2023/04/14 by Shunya Saito, Saito, Shunya
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #13C60 (Primary) #14H60 (Secondary) #16G50 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Alkaloids: synthesis and pharmacology #Commutative Algebra (math.AC) #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2304.06918
openalex publication_date 2023/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.