Verdier quotients of homotopy categories of rings and Gorenstein-projective precovers (Q6649841)

Relative homological algebra extends homological techniques to abelian categories without enough projective or injective objects by using a suitable class \(\mathcal{C}\) to compute resolutions, provided \(\mathcal{C}\)-precovers or \(\mathcal{C}\)-preenvelopes exist. Gorenstein homological algebra, a key example, replaces projective, injective, and flat modules with Gorenstein-projective, Gorenstein-injective, and Gorenstein-flat modules, which is particularly useful in commutative algebra for rings with finite Gorenstein global dimension. This approach allows derived functors to be computed in categories lacking enough projectives or injectives.\N\NThe existence of Gorenstein-projective precovers is established for certain classes of rings, such as left coherent and right \(n\)-perfect rings, where \textit{S. Estrada} et al. [Mediterr. J. Math. 14, No. 1, Paper No. 33, 10 p. (2017; Zbl 1368.18008)] extended Jørgensen's earlier homotopy-based results. Jørgensen's key insight assumes that the subcategory \(\mathcal{K}\) of \(K(\text{Proj})\) consisting of totally acyclic complexes is coreflective. Building on this, the author shows that if the Verdier quotient of \(K(\text{Proj})\) by \(\mathcal{K}\) has small Hom-sets, then \(\mathcal{K}\) is coreflective in \(K(\text{Proj})\), ensuring Gorenstein-projective precovers in \(\text{Mod-}R\). Additionally, they prove the existence of totally acyclic precovers in \(C(\text{Mod-}R)\).