## Logic Seminar

Sean Cox

VCU

Gorenstein Homological Algebra and elementary submodels

**Abstract:**The class of projective modules is central to classical homological algebra. Relative homological algebra attempts to use some class $\mathcal{G}$ and "do" homological algebra, but with $\mathcal{G}$ playing the same role that the class of projectives played in the original setting. However, an essential requirement for this to work is that $\mathcal{G}$ be a "precovering" class (also called a "right-approximating" class) of modules. There has been considerable work in the last 20 years on the question of whether the class of "Gorenstein Projective" modules is always a precovering class (over every ring). While the question remains open, it is now known that the answer is affirmative if there are enough large cardinals in the universe. This was first proved by Saroch, and then (independently) by me, using entirely different methods. I will discuss some of the key ideas of my proof, especially the use of (set-theoretic) "elementary submodel" arguments and Stationary Logic.

Tuesday February 2, 2021 at 4:00 PM in Zoom