Homological aspects of semidualizing modules. (English) Zbl 1193.13012

Let \(C\) be a semidualizing module over a commutative ring \(R\). An \(R\)-module is called \(C\)-projective if it has the form \(C \otimes_R P\) for some projective \(R\)-module \(P\). The class of such modules is denoted by \(\mathcal{P}_C\). The authors introduce for any \(R\)-module \(M\) its \(\mathcal{P}_C\)-projective dimension as the length of the shortest proper resolution of \(M\) by modules in \(\mathcal{P}_C\).
A number of interesting descriptions of the \(\mathcal{P}_C\)-projective dimension is provided, for example, the \(\mathcal{P}_C\)-projective dimension of a module \(M\) equals the (ordinary) projective dimension of Hom\(_R(C,M)\). Furthermore, the (ordinary) projective dimension of a module \(N\) equals the \(\mathcal{P}_C\)-projective dimension of \(C \otimes_R N\).
The authors also investigate the relative cohomology theory Ext\(_{\mathcal{P}_C}^*\) with respect to the precovering class \(\mathcal{P}_C\).
In parallel, a similar theory for \(C\)-injective modules and \(\mathcal{I}_C\)-injective dimension is developed.


13D05 Homological dimension and commutative rings
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
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