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"This is the International Edition. The content is in English, same as US version but different cover. Please DO NOT buy if you can not accept this difference.Ship from Shanghai China, please allow about 3 weeks on the way to US or Europe.Message me if you have any questions." ... Read more

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These notes were prepared for a series of ten lectures given at Regional Conference of the Conference Board of the Mathematical Sciences in June 1971. The bulk of the lectures were on projective dimensions of ``very large'' modules. Although the material in these notes is not new, there are several places where existing work has been simplified. For example, a commutative local nondomain of global dimension 3 is described without reference to analysis, and the dimension of a quotient field of a polynomial ring rather than a regular local ring is calculated. A derivation of Tor, one step at a time without the usual derived functor machinery, is also included. ... Read more

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This text examines the field of homology. ... Read more

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Homology is a powerful tool used by mathematicians to study the properties of spaces and maps that are insensitive to small perturbations. This book uses a computer to develop a combinatorial computational approach to the subject. The core of the book deals with homology theory and its computation. Following this is a section containing extensions to further developments in algebraic topology, applications to computational dynamics, and applications to image processing. Included are exercises and software that can be used to compute homology groups and maps. The book will appeal to researchers and graduate students in mathematics, computer science, engineering, and nonlinear dynamics.

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In spite of some recent applications of ultraproducts in algebra, they remain largely unknown to commutative algebraists, in part because they do not preserve basic properties such as Noetherianity. This work wants to make a strong case against these prejudices. More precisely, it studies ultraproducts of Noetherian local rings from a purely algebraic perspective, as well as how they can be used to transfer results between the positive and zero characteristics, to derive uniform bounds, to define tight closure in characteristic zero, and to prove asymptotic versions of homological conjectures in mixed characteristic. Some of these results are obtained using variants called chromatic products, which are often even Noetherian. This book, neither assuming nor using any logical formalism, is intended for algebraists and geometers, in the hope of popularizing ultraproducts and their applications in algebra. ... Read more

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This monograph develops projective geometries and provides asystematic treatment of morphisms. It is unique in that it does notconfine itself to isomorphisms. This work introduces a new fundamentaltheorem and its applications describing homogeneous coordinates asmorphisms of projective geometries by semilinear maps. Other topicstreated include three equivalent definitions of projective geometriesand their correspondence with certain lattices; quotients ofprojective geometries and isomorphism theorems; recent results indimension theory; morphisms and homomorphisms of projectivegeometries; special morphisms; duality theory; morphisms of affinegeometries; polarities; orthogonalities; Hilbertian geometries andpropositional systems. The book concludes with a large section ofexercises.

Audience: This volume will be of interest to mathematicians andresearchers whose work involves projective geometries and theirmorphisms, semilinear maps and sesquilinear forms, lattices, categorytheory, and quantum mechanics. This book can also be recommended as atext in axiomatic geometry. ... Read more

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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.

Customer Reviews (1)

Well, it is what it is!
I'm a math and physics double major, and I've been interested in category theory for a while. This book does exactly what it intends to do, which the authors state in the preface is to "...present categories, homological algebra and sheaves in a systematic and exhaustive manner starting from scratch and continuing with full proofs to an exposition of the most recent results in the literature and sometimes beyond."

It's a typical advanced math book...i.e. a seeming grocery list of definitions and lemmas, then theorems and proofs.

An advanced undergraduate may be interested, but it's a bit abstract (as all advanced math is!).

The first chapter is a decent introduction to categories. I would advise learning linear algebra and abstract algebra before reading this chapter, and of course read this chapter before the rest of the book!

Personally, without applications to reality, learning new math leaves me asking "So what?" The book has mathematical applications of category theory to "cast" abstract algebra and linear algebra in the language of categories instead of using the language of set theory.

So if you don't know linear algebra and abstract algebra, you'll be left asking "So what?"

Further, a lot of these concepts that are presented are hard...not in the sense "Solving this equation is hard!" But in the sense that it's deep, so it's hard like "Reading Hegel is hard!"

Overall, I think it's a great book and worth it's money. I wouldn't advise getting it without good knowledge of abstract algebra (since then the notion of a category of groups, for example, would be meaningless without knowledge of what a group is!) or linear algebra (which helps with the notion of morphisms, etc.).

Just my two cents... ... Read more