n-Strongly Gorenstein Projective and Injective Complexes

Abstract

In this paper, we introduce and study the notions of n-strongly Gorenstein projective and injective complexes, which are generalizations of n-strongly Gorenstein projective and injective modules, respectively. Further, we characterize the so-called notions and prove that the Gorenstein projective (resp., injective) complexes are direct summands of n-strongly Gorenstein projective (resp., injective) complexes. Also, we discuss the relationships between n-strongly Gorenstein injective and n-strongly Gorenstein flat complexes, and for any two positive integers n and m, we exhibit the relationships between n-strongly Gorenstein projective (resp., injective) and m-strongly Gorenstein projective (resp., injective) complexes.

1 Introduction

Throughout this paper, let R be an associative ring with identity and $\mathscr {C}$ be the abelian category of complexes of R-modules. Unless stated otherwise, a complex and an R-module will be understood to be a complex of left R-modules and a left R-module respectively.

Bennis and Mahdou [2] introduced the notions of strongly Gorenstein projective, injective and flat modules which are further studied and characterized by Liu [8]. Later, Bennis and Mahdou [3] generalized the notion of strongly Gorenstein projective modules to n-strongly Gorenstein projective modules and [11] Zhao studied the homological behaviors of n-strongly Gorenstein projective, injective and flat modules. Zhang et al. [10] studied the notions of strongly Gorenstein projective and injective complexes. Motivated by the above works in this article, we introduce and study the notions of n-strongly Gorenstein projective and injective complexes, which are generalizations of n-strongly Gorenstein projective and injective modules, respectively. In [2, Theorem 2.7], it is proved that a module is Gorenstein projective if and only if it is a direct summand of a strongly Gorenstein projective module. Using [9, Theorem 2.3], we prove the following.

Theorem

Let G be a complex. Then the following holds:

• (1) G is Gorenstein projective if and only if it is a direct summand of an n-SG-projective complex.

• (2) G is Gorenstein injective if and only if it is a direct summand of an n-SG-injective complex.

In [9, Theorem 3.1], the relationship between Gorenstein flat and Gorenstein injective complexes is given. In connection to [9, Theorems 3.1 and 3.3] and [7, Proposition 4.7], we have the following result.

Theorem

Let R be a left artinian ring and let the injective envelope of every simple left R-module be finitely generated. Then the following hold:

• (1) If a complex G of left R-modules is n-SG-injective, then $G^+$ is an n-SG-flat complex of right R-modules.

• (2) If a complex G of right R-modules is n-SG-flat, then $G^+$ is an n-SG-injective complex of left R-modules.

In Sect. 2, we recall some known definitions and terminologies which will be needed in the sequel.

In Sect. 3, we introduce and study the notions of n-strongly Gorenstein projective and injective complexes. We show that a complex is Gorenstein projective (resp., injective) if and only if it is a direct summand of an n-SG-projective (resp., injective) complex and prove that the modules in an n-SG-projective (resp., injective) complex are precisely the n-SG-projective (resp., injective) modules. Further, over a left artinian ring R, we discuss the relationships between n-SG-injective and n-SG-flat complexes.

In the last section, we study the relationships between n-SG-projective (resp., injective) and m-SG-projective (resp., injective) complexes for any two positive integers n and m.

2 Preliminaries

In this section, we first recall some known definitions and terminologies which we need in the sequel.

In this paper, a complex

$$\cdots \rightarrow C^{-1} \mathop {\rightarrow }\limits ^{\delta ^{-1}} C^0 \mathop {\rightarrow }\limits ^{\delta ^{0}} C^1 \mathop {\rightarrow }\limits ^{\delta ^{1}} \cdots $$

will be denoted by C or $(C, \delta )$. We will use subscripts to distinguish complexes. So if $\{ C_i\}_{i \in I}$ is a family of complexes, $C_i$ will be

$$\cdots \rightarrow C_i^{-1} \mathop {\rightarrow }\limits ^{\delta _i^{-1}} C_i^0 \mathop {\rightarrow }\limits ^{\delta _i^{0}} C_i^1 \mathop {\rightarrow }\limits ^{\delta _i^{1}} \cdots .$$

Given an R-module M, we will denote by $\overline{M}$ the complex

$$\cdots 0 \rightarrow 0 \rightarrow M \mathop {\rightarrow }\limits ^{id} M \rightarrow 0 \rightarrow 0 \cdots $$

with M in the 1st and 0th degrees. Similarly, we denote by $\underline{M}$ the complex with M in the 0th degree and 0 in the other places. Note that an R-module M is injective (resp., projective) if and only if the complex $\overline{M}$ is injective (resp., projective).

Given a complex C and an integer m, C[m] denotes the complex such that $C[m]^n=C^{m+n}$ and whose boundary operators are $(-1)^m \delta ^{m+n}$. The nth cycle of a complex C is defined as Ker$\delta ^n$ and is denoted by $Z^n C$. The nth boundary of C is defined as Im$\delta ^{n-1}$ and is denoted by $B^n C$.

Let C be a complex of left R-modules (resp., of right R-modules) and let D be a complex of left R-modules. We denote by Hom(C, D) (respectively, $C\otimes D$) the usual homomorphism complex (resp., tensor product) of the complexes C and D. The nth degree term of the complex Hom(C, D) is given by

$$\text {Hom}(C, D)^n= \prod \limits _{t \in \mathbb {Z}}\,\text {Hom}(C^t, D^{n+t})$$

and whose boundary operators are

$$(\delta ^n f)^m= \delta _D ^{n+m}f^m- (-1)^n f^{m+1} \delta _C^m.$$

The nth degree term of $C \otimes D$ is given by

$$(C \otimes D)^n= \bigoplus \limits _{{t \in \mathbb {Z}}}(C^t \otimes _R D^{n-t})$$

and

$$\delta (x \otimes y)= \delta _C ^t (x) \otimes y + (-1)^t x \otimes \delta _D ^{n-t}(y),$$

for $x \in C^t$ and $y \in D^{n-t}.$

For a complex C of left R-modules, we have a functor $- \otimes \mathscr {C}: \mathscr {C}_R \rightarrow \mathscr {C}_{\mathbb {Z}}$, where $\mathscr {C}_R$ denotes the category of right R-modules. The functor $- \otimes \mathscr {C}: \mathscr {C}_R \rightarrow \mathscr {C}_{\mathbb {Z}}$ being right exact, we can construct the left derived functors which we denote by $Tor_i (-, C)$. Given two complexes C and D of $\mathscr {C}$, we use $Ext^i (C, D)$ for $i \ge 0$ to denote the groups we obtain from the right derived functors of Hom and we use $C^+$ to denote the complex $\underline{Hom}(C, \overline{\mathbb {Q}/ \mathbb {Z}})$.

Recall that a complex C is projective (respectively, injective) if C is exact and $Z^n C$ is a projective (respectively, an injective) R-module for each $i \in \mathbb {Z}$. A complex C is flat if C is exact and $Z^n C$ is flat R-module for each $i \in \mathbb {Z}$. Equivalently, a complex C is projective (respectively, injective) if and only if Hom$(C,-)$ (respectively, Hom $(-, C)$) is exact. Also a complex C is flat if and only if $- \otimes C$ is exact. For unexplained terminologies and notations we refer to [1, 4–6].

Definition 2.1

([10]) A complex G is called strongly Gorenstein projective (for short SG-projective) if there exists an exact sequence of complexes

$$\mathbb {P}: \cdots \rightarrow P \mathop {\rightarrow }\limits ^{\delta } P \mathop {\rightarrow }\limits ^{\delta } P \mathop {\rightarrow }\limits ^{\delta } \cdots $$

such that (i) P is a projective complex;

(ii) Ker $\delta _0 \cong G$;

(iii) Hom$(\mathbb {P}, Q)$ is exact for any projective complex Q.

Similarly, the SG-injective complexes are defined.

Definition 2.2

([7]) A complex G of right R-modules is called strongly Gorenstein flat (for short SG-flat) if there exists an exact sequence of complexes of right R-modules

$$\mathbb {F}: \cdots \rightarrow F \mathop {\rightarrow }\limits ^{\delta } F \mathop {\rightarrow }\limits ^{\delta } F \mathop {\rightarrow }\limits ^{\delta } \cdots $$

such that (i) F is flat;

(ii) Ker $\delta _0 \cong G$;

(iii) $\mathbb {F}\otimes I$ is exact for any injective complex I.

Definition 2.3

([7]) Let n be a positive integer. A complex G of right R-modules is said to be an n-SG-flat if there exists an exact sequence of complexes

$$0 \rightarrow G \mathop {\rightarrow }\limits ^{\delta _{n+1}} F_n \mathop {\rightarrow }\limits ^{\delta _n} F_{n-1} \rightarrow \cdots \rightarrow F_1 \mathop {\rightarrow }\limits ^{\delta _1} G \rightarrow 0$$

with $F_i$ projective for any $1 \le i \le n,$ such that $- \otimes I$ leaves the sequence exact whenever I is an injective complex.

Next, we present the characterizations of n-SG-flat complexes in order to use it further.

Proposition 2.4

([7]) Let R be a right coherent ring and G be any complex of right R-modules. Then the following are equivalent;

• (1) G is n-SG-flat;

• (2) There exists an exact sequence of complexes of right R-modules

  $$ 0 \rightarrow G \mathop {\rightarrow }\limits ^{\delta _{n+1}} F_n \mathop {\rightarrow }\limits ^{\delta _n} F_{n-1} \rightarrow \cdots \rightarrow F_1 \mathop {\rightarrow }\limits ^{\delta _1} G \rightarrow 0$$

  with $F_i$ flat for any $1 \le i \le n,$ such that $\bigoplus \limits _{i=2}^{n+1}$Im $\delta _i$ is SG-flat;

• (3) There exists an exact sequence of complexes of right R-modules

  $$ 0 \rightarrow G \mathop {\rightarrow }\limits ^{\delta _{n+1}} F_n \mathop {\rightarrow }\limits ^{\delta _n} F_{n-1} \rightarrow \cdots \rightarrow F_1 \mathop {\rightarrow }\limits ^{\delta _1} G \rightarrow 0$$

  with $F_i$ flat for any $1 \le i \le n,$ such that $\bigoplus \limits _{i=2}^{n+1}$ Im $\delta _i$ is Gorenstein flat.

3 n-Strongly Gorenstein Projective and Injective Complexes

In this section, we introduce and study the n-SG-projective and injective complexes which are generalizations of SG-projective and injective modules, respectively. Also we extend the results in [3, 11] on n-strongly Gorenstein projective and injective modules to that of complexes.

Definition 3.1

Let n be a positive integer. A complex G is said to be an n-strongly Gorenstein projective (for short n-SG-projective) if there exists an exact sequence of complexes

$$0 \rightarrow G \mathop {\rightarrow }\limits ^{\delta _{n+1}} P_n \mathop {\rightarrow }\limits ^{\delta _n} P_{n-1} \rightarrow \cdots \rightarrow P_1 \mathop {\rightarrow }\limits ^{\delta _1} G \rightarrow 0$$

with $P_i$ projective for any $1 \le i \le n,$ such that Hom$(-, Q)$ leaves the sequence exact whenever Q is a projective complex.

Definition 3.2

Let n be a positive integer. A complex G is said to be an n-strongly Gorenstein injective (for short n-SG-injective) if there exists an exact sequence of complexes

$$0 \rightarrow G \mathop {\rightarrow }\limits ^{\alpha _{n+1}} I_n \mathop {\rightarrow }\limits ^{\alpha _n} I_{n-1} \rightarrow \cdots \rightarrow I_1 \mathop {\rightarrow }\limits ^{\alpha _1} G \rightarrow 0$$

with $I_i$ injective for any $1 \le i \le n,$ such that Hom$(E, -)$ leaves the sequence exact whenever E is an injective complex.

Note that 1-SG-projective (resp., injective) complexes are just SG-projective (resp., injective) complexes. It is also clear that for any i with $2 \le i \le n+1,$ the complex Im $\delta _i$ (resp., Im $\alpha _i$) in the above exact sequence is n-SG-projective (resp., injective). The following proposition shows that the class of all n-SG-projective (resp., injective) complexes is between the class of all SG-projective (resp., injective) complexes and the class of all Gorenstein projective (resp., injective) complexes.

Proposition 3.3

Let n be a positive integer. Then:

• (1) Every SG-projective (resp., injective) complex is an n-SG-projective (resp., injective) complex.

• (2) Every n-SG-projective (resp., injective) complex is a Gorenstein projective (resp., injective) complex.

Proof

Since the SG-injective complex is the dual notion of SG-projective, we prove the results for SG-projective case.

(1) Let G be an SG-projective complex. There exists an exact sequence of complexes

$$0 \rightarrow G \mathop {\rightarrow }\limits ^{f} P\mathop {\rightarrow }\limits ^{g} G \rightarrow 0,$$

where P is a projective complex, such that Hom$(-, Q)$ leaves the sequence exact whenever Q is a projective complex. Then we get an exact sequence of complexes of the form

$$X: 0 \rightarrow G \mathop {\rightarrow }\limits ^{f} P\mathop {\rightarrow }\limits ^{fg} P\mathop {\rightarrow }\limits ^{fg} \cdots \rightarrow P\mathop {\rightarrow }\limits ^{g}G \rightarrow 0$$

such that Hom(X, Q) is exact for any projective complex Q. Therefore G is an n-SG-projective complex.

(2) Let G be an n-SG-projective complex. There exists an exact sequence of complexes

$$Y: 0 \rightarrow G \mathop {\rightarrow }\limits ^{\delta _{n+1}} P_n \mathop {\rightarrow }\limits ^{\delta _n} P_{n-1} \rightarrow \cdots \rightarrow P_1 \mathop {\rightarrow }\limits ^{\delta _1} G \rightarrow 0$$

with $P_i$ projective for any $1 \le i \le n,$ such that Hom$(-, Q)$ leaves the sequence exact whenever Q is a projective complex. Thus, we get the following exact sequence of complexes

$$Y': \cdots \rightarrow P_1 \mathop {\rightarrow }\limits ^{\delta _{n+1} \delta _1} P_n \mathop {\rightarrow }\limits ^{\delta _n} P_{n-1} \rightarrow \cdots \rightarrow P_1 \mathop {\rightarrow }\limits ^{\delta _{n+1} \delta _1} P_n\mathop {\rightarrow }\limits ^{\delta _n} \cdots .$$

such that $Im(\delta _{n+1} \delta _1) \cong G$. Let Q be any projective complex. Then the exactness of Hom$(Y', Q)$ follows from the exactness of Hom(Y, Q) and hence G is a Gorenstein projective complex.   $\square$

Proposition 3.4

Let $\{G_i\}_I$ be any family of complexes. Then

• (1) If $G_i$ is n-SG-projective for every $i \in I$, then $\bigoplus \limits _I G_i$ is an n-SG-projective complex.

• (2) If $G_i$ is n-SG-injective for every $i \in I$, then $\prod \limits _I G_i$ is an n-SG-injective complex.

Proof

• (1) For each i in I there exists an exact sequence of complexes

  $$\mathbb {X}_i: 0 \rightarrow G_i \rightarrow P_{in} \rightarrow P_{i n-1} \rightarrow \cdots \rightarrow P_{i1} \rightarrow G_i \rightarrow 0$$

with $P_{ij}$ projective for $1 \le j \le n,$ such that Hom$(\mathbb {X}_i, Q)$ is exact for any projective complex Q. Since the direct sum of projective complexes is projective, we obtain the following exact sequence of complexes

$$\bigoplus \limits _{i \in I} \mathbb {X}_i: 0 \rightarrow \bigoplus \limits _{i \in I} G_i \rightarrow \bigoplus \limits _{i \in I} P_{in} \rightarrow \cdots \rightarrow \bigoplus \limits _{i \in I}P_{i1} \rightarrow \bigoplus \limits _{i \in I}G_i \rightarrow 0$$

with $\bigoplus \limits _{i \in I} P_{ij}$ projective for $1 \le j \le n.$ Let Q be any projective complex. Then Hom$(\bigoplus \limits \mathbb {X}_i, Q) \cong \prod$ Hom$(\mathbb {X}_i, Q)$ is exact, and hence $\bigoplus \limits G_i$ is an n-SG-projective complex.

(2) The proof is similar to (1).   $\square$

In [2, Theorem 2.7], it is proved that a module is Gorenstein projective if and only if it is a direct summand of a strongly Gorenstein projective module. Using [9, Theorem 2.3] and Proposition 3.3, we have the following.

Theorem 3.5

Let G be a complex. Then the following hold:

• (1) G is Gorenstein projective if and only if it is a direct summand of an n-SG-projective complex.

• (2) G is Gorenstein injective if and only if it is a direct summand of an n-SG-injective complex.

Proof

(1) Let G be a Gorenstein projective complex. Then it is a direct summand of an SG-projective complex by [10, Theorem 1]. Hence G is a direct summand of an n-SG-projective complex by Proposition 3.3. Conversely, let G be a direct summand of an n-SG-projective complex C. Then C is Gorenstein projective by Proposition 3.3 (2). Since the class of all Gorenstein projective complexes is closed under direct summands by [9, Theorem 2.3], it follows that G is Gorenstein projective.

(2) The proof is similar to (1).   $\square$

In [11, Theorem 3.9], Zhao and Huang have given some characterizations of n-SG-projective modules. Now, we have the similar characterization for n-SG-projective complexes in the following.

Proposition 3.6

Let G be any complex. Then the following are equivalent;

• (1) G is n-SG-projective;

• (2) There exists an exact sequence of complexes

  $$ 0 \rightarrow G \mathop {\rightarrow }\limits ^{\delta _{n+1}} P_n \mathop {\rightarrow }\limits ^{\delta _n} P_{n-1} \rightarrow \cdots \rightarrow P_1 \mathop {\rightarrow }\limits ^{\delta _1} G \rightarrow 0$$

  with $P_i$ projective for any $1 \le i \le n,$ such that $\bigoplus \limits _{i=2}^{n+1}$Im $\delta _i$ is SG-projective;

• (3) There exists an exact sequence of complexes

  $$ 0 \rightarrow G \mathop {\rightarrow }\limits ^{\delta _{n+1}} P_n \mathop {\rightarrow }\limits ^{\delta _n} P_{n-1} \rightarrow \cdots \rightarrow P_1 \mathop {\rightarrow }\limits ^{\delta _1} G \rightarrow 0$$

  with $F_i$ projective for any $1 \le i \le n,$ such that $\bigoplus \limits _{i=2}^{n+1}$ Im $\delta _i$ is Gorenstein projective.

Proof

(1) $\Rightarrow$ (2). Let G be an SG-projective complex. Then there exists an exact sequence of complexes

$$ 0 \rightarrow G \mathop {\rightarrow }\limits ^{\delta _{n+1}} P_n \mathop {\rightarrow }\limits ^{\delta _n} P_{n-1} \rightarrow \cdots \rightarrow P_1 \mathop {\rightarrow }\limits ^{\delta _1} G \rightarrow 0$$

with $P_i$ projective for any $1 \le i \le n,$ such that Hom$(-, Q)$ leaves the sequence exact whenever Q is a projective complex. Now for each i with $2 \le i \le n+1,$ we have an exact sequence of complexes

$$ 0 \rightarrow \,\text {Im}\,\delta _i \mathop {\rightarrow }\limits ^{\alpha _{i}} P_{i-1} \mathop {\rightarrow }\limits ^{\delta _{i-1}} \cdots \rightarrow P_1 \mathop {\longrightarrow }\limits ^{\delta _{n+1}\delta _1}P_{n}\mathop {\rightarrow }\limits ^{\delta _n} \cdots \rightarrow P_i \mathop {\rightarrow }\limits ^{\delta _i}\,\text {Im}\,\delta _i\rightarrow 0.$$

By adding these exact sequences, we obtain the following exact sequence

$$ 0 \rightarrow \bigoplus \limits _{i=2}^{n+1}\,\text {Im}\, \delta _i \mathop {\rightarrow }\limits ^{\alpha } \bigoplus \limits _{i=1}^{n}P_{i} \mathop {\rightarrow }\limits ^{\delta } \cdots \rightarrow P_n \oplus P_0 \oplus \cdots \oplus P_{n-1} \rightarrow \cdots $$

where $\alpha =$ diag$\{\alpha _1, \alpha _2,..., \alpha _n \}$ and $\delta =$ diag $\{\delta _{n+1}\delta _1, \delta _2,..., \delta _n \}$. Hence it is clear that Im $\delta \cong \bigoplus \limits _{i=2}^{n+1} \delta _i$ and $Ext_1 (\bigoplus \limits _{i=2}^{n+1}$ Im $\delta _i, Q) \cong \prod \limits _{i=2}^{n+1} Ext_1 ($ Im $\delta _i, Q)=0$ for any projective complex Q. Therefore $\bigoplus \limits _{i=2}^{n+1}$ Im $\delta _i$ is SG-flat.

(2) $\Rightarrow$ (3) It follows from the Proposition 3.3.

(3) $\Rightarrow$ (1) It is obvious.   $\square$

Similarly, we can characterize the n-SG-injective complexes.

In [9, Theorem 3.1], the relationship between Gorenstein flat and Gorenstein injective complexes is given. In connection to [9, Theorems 3.1 and 3.3] and Proposition 2.4, we have the following.

Theorem 3.7

Let R be a left artinian ring and let the injective envelope of every simple left R-module be finitely generated. Then the following hold:

• (1) If a complex G of left R-modules is n-SG-injective, then $G^+$ is an n-SG-flat complex of right R-modules.

• (2) If a complex G of right R-modules is n-SG-flat, then $G^+$ is an n-SG-injective complex of left R-modules.

Proof

(1) Let G be an n-SG-injective complex. Then using the characterization of n-SG-injective complexes similar to Proposition 3.6, we get an exact sequence of complexes

$$ \mathbf I : 0 \rightarrow G \mathop {\rightarrow }\limits ^{\delta _{n+1}} I_n \mathop {\rightarrow }\limits ^{\delta _n} I_{n-1} \rightarrow \cdots \rightarrow I_1 \mathop {\rightarrow }\limits ^{\delta _1} G \rightarrow 0$$

where $I_j$ is an injective complex for $1 \le j \le n$ and $\bigoplus \limits _{j=2}^{n+1}$ Im $\delta _j$ is Gorenstein injective.

Thus we have the following exact sequence of right R-modules

$$ \mathbf I ^+: 0 \rightarrow G^+ \mathop {\longrightarrow }\limits ^{\delta _1^+} I_1^+ \mathop {\longrightarrow }\limits ^{\delta _2^+} I_2^+ \rightarrow \cdots \rightarrow I_n^+ \mathop {\longrightarrow }\limits ^{\delta _{n+1}^+} G^+ \rightarrow 0$$

where $I_j^+$ is a flat complex for $1 \le j \le n$. Since G is Gorenstein injective by Proposition 3.3, we have that Im $\delta _1^+ \cong G^+$ is Gorenstein flat by [9, Theorem 3.5]. Since $\bigoplus \limits _{j=2}^{n+1}$ Im $\delta _j$ is Gorenstein injective, we get that Im $\delta _j$ is Gorenstein injective for $2 \le j \le n+1$ by [9, Theorem 2.10]. Thus for every j with $1 \le j \le n$, Im $\delta _j^+$ is Gorenstein flat by [9, Theorem 3.5]. Hence $\bigoplus \limits _{j=1}^{n}$ Im $\delta _j^+$ is Gorenstein flat since Gorenstein flat complexes are closed under direct sums. Therefore $G^+$ is n-SG-flat by Proposition 2.4.

(2) Let G be an n-SG-flat complex. Then by Proposition 2.4, we get an exact sequence of complexes of right R-modules

$$ \mathbf F : 0 \rightarrow G \mathop {\rightarrow }\limits ^{\delta _{n+1}} F_n \mathop {\rightarrow }\limits ^{\delta _n} F_{n-1} \rightarrow \cdots \rightarrow F_1 \mathop {\rightarrow }\limits ^{\delta _1} G \rightarrow 0$$

where $F_j$ is a flat complex for $1 \le j \le n$ and $\bigoplus \limits _{j=2}^{n+1}$ Im $\delta _j$ is Gorenstein flat. Thus we have the following exact sequence of complexes of R-modules

$$ \mathbf F ^+: 0 \rightarrow G^+ \mathop {\longrightarrow }\limits ^{\delta _1^+} F_1^+ \mathop {\longrightarrow }\limits ^{\delta _2^+} F_2^+ \rightarrow \cdots \rightarrow F_n^+ \mathop {\longrightarrow }\limits ^{\delta _{n+1}^+} G^+ \rightarrow 0$$

where $F_j^+$ is an injective complex for $1 \le j \le n$. Since G is Gorenstein flat by [7, Proposition 4.2], we have that Im $\delta _1^+ \cong G^+$ is Gorenstein injective by [9, Theorem 3.1]. Since $\bigoplus \limits _{j=2}^{n+1}$ Im $\delta _j$ is Gorenstein flat, we get that Im $\delta _j$ is Gorenstein flat for $2 \le j \le n+1$ by [9, Theorem 3.3]. Thus for every j with $1 \le j \le n$, Im $\delta _j^+$ is Gorenstein injective by [9, Theorem 3.1]. Hence $\bigoplus \limits _{j=1}^{n}$ Im $\delta _j^+$ is Gorenstein injective since Gorenstein injective complexes are closed under finite direct sums. Therefore $G^+$ is n-SG-injective by Proposition 3.3.   $\square$

Corollary 3.8

Let R be a left artinian ring and let the injective envelope of every simple left R-module be finitely generated. Then the following hold:

• (1) If a complex G of R-modules is n-SG-injective, then $G^{++}$ is an n-SG-injective complex.

• (2) If a complex G of right R-modules is n-SG-flat, then $G^{++}$ is an n-SG-flat complex.

Proof

The proof follows from Theorem 3.7.   $\square$

The following result shows the relationship between n-SG-projective complexes and n-SG-projective modules.

Proposition 3.9

Let G be a complex. If G is n-SG-projective, then $G^i$ is an n-SG-projective R-module for all $i \in \mathbb {Z}$.

Proof

Suppose G is an n-SG-projective complex. By Proposition 3.6, there exists an exact sequence of complexes

$$ 0 \rightarrow G \mathop {\rightarrow }\limits ^{\delta _{n+1}} P_n \mathop {\rightarrow }\limits ^{\delta _n} P_{n-1} \rightarrow \cdots \rightarrow P_1 \mathop {\rightarrow }\limits ^{\delta _1} G \rightarrow 0$$

where $P_j$ is a projective complex for $1 \le j \le n$ and $\bigoplus \limits _{j=2}^{n+1}$ Im $\delta _j$ is Gorenstein projective. Then for each $i \in \mathbb {Z}$, we get an exact sequence of modules

$$ 0 \rightarrow G^i \mathop {\longrightarrow }\limits ^{{\delta }_{n+1}^i} P_n^i \mathop {\longrightarrow }\limits ^{{\delta }_n^i} P_{n-1}^i \rightarrow \cdots \rightarrow P_1^i \mathop {\longrightarrow }\limits ^{{\delta }_1^i} G^i \rightarrow 0$$

such that $P_j ^i$ is a projective R-module for $1 \le j \le n$. Since $\bigoplus \limits _{j=2}^{n+1}$ Im $\delta _j$ is a Gorenstein projective complex if and only if Im $\delta _j$ is a Gorenstein projective complex for $2 \le j \le n+1$ by [9, Theorem 2.3]. Then by [9, Theorem 2.2], we have Im $\delta _j$ is a Gorenstein projective complex if and only if Im $\delta _j^i$ is a Gorenstein projective R-module for every $i \in \mathbb {Z}$ and $2 \le j \le n+1$. Thus we get that $\bigoplus \limits _{j=2}^{n+1}$ Im $\delta _j^i$ is a Gorenstein projective R-module since the class of all Gorenstein projective modules is closed under direct sums. Therefore the result follows from [11, Theorem 3.9].   $\square$

Corollary 3.10

Let M be an R-module. Then M is n-SG-projective if and only if the complex $\overline{M}$ is n-SG-projective.

Proof

Suppose M is an n-SG-projective module. Then there exists an exact sequence of R-modules

$$X: 0 \rightarrow M \rightarrow P_n \rightarrow P_{n-1} \rightarrow \cdots \rightarrow P_1\rightarrow M \rightarrow 0,$$

where $P_i$ is a projective R-module for $1 \le i \le n$, such that Hom$_R(-, Q)$ leaves the sequence exact for any projective module Q. Thus, we get an exact sequence of complexes

$$\overline{X}: 0 \rightarrow \overline{M} \rightarrow \overline{P}_n\rightarrow \overline{P}_{n-1}\rightarrow \cdots \rightarrow \overline{P}_1 \rightarrow \overline{M} \rightarrow 0$$

with $\overline{P}_i$ a projective complex for $1 \le i \le n$. Now let $Q'$ be any projective complex. Then it is a direct product of complexes of the form $\overline{P}[n]$ for some projective module P and $n \in \mathbb {Z}$. Then

$$\begin{aligned} Hom(\overline{X}, Q')\cong & {} Hom(\overline{X}, \prod _{n \in \mathbb {Z}} \overline{P}[n])\\\cong & {} \prod _{n \in \mathbb {Z}} Hom(\overline{X}, \overline{P}[n]) \end{aligned}$$

is exact for all $n \in \mathbb {Z}$ and hence $\overline{M}$ is n-SG-projective. The converse follows from Proposition 3.9.

The following example describes that there are 2-SG-projective complexes which are not necessarily 1-SG-projective.

Example 3.11

• (1) Let R be a local ring and consider the ring $S=R[X, Y]/ (XY)$. Let [X] and [Y] be the residue classes in S of X and Y respectively. Then by [3, Example 2.6], we observe that the R-modules [X] and [Y] are 2-SG-projective but are not 1-SG-projective. Then by Corollary 3.10, the complexes $\overline{[X]}$ and $\overline{[Y]}$ are 2-SG-projective but are not SG-projective.

• (2) In general, n-SG-projective complexes need not be m-SG-projective whenever $n \not \mid m$. Based on the assumptions in [11, Example 3.2], we observe that the modules $S_i~(1 \le i \le n)$ are n-strongly Gorenstein projective but are not m-strongly Gorenstein projective. Then by the Corollary 3.10, we see that the complexes $\overline{S_i}$ are n-SG-projective but are not m-SG-projective whenever $n \not \mid m.$

4 n-SG-Projective and m-SG-Projective Complexes

In this section, we study the relationships between n-SG-projective (resp., injective) and m-SG-projective (resp., injective) complexes for any two positive integers n and m.

Lemma 4.1

Let m, n and r be any positive integers such that $m=rn$. Then the class of all m-SG-projective (resp., injective) complexes contains the class of all n-SG-projective (resp., injective) complexes.

Proof

Let G be an n SG-projective complex. Then there exists an exact sequence of complexes

$$ \mathbf X :0 \rightarrow G \mathop {\rightarrow }\limits ^{\alpha _{n+1}} I_n \mathop {\rightarrow }\limits ^{\alpha _n} I_{n-1} \rightarrow \cdots \rightarrow I_1 \mathop {\rightarrow }\limits ^{\alpha _1} G \rightarrow 0$$

with $I_j$ injective for any $1 \le j \le n,$ such that $\bigoplus \limits _{j=2}^{n+1}$ Im $\alpha _j$ is a Gorenstein projective complex. So Im $\delta _j$ is Gorenstein projective for every $1 \le j \le n$ by [9, Theorem 2.3]. Using the exact sequence $\mathbf X$ for r times, we have the following exact sequence

$$ \mathbf Y :0 \rightarrow G \mathop {\rightarrow }\limits ^{\alpha _{n+1}} I_n \mathop {\rightarrow }\limits ^{\alpha _n} I_{n-1} \rightarrow \cdots \rightarrow I_1\mathop {\rightarrow }\limits ^{\delta } I_n \rightarrow \cdots I_1 \mathop {\rightarrow }\limits ^{\alpha _1} G \rightarrow 0$$

with $I_j$ injective for any $1 \le j \le n$ and $\delta = \alpha _{n+1}\alpha _1$. Then $\bigoplus \limits _{j=2}^{n+1}$ Im $\alpha _j$ is Gorenstein projective since Im $\alpha _j$ and G are Gorenstein projective.   $\square$

For any positive integer n, we use n-SG-Proj$(\mathscr {C})$ (resp., n-SG-Inj$(\mathscr {C})$) to denote the subcategory of $_R \mathscr {C}$ consisting of n-SG-projective (resp., injective) complexes of left R-modules. The following results extend [11, Proposition 3.4 (2) and Theorem 3.5] to that of complexes.

Proposition 4.2

Let n and m be positive integers. Then the following hold:

• (1) If $n \vert m$, then n-SG-Proj$(\mathscr {C})$ $\bigcap$ m-SG-Proj$(\mathscr {C})$ = n-SG-Proj$(\mathscr {C})$.

• (2) If $n \not \mid m$ and $m=kn+j$, where k is a positive integer and $0< j <n,$ then n-SG-Proj$(\mathscr {C})$ $\bigcap$ m-SG-Proj$(\mathscr {C})$ $\subseteq$ j-SG-Proj$(\mathscr {C})$.

Proof

(1) It follows from Lemma 4.1.

(2) By Lemma 4.1, we have that m-SG-Proj$(\mathscr {C})$ $\bigcap$ n-SG-Proj$(\mathscr {C})$ $\subseteq$ m-SG-Proj$(\mathscr {C})$ $\bigcap$ kn-SG-Proj$(\mathscr {C})$. Suppose that a complex G is in m-SG-Proj$(\mathscr {C})$ $\bigcap$ kn-SG-Proj$(\mathscr {C})$. Then there exists an exact sequence of complexes

$$\mathbb {P}: 0 \rightarrow G \rightarrow P_m \rightarrow \cdots \rightarrow P_2 \rightarrow P_1 \rightarrow 0$$

with $P_i$ projective for any $1 \le i \le m$. Put $L_i=$ Ker$(P_i \rightarrow P_{i-1})$ for any $2 \le i \le m$. Since G is kn-SG-projective, we see that G and $L_{kn}$ are projectively equivalent, i.e., there exist projective complexes P and Q in $\mathscr {C}$ such that $G \oplus P \cong Q \oplus L_{kn}$.

Now consider the following pullback diagram:

Then X is a projective complex. Next, consider the following pullback diagram

Hence Y is also projective. Combining the exact sequence $\mathbb {P}$ and the first row in the above diagram, we get the following exact sequence of complexes

$$ 0 \rightarrow G \rightarrow P_m \rightarrow \cdots \rightarrow P_{kn+1} \rightarrow Y \rightarrow G \rightarrow 0$$

such that Hom$(-, Q')$ leaves the sequence exact for any projective complex $Q'$. Thus G is j-SG-projective and hence n-SG-Proj$(\mathscr {C})$ $\bigcap$ m-SG-Proj$(\mathscr {C})$ $\subseteq$ j-SG-Proj$(\mathscr {C})$.   $\square$

Dually, we have the following result for n-SG-injective complexes.

Proposition 4.3

Let n and m be positive integers. Then the following hold:

• (1) If $n \vert m$, then n-SG-Inj$(\mathscr {C})$ $\bigcap$ m-SG-Inj$(\mathscr {C})$ = n-SG-Inj$(\mathscr {C})$.

• (2) If $n \not \mid m$ and $m=kn+j$, where k is a positive integer and $0< j <n,$ then n-SG-Inj$(\mathscr {C})$ $\bigcap$ m-SG-Inj$(\mathscr {C})$ $\subseteq$ j-SG-Inj$(\mathscr {C})$.

For any two positive integers m and n, we use (m, n) (resp., [m, n]) to denote the greatest common divisor (resp., least common multiple) of m and n.

Proposition 4.4

For any two positive integers m and n, we have the following:

• (1) m-SG-Proj$(\mathscr {C})$ $\bigcap$ n-SG-Proj$(\mathscr {C})$ $=$ (m, n)-SG-Proj$(\mathscr {C})$.

• (2) m-SG-Proj$(\mathscr {C})$ $\bigcap$ $(m+1)$-SG-Proj$(\mathscr {C})$ $=$ 1-SG-Proj$(\mathscr {C})$.

Proof

(1) If $n \vert m,$ then the result follows from Proposition 4.3 (1). Now suppose $n \not \mid m$ and $m= k_0n + j_0$, where $k_0$ is a positive integer and $0<j_0 < n.$ By Proposition 4.3 (2), we have that m-SG-Proj$(\mathscr {C})$ $\bigcap$ n-SG-Proj$(\mathscr {C})$ $\subseteq$ $j_0$-SG-Proj$(\mathscr {C})$. If $j_0 \not \mid n$ and $n=k_1 j_0 + j_1,$ with $0< j_1 < j_0,$ then by Proposition 4.3 (2) again, we have that m-SG-Proj$(\mathscr {C})$ $\bigcap$ n-SG-Proj$(\mathscr {C})$ $\subseteq$ n-SG-Proj$(\mathscr {C})$ $\bigcap$ $j_0$-SG-Proj$(\mathscr {C})$ $\subseteq$ $j_1$-SG-Proj$(\mathscr {C})$. Continuing the process, after finite steps, there exists a positive integer t such that $j_t = k_{t+2} j_{t+1}$ and $j_{t+1}=(m, n)$. Thus m-SG-Proj$(\mathscr {C})$ $\bigcap$ n-SG-Proj$(\mathscr {C})$ $\subseteq$ $j_t$-SG-Proj$(\mathscr {C})$ $\bigcap$ $j_{t+1}$-SG-Proj$(\mathscr {C})= j_{t+1}$-SG-Proj$(\mathscr {C})$=(m, n)-SG-Proj$(\mathscr {C})$. Then the result follows from the fact that (m, n)-SG-Proj$(\mathscr {C})$ $\subseteq$ m-SG-Proj$(\mathscr {C})$ $\bigcap$ n-SG-Proj$(\mathscr {C})$.

(2) It follows from (1).   $\square$

Corollary 4.5

For any two positive integers m and n, we have the following: m-SG-Proj$(\mathscr {C})$ $\bigcup$ n-SG-Proj$(\mathscr {C})$ $\subseteq$ [m, n]-SG-Proj$(\mathscr {C})$.

Proof

It is clear from the fact that every n-SG-projective complex is m-SG-projective whenever $n \vert m$.   $\square$

For the case of n-SG-injective complexes, we have the following.

Proposition 4.6

For any two positive integers m and n, we have the following:

• (1) m-SG-Inj$(\mathscr {C})$ $\bigcap$ n-SG-Inj$(\mathscr {C})$ $=$ (m, n)-SG-Inj$(\mathscr {C})$.

• (2) m-SG-Inj$(\mathscr {C})$ $\bigcap$ $(m+1)$-SG-Inj$(\mathscr {C})$ $=$ 1-SG-Inj$(\mathscr {C})$.

Proof

The proof is similar to Proposition 4.4.   $\square$

Corollary 4.7

For any two positive integers m and n, we have the following: m-SG-Inj$(\mathscr {C})$ $\bigcup$ n-SG-Inj$(\mathscr {C})$ $\subseteq$ [m, n]-SG-Inj$(\mathscr {C})$.

Proof

It is clear from the fact that every n-SG-injective complex is m-SG-injective whenever $n \vert m$.   $\square$