Search Results for "sheaf-theory-graduate-texts-in-mathematics"

Sheaf Theory

Sheaf Theory

  • Author: Glen E. Bredon
  • Publisher: Springer Science & Business Media
  • ISBN: 1461206472
  • Category: Mathematics
  • Page: 504
  • View: 6543
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Primarily concerned with the study of cohomology theories of general topological spaces with "general coefficient systems", the parts of sheaf theory covered here are those areas important to algebraic topology. Among the many innovations in this book, the concept of the "tautness" of a subspace is introduced and exploited; the fact that sheaf theoretic cohomology satisfies the homotopy property is proved for general topological spaces; and relative cohomology is introduced into sheaf theory. A list of exercises at the end of each chapter helps students to learn the material, and solutions to many of the exercises are given in an appendix. This new edition of a classic has been substantially rewritten and now includes some 80 additional examples and further explanatory material, as well as new sections on Cech cohomology, the Oliver transfer, intersection theory, generalised manifolds, locally homogeneous spaces, homological fibrations and p- adic transformation groups. Readers should have a thorough background in elementary homological algebra and in algebraic topology.

Sheaves in Topology

Sheaves in Topology

  • Author: Alexandru Dimca
  • Publisher: Springer Science & Business Media
  • ISBN: 3642188680
  • Category: Mathematics
  • Page: 240
  • View: 6027
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Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds. This introduction to the subject can be regarded as a textbook on modern algebraic topology, treating the cohomology of spaces with sheaf (as opposed to constant) coefficients. The author helps readers progress quickly from the basic theory to current research questions, thoroughly supported along the way by examples and exercises.

Cohomology of Sheaves

Cohomology of Sheaves

  • Author: Birger Iversen
  • Publisher: Springer Science & Business Media
  • ISBN: 3642827837
  • Category: Mathematics
  • Page: 464
  • View: 4552
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This text exposes the basic features of cohomology of sheaves and its applications. The general theory of sheaves is very limited and no essential result is obtainable without turn ing to particular classes of topological spaces. The most satis factory general class is that of locally compact spaces and it is the study of such spaces which occupies the central part of this text. The fundamental concepts in the study of locally compact spaces is cohomology with compact support and a particular class of sheaves,the so-called soft sheaves. This class plays a double role as the basic vehicle for the internal theory and is the key to applications in analysis. The basic example of a soft sheaf is the sheaf of smooth functions on ~n or more generally on any smooth manifold. A rather large effort has been made to demon strate the relevance of sheaf theory in even the most elementary analysis. This process has been reversed in order to base the fundamental calculations in sheaf theory on elementary analysis.

Categories and Sheaves

Categories and Sheaves

  • Author: Masaki Kashiwara,Pierre Schapira
  • Publisher: Springer Science & Business Media
  • ISBN: 3540279504
  • Category: Mathematics
  • Page: 498
  • View: 5166
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Categories and sheaves appear almost frequently in contemporary advanced mathematics. This book covers categories, homological algebra and sheaves in a systematic manner starting from scratch and continuing with full proofs to the most recent results in the literature, and sometimes beyond. The authors present the general theory of categories and functors, emphasizing inductive and projective limits, tensor categories, representable functors, ind-objects and localization.

Foundations of Differentiable Manifolds and Lie Groups

Foundations of Differentiable Manifolds and Lie Groups

  • Author: Frank W. Warner
  • Publisher: Springer Science & Business Media
  • ISBN: 1475717997
  • Category: Mathematics
  • Page: 276
  • View: 7799
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Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. Coverage includes differentiable manifolds, tensors and differentiable forms, Lie groups and homogenous spaces, and integration on manifolds. The book also provides a proof of the de Rham theorem via sheaf cohomology theory and develops the local theory of elliptic operators culminating in a proof of the Hodge theorem.

Sheaf Theory

Sheaf Theory

  • Author: B. R. Tennison
  • Publisher: Cambridge University Press
  • ISBN: 0521207843
  • Category: Mathematics
  • Page: 164
  • View: 2100
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Sheaf theory provides a means of discussing many different kinds of geometric objects in respect of the connection between their local and global properties. It finds its main applications in topology and modern algebraic geometry where it has been used as a tool for solving, with great success, several long-standing problems. This text is based on a lecture course for graduate pure mathematicians which builds up enough of the foundations of sheaf theory to give a broad definition of manifold, covering as special cases the algebraic geometer's schemes as well as the topological, differentiable and analytic kinds, and to define sheaf cohomology for application to such objects. Exercises are provided at the end of each chapter and at various places in the text. Hints and solutions to some of them are given at the end of the book.

Algebraic Geometry and Arithmetic Curves

Algebraic Geometry and Arithmetic Curves

  • Author: Qing Liu,Reinie Erne
  • Publisher: Oxford University Press
  • ISBN: 0191547808
  • Category: Mathematics
  • Page: 592
  • View: 2921
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This book is a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck's duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo's criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the funadmental theorem of stable reduction of Deligne-Mumford. The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly 600 exercises.

Deformation Theory

Deformation Theory

  • Author: Robin Hartshorne
  • Publisher: Springer Science & Business Media
  • ISBN: 1441915966
  • Category: Mathematics
  • Page: 234
  • View: 3945
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The basic problem of deformation theory in algebraic geometry involves watching a small deformation of one member of a family of objects, such as varieties, or subschemes in a fixed space, or vector bundles on a fixed scheme. In this new book, Robin Hartshorne studies first what happens over small infinitesimal deformations, and then gradually builds up to more global situations, using methods pioneered by Kodaira and Spencer in the complex analytic case, and adapted and expanded in algebraic geometry by Grothendieck. The author includes numerous exercises, as well as important examples illustrating various aspects of the theory. This text is based on a graduate course taught by the author at the University of California, Berkeley.

Sheaves on Manifolds

Sheaves on Manifolds

With a Short History. «Les débuts de la théorie des faisceaux». By Christian Houzel

  • Author: Masaki Kashiwara,Pierre Schapira
  • Publisher: Springer Science & Business Media
  • ISBN: 9783540518617
  • Category: Mathematics
  • Page: 512
  • View: 8295
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Sheaf Theory is modern, active field of mathematics at the intersection of algebraic topology, algebraic geometry and partial differential equations. This volume offers a comprehensive and self-contained treatment of Sheaf Theory from the basis up, with emphasis on the microlocal point of view. From the reviews: "Clearly and precisely written, and contains many interesting ideas: it describes a whole, largely new branch of mathematics." –Bulletin of the L.M.S.

The Geometry of Schemes

The Geometry of Schemes

  • Author: David Eisenbud,Joe Harris
  • Publisher: Springer Science & Business Media
  • ISBN: 0387226397
  • Category: Mathematics
  • Page: 300
  • View: 5129
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Grothendieck’s beautiful theory of schemes permeates modern algebraic geometry and underlies its applications to number theory, physics, and applied mathematics. This simple account of that theory emphasizes and explains the universal geometric concepts behind the definitions. In the book, concepts are illustrated with fundamental examples, and explicit calculations show how the constructions of scheme theory are carried out in practice.

Several Complex Variables with Connections to Algebraic Geometry and Lie Groups

Several Complex Variables with Connections to Algebraic Geometry and Lie Groups

  • Author: Joseph L. Taylor
  • Publisher: American Mathematical Soc.
  • ISBN: 082183178X
  • Category: Mathematics
  • Page: 507
  • View: 549
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This text presents an integrated development of the theory of several complex variables and complex algebraic geometry, leading to proofs of Serre's celebrated GAGA theorems relating the two subjects, and including applications to the representation theory of complex semisimple Lie groups. It includes a thorough treatment of the local theory using the tools of commutative algebra, an extensive development of sheaf theory and the theory of coherent analytic and algebraic sheaves, proofs of the main vanishing theorems for these categories of sheaves, and a complete proof of the finite dimensionality of the cohomology of coherent sheaves on compact varieties. The vanishing theorems have a wide variety of applications and these are covered in detail. Of particular interest are the last three chapters, which are devoted to applications of the preceding material to the study of the structure and representations of complex semisimple Lie groups.Included in this text are introductions to harmonic analysis, the Peter-Weyl theorem, Lie theory and the structure of Lie algebras, semisimple Lie algebras and their representations, algebraic groups and the structure of complex semisimple Lie groups. All of this culminates in Milicic's proof of the Borel-Weil-Bott theorem, which makes extensive use of the material developed earlier in the text. There are numerous examples and exercises in each chapter. This modern treatment of a classic point of view would be an excellent text for a graduate course on several complex variables, as well as a useful reference for the expert.

Elements of Homotopy Theory

Elements of Homotopy Theory

  • Author: George W. Whitehead
  • Publisher: Springer Science & Business Media
  • ISBN: 1461263182
  • Category: Mathematics
  • Page: 746
  • View: 3813
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As the title suggests, this book is concerned with the elementary portion of the subject of homotopy theory. It is assumed that the reader is familiar with the fundamental group and with singular homology theory, including the Universal Coefficient and Kiinneth Theorems. Some acquaintance with manifolds and Poincare duality is desirable, but not essential. Anyone who has taught a course in algebraic topology is familiar with the fact that a formidable amount of technical machinery must be introduced and mastered before the simplest applications can be made. This phenomenon is also observable in the more advanced parts of the subject. I have attempted to short-circuit it by making maximal use of elementary methods. This approach entails a leisurely exposition in which brevity and perhaps elegance are sacrificed in favor of concreteness and ease of application. It is my hope that this approach will make homotopy theory accessible to workers in a wide range of other subjects-subjects in which its impact is beginning to be felt. It is a consequence of this approach that the order of development is to a certain extent historical. Indeed, if the order in which the results presented here does not strictly correspond to that in which they were discovered, it nevertheless does correspond to an order in which they might have been discovered had those of us who were working in the area been a little more perspicacious.

Algebraic Geometry: Sheaves and cohomology

Algebraic Geometry: Sheaves and cohomology

  • Author: 健爾·上野
  • Publisher: American Mathematical Soc.
  • ISBN: 9780821813577
  • Category: Mathematics
  • Page: 184
  • View: 4361
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Modern algebraic geometry is built upon two fundamental notions: schemes and sheaves. The theory of schemes is presented in the first part of this book (Algebraic Geometry 1: From Algebraic Varieties to Schemes, AMS, 1999, Translations of Mathematical Monographs, Volume 185). In the present book, the author turns to the theory of sheaves and their cohomology. Loosely speaking, a sheaf is a way of keeping track of local information defined on a topological space, such as the local algebraic functions on an algebraic manifold or the local sections of a vector bundle. Sheaf cohomology is a primary tool in understanding sheaves and using them to study properties of the corresponding manifolds. The text covers the important topics of the theory of sheaves on algebraic varieties, including types of sheaves and the fundamental operations on them, such as coherent and quasicoherent sheaves, direct and inverse images, behavior of sheaves under proper and projective morphisms, and Cech cohomology. The book contains numerous problems and exercises with solutions. It would be an excellent text for the second part of a course in algebraic geometry.

Algebraic Geometry

Algebraic Geometry

  • Author: Robin Hartshorne
  • Publisher: Springer Science & Business Media
  • ISBN: 1475738498
  • Category: Mathematics
  • Page: 496
  • View: 2097
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An introduction to abstract algebraic geometry, with the only prerequisites being results from commutative algebra, which are stated as needed, and some elementary topology. More than 400 exercises distributed throughout the book offer specific examples as well as more specialised topics not treated in the main text, while three appendices present brief accounts of some areas of current research. This book can thus be used as textbook for an introductory course in algebraic geometry following a basic graduate course in algebra. Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. He is the author of "Residues and Duality", "Foundations of Projective Geometry", "Ample Subvarieties of Algebraic Varieties", and numerous research titles.

D-Modules, Perverse Sheaves, and Representation Theory

D-Modules, Perverse Sheaves, and Representation Theory

  • Author: Ryoshi Hotta,Toshiyuki Tanisaki
  • Publisher: Springer Science & Business Media
  • ISBN: 081764363X
  • Category: Mathematics
  • Page: 412
  • View: 1412
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D-modules continues to be an active area of stimulating research in such mathematical areas as algebraic, analysis, differential equations, and representation theory. Key to D-modules, Perverse Sheaves, and Representation Theory is the authors' essential algebraic-analytic approach to the theory, which connects D-modules to representation theory and other areas of mathematics. To further aid the reader, and to make the work as self-contained as possible, appendices are provided as background for the theory of derived categories and algebraic varieties. The book is intended to serve graduate students in a classroom setting and as self-study for researchers in algebraic geometry, representation theory.

Combinatorial Convexity and Algebraic Geometry

Combinatorial Convexity and Algebraic Geometry

  • Author: Günter Ewald
  • Publisher: Springer Science & Business Media
  • ISBN: 1461240441
  • Category: Mathematics
  • Page: 374
  • View: 4032
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The book is an introduction to the theory of convex polytopes and polyhedral sets, to algebraic geometry, and to the connections between these fields, known as the theory of toric varieties. The first part of the book covers the theory of polytopes and provides large parts of the mathematical background of linear optimization and of the geometrical aspects in computer science. The second part introduces toric varieties in an elementary way.

Graph Theory

Graph Theory

  • Author: B. Bollobás
  • Publisher: Elsevier
  • ISBN: 9780080871738
  • Category: Mathematics
  • Page: 200
  • View: 572
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The Cambridge Graph Theory Conference, held at Trinity College from 11 to 13 March 1981, brought together top ranking workers from diverse areas of the subject. The papers presented were by invitation only. This volume contains most of the contniutions, suitably refereed and revised. For many years now, graph theory has been developing at a great pace and in many directions. In order to emphasize the variety of questions and to preserve the freshness of research, the theme of the meeting was not restricted. Consequently, the papers in this volume deal with many aspects of graph theory, including colouring, connectivity, cycles, Ramsey theory, random graphs, flows, simplicial decompositions and directed graphs. A number of other papers are concerned with related areas, including hypergraphs, designs, algorithms, games and social models. This wealth of topics should enhance the attractiveness of the volume.

Diophantine Geometry

Diophantine Geometry

An Introduction

  • Author: Marc Hindry,Joseph H. Silverman
  • Publisher: Springer Science & Business Media
  • ISBN: 1461212103
  • Category: Mathematics
  • Page: 561
  • View: 8455
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This is an introduction to diophantine geometry at the advanced graduate level. The book contains a proof of the Mordell conjecture which will make it quite attractive to graduate students and professional mathematicians. In each part of the book, the reader will find numerous exercises.

Categorical Foundations

Categorical Foundations

Special Topics in Order, Topology, Algebra, and Sheaf Theory

  • Author: Maria Cristina Pedicchio,Walter Tholen
  • Publisher: Cambridge University Press
  • ISBN: 9780521834148
  • Category: Mathematics
  • Page: 417
  • View: 6053
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The book offers categorical introductions to order, topology, algebra and sheaf theory, suitable for graduate students, teachers and researchers of pure mathematics.

Measure Theory

Measure Theory

  • Author: Paul R. Halmos
  • Publisher: Springer
  • ISBN: 1468494406
  • Category: Mathematics
  • Page: 304
  • View: 8838
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Useful as a text for students and a reference for the more advanced mathematician, this book presents a unified treatment of that part of measure theory most useful for its application in modern analysis. Coverage includes sets and classes, measures and outer measures, Haar measure and measure and topology in groups. From the reviews: "Will serve the interested student to find his way to active and creative work in the field of Hilbert space theory." --MATHEMATICAL REVIEWS