Weakly Ding injective complexes

Working over a (left) coherent ring, we consider the class of weakly Ding injective complexes. These are the cycles of the exact complexes of FP-injective complexes that stay exact when applying $\Hom(A,-)$ for any FP-injective complex $A$. We study the cotorsion pair generated by the class of all such complexes, and exhibit it as part of an abelian model structure. As an application we show that when $R$ is a Ding-Chen ring, its stable chain complex category is compactly generated and triangle equivalent to the stable category of four Frobenius categories. They are the categories of all (i) complexes of Gorenstein injective modules, (ii) complexes of Gorenstein projective modules, (iii) complexes of Gorenstein flat-cotorsion modules, and (iv) complexes of Gorenstein FP-pro-injective modules.