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Chapters: Analytic Geometry, Homological Algebra, Bézout's Theorem, Generalized Riemann Hypothesis, Motive, Dessin D'enfant, Glossary of Arithmetic and Diophantine Geometry, Hodge Structure, Smooth Function, Nakayama Lemma, Resolution of Singularities, Algebraic K-Theory, Grassmannian, Valuation, Gröbner Basis, Calabi-yau Manifold, Chow Ring, Cotangent Complex, Algebraic Variety, Crystalline Cohomology, Esquisse D'un Programme, Macdonald Polynomial, Hilbert Scheme, Flat Module, Algebraic Geometry and Analytic Geometry, Homogeneous Polynomial, Quantum Cohomology, Gromov-witten Invariant, Drinfel'd Module, General Position, Ample Line Bundle, Milnor Number, Flip, Generic Property, Minimal Model Program, Hilbert's Arithmetic of Ends, Regular Local Ring, Adjunction Formula, Néron-tate Height, Pseudoholomorphic Curve, Codimension, Injective Sheaf, Characteristic Set, Kähler Manifold, Conifold, Dévissage, Rational Mapping, Arakelov Theory, Hasse-weil Zeta Function, Cohen-macaulay Ring, Algebraic Stack, List of Algebraic Geometry Topics, Lefschetz Theorem on (1,1)-Classes, A¹ Homotopy Theory, Adequate Equivalence Relation, Italian School of Algebraic Geometry, Fondements de La Géometrie Algébrique, Standard Conjectures on Algebraic Cycles, Shimura Variety, Algebraic Space, Cone of Curves, Hyperkähler Manifold, Deformation Theory, Noncommutative Algebraic Geometry, Flat Topology, Descent, Excellent Ring, Stable Vector Bundle, Finite Morphism, Amoeba, Tropical Geometry, Canonical Singularity, Complete Intersection, Tangent Cone, N!-Conjecture, Conic Bundle, Irreducible Component, Canonical Ring, Residue Field, Abstract Algebraic Variety, Inverse Image Functor, Kähler Differential, Formal Scheme, Grothendieck's Galois Theory, Mnev's Universality Theorem, Jacobian Conjecture, Local Parameter, Period, Differential of the First Kind, Positive Form, Zariski Tangent Space, Schubert Variety, Generic Point, Noetherian Topological Space, Néron ...More: http://booksllc.net/?id=1997 ... Read more

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This is a monograph in semi-infinite homological algebra, concentrated mostly on the semi-infinite theory of associative algebraic structures, but including also some material on the semi-infinite homology and cohomology of Lie algebras and topological groups. The main objects of study are the double-sided derived functors SemiExt and SemiTor, and the phenomenon of comodule-contramodule correspondence, connecting them with the more conventional, one-sided Ext and CtrTor. Contramodules, introduced originally by Eilenberg and Moore in 1960's but almost forgotten for four decades, play a very prominent role in this book, with many versions of them introduced and discussed. Among other topics considered in the monograph there are the semi-infinite model category structures and the relative nohomogeneous Koszul duality.

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In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects liked in diagrams by morphisms or arrows. One of the simplest examples of a category (which is a very important concept in topology) is that of groupoid, defined as a category whose arrows or morphisms are all invertible. Categories now appear in most branches of mathematics, some areas of theoretical computer science where they correspond to types, and mathematical physics where they can be used to describe vector spaces. Category theory provides both with a unifying notion and terminology. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942?45, in connection with algebraic topology. Category theory has several faces known not just to specialists, but to other mathematicians. A term dating from the 1940s, "general abstract nonsense", refers to its high level of abstraction, compared to more classical branches of mathematics. Homological algebra is category theory in its aspect of organising and suggesting manipulations in abstract algebra. ... Read more

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L. Badescu: Sur certaines singularités des variétés algébriques.- D.A. Buchsbaum: Homological and commutative algebra.- S. Greco: Anelli Henseliani.- C. Lair: Morphismes et structures algébriques.- B.A. Mitchell: Introduction to category theory and homological algebra.- R. Rivet: Anneaux de séries formelles et anneaux henseliens.- P. Salmon: Applicazioni della K-teoria all’algebra commutativa.- M. Tierney: Axiomatic sheaf theory: some constructions and applications.-C.B. Winters: An elementary lecture on algebraic spaces. ... Read more

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Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is considered one of the greatest mathematicians of the 20th century. He is most famous for his revolutionary advances in algebraic geometry, but he has also made major contributions to algebraic topology, number theory, category theory, Galois theory, descent theory, commutative homological algebra and functional analysis. He was awarded the Fields Medal in 1966, and was co-awarded the Crafoord Prize with Pierre Deligne in 1988. He declined the latter prize on ethical grounds in an open letter to the media. He is noted for his mastery of abstract approaches to mathematics, and his perfectionism in matters of formulation and presentation. In particular, he demonstrated the ability to derive concrete results using only very general methods. Relatively little of his work after 1960 was published by the conventional route of the learned journal, circulating initially in duplicated volumes of seminar notes; his influence was to a considerable extent personal, on French mathematics and the Zariski school at Harvard University. ... Read more

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This book contains two contributions: "Combinatorial and Asymptotic Methods in Algebra" by V.A. Ufnarovskij is a survey of various combinatorial methods in infinite-dimensional algebras, widely interpreted to contain homological algebra and vigorously developing computer algebra, and narrowly interpreted as the study of algebraic objects defined by generators and their relations. The author shows how objects like words, graphs and automata provide valuable information in asymptotic studies. The main methods emply the notions of Gröbner bases, generating functions, growth and those of homological algebra. Treated are also problems of relationships between different series, such as Hilbert, Poincare and Poincare-Betti series. Hyperbolic and quantum groups are also discussed. The reader does not need much of background material for he can find definitions and simple properties of the defined notions introduced along the way. "Non-Associative Structures" by E.N.Kuz'min and I.P.Shestakov surveys the modern state of the theory of non-associative structures that are nearly associative. Jordan, alternative, Malcev, and quasigroup algebras are discussed as well as applications of these structures in various areas of mathematics and primarily their relationship with the associative algebras. Quasigroups and loops are treated too. The survey is self-contained and complete with references to proofs in the literature. The book will be of great interest to graduate students and researchers in mathematics, computer science and theoretical physics. ... Read more

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one of the best books about Combinatorial Methods
I found all what I need in the book. Very interesting discussion of ideas in different areas of algebra. ... Read more

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Categorical algebra and its applications contain several fundamental papers on general category theory, by the top specialists in the field, and many interesting papers on the applications of category theory in functional analysis, algebraic topology, algebraic geometry, general topology, ring theory, cohomology, differential geometry, group theory, mathematical logic and computer sciences. The volume contains 28 carefully selected and refereed papers, out of 96 talks delivered, and illustrates the usefulness of category theory today as a powerful tool of investigation in many other areas. ... Read more

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The theory of finite-dimensional algebras is one of the mostfundamentaldomains of modern algebra, applied in severalother parts of mathematics andin theoretical physics. Thisbook, written by two of the leading researchersin the fieldand revised and augmented for the English edition, wastranslated from the Russian by a third leading specialist,who hascontributed to it an appendix.The book presents both the basic classical theory and morerecent results closely related to current research(somecategory theory including Morita's theorem, schemes ofquivers andtensor algebras, duality, quasi-Frobenius,hereditary, serial algebras). Theonly prior knowledgeassumed of the reader is linear algebra and, in places,alittle group theory. Each chapter includes a series ofexercises, illustrating the content and introducing morerefined results: for the more complicated ones, hints forthe solution are given - thus the book can be used as atextbook in class or for self-study, and as anup-to-datereference to the field. ... Read more

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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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High Quality Content by WIKIPEDIA articles! In the area of mathematics known as homological algebra, especially in algebraic topology and group cohomology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by Jean Leray in 1946, they have become an important research tool, particularly in homotopy theory. However, they have a reputation for being abstruse and difficult to comprehend. ... Read more