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Abstract

We prove that the stable endomorphism rings of rigid objects in a suitable
Frobenius category have only finitely many basic algebras in their derived
equivalence class and that these are precisely the stable endomorphism rings of
objects obtained by iterated mutation. The main application is to the Homological Minimal Model Programme. For a 3-fold flopping contraction $f \colon X \to \mathrm{Spec}\ R$, where $X$ has only Gorenstein terminal singularities, there is an associated finite dimensional algebra
$A_{\mathrm{con}}$ known as the contraction algebra. As a corollary of our main result, there are only finitely many basic algebras in the derived equivalence
class of $A_{\mathrm{con}}$ and these are precisely the contraction algebras of
maps obtained by a sequence of iterated flops from $f$. This provides evidence
towards a key conjecture in the area.