Article

Mathematische Zeitschrift

, Volume 254, Issue 1, pp 133-171

Hereditary noetherian categories of positive Euler characteristic

  • Helmut LenzingAffiliated withInstitut für Mathematik, Universität Paderborn Email author 
  • , Idun ReitenAffiliated withDepartment of Mathematical Sciences, Norwegian University of Science and Technology

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Abstract

We develop the fundamentals of hereditary noetherian categories with Serre duality over an arbitrary field k, where the category of coherent sheaves over a smooth projective curve over k serves as the prime example and others are coming from the representation theory of finite dimensional algebras. The proper way to view such a category is to think of coherent sheaves on a possibly non-commutative smooth projective curve. We define for each such category notions like function field and Euler characteristic, determine its Auslander-Reiten components and study stable and semistable bundles for an appropriate notion of degree. We provide a complete classification of hereditary noetherian categories for the case of positive Euler characteristic by relating these to finite dimensional representations of (locally bounded) hereditary k-algebras whose underlying valued quiver admits a positive additive function.