2017/07/31 by Pavel Čoupek, Jan Šťovíček
Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Derived category #Discrete mathematics #Disjoint sets #Homotopy and Cohomology in Algebraic Topology #Indecomposable module #Injective function #Injective module #Mathematics #Noetherian #Pure mathematics #Torsion (gastropod) #Triangulated category #math.AG #math.CT #math.RT
paper · pdf · doi:10.1007/s00209-019-02404-8
published as Math. Z. 296 (2020), no. 1-2, 275-312 · 39 pages; version 2: improvements in Section 3 (Theorem 3.10 characterizes cotilting torsion-free classes in arbitrary Grothendieck categories) and Section 6 (we show that any Noetherian scheme has a torsion-free generator), Remark 2.2 and some misprints corrected, references updated
arxiv created 2019/04/16 · openalex publication_date 2019/11/19 · arxiv updated 2021/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We develop theory of (possibly large) cotilting objects of injective dimension at most one in general Grothendieck categories. We show that such cotilting objects are always pure-injective and that they characterize the situation where the Grothendieck category is tilted using a torsion pair to another Grothendieck category. We prove that for Noetherian schemes with an ample family of line bundles a cotilting class is closed under injective envelopes if and only if it is invariant under twists by line bundles, and that such cotilting classes are parametrized by specialization closed subsets disjoint from the associated points of the scheme. Finally, we compute the cotilting sheaves of the latter type explicitly for curves as products of direct images of indecomposable injective modules or completed canonical modules at stalks.