Closure under transfinite extensions (Q2382962)

Let \(\mathcal{A}\) be a Grothendieck category with a fixed projective generator \(U\) and \(\mathcal{L}\) a class of objects of \(\mathcal{A}\) which is closed under isomorphisms. An object \(X\) of \(\mathcal{A}\) is said to be a direct (inverse) transfinite extension of objects of \(\mathcal{L}\) if \(X=\varinjlim X_{\alpha }\) (or \(X=\varprojlim X_{\alpha }\)) for a continuous direct (inverse) system \((X_{\alpha }:\alpha \leq \lambda )\) of monomorphisms (epimorphisms) such that \(\text{Coker}(X_{\alpha }\rightarrow X_{\alpha +1})\) (\(\text{Ker}(X_{\alpha +1}\rightarrow X_{\alpha })\)) is in \(\mathcal{L}\) whenever \( \alpha +1\leq \lambda .\) \(\mathcal{L}\) is said to be closed under direct (inverse) extensions if each direct (inverse) transfinite extension of objects in \(\mathcal{L}\) is also in \(\mathcal{L}.\) In this paper the authors prove that several important classes are closed under transfinite extensions (as the classes of exact complexes, complete projective (injective) resolutions in \(\mathcal{C(A)}=\) the category of complexes of objects of \(\mathcal{A}\) and the class \(\mathcal{L}\) of Gorenstein projective (injective) objects of \(\mathcal{A}\)).