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In this paper, we classify several subcategories of the category of coherentsheaves on a noetherian divisorial scheme (e.g. a quasi-projective scheme overa 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 theclassification of torsionfree (resp. torsion) classes of the category offinitely generated modules over a commutative noetherian ring by Takahashi(resp. Stanley--Wang).  Furthermore, we classify the Serre subcategories of a torsionfree class (inthe sense of Quillen's exact categories) by using the above classifications,which gives a certain generalization of Gabriel's classification of Serresubcategories. As explicit applications, we classify the Serre subcategories ofthe category of maximal pure sheaves, which are a natural generalization ofvector bundles for reducible schemes, on a reduced projective curve over afield, and the category of maximal Cohen-Macaulay modules over aone-dimensional Cohen-Macaulay ring.

Classifying torsionfree classes of the category of coherent sheaves and their Serre subcategories