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.
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Dedicated to Otto Kerner on the occasion of his 60th birthday
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Lenzing, H., Reiten, I. Hereditary noetherian categories of positive Euler characteristic. Math. Z. 254, 133–171 (2006). https://doi.org/10.1007/s00209-006-0938-6
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DOI: https://doi.org/10.1007/s00209-006-0938-6