitrary ring [11] Another extension of the G-dimension is based on Gorenstein flat modules These modules were introduced by Enochs et al [13] Gorenstein homological algebra is the relative version of homological algebra that uses the Gorenstein projective (Gorenstein injective, Gorenstein flat respectively) modules or complexes instead of the usual projective (injective, flat respectively) modules or complexes It was found that most results from traditional homological algebra have an analog in Gorenstein homological algebra, particularly over a Gorenstein ring R (that is, R is a two-sided Noetherian ring with finite left and right self-injective dimensions) The book [12] is a standard reference, though many authors have already studied the subject Since it is well known that any projective module is always flat in homological algebra, naturally, one wants to know whether any Gorenstein projective module is Goresntein flat or not in Gorenstein homological algebra There is positive answer to it over Goresntein rings [12] To study this problem over more general rings, Ding et al [3] introduced strongly Gorenstein flat modules As a dual notion, Gorenstein FP-injective modules were also defined [25] In [17] Gillespie renamed these two classes of modules as Ding projective and Ding injective modules respectively Ding projective modules and Ding injective modules over coherent rings possess Received October 1, 217, accepted March 12, 218 Supported by National Natural Science Foundation of China (Grant Nos and ) and Natural Science Foundation of Gansu Province of China (Grant No 17JR5RA91)

2 On Ding Projective Complexes 1719 many nice properties analogous to Gorenstein projective and Gorenstein injective modules over Noetherian rings, for examples see [17, 21] The study of relationships between complexes and their terms is an important research subject For instance, the well-known result that a complex δ X m+1 δ X m+1 X X m m X m 1 of R-modules is projective (respectively, injective, flat) if and only if each R-module Ker(δm) X is projective (respectively, injective, flat) in R-Mod and X is acyclic Being motivated and led partly, several authors investigated Gorenstein projective, injective and flat complexes [6, 9, 15, 23, 24, 29, 3], and Ding projective and injective complexes [18, 32] In [32], Yang et al showed that Ding projective complexes are precisely the complexes X for which each X i is Ding projective and any morphism X F is null homotopic whenever F is a flat complex It was shown recently by Gillespie in [18] that the null homotopic condition is automatically satisfied when the ring R is Ding Chen (ie, R is both left and right coherent and FP-id( R R)andFP-id(R R ) are both less than or equal to n for some non-negative integer n) In the present article, we will remove thoroughly the Ding Chen condition of the ring R In particular, we prove the following result Theorem 11 A complex G C(R) is Ding projective if and only if each term G m is Ding projective in R-Mod We should point out that our proof of the result depends heavily on Neeman s characterization of the pure acyclic complexes of flat modules in terms of the complexes of projective modules [26] (also see [5] and [27]) 2 Preliminaries Throughout this paper, R is an associate ring with identity and R-Mod is the category of left R-modules As usual, we use C(R) to denote the category of complexes of left R-modules When we say R-module, without an adjective, we mean left R-module δi+1 X A complex X of R-modules is a sequence X i+1 X i modules and R-homomorphisms such that δi XδX i+1 δ X i X i 1 of R- foralli Z A complex X is said to be bounded above if X i holdsfori, bounded below if X i holdsfori, and bounded if it is bounded above and below, ie X i holdsfor i Let X be a complex and let m be an integer The m-th cycle module is defined as Ker(δ X m) and is denoted by Z m X The m-th boundary module is Im(δ X m+1) and is denoted by B m XThem-th homology module of X is the module H m (X) Z m X/B m X The complex X is said to be acyclic (exact) ifits homology module H i (X) for all i Z The m-fold shift of X is the complex Σ m X given by (Σ m X) i X i m and δ Σm X i ( 1) m δi m X Usually, Σ1 X is denoted simply by ΣX Let X and Y be two complexes We will let Hom R (X, Y ) denote the hom-complex of Z- modules with m-th component Hom R (X, Y ) m i Z Hom R(X i,y i+m ) and differential (δ(g)) i δi+m Y g i ( 1) m g i 1 δi X for g (g i ) i Z i Z Hom R(X i,y i+m ) By a morphism f : X Y we mean a sequence f i : X i Y i such that δi Y f i f i 1 δi X for all i Z A morphism f : X Y is said to be null homotopic if there exists a sequence s i : X i Y i+1 such that f i δi+1 Y s i + s i 1 δi X for all i Z The mapping Cone Cone(f) of a morphism f : X Y is

3 172 Yang G and Da X S defined as Cone(f) i Y i X i 1 with δ Cone(f) i ( δ Y i f i 1 δ X i 1 If M is an R-module, then we denote the complex M with M in the m-th degree by S m (M), and denote the complex M Id M with M in the m 1andm-th degrees by D m (M) Usually, S (M) is denoted simply by M We use Hom(X, Y ) to present the group of all morphisms from X to Y Recall that a complex P is projective if the functor Hom(P, ) is exact Equivalently, P is projective if and only if P is acyclic and Z i P is a projective R-module for each i Z For example, if M is a projective R-module, then each complex D m (M) is projective A injective complex is defined dually Thus the category of complexes C(R) ofr-modules has enough projectives and injectives, we can compute right derived functors Ext i (X, Y )ofhom(, ) In particular, Ext 1 (X, Y ) will denote the group of (equivalent classes) of short exact sequences Y Z X under the Baer sum operation There is a subgroup Ext 1 dw(x, Y ) Ext 1 (X, Y ) consisting of the degreewise split short exact sequences That is, those for which each Y m Z m X m is split exact The following lemma gives a well-known connection between Ext 1 dw(, ) andtheabove hom-complex Hom R (, ) Lemma 21 Let X and Y be two complexes Then there are isomorphisms Ext 1 dw(x, Σ m 1 Y ) H m Hom R (X, Y )Hom K(R) (X, Σ m Y ), where K(R) is the homotopy category of complexes In particular, the hom-complex Hom R (X, Y ) is acyclic if and only if for each m Z, any morphism f :Σ m X Y (or f : X Σ m Y ) is homotopic to Recall from [11] that an R-module M is Gorenstein projective if there exists an exact sequence P 1 P P P 1 of projective R-modules with M Ker(P P 1 ) such that Hom R (,P) leaves the sequence exact whenever P is a projective R-module Gorenstein injective modules are defined dually As special cases of the Gorenstein projective and Gorenstein injective modules, the Ding projective and Ding injective modules (renamed by Gillespie [17]) were introduced and studied by Ding, Mao and co-authors in [3] and [25] as strongly Gorenstein flat and Gorenstein FP-injective modules respectively Definition 22 ([3]) An R-module M is called Ding projective if there exists an exact sequence of projective R-modules P 1 P P P 1 with M Ker(P P 1 ) and which remains exact after applying Hom R (,F) for any flat R-module F Definition 23 ([25]) An R-module N is called Ding injective if there exists an exact sequence of injective R-modules I 1 I I I 1 with N Ker(I I 1 ) and which remains exact after applying Hom R (J, ) for any FPinjective R-module J Note that every Ding projective (respectively, Ding injective) R-module is clearly Gorenstein projective (respectively, Gorenstein injective), and if R is Ding Chen, then every Gorenstein )

4 On Ding Projective Complexes 1721 projective (respectively, Gorenstein injective) R-module is Ding projective (respectively, Gorenstein injective) [18, Theorem 11] By [31, Theorem 26], the class of Ding projective modules is projectively resolving, that is, all projective modules are Ding projective, and for every short exact sequence L M N of modules with N being Ding projective, L is Ding projective if and only if M is Ding projective The class of Ding projective modules is easily seen to be closed under arbitrary direct sums by the definition Recall from [8] that a complex F is flat if it is acyclic and for each i Z, Z i F is a flat R-module (also see [7, 16]) A complex J is FP-injective if it is acyclic and for each i Z, Z i J is an FP-injective R-module (see [32]) Definition 24 A complex G is called Ding projective if there exists an exact sequence of projective complexes P 1 P P P 1 with G Ker(P P 1 ) and which remains exact after applying Hom(,F) for any flat complex F Definition 25 A complex H is called Ding injective if there exists an exact sequence of injective complexes I 1 I I I 1 with H Ker(I I 1 ) and which remains exact after applying Hom(J, ) for any FP-injective complex J In the following, we will denote the subcategories of projective modules, injective modules, and flat modules by P, I, andf respectively we use DP and DI to denote the subcategories of Ding projective modules and Ding injective modules respectively If A is a subcategory in R-Mod, then we use C(A) (respectively, C (A)) to denote the subcategory of unbounded complexes (respectively, bounded above complexes) with all terms in A 3 Main Results This section is devoted to studying Ding projective modules and complexes of Ding projective modules Recall from [2] that an R-module M is said to be of type FP (or super finitely presented) ifm has a projective resolution P n P n 1 P 1 P M with each P i being finitely generated We begin with the following result Theorem 31 Let M be an R-module of type FP If M is Gorenstein projective, then M is Ding projective Proof Since M is of type FP and Gorenstein projective, there is a projective resolution P 2 P 1 P M, ( ) of M, which remains exact after applying the functor Hom R (,Q) for any projective module Q, whereeachp i is finitely generated Because M is Gorenstein projective, one has an exact sequence M P L, where P is a projective module and L is a Gorenstein projective module Now it follows from [26, Lemma 32] that M P factors as M g P 1 h P,

5 1722 Yang G and Da X S where P 1 is finitely generated and projective since M is of type FP (of course, finitely presented) and P is flat Note that g : M P 1 is injective, and so there is an exact sequence M P 1 N such that N Coker(g) is easily seen of type FP For any projective module Q, sincethe sequence M P L remains exact after applying the functor Hom R (,Q), it is easily seen that M P 1 N remains exact after applying Hom R (,Q) This implies that Ext 1 R(N,Q) andson is Gorenstein projective [2, Corollary 211] Now we have shown that the module N has the same properties as M Then, one can use the same procedure to construct an exact sequence M P 1 P 2, ( ) which remains exact after applying the functor Hom R (,Q) for any projective module Q, where each P i is finitely generated and projective Assembling the sequences ( ) and( ), we get an exact sequence of projective modules P 2 P 1 P P 1 P 2, (#) with M Ker(P 1 P 2 )andeachp i finitely generated and which remains exact after applying the functor Hom R (,Q) for any projective module Q Let F be a flat module, and {Q α } α Λ be a direct system of projective modules such that F lim Q α Then, by [19, Lemma 316], we have a commutative diagram Hom R (P 1, lim Q α ) Hom R (P, lim Q α ) Hom R (P 1, lim Q α ) lim Hom R (P 1,Q α ) lim Hom R (P,Q α ) lim Hom R (P 1,Q α ) with all vertical arrows isomorphisms It follows from exactness of the lower row that the upper row is also exact, that is, the sequence (#) remains exact after applying the functor Hom R (,F) for any flat module F ThisprovesthatM is a Ding projective module In the following, we will be concerned with direct and inverse limits of systems of objects of R-Mod or C(R) Let λ be an ordinal Recall from [1] that a direct (inverse) system {(X α,f αβ ) α β λ} is said to be continuous if X and if for each limit ordinal β λ we have X β lim X α (or X β lim X α ) with the limit over the α<β The direct (inverse) system {(X α,f αβ ) α β λ} is said to be a system of monomorphisms (epimorphisms) if all the morphisms f αβ in the system are monomorphisms (epimorphisms) In the category R-Mod or C(R), we note if {(X α,f αβ ) α β λ} is a continuous direct system such that each f α,α+1 : X α X α+1 is a monomorphism whenever α +1 λ, then {(X α,f αβ ) α β λ} is a system of monomorphisms If {(X α,f βα ) α β λ} is a continuous inverse system such that each f α+1,α : X α+1 X α is an epimorphism, then {(X α,f βα ) α β λ} is a system of epimorphisms

6 On Ding Projective Complexes 1723 Let L be a class of objects of R-Mod or C(R) which is closed under isomorphisms Recall from [1] that an object X is said to be a direct (inverse) transfinite extensions of objects of L if X lim X α (or X lim X α ) for a continuous direct (inverse) system {(X α,f αβ ) α β λ} of monomorphisms (epimorphisms) such that Coker(X α X α+1 )(Ker(X α+1 X α )) is in L whenever α +1 λ L is said to be closed under direct (inverse) extensions if each direct (inverse) transfinite extensions of objects in L is also in L Itwasshownthattheclass L {M Ext 1 (M,L) for any object L L}is closed under direct transfinite extensions [4, Theorem 12], and the class L {N Ext 1 (L, N) for any object L L}is closed under inverse transfinite extensions [28, Theorem 23] Enochs et al [1] proved that the class of all Gorenstein projective modules is closed under direct transfinite extensions, and the class of all Gorenstein injective modules is closed under inverse transfinite extensions One can see easily that the classes of Ding projective and Ding injective modules share these properties too Proposition 32 The class DP (respectively, DI) of Ding projective (respectively, Ding injective) modules is closed under direct (respectively, inverse) transfinite extensions Proof The proof is similar to that of [1, Theorem 32] Lemma 33 Assume that G C (DP) is an acyclic complex If the sequence Hom R (G, F ) of abelian groups is acyclic for any flat module F,thenExt 1 (G, F )for any F C(F) Proof Use [22, Lemma 31] Definition 34 Let X be a complex and let m be an integer The hard truncation above of X at m, denotedx m,isthecomplex X m X m δm X δm 1 X X m 1 X m 2 Similarly, the hard truncation below of X at m, denotedx m,isthecomplex δ X m+2 δ X m+1 X m X m+2 X m+1 X m In the next, we will show that any (bounded above) complex of Ding projective modules can be approximated by a (bounded above) complex of projective modules Lemma 35 If G C (DP), then there exists an exact sequence G P C such that the following conditions are satisfied: (1) P C (P), (2) C C (DP) is acyclic, (3) Hom R (C, F) is acyclic for any flat module F Proof If G C (DP), then we should assume without loss of generality that G : G G 1 G 2 Now we denote G(n) G n, the hard truncation below of G at n for each n Then {(G(n),α mn ) m >n } forms an inverse system in C(R) andg lim G(n), where α mn :

7 1724 Yang G and Da X S G(m) G(n) is the following natural projection for any m n G(m) G G 1 G n G m α mn G(n) G G 1 G n We will show by induction on n Forn,sinceG is Ding projective, there exists an exact f sequence G P P 1 P 2 with each P i projective and which remains exact after applying the functor Hom R (,F) for any flat module F LetP() : P P 1 P 2, and consider the following monomorphism of complexes φ() : G() P () G() G φ() f P () P P 1 P 2 Let C() Coker(φ()), that is C() : G P 1 P 2 with G Coker(f) Clearly, one has P () C (P), C() C (DP) is acyclic, and Hom R (C(),F) is acyclic for any flat module F Now suppose that there is a monomorphism φ(n) :G(n) P (n) withp (n) C (P) and C(n) Coker(φ(n)) C (DP) forn as follows G(n) G G 1 G n φ(n) f f 1 f n P (n) P δ P 1 δ 1 P n δ n P n 1 where C(n) is acyclic, and Hom R (C(n),F) is acyclic for any flat module F Let G(n +1): d G d 1 d n g G 1 G n G n 1,andlet G n 1 Q n 1 Q n 2 Q n 3 be an exact sequence with each Q i projective and it remains exact after applying the functor Hom R (,F) for any flat module F We denote by Q the complex Q n 1 Q n 2 Q n 3 with Q n 1 in the ( n 1)-th degree By the above proof, we have a monomorphism ι : S n 1 (G n 1 ) Q such that Coker(ι) C (DP) is acyclic, and also Hom R (Coker(ι),F) is acyclic for any flat module F Let μ :Σ 1 G(n) S n 1 (G n 1 ) be the following morphism Σ 1 G(n) G d G n+1 G n μ S n 1 (G n 1 ) G n 1 Note that the sequence d n Σ 1 Σ 1φ(n) G(n) Σ 1 Σ 1π(n) P (n) Σ 1 C(n) is exact Then it follows from Lemma 33 that the sequence Hom(Σ 1 C(n),Q) Hom(Σ 1 P (n),q) Hom(Σ 1 G(n),Q) Ext 1 (Σ 1 C(n),Q)

8 On Ding Projective Complexes 1725 is exact, and so there exists a morphism ν :Σ 1 P (n) Q such that the following diagram commutes Σ 1 Σ 1 G(n) φ(n) Σ 1 P (n) μ S n 1 (G n 1 ) ι Q ν Thus one gets by the factor lemma a commutative diagram Σ 1 G(n) Σ 1 φ(n) Σ 1 P (n) Σ 1 π(n) Σ 1 C(n) μ S n 1 (G n 1 ) ι Q ν q ω Coker(ι) with exact rows, where q is the natural projection It is easily checked that the sequence Cone(μ) ι φ(n) Cone(ν) q π(n) Cone(ω) is exact Note that G(n+1) Cone(μ) If we put P (n+1) Cone(ν) andc(n+1) Cone(ω), then we have an exact sequence G(n +1) φ(n+1) P (n +1) π(n+1) C(n +1) with φ(n +1) ( ) ( ) ι q φ(n) and π(n +1) π(n) On one hand, exactness of the sequence Q P (n +1) P (n) implies that P (n +1) C (P) sincep (n) andq are in C (P), and P (n +1) k P (n) k for k n On the other hand, exactness of the sequence Coker(ι) C(n +1) C(n) implies that C(n+1) C (DP) is acyclic, and Hom R (C(n+1),F) is acyclic for any flat module F since Coker(ι) andc(n) are so Clearly, one has C(n +1) k C(n) k for k n Note that every morphism G(n +1) G(n) is surjective By [12, Theorem 1513], the sequence G lim G(n) lim φ(n) lim P (n) lim C(n) is exact Let P lim P (n), and C lim C(n) Then P k lim P (n) k P (k) k for any k andp k foranyk 1, C k lim C(n) k C(k) k for any k andc k for any k 1 Thus one can check easily that P C (P), C C (DP) is acyclic, and Hom R (C, F) is acyclic for any flat module F, as desired The following lemma is of great importance for proving our main result Lemma 36 If G C(DP), then there exists an exact sequence G P C such that the following conditions are satisfied:

9 1726 Yang G and Da X S (1) P C(P); (2) C C(DP) is acyclic; (3) Ext 1 (C, F) for any complex F C(F) Proof If we write G(n) G n for each n, then ((G(n)), (α mn )) n is a direct system in C(R) and limg(n) G, whereα mn : G(m) G(n) is the following natural injection for any m<n G(m) G m G m 1 G m 2 α mn G(n) G n G n 1 G m G m 1 G m 2 By Lemma 35, there exists an exact sequence G() η P () C() such that P () C (P), C() C (DP) is acyclic, and Hom R (C(),F ) is acyclic for any flat module F By Lemma 33, one gets that Ext 1 (C(),F) for any F C(F) Consider the push-out diagram of morphisms η : G() P () and α 1 : G() G(1) G() η P () C() α 1 G(1) μ U λ C() S 1 (G 1 ) S 1 (G 1 ) It is clear that U C (DP) sincep () and S 1 (G 1 )areinc (DP) By Lemma 35 again, we get that there exists an exact sequence U ν P (1) L(1) such that P (1) C (P), L(1) C (DP) is acyclic, and Hom R (L(1),F ) is acyclic for any flat module F By Lemma 33, one gets that Ext 1 (L(1),F) for any complex F C(F) Consider the push-out diagram of morphisms U C() and ν : U P (1) G(1) μ U C() G(1) P (1) ν L(1) V L(1)

10 On Ding Projective Complexes 1727 The exactness of the rightmost column implies that V C (DP) is acyclic, and Hom R (V,F ) is acyclic for any flat module F By Lemma 33 again, one gets that Ext 1 (V,F) for any complex F C(F) Let C(1) V,andβ 1 νλ Therefore we get, by the construction above, a commutative diagram with exact rows and columns G() η P () C() α 1 G(1) β 1 η 1 P (1) γ 1 C(1) S 1 (G 1 ) N(1) L(1) One gets easily from the lower row of the above diagram that N(1) C (DP) The middle column implies that N(1) i has finitely projective (flat) dimension for each i Z, andson(1) C (P) If we continue this process, then we get a commutative diagram with exact rows as follows G() η P () C() α 1 G(1) β 1 η 1 P (1) γ 1 C(1) α 12 G(2) β 12 η 2 P (2) γ 12 C(2) α 23 β 23 γ 23 For each n, we have that P (n) C (P), C(n) C (DP) is acyclic, and Hom R (C(n),F ) is acyclic for any flat module F By Lemma 33 again, one gets that Ext 1 (C(n),F)for any complex F C(F) Also for each row G(n) η n P (n) C(n),

11 1728 Yang G and Da X S there is a commutative diagram G(n) η n P (n) C(n) α n,n+1 G(n +1) β n,n+1 η n+1 P (n +1) γ n,n+1 C(n +1) S n+1 (G n+1 ) N(n +1) L(n +1) with exact rows and columns, where N(n +1) C (P), L(n +1) C (DP) is acyclic, and Hom R (L(n +1),F ) is acyclic for any flat module F By Lemma 33 again, one gets that Ext 1 (L(n +1),F) for any complex F C(F) Now except for ((G(n)), (α mn )) n, we have another two continuous direct systems of monomorphisms in C(R) The first one is ((P (n)), (β mn )) n, which is in the class C(P) with Coker(β n,n+1 )N(n +1) C(P) foreachn This yields that lim P (n) C(P) sincethe class P of projective modules is closed under direct transfinite extensions [1, Proposition 3] The second one is ((C(n)), (γ mn )) n, which is in the class C(DP) withcoker(γ n,n+1 )L(n +1) C(DP) foreachn One gets by Proposition 32 that limc(n) C(DP) Note that each C(n) is acyclic and the class of acyclic complexes is a left side of a cotorsion pair [14], we getthatlim C(n) is acyclic by [4, Theorem 12] By the above construction, one gets that Ext 1 (lim C(n),F) for each complex F C(F) since the left orthogonal class of any class is closed under direct transfinite extensions (see [4, Theorem 12] and [1, Theorem 15]) In fact, the direct limit of the system ( G(n) P (n) C(n) ) n of short exact sequences produces our desired sequence G P C inwhichg limg(n), P limp (n) and C limc(n) This completes the proof It was shown recently in [33] that every complex of Gorenstein projective modules admits a special C(P)-preenvelope Since any Ding projective module is always a Gorenstein projective module, it is obvious that every complex of Ding projective modules admits a special C(P)- preenvelope, also this is a direct consequence of Lemma 36 Lemma 37 ([32, Theorem 37]) Let R be any ring and G C(R) a complex Then G is Ding projective in C(R) if and only if each term G m is Ding projective in R-Mod and Hom R (G, F ) is acyclic for any flat complex F It was shown recently by Gillespie in [18] that Ding projective complexes are precisely the complexes X for which all terms X m are Ding projective R-modules whenever R is a Ding Chen ring Now we will remove the Ding Chen condition of the ring R Theorem 38 Let R be any ring and G C(R) a complex Then G is Ding projective in C(R) if and only if each term G m is Ding projective in R-Mod

12 On Ding Projective Complexes 1729 Proof The necessity follows from Lemma 37 For the sufficiency, we need only to show that any morphism f : G F is null homotopic for any flat complex F by Lemmas 21 and 37 Let f : G F be any morphism with F a flat complex By Lemma 36, there exists an exact sequence G ρ P C ( ) such that P C(P) and Ext 1 (C, F) for any complex F C(F) This implies that the sequence ( ) remains exact after applying the functor Hom(,F) for any flat complex F So we have a commutative diagram G f g ρ P C F That is to say, f canbefactoredasf gρ However, it follows from [26] (also see [5, 27]) that g is null homotopic, and so f is null homotopic too, as desired Since all projective R-modules are Ding projective, we have the following result Corollary 39 Any complex G C(P) is Ding projective in C(R) We hope that a dual version of Theorem 38 holds We formulate it as the following conjecture Conjecture 31 Let R be any ring and G C(R) a complex Then G is Ding injective in C(R) if and only if each term G m is Ding injective in R-Mod We finish the paper by pointing out that the following interesting problem remains still open Problem 311 Let R be any ring and I C(I) a complex of injective modules Then any morphism f : J I for any FP-injective complex J is null homotopic Acknowledgements valuable comments We thank the referees for careful reading of the paper and for many References [1] Auslander, M: On the dimension of modules and algebras III Global dimension Nagoya Math J, 9, (1955) [2] Bravo, D, Gillespie, J, Hovey, M: The stable module category of a general ring arxiv: v1 [mathra]22 May 214 [3] Ding, N Q, Li, Y L, Mao, L X: Strongly Gorenstein flat modules J Aust Math Soc, 86, (29) [4] Eklof, P C: Homological algebra and set theory Trans Amer Math Soc, 227, (1977) [5] Emmanouil, I: On acyclic complexes J Algebra 465, (216) [6] Enochs, E E, Estrada, S, Iacob, A: Gorenstein projective and flat complexes over noetherian rings Math Nachr, 285, (212) [7] Enochs, E E, García Rozas, J R: Tensor products of complexes Math J Okayama Univ, 39, (1997) [8] Enochs, E E, García Rozas, J R: Flat covers of complexes J Algebra, 21, (1998) [9] Enochs, E E, García Rozas, J R: Gorenstein injective and projective complexes Comm Algebra, 26, (1998)

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