Injective resolutions of Noetherian rings and cogenerators

Miyachi, Jun-ichi

doi:10.1090/S0002-9939-00-05305-3

Injective resolutions of Noetherian rings and cogenerators
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Proc. Amer. Math. Soc. 128 (2000), 2233-2242 Request permission

Abstract:

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 $\operatorname {E}(M)$.

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