On classical and Gorenstein homological invariants of rings
We prove that, over any ring $R$, the supremum of the projective dimensions of the flat left $R$-modules coincides with the supremum of the Gorenstein projective dimensions of the Gorenstein flat left $R$-modules. As a consequence, we obtain new characterizations of left $n$-perfect rings in terms of Gorenstein projective, Ding projective, and projectively coresolved Gorenstein flat dimensions, extending results by Emmanouil and Dalezios and by Christensen, Estrada, and Thompson. We also introduce Gorenstein analogues of several classical homological invariants and study their relationships with the classical ones, identifying conditions under which they coincide with the classical invariants. Finally, we obtain characterizations of left weakly $n$-$Σ$-cotorsion rings introduced by Cortés-Izurdiaga, Estrada and Fresneda in terms of Gorenstein classes of modules.