Abstract. We use semi-invariant pictures to prove two conjectures about maximal green sequences. First: if Q is any acyclic valued quiver with an arrow j → i of infinite type then any maximal green sequence for Q must mutate at i before mutating at j. Second: for any quiver Q obtained by mutating an acyclic valued quiver Q of tame type, there are only finitely many maximal green sequences for Q . Both statements follow from the Rotation Lemma for reddening sequences and this in turn follows from the Mutation Formula for the semi-invariant picture for Q.

The "No Gap Conjecture" of Brüstle-Dupont-Pérotin states that the set of lengths of maximal green sequences for hereditary algebras over an algebraically closed field has no gaps. This follows from a stronger conjecture that any two maximal green sequences can be "polygonally deformed" into each other. We prove this stronger conjecture for all tame hereditary algebras over any field.2010 Mathematics Subject Classification. 16G20; 20F55.

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Abstract. We compute the minimal model for Ginzburg algebras associated to acyclic quivers Q. In particular, we prove that there is a natural grading on the Ginzburg algebra making it formal and quasi-isomorphic to the preprojective algebra in non-Dynkin type, and in Dynkin type is quasi-isomorphic to a twisted polynomial algebra over the preprojective with a unique higher A∞-composition. To prove these results, we construct and study the minimal model of an A∞-envelope of the derived category D b (Q) whose higher compositions encode the triangulated structure of D b (Q).

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