Algebraický seminář

Simion Breaz (Univ. Cluj): On the exchange property and the Krull-Remak-Schmidt-Azumaya Theorem in idempotent complete categories
Aurelian Minuta (Univ. Cluj): G-graded Morita theory over a G-graded G-acted algebra

Abstract (Simion Breaz): I will show that some results and techniques already known for categories with kernels and which satisfy a weak Grothendieck condition can be extended to additive categories with enough compact objects. In particular, the Krull-Remak-Schmidt-Azumaya unique decomposition theorem holds in idempotent complete additive categories with enough compact objects (hence in compactly generated triangulated categories). If time permits I will give an example of an object with the finite exchange property that does not have the exchange property.
Abstract (Aurelian Minuta): In the representation theory of finite groups, an important strategy used in the verification of local-global conjectures is to reduce their statements to some inductive conditions on simple groups. In order to obtain such reduction theorems, the language of character triples and the study of the relations between them have proven useful, as demonstrated by Britta Späth.
According to the convictions expressed primarily in the work of Michel Broué, which posits that character correspondences with good properties are consequences of categorical equivalences, together with Andrei Marcus, we have proven that there is a link between character triples and group graded Morita equivalences with some additional properties. Namely, these equivalences are induced by group graded bimodules that act over a group graded group acted algebra.
In this presentation, I will delve into the construction of such equivalences, their properties, and different results we have obtained, such as various analogues of the so-called ``butterfly theorem'' and a notion of a group graded Picent group, which gives rise to an exact sequence resembling the Beattie--del Río sequence.