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Acyclic Calabi-Yau categories. (English) Zbl 1171.18008

The main theorem gives a necessary and sufficient condition for a 2-Calabi-Yau category (defined over an algebraically closed field) to be a cluster category. It is shown that a 2-Calabi-Yau category is a cluster category if and only if it contains a cluster titling subcategory whose quiver has no oriented cycles. This theorem is then used to produce new results in commutative algebra and in combinatorics quiver mutations. The paper ends with an appendix where Michel Van den Bergh gives an alternative proof of the main theorem by appealing to the universal property of the triangulated orbit category.

MSC:

18E30 Derived categories, triangulated categories (MSC2010)
16D90 Module categories in associative algebras
18G40 Spectral sequences, hypercohomology
18G10 Resolutions; derived functors (category-theoretic aspects)
55U35 Abstract and axiomatic homotopy theory in algebraic topology
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