Two Kinds of Derived Categories, Koszul Duality, and Comodule-Contramodule Correspondence - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Two Kinds of Derived Categories, Koszul Duality, and Comodule-Contramodule Correspondence write by Leonid Positselski. This book was released on 2011. Two Kinds of Derived Categories, Koszul Duality, and Comodule-Contramodule Correspondence available in PDF, EPUB and Kindle. "July 2011, volume 212, number 996 (first of 4 numbers)."
Semi-Infinite Algebraic Geometry of Quasi-Coherent Sheaves on Ind-Schemes
Semi-Infinite Algebraic Geometry of Quasi-Coherent Sheaves on Ind-Schemes - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Semi-Infinite Algebraic Geometry of Quasi-Coherent Sheaves on Ind-Schemes write by Leonid Positselski. This book was released on 2023-10-16. Semi-Infinite Algebraic Geometry of Quasi-Coherent Sheaves on Ind-Schemes available in PDF, EPUB and Kindle. 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.
Relative Nonhomogeneous Koszul Duality
Relative Nonhomogeneous Koszul Duality - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Relative Nonhomogeneous Koszul Duality write by Leonid Positselski. This book was released on 2022-02-10. Relative Nonhomogeneous Koszul Duality available in PDF, EPUB and Kindle. This research monograph develops the theory of relative nonhomogeneous Koszul duality. Koszul duality is a fundamental phenomenon in homological algebra and related areas of mathematics, such as algebraic topology, algebraic geometry, and representation theory. Koszul duality is a popular subject of contemporary research. This book, written by one of the world's leading experts in the area, includes the homogeneous and nonhomogeneous quadratic duality theory over a nonsemisimple, noncommutative base ring, the Poincare–Birkhoff–Witt theorem generalized to this context, and triangulated equivalences between suitable exotic derived categories of modules, curved DG comodules, and curved DG contramodules. The thematic example, meaning the classical duality between the ring of differential operators and the de Rham DG algebra of differential forms, involves some of the most important objects of study in the contemporary algebraic and differential geometry. For the first time in the history of Koszul duality the derived D-\Omega duality is included into a general framework. Examples highly relevant for algebraic and differential geometry are discussed in detail.
Homological Algebra of Semimodules and Semicontramodules
Homological Algebra of Semimodules and Semicontramodules - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Homological Algebra of Semimodules and Semicontramodules write by Leonid Positselski. This book was released on 2010-09-02. Homological Algebra of Semimodules and Semicontramodules available in PDF, EPUB and Kindle. This book provides comprehensive coverage on semi-infinite homology and cohomology of associative algebraic structures. It features rich representation-theoretic and algebro-geometric examples and applications.
Bousfield Classes and Ohkawa's Theorem
Bousfield Classes and Ohkawa's Theorem - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Bousfield Classes and Ohkawa's Theorem write by Takeo Ohsawa. This book was released on 2020-03-18. Bousfield Classes and Ohkawa's Theorem available in PDF, EPUB and Kindle. This volume originated in the workshop held at Nagoya University, August 28–30, 2015, focusing on the surprising and mysterious Ohkawa's theorem: the Bousfield classes in the stable homotopy category SH form a set. An inspiring, extensive mathematical story can be narrated starting with Ohkawa's theorem, evolving naturally with a chain of motivational questions: Ohkawa's theorem states that the Bousfield classes of the stable homotopy category SH surprisingly forms a set, which is still very mysterious. Are there any toy models where analogous Bousfield classes form a set with a clear meaning? The fundamental theorem of Hopkins, Neeman, Thomason, and others states that the analogue of the Bousfield classes in the derived category of quasi-coherent sheaves Dqc(X) form a set with a clear algebro-geometric description. However, Hopkins was actually motivated not by Ohkawa's theorem but by his own theorem with Smith in the triangulated subcategory SHc, consisting of compact objects in SH. Now the following questions naturally occur: (1) Having theorems of Ohkawa and Hopkins-Smith in SH, are there analogues for the Morel-Voevodsky A1-stable homotopy category SH(k), which subsumes SH when k is a subfield of C?, (2) Was it not natural for Hopkins to have considered Dqc(X)c instead of Dqc(X)? However, whereas there is a conceptually simple algebro-geometrical interpretation Dqc(X)c = Dperf(X), it is its close relative Dbcoh(X) that traditionally, ever since Oka and Cartan, has been intensively studied because of its rich geometric and physical information. This book contains developments for the rest of the story and much more, including the chromatics homotopy theory, which the Hopkins–Smith theorem is based upon, and applications of Lurie's higher algebra, all by distinguished contributors.
