Reflexivity and ring homomorphisms of finite flat dimension. (English) Zbl 1118.13015
The paper focuses on the study of reflexivity properties of homologically finite complexes with respect to semidualizing complexes, in the case of non-local rings. A main theme is the descent of these properties over ring morphisms of finite flat dimension, presented in terms of inequalities between generalized G-dimensions. More precisely, first some results already proved by L. W. Christensen [Trans. Am. Math. Soc. 353, 1839–1883 (2001; Zbl 0969.13006)] in the set-up of local rings and local morphisms are extended to the non-local case. Then a non-local version of the amplitude inequality of Iversen, Foxby and Iyengar is proved. This result is exploited in order to prove converses for a number of results of L. W. Christensen [loc. cit.] and to extend the results and their converses to the global case. A multitude of examples are spread throughout the paper, showing that the results obtained by the author are optimal. Several open questions are considered.
Reviewer: Christodor-Paul Ionescu (Bucureşti)
MSC:
| 13D25 | Complexes (MSC2000) |
| 13B10 | Morphisms of commutative rings |
| 13D05 | Homological dimension and commutative rings |
Citations:
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