Simple tilts of length hearts and simple-minded mutation

Abstract

We characterise when a simple Happel-Reiten-Smalo tilt of a length heart is again a length heart in terms of approximation theory and the existence of a stability condition with a phase gap. We apply simple-minded reduction to provide a sufficient condition for infinite iterability of simple-minded mutation/simple tilting. We use simple-minded mutation pairs to provide a common framework to show that mutation of simple-minded collections (resp. $w$-simple-minded systems, for $w\ge 1$) gives simple-minded collections (resp. $w$-simple-minded systems) under mild conditions, in the process providing a unified proof of results of Alex Dugas and Peter Jorgensen. Finally, we show that under mild conditions, mutation of simple-minded collections is compatible with mutation of $w$-simple-minded systems via a singularity category construction due to Haibo Jin.

29 pages; 3 figures; updates after peer review; journal accepted version