Characterizations of Ding injective complexes (Q1988554)

Let \(R\) be an arbitrary associative ring. Denote by \(Ch(R)\) the category of chain complexes of \(R\)-modules. Chain complex \(X\in Ch(R)\) is called acyclic if it is exact. An \(R\)-module \(M\) is called Ding injective [\textit{N. Ding} et al., J. Aust. Math. Soc. 86, No. 3, 323--338 (2009; Zbl 1200.16010); \textit{L. Mao} and \textit{N. Ding}, J. Algebra Appl. 7, No. 4, 491--506 (2008; Zbl 1165.16004)] complex \(X\) of injective \(R\)-modules \[ \cdots \to X_1 \overset{d_1}\rightarrow X_0 \overset{d_0}\rightarrow X_{-1} \to \cdots \] with \(M\cong\mathrm{Ker }d_0\) and which remains exact after applying \(\mathrm{Hom}_R(J,-)\) for any FP-injective \(R\)-module \(J\). The category \(Ch(R)\) is abelian and has enough injectives and projectives. This allows us to consider extension groups \(\mathrm{Ext}^n(X,Y\) for arbitrary chain complexes \(X\) and \(Y\). The group \(\mathrm{Ext}^0(X,Y)\) is equal to the group \(\mathrm{Hom}(X,Y)\) of chain homomorphisms. Definition 2.2. A chain complex \(J\) is FP-injective if \(\mathrm{Ext}^1(A,J)=0\) for any finitely presented chain complex \(A\). Definition 2.3. A chain complex \(X\) is called Ding injective if there exists an exact sequence of injective chain complexes \[ \cdots \to I_1 \to I_0 \to I^0 \to I^1 \to \cdots \] with \(X \cong\mathrm{Ker}(I^0\to I^1)\) and which remains exact after applying \(\mathrm{Hom}(J, -)\) for any FP-injective chain complex \(J\). Theorem 3.4 The following conditions are equivalent for any ring \(R\). (1) Every bounded below chain complex of Ding injective \(R\)-modules is Ding injective in \(Ch(R)\). (2) Every bounded below chain complex of injective \(R\)-modules is Ding injective in \(Ch(R)\). (3) Every chain complex of Ding injective \(R\)-modules is Ding injective in \(Ch(R)\). (4) Every chain complex of injective \(R\)-modules is Ding injective in \(Ch(R)\). (5) Every acyclic chain complex of Ding injective \(R\)-modules is Ding injective in \(Ch(R)\). (6) Every acyclic chain complex of injective \(R\)-modules is Ding injective in \(Ch(R)\). Theorem 4.1 Let \(R\) be a left coherent ring and \(G \in Ch(R)\) a chain complex. Then \(G\) is Ding injective in \(Ch(R)\) if and only if each term \(G_m\) is Ding injective in \(R-\mathrm{Mod}\).