Seminari del Dipartimento

Algebra Commutativa

According to the classical Krull-Schmidt Theorem for modules, any module of finite composition length decomposes as a direct sum of indecomposable modules in an essentially unique way, that is, unique up to isomorphism of the indecomposable summands and a permutation of the summands. In 1975, Warfield proved that every finitely presented module over a serial ring is a finite direct sum of uniserial modules and posed a problem, essentially asking whether the Krull-Schmidt Theorem holds for finite direct sums of uniserial modules. The negative answer to this question was given by A. Facchini in 1996. He showed that even though the Krull-Schmidt Theorem does not hold for serial modules, it is possible to prove a weak version of it. This phenomenon can be found not only for serial modules, but also for other classes of modules like cyclically presented modules over a local ring, kernels of morphisms between indecomposable injective modules, couniformly presented modules, and more generally, for a number of classes of modules whose endomorphism ring has at most two maximal right ideals. In all these example, direct-sum decompositions are described by two invariants (depending on the specific case). After a brief discussion about the theme of finite direct-sum decompositions of modules, we consider weak forms of the Krull-Schmidt Theorem in additive categories. We provide several examples of additive categories in which weak forms of the Krull-Schmidt Theorem hold, showing that the number of invariants needed to describe finite direct-sum decompositions can be arbitrarily large. Finally, we explain how to find a general pattern that allow to treat all our examples at the same time. This talk is based on a joint work with Alberto Facchini
org: TARTARONE Francesca