Categorical compactness for rings

Third Mac Lane Cohomology via Categorical Rings

In the fifties, Saunders Mac Lane invented a cohomology theory of rings using the cubical construction introduced earlier by Eilenberg and himself to calculate stable homology of Eilenberg-Mac Lane spaces. As shown in [9], this theory coincides with the topological Hochschild cohomology for Eilenberg-Mac Lane ring spectra. In particular, the third dimensional cohomology group is expected to pro...

Generically stable, ω-categorical groups and rings

In two recent papers with Krzysztof Krupi´nski, we proved that every ω-categorical, generically stable group is nilpotent-by-finite and that every ω-categorical, generically stable ring is nilpotent-by-finite. During the lecture, after a brief overview of well-known results on ω-categorical groups and rings, I will explain the main ideas of the proofs of our results.

On ω-categorical groups and rings with NIP

We show that ω-categorical rings with NIP are nilpotent-by-finite. We prove that an ω-categorical group with NIP and fsg is nilpotent-by-finite. We also notice that an ω-categorical group with at least one strongly regular type is abelian. Moreover, we get that each ω-categorical, characteristically simple p-group with NIP has an infinite, definable abelian subgroup. Assuming additionally the e...

On ω-categorical, generically stable groups and rings

We prove that every ω-categorical, generically stable group is nilpotent-byfinite and that every ω-categorical, generically stable ring is nilpotent-by-finite.

Compactness for Regionalization and its Application in Land-Use Planning