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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 Program. For a 3-fold flopping contraction f : X → Spec R, where X has only Gorenstein terminal singularities, there is an associated finite dimensional algebra Acon 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 Acon 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.

We construct Grassmannian categories of infinite rank, providing an infinite analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. Each Grassmannian category of infinite rank is given as the category of graded maximal Cohen–Macaulay modules over a certain hypersurface singularity. We show that generically free modules of rank $1$ in a Grassmannian category of infinite rank are in bijection with the Plücker coordinates in an appropriate Grassmannian cluster algebra of infinite rank. Moreover, this bijection is structure preserving, as it relates rigidity in the category to compatibility of Plücker coordinates. Along the way, we develop a combinatorial formula to compute the dimension of the $\textrm {Ext}^{1}$-spaces between any two generically free modules of rank $1$ in the Grassmannian category of infinite rank.

To every minimal model of a complete local isolated cDV singularity Donovan-Wemyss associate a finite dimensional symmetric algebra known as the contraction algebra. We construct the first known standard derived equivalences between these algebras and then use the structure of an associated hyperplane arrangement to control the compositions, obtaining a faithful group action on the bounded derived category. Further, we determine precisely those standard equivalences which are induced by two-term tilting complexes and show that any standard equivalence between contraction algebras (up to algebra automorphism) can be viewed as the composition of our constructed functors. Thus, for a contraction algebra, we obtain a complete picture of its derived equivalence class and, in particular, of its derived autoequivalence group.

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