Ramji Lal - Algebra 3: Homological Algebra and Its Applications

The Infosys Science Foundation SeriesThe Infosys Science Foundation Series is a collaborative effort between the Infosys Science Foundation and Springer to publish high quality content. All content published in the Series is written, edited, or vetted by the laureates or jury members of the Infosys Prize. This umbrella series publishes content across academic domains, including, but not limited to, Applied Sciences, Engineering, Pure Sciences, and Mathematics. With this Series, Springer and the Infosys Science Foundation hope to provide readers with textbooks, monographs, handbooks, and professional books of the highest academic quality. Literature in this Series will appeal to a wide audience of researchers, students, educators, and professionals across disciplines.

in Mathematical Sciences, a Scopus-indexed book series, is a sub-series of the Infosys Science Foundation Series. This sub-series focuses on high-quality content in the domain of mathematical sciences and various disciplines of mathematics, statistics, bio-mathematics, financial mathematics, applied mathematics, operations research, applied statistics and computer science. All content published in the sub-series are written, edited, or vetted by the laureates or jury members of the Infosys Prize. With this series, Springer and the Infosys Science Foundation hope to provide readers with monographs, handbooks, professional books and textbooks of the highest academic quality on current topics in relevant disciplines. Literature in this sub-series will appeal to a wide audience of researchers, students, educators, and professionals across mathematics, applied mathematics, statistics and computer science disciplines.

More information about this subseries at http://www.springer.com/series/13817

This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd.

The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Algebra has played a central and a decisive role in formulating and solving the problems in all branches of mathematics, science, and engineering. My earlier plan was to write a series of three volumes on algebra covering a wide spectrum to cater the need of students and researchers at various levels. The two initial volumes have already appeared. However, looking at the size and the contents to be covered, we decided to split the third volume into two volumes, Algebra 3 and Algebra 4. Algebra 3 concentrates on the homological algebra together with its important applications in mathematics, whereas Algebra 4 is about Lie algebras, Chevalley groups, and their representation theory.

Homological algebra has played and is playing a pivotal role in understanding and classifying (up to certain equivalences) the mathematical structures such as topological, geometrical, arithmetical, and the algebraic structures by associating computable algebraic invariants to these structures. Indeed, it has also shown its deep intrinsic presence in dealing with the problems in physics, in particular, in string theory and quantum theory. The present volume, Algebra 3, the third volume in the series, is devoted to introduce the homological methods and to have some of its important applications in geometry, topology, algebraic geometry, algebra, and representation theory. It contains category theory, abelian categories, and homology theory in abelian categories, the -fold extension functors , the torsion functors , the theory of derived functors, simplicial and singular homology theories with their applications, co-homology of groups, sheaf theory, sheaf co-homology, some amount of algebraic geometry, tale sheaf theory and co-homology, and the -adic co-homology with a demonstration showing its application in the representation theory. The book can act as a text for graduate and advance graduate students specializing in mathematics.