Dolors Herbera (Universitat Autònoma de Barcelona)
Torsion free modules over commutative domains of Krull dimension 1
Abstract: Let $R$ be a commutative domain. Let $\mathcal F$ be the class of $R$-modules that are infinite direct sums of finitely generated torsion-free modules. In the talk we will discuss the question of whether $\mathcal F$ is closed under direct summands.
If $R$ is local of Krull dimension 1, $\mathcal F$ being closed under direct summands is equivalent to saying that any indecomposable, finitely generated torsion-free module has local endomorphism ring.
For the global case, we show also in the case of Krull dimension 1 that the property on $\mathcal F$ is inherited by the localization at a maximal ideal. Moreover, there is an interesting relation between ranks of indecomposable modules over such localizations.
The machinery we use to prove these results was explained in Roman Álvarez’s talk, in the previous session of the seminar.
Time permitting, we will also discuss the property `being locally a direct summand’ versus `being a direct summand’ in the setting of our problem. The results we obtain allows us to give a complete answer to the initial problem in some particular cases.
The talk is based on a joint work with Roman Álvarez and Pavel Příhoda.