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Homological Algebra of Semimodules and Semicontramodules

2010-09-02 Mathematics
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Release : 2010-09-02
Genre : Mathematics
Kind : eBook
Book Rating : 36X/5 ( reviews)

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Book Synopsis Homological Algebra of Semimodules and Semicontramodules by : Leonid Positselski

Download or read book Homological Algebra of Semimodules and Semicontramodules written by Leonid Positselski. This book was released on 2010-09-02. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Homological Algebra

1999-12-19 Mathematics
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Release : 1999-12-19
Genre : Mathematics
Kind : eBook
Book Rating : 915/5 ( reviews)

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Book Synopsis Homological Algebra by : Henri Cartan

Download or read book Homological Algebra written by Henri Cartan. This book was released on 1999-12-19. Available in PDF, EPUB and Kindle. Book excerpt: When this book was written, methods of algebraic topology had caused revolutions in the world of pure algebra. To clarify the advances that had been made, Cartan and Eilenberg tried to unify the fields and to construct the framework of a fully fledged theory. The invasion of algebra had occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. This book presents a single homology (and also cohomology) theory that embodies all three; a large number of results is thus established in a general framework. Subsequently, each of the three theories is singled out by a suitable specialization, and its specific properties are studied. The starting point is the notion of a module over a ring. The primary operations are the tensor product of two modules and the groups of all homomorphisms of one module into another. From these, "higher order" derived of operations are obtained, which enjoy all the properties usually attributed to homology theories. This leads in a natural way to the study of "functors" and of their "derived functors." This mathematical masterpiece will appeal to all mathematicians working in algebraic topology.

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

2023-10-16 Mathematics
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Release : 2023-10-16
Genre : Mathematics
Kind : eBook
Book Rating : 055/5 ( reviews)

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Book Synopsis Semi-Infinite Algebraic Geometry of Quasi-Coherent Sheaves on Ind-Schemes by : Leonid Positselski

Download or read book Semi-Infinite Algebraic Geometry of Quasi-Coherent Sheaves on Ind-Schemes written by Leonid Positselski. This book was released on 2023-10-16. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Homological Algebra

2013 Mathematics
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Release : 2013
Genre : Mathematics
Kind : eBook
Book Rating : 923/5 ( reviews)

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Book Synopsis Homological Algebra by : Marco Grandis

Download or read book Homological Algebra written by Marco Grandis. This book was released on 2013. Available in PDF, EPUB and Kindle. Book excerpt: This book proposes a study of semi-exact homological categories as a basis for a generalized homological algebra. The aim is to extend homological notions to deeply non-abelian situations, where satellites and spectral sequences produced by unstable homotopy can still be studied.

Relative Nonhomogeneous Koszul Duality

2022-02-10 Mathematics
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Release : 2022-02-10
Genre : Mathematics
Kind : eBook
Book Rating : 408/5 ( reviews)

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Book Synopsis Relative Nonhomogeneous Koszul Duality by : Leonid Positselski

Download or read book Relative Nonhomogeneous Koszul Duality written by Leonid Positselski. This book was released on 2022-02-10. Available in PDF, EPUB and Kindle. Book excerpt: 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.

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