Gorenstein derived categories
Let \(\mathcal A\) be an abelian category with enough projective objects. An object \(G\) of \(\mathcal A\) is Gorenstein-projective if there is an exact sequence \(\dots \rightarrow P^1\rightarrow P^0\rightarrow P_0\rightarrow P_1\rightarrow\dots\) of projective objects of \(\mathcal A\) which stays exact under Hom\(_{\mathcal A}(-,P)\) for each projective objects \(P\) of \(\mathcal A\) and so that \(G\) is isomorphic to the image of \(P^0\rightarrow P_0\). The full subcategory of \(\mathcal A\) of Gorenstein-projective objects is denoted by \({\mathcal G}P\). The authors define the derived category \(D^*( {\mathcal G}P)\) in an appropriate way and study its properties, as well as related constructions such as homotopy category constructions. In particular a Morita theorem in the sense of Rickard is proven for equivalences between \(D^*( {\mathcal G}P)\) for two abelian categories \({\mathcal A}=A-{\mathcal M}od\) and \({\mathcal B}=B-{\mathcal M}od\). The relation to the usual derived category \(D^*({\mathcal A})\) is studied systematically.
scientific article; zbMATH DE number 5696937
scientific article