Direct-sum decompositions in additive categories

Alberto Facchini

University of Padova

Recent progress has been done recently in the study of direct-sum decompositions in the category of modules. We will try to extend these results to the setting of arbitrary additive categories, or, better to additive categories in which idempotents have kernels, equivalently, categories in which idempotents split. For these categories, called amenable by Freyd, the Krull-Schmidt Theorem holds for objects with a local endomorphism ring, as was observed by Bass. Our main ingredients will be valuations of additive categories and categories with invariant basis number.