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

Quasi-Coherent Torsion Sheaves, the Semiderived Category, and the Semitensor Product
2023
ISBN: 978-3-031-37907-9
Verlag: Springer Nature Switzerland

Buch, Englisch, 216 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 365 g

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 semitensorproduct, 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.

Zielgruppe

Research

Autoren/Hrsg.

Positselski, Leonid

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Homologische Algebra
• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie
• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie

Inhaltsverzeichnis

- 1. Ind-Schemes and Their Morphisms. - 2. Quasi-Coherent Torsion Sheaves. - 3. Flat Pro-Quasi-Coherent Pro-Sheaves. - 4. Dualizing Complexes on Ind-Noetherian Ind-Schemes. - 5. The Cotensor Product. - 6. Ind-Schemes of Ind-Finite Type and the factorial !-Tensor Product. - 7. X-Flat Pro-Quasi-Coherent Pro-Sheaves on Y. - 8. The Semitensor Product. - 9. Flat Affine Ind-Schemes over Ind-Schemes of Ind-Finite Type. - 10. Invariance Under Postcomposition with a Smooth Morphism. - 11. Some Infinite-Dimensional Geometric Examples.

Über Autor(innen)

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).