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

0576365 - MÚ 2024 RIV CH eng B - Monography
Positselski, Leonid
Semi-Infinite Algebraic Geometry of Quasi-Coherent Sheaves on Ind-Schemes: Quasi-Coherent Torsion Sheaves, the Semiderived Category, and the Semitensor Product.
1. - Cham: Birkhäuser, 2023. 216 s. ISBN 978-3-031-37904-8
R&D Projects: GA ČR(CZ) GA20-13778S
Institutional support: RVO:67985840
Keywords : semi-infinite tensor * semi-infinite algebraic geometry * algebraic geometry
OECD category: Pure mathematics
https://doi.org/10.1007/978-3-031-37905-5

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.
Permanent Link: https://hdl.handle.net/11104/0345934