Book Description
This is an updated English translation of "Cohomologie Galoisienne", published more than 30 years ago as one of the very first Lecture Notes in Mathematics (LNM 5). It includes a reproduction of an influential paper of R. Steinberg, together with some new material and an expanded bibliography.
Book Description
This book is designed to be an introduction to some of the basic ideas in the field of algebraic topology. In particular, it is devoted to the foundations and applications of homology theory. The only prerequisite for the student is a basic knowledge of abelian groups and point set topology. The essentials of singular homology are given in the first chapter, along with some of the most important applications. In this way the student can quickly see the importance of the material. The successive topics include attaching spaces, finite CW complexes, the Eilenberg-Steenrod axioms, cohomology products, manifolds, Poincaré duality, and fixed point theory. Throughout the book the approach is as illustrative as possible, with numerous examples and diagrams. Extremes of generality are sacrificed when they are likely to obscure the essential concepts involved. The book is intended to be easily read by students as a textbook for a course or as a source for individual study. The second edition has been substantially revised. It includes a new chapter on covering spaces in addition to illuminating new exercises.
Customer Reviews:
Has the good and bad.......2005-12-30
This is a terrific book on homology theory, covering all the standard topics, plus some nice topics that are hard to find in other introductory books. The motivation for theory is presented in both algebraic/categorical and geometric flavors. The structure of the book is mostly solid, getting straight to the point with singular homology instead of wasting time with simplicial homology and its results (a rarity with algebraic topology books). My only complaints are that the book is riddled with typos and chapter 5 (on products in homology and cohomology) is quite messy.
Masterful.......2000-10-01
This introduction to singular homology combines a strong historical sense with an easy mastery of modern methods. The massive contributions of Poincare and Brouwer are credited, and their geometrical motivations are clear. At the same time the book neither minimizes nor apologizes for modern algebraic machinery, but treats categories and acyclic models and more as natural means to simplify the subject. The book goes through Poincare duality and a good account of the Lefschetz fixed point theorems. It is at once very visual and algebraically slick. The only problem with this approach is that the author seems a bit uncomfortable descending into the nuts and bolts of the longer proofs of two key results (the acyclic model theorem, and the duality theorem). He handles the details unevenly and makes some actual mis-statements. Here the reader needs the experience and confidence to make some corections.
Average customer rating:
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Local Cohomology: An Algebraic Introduction with Geometric Applications (Cambridge Studies in Advanced Mathematics)
M. P. Brodmann , and
R. Y. Sharp
Manufacturer: Cambridge University Press
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ASIN: 0521372860 |
Book Description
This book provides a careful and detailed algebraic introduction to Grothendieck's local cohomology theory, and illustrates many applications for the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties. Topics covered include Castelnuovo-Mumford regularity, the Fulton-Hansen connectedness theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology. It is designed for graduate students who have some experience of basic commutative algebra and homological algebra, and also for experts in commutative algebra and algebraic geometry.
Book Description
This is a translation of Corps Locaux, supplemented with an updated bibliography and some new exercises. This book is accessible to graduate students, and can be used as a reference source by research mathematicians in algebra and number theory.
Average customer rating:
- very comprehensive treatment on the subject
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Cyclic Homology (Grundlehren der mathematischen Wissenschaften)
Jean-Louis Loday
Manufacturer: Springer
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A User's Guide to Spectral Sequences (Cambridge Studies in Advanced Mathematics)
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Counterexamples in Topology
ASIN: 3540630740 |
Book Description
From the reviews: "This is a very interesting book containing material for a comprehensive study of the cyclid homological theory of algebras, cyclic sets and S1-spaces. Lie algebras and algebraic K-theory and an introduction to Connes'work and recent results on the Novikov conjecture. The book requires a knowledge of homological algebra and Lie algebra theory as well as basic technics coming from algebraic topology. The bibliographic comments at the end of each chapter offer good suggestions for further reading and research. The book can be strongly recommended to anybody interested in noncommutative geometry, contemporary algebraic topology and related topics." European Mathematical Society Newsletter
In this second edition the authors have added a chapter 13 on MacLane (co)homology.
Customer Reviews:
very comprehensive treatment on the subject.......2004-02-21
I think that this is probably the only comprehensive book on the subject. There are several lecture notes published, but no other books have collected all up to date recent research results on the subject like this book did. Probably must-read classical book if you are interested in cyclic homology, or if you want to use it for you research. Very unfortunate fact is that this book is very rare, so that very hard to obtain. You may have lots of trouble buying this book.
Book Description
This book covers elementary aspects of category theory and topos theory. It has few mathematical prerequisites and uses categorical methods throughout rather than beginning with set theoretic foundations. It works with key notions such as cartesian closedness, adjunctions, regular categories, and the internal logic of a topos. Full statements and elementary proofs are given for the central theorems, including the fundamental theorem of toposes, the sheafification theorem, and the constriction of Grothendieck toposes over any topos as base. Three chapters discuss applications of toposes in detail, namely to sets, to basic differential geometry, and to recursive analysis.
Customer Reviews:
Very good if you need no slack.......2004-10-28
I've been working my way through McLarty's book off and on for several months now. It is a tremendously clear and well-organized book, and you can learn a lot from it. HOWEVER: it is a "math book" in the strictest sense of the word. Exposition is kept to a bare minimum, and you have to actually work your way through the material (AND the exercises, since many of the definitions are given in them) in order to learn anything. He could have easier doubled or tripled the amount of exposition and still have produced a lean, mean textbook. This is a really good book if you need to learn category theory and you already know why. The only extensive example is a short chapter on group theory. After reading his article on category theory in the Routledge encyclopedia of philosophy I expected rather more in the way of theorizing. Be that as it may: everything you need to know about categories and toposes is in here, and nothing else. The best math book I've read in a long time.
Average customer rating:
- Algebraic topology can be done in algebraic geometry
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Cycles, Transfers, and Motivic Homology Theories
Vladimir Voevodsky ,
Andrei Suslin , and
Eric M. Friedlander
Manufacturer: Princeton University Press
ProductGroup: Book
Binding: Paperback
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ASIN: 0691048150 |
Book Description
The original goal that ultimately led to this volume was the construction of "motivic cohomology theory," whose existence was conjectured by A. Beilinson and S. Lichtenbaum. This is achieved in the book's fourth paper, using results of the other papers whose additional role is to contribute to our understanding of various properties of algebraic cycles. The material presented provides the foundations for the recent proof of the celebrated "Milnor Conjecture" by Vladimir Voevodsky.
The theory of sheaves of relative cycles is developed in the first paper of this volume. The theory of presheaves with transfers and more specifically homotopy invariant presheaves with transfers is the main theme of the second paper. The Friedlander-Lawson moving lemma for families of algebraic cycles appears in the third paper in which a bivariant theory called bivariant cycle cohomology is constructed. The fifth and last paper in the volume gives a proof of the fact that bivariant cycle cohomology groups are canonically isomorphic (in appropriate cases) to Bloch's higher Chow groups, thereby providing a link between the authors' theory and Bloch's original approach to motivic (co-)homology.
Customer Reviews:
Algebraic topology can be done in algebraic geometry.......2002-09-02
Beginning with the work of Grothendiek, the theory of motives is, very loosely speaking, an attempt at a "unified theory" of number theory and algebraic geometry. This years Fields Medals reflect the interest in motives, as one of the authors in this book (Voevodsky) was awarded for his research in this area. In a gist, this book tries to see how much of (standard) algebraic topology can be carried over to the study of algebraic varieties and schemes. By defining a new topology on algebraic cycles, the "qfh topology", Voevodsky showed that the techniques of sheaf theory can be used to study them from the standpoint of algebraic topology. This topology is finer than the etale topology and allows one to use sheaf cohomology to study algebraic cycles. The reader will be expected to have a substantial background in the theory of schemes, higher K-theory, algebraic topology, and sheaf theory. Reading this book will give one a deep appreciation of how difficult it is to do algebraic topology in algebraic geometry, requiring formidable technical machinery.
The use of K-theory in topology and algebra goes back half a century, beginning with the K-theory of CW-complexes and the construction of Atiyah and Hirzebruch of spectral sequences relating singular cohomology to topological K-theory. The K-theory of algebraic varieties is a little more subtle, and involves looking at the isomorphism classes of algebraic vector bundles on the variety. These form an abelian group with the group operation being defined via the existence of an exact sequence between the isomorphism classes.
As a warm-up to the scheme-theoretic setting, the K-theory of an arbitrary ring proceeds by analogy with the simplicial setting, the latter of which involves the classifying space of homotopy maps of the complex and the notion of stable equivalence. But for a general ring, the unit interval used in the definition of homotopy is replaced by the affine line. The work of Karoubi and Villamayor, and Quillen defined precisely higher algebraic K-theory for rings, the former using this simplicial motivation, the latter using what is called a "Q-construction". The definitions coincide for regular schemes but not for singular ones.
Motivic cohomology, which is an algebraic analog of singular cohomology, arose in the setting of the Chow ring of algebraic cycles modulo rational equivalence. A homology theory of the free abelian group of algebraic cycles of a variety, with the replacement of the unit interval with the affine line, was developed. The products existing in cohomology arise from the consideration of the intersection of subvarieties, leading to the familiar Chow ring. The Chow ring is functorial under pull-backs, and can be related to the zeroth K-group via the use of the Chern class and the Riemann-Roch theorem. The higher K-groups of Quillen give the desired long exact sequence of K-groups.
Bloch then defined motivic cohomology via the construction of higher Chow groups, again by analogy to the simplicial theory, and with a careful definition of intersection product, so as to insure the algebraic cycles intersect the faces in the correct codimension. It was then shown that the higher Chow groups are related to the the higher K-groups for a variety which is smooth over a field.
One of the authors (Frielander) and Dwyer, using the etale cohomology of Grothendieck, gave a mod-n topological K-theory, called etale K-theory, which led to the work of Suslin and Voevodsky on the motivic homology of algebraic cycles, which is the main focus of this book.
After a brief introduction to motivic cohomology in chapter 1 and an historical introduction, the second chapter deals with relative cycles on schemes and Chow sheaves. Relative cycles are defined for schemes of finite type over a Noetherian (base) scheme and are well-behaved for morphisms of of the base scheme. The authors concentrate most of their attention not to general schemes but to varieties over a field. The cdh-topology is introduced here as one which allows the construction of long exact sequences for sheaves of relative cycles.
Chapter 3 overviews the cohomological theory of presheaves and defines the notion of a transfer map. For smooth schemes over a field, these maps are used to define a "pretheory" over the field, and homotopy invariance of pretheories can then be defined. Examples of pretheories include etale cohomology, algebraic K-theory, and algebraic de Rham cohomology. The Mayer-Vietoris exact sequence for the Suslin homology is proven, giving another analogue of ordinary algebraic topology.
In chapter 4 the authors consider the generalization of the duality property of homology and cohomology in algebraic topology using bivariant cycle cohomology. The bivariant cycle cohomology groups are defined for schemes of finite type over a field in terms of the higher Chow groups. They have the origin in the generalization of the simplicial theory to the algebraic geometry setting. Homotopy invariance, suspension maps, and the Gysin sequence find their place here also. The authors detail to what extent the higher Chow groups can be considered to be a motivic cohomology theory. Motivic homology, motivic cohomology, and Borel-Moore motivic cohomology are shown to be related to the bivariant cycle cohomology and their algebraic topological properties discussed briefly.
Chapter 5 studies algebraic cycle cohomology theories categorically via the construction of triangulated categories of motives. This is the key step in allowing the techniques of (ordinary) sheaf cohomology to be applied to the category of motives. The discussion is done in the context of smooth schemes, but it would be interesting if the authors would have given some concrete examples, possibly with elliptic curves, showing how these constructions come into play for elementary algebraic varieties.
The book ends with a discussion of the higher Chow groups and how they relate to etale cohomology. A relatively concrete presentation, the author proves the equality between the higher Chow groups and etale cohomology with compact supports for quasiprojective schemes over algebraically closed fields of characteristic zero.
Book Description
As a second year graduate textbook, Cohomology of Groups introduces students to cohomology theory (involving a rich interplay between algebra and topology) with a minimum of prerequisites. No homological algebra is assumed beyond what is normally learned in a first course in algebraic topology. The basics of the subject are given (along with exercises) before the author discusses more specialized topics.
Book Description
From the reviews:
"The author has attempted an ambitious and most commendable project. He assumes only a modest knowledge of algebraic topology on the part of the reader to start with, and he leads the reader systematically to the point at which he can begin to tackle problems in the current areas of research centered around generalized homology theories and their applications. ... The author has sought to make his treatment complete and he has succeeded. The book contains much material that has not previously appeared in this format. The writing is clean and clear and the exposition is well motivated. ... This book is, all in all, a very admirable work and a valuable addition to the literature...
(S.Y. Husseini in Mathematical Reviews, 1976)
Customer Reviews:
This might take a while..........2002-06-09
The earlier chapters are quite good; however, some of the advanced topics in this book are better approached (appreciated) after one has learned about them elsewhere, at a more leisurely pace. For instance, this isn't the best place to first read about characteristic classes and topological K theory (I would recommend, without much hesitation, the books by Atiyah and Milnor & Stasheff, instead). Much to my disappointment, the chapter on spectral sequences is quite convoluted. Parts of 'user's guide' by Mcleary would certainly come in handy here (which sets the stage rather nicely for applications).
So it turns out that supplemental reading (exluding Whitehead's massive treatise) is necessary to achieve a better understanding of algebraic topology at the level of this book. The homotopical view therein will be matched (possibly superseded) by Aguilar's book (forthcoming, to which I am very much looking forward).
Good luck!
Book Description
Now in paperback, this book provides a self-contained introduction to the cohomology theory of Lie groups and algebras and to some of its applications in physics. No previous knowledge of the mathematical theory is assumed beyond some notions of Cartan calculus and differential geometry (which are nevertheless reviewed in the book in detail). The examples, of current interest, are intended to clarify certain mathematical aspects and to show their usefulness in physical problems. The topics treated include the differential geometry of Lie groups, fiber bundles and connections, characteristic classes, index theorems, monopoles, instantons, extensions of Lie groups and algebras, some applications in supersymmetry, Chevalley-Eilenberg approach to Lie algebra cohomology, symplectic cohomology, jet-bundle approach to variational principles in mechanics, Wess-Zumino-Witten terms, infinite Lie algebras, the cohomological descent in mechanics and in gauge theories and anomalies. This book will be of interest to graduate students and researchers in theoretical physics and applied mathematics.
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Local Fields (Graduate Texts in Mathematics)
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