Giovanna Le Gros, University of Stuttgart

Title: Tilting cotorsion pairs over commutative rings.

Abstract: Tilting classes over commutative rings were neatly classified by Hrbek in the 1-dimensional case, and by Hrbek-Stovicek in the n-dimensional case in terms of sequences of length n-1 of certain faithful finite generated Gabriel topologies. In the 1-dimensional case, the modules in the tilting class are exactly the modules which are divisible by the ideals in the associated Gabriel topology.
 This useful classification has paved the way for the study of tilting cotorsion pairs over commutative rings. Specifically, the classification of the rings over which a 1-tilting cotorsion pair provides envelopes or covers, and well-behaved n-tilting cotorsion pairs (that is with the associated Gabriel topologies) that provide envelopes. Moreover, after Hrbek's characterisation of 1-tilting classes, he considered when tilting classes are closed under flat covers. This is a natural question to ask since the analogous statement, that torsion-free modules are closed under injective envelopes, is always true.
In this talk, we will give an overview of these results and discuss some progress, which are based on ongoing joint works with Silvana Bazzoni and Dolors Herbera.