Locally Type \(\text {FP}_{{\varvec{n}}}\) and \({\varvec{n}}\)-Coherent Categories

We study finiteness conditions in Grothendieck categories by introducing the concepts of objects of type \(\textrm{FP}_n\) and studying their closure properties with respect to short exact sequences. This allows us to propose a notion of locally type \(\textrm{FP}_n\) categories as a generalization of locally finitely generated and locally finitely presented categories. We also define and study the injective objects that are Ext-orthogonal to the class of objects of type \(\textrm{FP}_n\), called \(\textrm{FP}_n\)-injective objects, which will be the right half of a complete cotorsion pair. As a generalization of the category of modules over an n-coherent ring, we present the concept of n-coherent categories, which also recovers the notions of locally noetherian and locally coherent categories for \(n = 0, 1\). Such categories will provide a setting in which the \(\textrm{FP}_n\)-injective cotorsion pair is hereditary, and where it is possible to construct (pre)covers by \(\textrm{FP}_n\)-injective objects.