Semi-Infinite Algebraic Geometry of Quasi-Coherent Sheaves on Ind-Schemes: Quasi-Coherent Torsion Sheaves, the Semiderived Category, and the Semitensor Product

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