Ding injective envelopes in the category of complexes

TL;DR

This paper introduces Ding injective complexes, characterizes them as complexes of Ding injective modules, and proves that this class is enveloping over any ring, advancing the understanding of homological properties in complex categories.

Contribution

It establishes the equivalence between Ding injective complexes and complexes of Ding injective modules, and shows the class is enveloping over any ring.

Findings

Ding injective complexes are characterized as complexes of Ding injective modules.

The class of Ding injective complexes is enveloping over any ring.

The paper extends the theory of Ding injective modules to complexes.

Abstract

A complex $X$ is called Ding injective if there exists an exact sequence of injective complexes $\ldots \rightarrow E_1 \rightarrow E_0 \rightarrow E_{-1} \rightarrow \ldots$ such that $X = Ker(E_0 \rightarrow E_{-1})$, and the sequence remains exact when the functor $Hom(A,-)$ is applied to it, for any $FP$-injective complex $A$. We prove that, over any ring $R$, a complex is Ding injective if and only if it is a complex of Ding injective modules. We use this to show that the class of Ding injective complexes is enveloping over any ring.