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Abstract
Let R → S be a ring homomorphism. We consider the relationships of the Gorenstein dimensions of an R-complex X (possibly unbounded) with those of the S-complexes RHom
R
(S, X) and . More generally, the Gorenstein injective dimension of RHom
R
(U, X) is considered where U is an S-complex with finite projective dimension. As an application, it is shown that if R is a local noetherian ring, then a complex X of R-modules has finite Gorenstein projective dimension if and only if it has finite Gorenstein flat dimension if and only if
belongs to the Auslander category
. This gives a resolution-free characterizations of complexes for which their Gorenstein projective dimensions are finite.
ACKNOWLEDGMENT
The authors wish to express their sincere thanks to the referee for his/her valuable suggestions and comments which have greatly improved the paper, especially simplified the proof of Lemmas 4 and 5.
This work was supported by National Natural Science Foundation of China (No. 11261050) and the Program of Science and Technique, Gansu Province (No. 1010RJZA025).
Notes
Communicated by U. Walther.