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Classifying torsionfree classes of the category of coherent sheaves and their Serre subcategories (2304.06918v2)

Published 14 Apr 2023 in math.RT, math.AC, and math.AG

Abstract: In this paper, we classify several subcategories of the category of coherent sheaves on a noetherian divisorial scheme (e.g. a quasi-projective scheme over a commutative noetherian ring). More precisely, we classify the torsionfree (resp. torsion) classes closed under tensoring with line bundles by the subsets (resp. specialization-closed subsets) of the scheme, which generalizes the classification of torsionfree (resp. torsion) classes of the category of finitely generated modules over a commutative noetherian ring by Takahashi (resp. Stanley--Wang). Furthermore, we classify the Serre subcategories of a torsionfree class (in the sense of Quillen's exact categories) by using the above classifications, which gives a certain generalization of Gabriel's classification of Serre subcategories. As explicit applications, we classify the Serre subcategories of the category of maximal pure sheaves, which are a natural generalization of vector bundles for reducible schemes, on a reduced projective curve over a field, and the category of maximal Cohen-Macaulay modules over a one-dimensional Cohen-Macaulay ring.

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