Jan Šaroch, Charles University

Title: Utilization of $\lambda$-pure-injective modules

Abstract: Let $\lambda$ be a regular infinite cardinal number.
The notion of a $\lambda$-pure-injective module forms a natural
generalization of ($\aleph_0$-)pure-injectivity: instead of being
injective with respect to all pure embeddings, we require that a
given module be injective with respect to $\lambda$-pure
embeddings only which is, in general, a smaller class of
embeddings if $\lambda$ is uncountable. The utilization of this
concept in the literature has been very sparse so far, mostly
because of the lack of explicit examples.

In my talk, I would like to present some interesting properties
of $\lambda$-pure-injective modules, together with a couple of
applications which make use of additional set-theoretic axioms:
one (rather short) towards Gorenstein homological algebra, the
other one dealing with the question whether the category Mod-$R$
possesses enough $\lambda$-pure-injective objects. The talk will
be mostly based on the sixth section of my recent joint paper with
Manuel Cortés-Izurdiaga. Its preprint version can be accessed via
the link https://arxiv.org/abs/2104.08602