Reconstructing projective schemes from Serre subcategories

2006/08/23 by Grigory Garkusha, Garkusha, Grigory, Mike Prest +1
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #math.AG #math.AT

paper · pdf · doi:10.48550/arxiv.math/0608574

some minor corrections made

openalex publication_date 2006/08/23 · arxiv created 2007/04/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a positively graded commutative coherent ring A which is finitely generated as an A₀-algebra, a bijection between the tensor Serre subcategories of qgr A and the set of all subsets Y⊆ Proj A of the form Y=\bigcup_i∈ΩY_i with quasi-compact open complement Proj A\Y_i for all i∈Ωis established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of isomorphism classes of indecomposable injective graded modules are used in an essential way. Also, there is constructed an isomorphism of ringed spaces (Proj A,O_{Proj A}) --> (Spec(qgr A),O_{qgr A}), where (Spec(qgr A),O_{qgr A}) is a ringed space associated to the lattice L_serre(qgr A) of tensor Serre subcategories of qgr A.

Related

• Classifying torsionfree classes of the category of coherent sheaves and their Serre subcategories
• Serre's theorem for coherent sheaves via Auslander's techniques
• Blowups and fibers of morphisms
• Note on the Serre-Swan Theorem
• The Ziegler spectrum for enriched ringoids and schemes