Functorially finite hearts, simple-minded systems in negative cluster categories, and noncrossing partitions

Abstract

Let Q be an acyclic quiver and w 1 be an integer. Let C−w(kQ) be the (−w)-cluster category of kQ. We show that there is a bijection between simple-minded collections in Db(kQ) lying in a fundamental domain of C−w(kQ) and w-simple-minded systems in C−w(kQ). This generalises the same result of Iyama–Jin in the case that Q is Dynkin. A key step in our proof is the observation that the heart H of a bounded t-structure in a Hom-finite, Krull–Schmidt, k-linear saturated triangulated category D is functorially finite in D if and only if H has enough injectives and enough projectives. We then establish a bijection between w-simple-minded systems in C−w(kQ) and positive w-noncrossing partitions of the corresponding Weyl group WQ.