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Comparison of relative cohomology theories with respect to semidualizing modules. (English) Zbl 1190.13007

The authors study and compare a number of relative Ext groups which naturally arise from a semidualizing module.
More precisely, let \(R\) be a commutative ring, and let \(C\) be a semidualizing \(R\)-module. Denote by \(\mathcal{P}_C\) the class of \(C\)-projective modules, and by \(\mathcal{I}_C\) the class of \(C\)-injective modules over \(R\). Furthermore, let \(\mathcal{GP}_C\) be class of \(\text{G}_C\)-projective modules, and let \(\mathcal{GI}_C\) be the class of \(\text{G}_C\)-injective modules over \(R\) (the “G” is for Gorenstein). These four classes of modules have previously been studied by the authors, by the reviewer and P. Jørgensen [J. Pure Appl. Algebra 205, No. 2, 423–445 (2006; Zbl 1094.13021)], and by others.
Using standard constructions from relative homological algebra, one can (under suitable assumptions) define relative Ext groups, \(\text{Ext}^*_{\mathcal{P}_C}(-,-)\), \(\text{Ext}^*_{\mathcal{I}_C}(-,-)\), \(\text{Ext}^*_{\mathcal{GP}_C}(-,-)\), and \(\text{Ext}^*_{\mathcal{GI}_C}(-,-)\). The authors prove that, in general, these four functors are pairwise non-isomorphic. However, if \(R\) is Cohen–Macaulay with a dualizing module \(D\), and \(C^\dagger\) is defined to be \(\text{Hom}_R(C,D)\), then one has \(\text{Ext}^*_{\mathcal{P}_C}(M,N) \cong \text{Ext}^*_{\mathcal{I}_{C^\dagger}}(M,N)\) if \(M\) has finite \(\mathcal{P}_C\)-dimension and \(N\) has finite \(\mathcal{I}_{C^\dagger}\)-dimension. Similar results are proved for the “Gorenstein versions” of these Ext groups. Finally, the authors give a Yoneda-type description of a general relative \(\text{Ext}^1(M,N)\) group.

MSC:

13D02 Syzygies, resolutions, complexes and commutative rings
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)

Citations:

Zbl 1094.13021
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References:

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