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Homological Theory of Representations

Henning Krause
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
James Turner
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Representation theory, as an area of mathematics, is not only a subject unto itself, but finds important applications in algebra, algebraic geometry, algebraic topology, combinatorics, differential geometry, as well as in many areas of physics such as quantum mechanics.  Classically, it has its roots in the theory of linear representations of finite groups. This is the study of homomorphisms of finite groups into the general group of invertible matrices over a division ring (e.g. a field). On a basic level, one of the goals of this theory is to find those representations that provide the means to effectively compute such groups and further prove theorems about them. One way to view such a search is to develop for groups the analogs of Jordan normal or the lower-upper decomposition for matrices. Now, a linear representation of a group G over a (division) ring R can be understood, equivalently, as a free R-module equipped with an action by G. Furthermore, this latter action can be extended to a module structure over the corresponding group ring RG. Thus the representation theory of (finite) groups can be embedded into the theory of modules over group rings, or, more generally, into the theory of modules over associative algebras. This last step in generality opens the door to bringing in powerful tools from the category theory of associative algebras. In particular, Morita theory provides an explicit means for identifying and transferring information from one module category to an equivalent one. The necessary and sufficient condition for a category of R-modules to be equivalent to a category of S-modules is that the ring S be isomorphic to the ring of endomorphisms on a finitely generated projective R-module.  Moreover, if R is a Artin ring then S is isomorphic to a direct sum of algebras, each being a matrix algebra on a division algebra, thus providing the criteria for a representation to be within the classical setting. Furthermore, the entrance of projective modules opens the door to a homological perspective for representation theory.
From the standpoint of finite groups, full blooded homological methods enter into representation theory through the tools of group homology and cohomology of modules. These are sequences of R-modules, indexed by non-negative integers, associated to an RG-module for a finite group G. In turn, each of these can be understood as a special case of associative algebra (co-)homology for modules. They serve to house important data that disclose deeper structures in these modules, such as torsion, extensions, and their higher order counterparts.  Now, from a categorical perspective, associative algebra (co-)homology of modules arise as special cases of homology functors on the category of chain complexes on abelian categories (such as that of R-modules or its opposite category) through the method of resolutions by projective or injective modules. Since what is sought is the data housed in and interpreted through the homology modules, such information is invariant up to isomorphism on them.  On the level of chain complexes, isomorphism can be softened to quasi-isomorphism (chain morphisms that become isomorphisms through the homology functors). By a process known as localization, quasi-isomorphisms may be inverted in the category of chain complexes to form the associated derived category. These latter categories are then the cause célèbre and the central focus of the homological approach to representation theory.
Krause’s book is a tour de force, providing the current state-of-the-art methodology in homological algebra as it pertains to representation theory. As such, it is divided into two main points of focus. The first is a detailed exposition of major results on the derived module categories associated to a ring. One major organizing principle, in this regard, is the notion of orthogonal decompositions, which can be seen as a type of extension of the result mentioned above for Artin rings. Another important tool that is heavily utilized in this context is the notion of tilting modules. These provide the means to uncover valuable information about and establish correspondences between deep categorical properties within derived categories. Further, the book explores in great detail Morita-type results pertaining to equivalences between derived module categories.
The second major focus in this work explores representations of abelian categories in functor categories through the notion of purity. Here, the main ideas orbit around the identification and characterization of pure category theory, in general, and pure-injective objects specifically. One particular important focus is on special sets of indecomposable versions of pure-injective objects and characterizing them through special subcategories, called definable subcategories, which have rich closure properties with respect to purity.  Finally, another beautiful result of this part of the book is a generalization of the Hilbert basis theorem through the notion of Grobner categories.
In conclusion, this book is a gem. With sufficient background in homological algebra and representation theory, the reader will find a nicely self contained resource on the subject matter. The book is broken into four parts. After a brief yet very helpful summary of major results found in the book, the first part gives a thorough survey of the foundational material in category theory and homological algebra. This, in and of itself, is worth its weight in gold for anybody working in these areas. In every part, each chapter begins with a helpful table of contents and closes with a nice overview providing a historical context. The subject matter in each chapter is nicely developed and motivated. Detailed proofs can be found for significant results, with helpful summaries and references when proofs of other results are not given. The book closes with a very helpful glossary of terms and a thorough bibliography.  The one (minor) criticism I would make is that the index could be more thorough.  Overall, though, this is a fine and beautiful resource for a graduate student or researcher to utilize in order to enter into the cutting-edge research in representation theory from a homological perspective.

James Turner is a Professor of Mathematics at Calvin University.