January 31
Chris Lambie-Hanson, Virginia Commonwealth University
Set theoretic compactness and higher derived limits
Issues of set theoretic compactness frequently arise when considering questions from homological algebra about derived functors. In particular, the non-vanishing of such derived functors is often witnessed by a concrete combinatorial instance of set theoretic incompactness, so that homological questions can be translated into questions about combinatorial set theory. In this talk, we will discuss some recent results about the derived functors of the inverse limit functor. We will focus on a specific inverse system of abelian groups, $\mathbf{A}$, that arose in Mardešić and Prasolov's work on the additivity of strong homology and has since arisen independently in a number of contexts. Our main result states that, relative to the consistency of a weakly compact cardinal, it is consistent that the $n$-th derived limits $\lim^n \mathbf{A}$ vanish simultaneously for all $n \geq 1$. We will sketch a proof of this theorem and then discuss the extent to which certain generalizations of the result can hold. The arguments will be purely set theoretic, and no knowledge of homological algebra will be assumed. This is joint work with Jeffrey Bergfalk.