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Semi-Infinite Algebraic Geometry of Quasi-Coherent Sheaves on Ind-Schemes

Quasi-Coherent Torsion Sheaves, the Semiderived Category, and the Semitensor Product

  • Book
  • © 2023

Overview

Authors:
  1. Leonid Positselski

    1. Institute of Mathematics, Czech Academy of Sciences, Praha 1, Czech Republic

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  • First monograph on quasi-coherent torsion sheaves on ind-schemes

  • Introduces novel algebraic structures which will play an important role in algebraic geometry to come

  • Explores the semi-infinite tensor product and other elements of interest for the field

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xix
    Download chapter PDF

  2. Ind-Schemes and Their Morphisms

    • Leonid Positselski
    Pages 1-9

  3. Quasi-Coherent Torsion Sheaves

    • Leonid Positselski
    Pages 11-25

  4. Flat Pro-Quasi-Coherent Pro-Sheaves

    • Leonid Positselski
    Pages 27-41

  5. Dualizing Complexes on Ind-Noetherian Ind-Schemes

    • Leonid Positselski
    Pages 43-61

  6. The Cotensor Product

    • Leonid Positselski
    Pages 63-80

  7. Ind-Schemes of Ind-Finite Type and the \(!\)-Tensor Product

    • Leonid Positselski
    Pages 81-103

  8. \({\mathfrak X}\)-Flat Pro-Quasi-Coherent Pro-Sheaves on \({\boldsymbol {\mathfrak Y}}\)

    • Leonid Positselski
    Pages 105-120

  9. The Semitensor Product

    • Leonid Positselski
    Pages 121-137

  10. Flat Affine Ind-Schemes over Ind-Schemes of Ind-Finite Type

    • Leonid Positselski
    Pages 139-156

  11. Invariance Under Postcomposition with a Smooth Morphism

    • Leonid Positselski
    Pages 157-177

  12. Some Infinite-Dimensional Geometric Examples

    • Leonid Positselski
    Pages 179-195

  13. Back Matter

    Pages 197-216
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Keywords

About this book

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.

Authors and Affiliations

  • Institute of Mathematics, Czech Academy of Sciences, Praha 1, Czech Republic

    Leonid Positselski

About the author

Leonid Positselski did his undergraduate studies in Moscow in 1988-93 and received his Ph.D. in Mathematics from Harvard University in 1998. After a series of postdoctoral positions in the U.S. and Europe, he returned to Moscow in 2003. His first book was published in 2005 and the second one in 2010. In 2011-14 he taught as an Associate Professor at the Faculty of Mathematics of the Higher School of Economics in Moscow. Positselski emigrated from Russia to Israel in Spring 2014. Since 2018, he works as a researcher at the Institute of Mathematics of the Czech Academy of Sciences in Prague.
Positselski is an algebraist specializing in Homological Algebra and homological aspects of various branches of the "algebraic half" of mathematics, including algebraic geometry, representation theory, commutative algebra, and algebraic K-theory.  He is known for his work on Koszul algebras and derived Koszul duality, as well as curved DG-rings, coderived and contraderived categories, and contramodules. Positselski penned more than 45 research papers and two survey papers. He is the author of five books and memoirs, including "Quadratic Algebras" (joint with A. Polishchuk, AMS University Lecture Series, 2005), "Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures" (Monografie Matematyczne IMPAN, Birkhäuser Basel, 2010), "Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence" (AMS Memoir, 2011), "Weakly curved A-infinity algebras over a topological local ring" (SMF Memoir, 2018-19), and "Relative nonhomogeneous Koszul duality" (Frontiers in Mathematics, Birkhäuser Switzerland, 2021-22).

Bibliographic Information

  • Book Title: Semi-Infinite Algebraic Geometry of Quasi-Coherent Sheaves on Ind-Schemes

  • Book Subtitle: Quasi-Coherent Torsion Sheaves, the Semiderived Category, and the Semitensor Product

  • Authors: Leonid Positselski

  • DOI: https://doi.org/10.1007/978-3-031-37905-5

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

  • Hardcover ISBN: 978-3-031-37904-8Published: 16 September 2023

  • Softcover ISBN: 978-3-031-37907-9Due: 30 September 2024

  • eBook ISBN: 978-3-031-37905-5Published: 14 September 2023

  • Edition Number: 1

  • Number of Pages: XIX, 216

  • Topics: Algebraic Geometry, Commutative Rings and Algebras, Category Theory, Homological Algebra

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