A survey of Morita theorems for associative algebras
Ulrik Sørgaard Djupvik
Tallinn University of Technology
Abstract:
In this talk I will discuss the main results of my master's thesis, which was mainly based on classical Morita theories, including derived Morita equivalence. This talk is meant to be introductory, and as such I will assume a fairly limited familiarity with most of the algebraic notions I will be using.
The mathematical setting is that we have a commutative ring
K, and
K-algebras
Λ and
Γ, and we wish to quantify similarities between the representations of these two algebras. In the setting of classical Morita theory, this amounts to describing when their module categories are equivalent, which turns out to hinge on the existence of so-called progenerators. While in the setting of classical tilting theory, the similarities are quantified by a process known as tilting, which hinges on the existence of so-called (generalized) tilting modules, which generalize the notion of progenerators. It turns out that the tilting process is actually an artifact of an equivalence between the derived categories of the respective module categories, which in turn hinges on the existence of so-called tilting complexes. In order to properly give this last statement, we will require the notions of derived categories and triangulated categories, which I will promptly introduce.