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Derived equivalences for group rings. (English) Zbl 0898.16002

Lecture Notes in Mathematics. 1685. Berlin: Springer. x, 246 p. (1998).
The volume under review is a very welcome addition to the recent literature on representation theory of finite groups, finite dimensional algebras and orders, being linked with D. Happel [Triangulated categories in the representation theory of finite dimensional algebras (Lond. Math. Soc. Lect. Note Ser. 119, 1988; Zbl 0635.16017)], K. Erdmann [Blocks of tame representation type and related algebras (Lect. Notes Math. 1428, 1990; Zbl 0696.20001)], D. J. Benson [Representations and cohomology I, II (Cambridge Univ. Press 1991; Zbl 0718.20001 and Zbl 0731.20001)], P. Gabriel and A. Roiter [Representations of finite dimensional algebras (Springer 1992; Zbl 0839.16001)]; C. A. Weibel [An introduction to homological algebra (Cambridge Univ. Press 1994; Zbl 0797.18001)], J. Thévenaz [\(G\)-algebras and modular representation theory (Oxford, Clarendon Press 1995; Zbl 0837.20015)], S. I. Gelfand and Yu. I. Manin [Methods of homological algebra (Springer 1996; Zbl 0855.18001)], J. F. Carlson [Modules and group algebras (Birkhäuser 1996; Zbl 0883.20006)]. This book is the first comprehensive presentation of the developments arising from J. Rickard’s fundamental work on derived categories [J. Lond. Math. Soc., II. Ser. 39, No. 3, 436-456 (1989; Zbl 0642.16034)]and the formulation of M. Broué’s conjectures [Astérisque 181-182, 61-92 (1990; Zbl 0704.20010)]. Most of the material included in this volume was lectured at the summer school and workshop on “Derived equivalences for group rings”, Pappenheim, Germany, September 26-October 2, 1994.
The book is divided into 11 chapters. Chapter 1 is a detailed presentation of its content, and Chapter 2 is a brief introduction to triangulated categories, stable categories and derived categories, focussing mainly on the derived category of the module category over a ring \(R\), and giving many examples. Chapter 3 is a careful exposition of Rickard’s theorem characterizing equivalences between the bounded derived categories \({\mathbf D}^{\mathbf b}(\Lambda)\) and \({\mathbf D}^{\mathbf b}(\Gamma)\) (regarded as triangulated categories) via one-sided tilting complexes, where \(\Lambda\) and \(\Gamma\) are two rings. This is a “Morita theory” for derived categories, and it is pointed out how the road from the original Morita theory to the Rickard theory was paved by results of Gelfand-Bernstein-Ponomarev, Auslander-Platzeck-Reiten, Brenner-Butler, Happel and Ringel.
Chapter 4 introduces needed facts on modular and local representation theory, presenting the Cartan-Brauer triangle, correspondences of Green and Brauer and Green’s theorem on Brauer tree algebras. Detailed description and structure theorem for Roggenkamp’s Green orders are also given.
Chapter 5 is devoted to effective construction of tilting complexes and computation of their endomorphism rings. The general construction is given in the context of Gorenstein orders, and covers many important particular cases including Green orders, Brauer tree algebras, blocks of tame representation type.
Chapter 6 presents another fundamental result of Rickard. If \(A\) and \(B\) are \(R\)-projective \(R\)-algebras, then any derived equivalence between \(A\) and \(B\) is of the form \(X\otimes_B^A-\colon D^b(B)\to D^b(A)\) where \(X\) is a complex in \(D^b(A\otimes_R B)\), called a two-sided tilting complex. If \(R\) is a complete discrete valuation ring of characteristic zero and algebraically closed residue field of characteristic \(p>0\), then Broué’s conjecture states that if \(A\) is a block of the group algebra \(RG\) with abelian defect group \(D\) and \(B\) is its Brauer correspondent in \(RN_G(D)\), then \(A\) and \(B\) should be derived equivalent. Consequences at the character level of the existence of a two-sided tilting complex for \(A,B\)-bimodules are given. The chapter ends with a construction of a two-sided tilting complex for Green orders. A historical account and a useful guide to the literature are given in Chapter 7.
The last chapters are contributions by B. Keller, J. Rickard, R. Rouquier and M. Linckelmann. B. Keller presents his alternative approach to Rickard’s theorems. His techniques involve differential graded algebras and unbounded complexes. He also explains how derived equivalences preserve cyclic homology. In chapter 9 J. Rickard shows that there are certain derived equivalences, called splendid, which are compatible with the local structure of blocks of group algebras. He also presents some new techniques in representation theory and cohomology of finite groups, which involve, somewhat surprisingly, infinite dimensional modules. Chapter 10 by R. Rouquier is a self-contained account of the theory of blocks with cyclic defect groups, leading to a proof in this case of the strongest form of Broué’s conjecture: the existence of a splendid two-sided tilting complex. In the last chapter M. Linckelmann surveys some of his results on stable equivalences of Morita type between blocks of group algebras.
The authors aimed to introduce the reader into this fascinating subject and to help him to understand the already rich literature. They definitely succeeded, and this volume will be inspirational for further discoveries.

MSC:

16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras
16D90 Module categories in associative algebras
16G30 Representations of orders, lattices, algebras over commutative rings
18-02 Research exposition (monographs, survey articles) pertaining to category theory
20-02 Research exposition (monographs, survey articles) pertaining to group theory
16E20 Grothendieck groups, \(K\)-theory, etc.
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16G10 Representations of associative Artinian rings
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
18E30 Derived categories, triangulated categories (MSC2010)
16S34 Group rings
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20C20 Modular representations and characters
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
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