A note on generic objects and locally finite triangulated categories

TL;DR

This paper explores the conditions under which the homotopy category of injective modules and the derived category of all modules are generically trivial, linking these properties to concepts like local finiteness and Krull-Gabriel dimension.

Contribution

It establishes an equivalence between the generic triviality of the homotopy category of injective modules and the derived category for an algebra, and connects generic objects with local finiteness and Krull-Gabriel dimension.

Findings

Homotopy category of injective modules is generically trivial iff the derived category is.

Connections between generic objects, local finiteness, and Krull-Gabriel dimension are identified.

Abstract

We show that the homotopy category of injective $A$-modules is generically trivial if and only if the derived category of all modules is generically trivial for an algebra $A$. Moreover we show some connections between the generic objects, locally finiteness and Krull-Gabriel dimension.