March 31, 2025, 16:30 +0100
Padova, Torre Archimede 2AB40 and online

On a problem raised by Cuadra and Simson

Manuel Saorín, University of Murcia

Abstract

In a paper of 2007 Juan Cuadra and Daniel Simson asked whether a locally finitely presented Grothendieck category \(\mathcal{G}\) with enough flats has enough projectives. In an ongoing joint work with Carlos Parra and Lorenzo martini, we have proved that such a \(\mathcal{G}\) is \(\mathsf{AB}6\) and \(\mathsf{AB}4^\ast\), which, by a little variation of a result of Roos, means that \(\mathcal{G}\) is equivalent to \(\mathrm{Mod}-\mathcal{A}/\mathcal{T}\), where \(\mathcal{A}\) is a small preadditive category and \(\mathcal{T}\) is a TTF class in \(\mathrm{Mod}-\mathcal{A}\). In particular, by ( a generalization of) a result of Jans, there is a uniquely determined idempotent ideal \(\mathcal{I}\) of \(\mathcal{A}\) such that \(\mathcal{T}\) consists of the right \(\mathcal{A}\)-modules (=additive functor \(\mathcal{A}^{op}\to \mathrm{Ab}\)) annihilated by \(\mathcal{I}\). We will show in the talk that Cuadra-Simson problem is then a particular case of a classical problem, first apparently raised when \(\mathcal{A}\) is a ring (=preadditive category with just one object) by Miller in 1975. This problem asks when an idempotent ideal is the trace of a projective \(\mathcal{A}\)-module and, as shown by Krause in 200, also includes the verification-refutation of the telescope conjecture for a given compactly generated triangulated category as a particular case.

We will end the talk by showing some affirmative answers to Cuadra-Simson problem.