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Faculty of Science News
Published in The Rigakubu News January 2025
Science Bookshelf
Layers and Homological Algebra
Atsushi Shiho (Professor, Graduate School of Mathematical Sciences / Department of Mathematics)

By Jun Shifu "Layers and Homological Algebras" |
Kyoritsu Publishing (2016) |
It is difficult to show directly that n-dimensional and m-dimensional coordinate spaces are not the same as topological spaces (not as linear spaces) for different natural numbers n and m, but it is easy to show by comparing the algebraic object of homology groups of spaces with one point removed.
In this way, it has been done in many fields of mathematics to successfully define an algebraic object such as a homology group from a difficult object and to study the original object by examining it. The above was an example in topology, but de Rham cohomology is used in differential geometry, and Galois cohomology is used in number theory. The algebraic part of how these (co)homology groups are defined is abstracted and summarized as homology algebra. The algebraic object that abstracts the pasting property of functions is a layer, and cohomology theories often consider a layer as a cohomology group whose coefficients are coefficients.
Layers and Homological Algebras" is a book about homological algebras and layers, and is based on my experience in teaching a course for fourth-year and graduate students in the Department of Mathematics, which is given once every two years. I have tried to present only the basics, so that little prior knowledge is required. In recent years, homological algebra has been elevated to homotopy algebra, and the content of this book may now be classical, but it is essential for a serious understanding of mathematics. As one of the mathematics courses beyond the first semester of College of Arts and Sciences, I recommend this book to students who are considering entering various fields that use mathematics or mathematics in a serious way.