Resolutions by Gorenstein injective and projective modules and modules of finite injective dimension over Gorenstein rings. (English) Zbl 0823.16003
Let \(R\) be an \(n\)-Gorenstein ring, \(\mathcal L\) denotes the class of all \(R\)- modules of finite injective dimension, \({\mathcal L}^ \perp\) denotes the class of Gorenstein injective modules, i.e. the class of all modules \(K\) such that \(\text{Ext}^ 1 (N,K) = 0\) for all \(N \in {\mathcal L}\), whilst \(^ \perp {\mathcal L}\) denotes the class of Gorenstein projectives i.e. satisfying \(\text{Ext}^ 1 (K,N) = 0\) for all \(N \in \mathcal L\). In this paper the authors show that the measure of Gorenstein injectivity or projectivity of a module is concentrated in the zeroth term of the minimal resolutions in the sense that the resolutions of the first cosyzygy and syzygy are injective and projective resolutions respectively. This is then applied to results on \({\mathcal L}^ \perp\)- injective dimension and dually to \(^ \perp{\mathcal L}\)-projective dimension when \(R\) is commutative and local.
Reviewer: T.Porter (Bangor)
MSC:
16D50 | Injective modules, self-injective associative rings |
16D40 | Free, projective, and flat modules and ideals in associative algebras |
16E10 | Homological dimension in associative algebras |
16P40 | Noetherian rings and modules (associative rings and algebras) |
13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
13D02 | Syzygies, resolutions, complexes and commutative rings |
13D03 | (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) |
Keywords:
\(n\)-Gorenstein rings; \(R\)-modules of finite injective dimension; Gorenstein injective modules; Gorenstein projectives; minimal resolutions; cosyzygy; syzygy; projective resolutionsReferences:
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