Injective resolutions of Noetherian rings and cogenerators

We give new construction of injective resolutions of complexes and bimodules. Applying this construction to an injective resolution of a Noetherian ring, we construct a $\Sigma$-embedding cogenerator for the category of modules of projective dimension $\leq n$. Moreover, for a Noetherian projective $k$-algebra $R$, we show that $R$ satisfies the Auslander condition if and only if the flat dimension of every $R$-module $M$ is equal to or larger than the one of the injective hull $\operatorname{E}(M)$.