Alessandro Vignati "Nonvanishing higher derived limits"

jeudi 3 mars 2022, 11:00 → 12:00 Europe/Paris

Fokko

Title : Nonvanishing higher derived limits.

Abstract : In homological algebra one considers directed inverse systems of abelian groups indexed by a directed set. One is interested in computing higher derived limits of these systems. Connections with set theory were known from the 1960s and 1970s due to the work of Kaplansky, Osofsky, Mitchell, and others, but at first were largely ignored by set
theorists. In the 80's Mardešić and Prasolov isolated a certain inverse system of abelian groups $A$ indexed by elements of $\omega^\omega$, connecting the study of its higher derived limits with additivity of strong homology of large classes of topological spaces. They also showed that the Continuum Hypothesis implies that the $\lim^1 (A)$ is nonzero.
Later work of Dow, Simon and Vaughan showed that $lim^1(A)=0$ under Forcing Axioms. More recently, Bergfalk gave the first model in which $lim^2 (A)$ is nontrivial, leaving open whether higher derived limits of $A$ can be nontrivial.
With Veličković, we prove that for each $n$ there is a model in which $\lim^n (A)$ is nontrivial. In this talk we review the main concepts of interest, we sketch the proof of our main result, and leave on the table a list of open questions. This is joint work with Boban Veličković.