Locally simple objects

Abstract

Let A be an abelian category. The Ziegler spectrum of A is used to introduce the notion of a locally simple object of A, which is analyzed in terms of the transitive relation A ⊏ B on A, defined to hold whenever the object A occurs at least twice as a subquotient of B. It is shown that a nonzero object is locally simple if and only if it is minimal with respect to ⊏. By analogy to the notion of simplicity, the Serre subcategory generated by the locally simple objects is used to define local Krull-Gabriel dimension. It is proved that an object has local Krull-Gabriel dimension if and only if it satisfies the ⊏-descending chain condition and that the Serre subcategory of objects with local Krull-Gabriel dimension corresponds in the Ziegler spectrum to the interior of the set of points E satisfying Prest’s condition (A): The localization A/S(E) is local, where S(E) denotes the Serre annihilator of E in A. We also show that a locally finite length object satisfies Fitting’s Lemma.KeywordsSimple ObjectEndomorphism RingAbelian CategoryLondon Mathematical Society Lecture NoteInjective EnvelopeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.