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Reducing invariants and total reflexivity. (English) Zbl 1441.13039
Let \(R\) denote a commutative Noetherian local ring with unique maximal ideal \(\mathfrak{m}\) and residue field \(k\). In the paper under review, the authors introduce reducing versions of homological invariants (in particular those of homological dimensions) of \(R\)-modules. They show that a module can have finite reducing Gorenstein dimension, even if it has infinite Gorenstein dimension and they give such examples. As a main result the authors prove the following.
Theorem. Let \(M\) be an \(R\)-module which has finite reducing Gorenstein dimension. Then \(\mathrm{G-dim}_R (M) = sup\{i\in \mathbb{Z}| \mathrm{Ext}^i_R(M,R)\neq 0\}\).
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
13C13 Other special types of modules and ideals in commutative rings
13C14 Cohen-Macaulay modules
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
Full Text: DOI Euclid
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