“Homological algebra: deforming abelian groups using torsion pairs”
Wednesday 18 November 2020 h. 14:30 - Zoom - Sergio Pavon (Padova, Dip. Mat.)
Given an abelian category, one can (often) construct its derived category. We illustrate this process for the familiar category Ab of abelian groups and group homomorphisms. We also give some motivation for this construction, as we briefly see that derived categories provides the “correct” environment for the many (co)homological theories found throughout mathematics.
Having understood what the derived category of an abelian category is, we address the natural question of “derived equivalences”: when do two abelian categories have the same derived category? This turns out to be an interesting but very difficult question. We then restrict ourselves to abelian categories obtained by “deforming” the category Ab of abelian groups: a major role in this construction is played by torsion pairs, whose definition we will illustrate with some examples. We will conclude with a nice theorem, which states that deforming Ab with any torsion pair enjoing a certain property (“hereditarity”) always yields an abelian category derived equivalent to Ab.
This seminar is based on a joint work with J. Vitória.