Semi-Infinite Algebraic Geometry of Quasi-Coherent Sheaves on Ind-Schemes [[electronic resource] ] : Quasi-Coherent Torsion Sheaves, the Semiderived Category, and the Semitensor Product / / by Leonid Positselski

• Autore: Positselski Leonid
• Pubblicazione: Cham : , : Springer Nature Switzerland : , : Imprint : Birkhäuser, , 2023
• Edizione: 1st ed. 2023.
• Descrizione fisica: 1 online resource (225 pages)
• Disciplina: 516.35
• Soggetto topico: Algebraic geometry; Commutative algebra; Commutative rings; Algebra, Homological; Algebraic Geometry; Commutative Rings and Algebras; Category Theory, Homological Algebra
• Nota di contenuto: Intro -- Introduction -- Acknowledgement -- Contents -- 1 Ind-Schemes and Their Morphisms -- 1.1 Ind-Objects -- 1.2 Ind-Schemes -- 1.3 Morphisms of Ind-Schemes -- 1.4 Ind-Affine Examples -- 2 Quasi-Coherent Torsion Sheaves -- 2.1 Reasonable Ind-Schemes -- 2.2 Quasi-Coherent Sheaves and Functors -- 2.3 Quasi-Coherent Torsion Sheaves -- 2.4 Ind-Affine Examples -- 2.5 Direct Limits -- 2.6 Direct Images -- 2.7 -Systems -- 2.8 Inverse Images -- 2.9 Injective Quasi-Coherent Torsion Sheaves -- 3 Flat Pro-Quasi-Coherent Pro-Sheaves -- 3.1 Pro-Quasi-Coherent Pro-Sheaves -- 3.2 Action of Pro-Sheaves in Torsion Sheaves -- 3.3 Inverse and Direct Images -- 3.4 Flat Pro-Quasi-Coherent Pro-Sheaves -- 3.5 Coproducts and Colimits -- 3.6 Pro-Quasi-Coherent Commutative Algebras -- 4 Dualizing Complexes on Ind-Noetherian Ind-Schemes -- 4.1 Ind-Noetherian Ind-Schemes -- 4.2 Definition of a Dualizing Complex -- 4.3 Derived Categories of Flat Sheaves and Flat Pro-Sheaves -- 4.4 Coderived Category of Torsion Sheaves -- 4.5 The Triangulated Equivalence -- 5 The Cotensor Product -- 5.1 Construction of Cotensor Product -- 5.2 Ind-Artinian Examples -- 6 Ind-Schemes of Ind-Finite Type and the !-Tensor Product -- 6.1 External Tensor Product of Quasi-Coherent Sheaves -- 6.2 External Tensor Product of Pro-Sheaves -- 6.3 External Tensor Product of Torsion Sheaves -- 6.4 Derived Restriction with Supports -- 6.5 Rigid Dualizing Complexes -- 6.6 Covariant Duality Commutes with External Tensor Products -- 6.7 The Cotensor Product as the !-Tensor Product -- 7 X-Flat Pro-Quasi-Coherent Pro-Sheaves on Y -- 7.1 Semiderived Category of Torsion Sheaves -- 7.2 Pro-sheaves Flat over the Base -- 7.3 The Triangulated Equivalence -- 8 The Semitensor Product -- 8.1 Underived Tensor Products in the Relative Context -- 8.2 Relatively Homotopy Flat Resolutions. ; 8.3 Left Derived Tensor Products for Pro-sheaves Flatover a Base -- 8.4 Construction of Semitensor Product -- 8.5 The Ind-Artinian Base Example -- 9 Flat Affine Ind-Schemes over Ind-Schemes of Ind-Finite Type -- 9.1 Derived Inverse Image of Pro-sheaves -- 9.2 Derived Inverse Image of Torsion Sheaves -- 9.3 Derived Restriction with Supports in the Relative Context -- 9.4 Composition of Derived Inverse Images of Pro-sheaves -- 9.5 External Tensor Products in the Relative Context -- 9.6 Derived Tensor Product of Pro-sheaves as Derived Restriction to the Diagonal -- 9.7 Semiderived Equivalence and Change of Fiber -- 9.8 Semiderived Equivalence and Base Change -- 9.9 Semiderived Equivalence and External Tensor Product -- 9.10 The Semitensor Product Computed -- 10 Invariance Under Postcomposition with a Smooth Morphism -- 10.1 Weakly Smooth Morphisms -- 10.2 Flat and Injective Dimension Under Weakly Smooth Morphisms -- 10.3 Preservation of the Derived Category of Pro-sheaves -- 10.4 Preservation of the Semiderived Category of Torsion Sheaves -- 10.5 Derived Restriction with Supports Commutes with Flat Pullback -- 10.6 Preservation of the Semiderived Equivalence -- 10.7 Preservation of the Semitensor Product -- 11 Some Infinite-Dimensional Geometric Examples -- 11.1 The Tate Affine Space Example -- 11.2 Cotangent Bundle to Discrete Projective Space -- 11.3 Universal Fibration of Quadratic Cones in Linearly Compact Vector Space -- 11.4 Loop Group of Affine Algebraic Group -- A The Semiderived Category for a Nonaffine Morphism -- A.1 Becker's Coderived Category -- A.2 Locality of Coacyclity on Schemes -- A.3 The Semiderived Category for a Nonaffine Morphismof Schemes -- A.4 Direct Images of Restrictions of Injective Sheaves -- A.5 The Semiderived Category for a Morphism of Ind-schemes -- References.
• Sommario/riassunto: Semi-Infinite Geometry is a theory of "doubly infinite-dimensional" geometric or topological objects. In this book the author explains what should be meant by an algebraic variety of semi-infinite nature. Then he applies the framework of semiderived categories, suggested in his previous monograph titled Homological Algebra of Semimodules and Semicontramodules, (Birkhäuser, 2010), to the study of semi-infinite algebraic varieties. Quasi-coherent torsion sheaves and flat pro-quasi-coherent pro-sheaves on ind-schemes are discussed at length in this book, making it suitable for use as an introduction to the theory of quasi-coherent sheaves on ind-schemes. The main output of the homological theory developed in this monograph is the functor of semitensor product on the semiderived category of quasi-coherent torsion sheaves, endowing the semiderived category with the structure of a tensor triangulated category. The author offers two equivalent constructions of the semitensor product, as well as its particular case, the cotensor product, and shows that they enjoy good invariance properties. Several geometric examples are discussed in detail in the book, including the cotangent bundle to an infinite-dimensional projective space, the universal fibration of quadratic cones, and the important popular example of the loop group of an affine algebraic group.
• Titolo autorizzato: Semi-Infinite Algebraic Geometry of Quasi-Coherent Sheaves on Ind-Schemes
• ISBN: 3-031-37905-5
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910744508803321
• Lo trovi qui: Univ. Federico II