Let R be a ring, Proj be the class of all the projective right R-modules, K be the full subcategory of the homotopy category K(Proj) whose class of objects consists of all the totally acyclic complexes, and MorK be th...Let R be a ring, Proj be the class of all the projective right R-modules, K be the full subcategory of the homotopy category K(Proj) whose class of objects consists of all the totally acyclic complexes, and MorK be the class of all the morphisms in K(Proj) whose cones belong to K. We prove that if K(Proj) has enough MorK-injective objects, then the Verdier quotient K(Proj)/K has small Hom-sets, and this last condition implies the existence of Gorenstein-projective precovers in Mod-R and of totally acyclic precovers in C(Mod-R).展开更多
基金supported by the Spanish Government (Grant No. PID2020-113206GBI00, funded by MCIN/AEI/10.13039/501100011033)Junta de Andalucia (Grant No. P20-00770)。
文摘Let R be a ring, Proj be the class of all the projective right R-modules, K be the full subcategory of the homotopy category K(Proj) whose class of objects consists of all the totally acyclic complexes, and MorK be the class of all the morphisms in K(Proj) whose cones belong to K. We prove that if K(Proj) has enough MorK-injective objects, then the Verdier quotient K(Proj)/K has small Hom-sets, and this last condition implies the existence of Gorenstein-projective precovers in Mod-R and of totally acyclic precovers in C(Mod-R).
