##
**Triangulated categories in the representation theory of finite dimensional algebras.**
*(English)*
Zbl 0635.16017

London Mathematical Society Lecture Note Series, 119. Cambridge etc.: Cambridge University Press. IX, 208 p.; £13.95; $ 24.95 (1988).

The aim of this book is to show that the concept of triangulated categories, due to Grothendieck and Verdier, is important for studying modules over finite dimensional algebras and conversely, that the methods of the representation theory of finite dimensional algebras might be useful for the description of certain derived categories.

The book is divided into five chapters. Chapter I is an introduction to the theory of triangulated categories. The author shows that the stable categories associated with the Frobenius categories admit triangulated structures. In particular, it is shown that the following two important categories, associated with any finite dimensional algebra A over an algebraically closed field k, are triangulated categories: (1) the derived category \(D^ b(A)=D^ b(mod A)\) of bounded complexes over the category mod A of finitely generated left A-modules, and (2) the stable category \({}^ ZT(A)\) of the category \(mod^ ZT(A)\) of finitely generated Z-graded modules over the trivial extension \(T(A)=A\ltimes D(A)\) of A by the injective cogenerator \(D(A)=Hom_ k(A,k)\). The author introduces the Auslander-Reiten triangles in a triangulated category and shows that \(D^ b(A)\) has Auslander-Reiten triangles in case A has finite global dimension. Moreover, for A hereditary, a complete description of \(D^ b(A)\) is presented.

In Chapter II the author constructs a full, faithful, exact functor \(F: D^ b(A)\to \underline{mod}^ ZT(A)\) of triangulated categories which is dense if A has finite global dimension. This is a very important result which allows to investigate the derived categories \(D^ b(A)\) using methods of the representation theory of algebras. Applying this result, the reviewer and I. Assem [Math. Ann. (to appear; Zbl 0617.16017)] classified all derived categories \(D^ b(A)\) whose oriented cycles of indecomposable complexes lie in tubes.

Chapter III is devoted to the tilting theory. Almost all important results of this theory obtained in recent years as well as their applications are presented. It is shown that, if an algebra B is tiltable to an algebra A of finite global dimension, then \(D^ b(B)\) and \(D^ b(A)\) are triangle-equivalent. Moreover, a triangle-equivalence of \(D^ b(A)\) and \(D^ b(B)\) implies an isometry of the Grothendieck groups \(K_ 0(A)\) and \(K_ 0(B).\)

Chapter IV is devoted to the detailed study of algebras A wh(B) implies \(s=s!\) Let m be the linear space of real bounded sequences and let w be the maximal set of almost positive, regular methods such that for any pair of methods (A) and (B) the methods (AB) and (BA) are absolutely equivalent. For \(x\in m\) let \(V(x)=\inf_{A\in W}\nu_ A(x)\), where \(\nu_ A:m\to {\mathbb{R}}\) is defined by \(\nu_ A(x)=\overline{\lim}_{n}\sup_{i}\sum_{k}a_{nk}(i)x_ k.\) The authors prove the following: (1) \(V(Ax)=V(x)\) for all \(x\in m\) and \(A\in W\). (2) In order that \(\phi\) in the algebraic dual of m satisfies \(\phi (x)\leq \overline{\lim}_{p}\sup_{n}(p+1)^{- 1}\sum^{p}_{i=0}x_{n+i}\) and \(\phi (Ax)=\phi (x)\), \(A\in W\) it is necessary and sufficient that \(\phi\) (x)\(\leq V(x)\), \(x\in m\). (3) If W contains a translative method on m, then V generates Banach limits which are invariant under W. (4) W consists of consistent methods for bounded sequences.

The book is divided into five chapters. Chapter I is an introduction to the theory of triangulated categories. The author shows that the stable categories associated with the Frobenius categories admit triangulated structures. In particular, it is shown that the following two important categories, associated with any finite dimensional algebra A over an algebraically closed field k, are triangulated categories: (1) the derived category \(D^ b(A)=D^ b(mod A)\) of bounded complexes over the category mod A of finitely generated left A-modules, and (2) the stable category \({}^ ZT(A)\) of the category \(mod^ ZT(A)\) of finitely generated Z-graded modules over the trivial extension \(T(A)=A\ltimes D(A)\) of A by the injective cogenerator \(D(A)=Hom_ k(A,k)\). The author introduces the Auslander-Reiten triangles in a triangulated category and shows that \(D^ b(A)\) has Auslander-Reiten triangles in case A has finite global dimension. Moreover, for A hereditary, a complete description of \(D^ b(A)\) is presented.

In Chapter II the author constructs a full, faithful, exact functor \(F: D^ b(A)\to \underline{mod}^ ZT(A)\) of triangulated categories which is dense if A has finite global dimension. This is a very important result which allows to investigate the derived categories \(D^ b(A)\) using methods of the representation theory of algebras. Applying this result, the reviewer and I. Assem [Math. Ann. (to appear; Zbl 0617.16017)] classified all derived categories \(D^ b(A)\) whose oriented cycles of indecomposable complexes lie in tubes.

Chapter III is devoted to the tilting theory. Almost all important results of this theory obtained in recent years as well as their applications are presented. It is shown that, if an algebra B is tiltable to an algebra A of finite global dimension, then \(D^ b(B)\) and \(D^ b(A)\) are triangle-equivalent. Moreover, a triangle-equivalence of \(D^ b(A)\) and \(D^ b(B)\) implies an isometry of the Grothendieck groups \(K_ 0(A)\) and \(K_ 0(B).\)

Chapter IV is devoted to the detailed study of algebras A wh(B) implies \(s=s!\) Let m be the linear space of real bounded sequences and let w be the maximal set of almost positive, regular methods such that for any pair of methods (A) and (B) the methods (AB) and (BA) are absolutely equivalent. For \(x\in m\) let \(V(x)=\inf_{A\in W}\nu_ A(x)\), where \(\nu_ A:m\to {\mathbb{R}}\) is defined by \(\nu_ A(x)=\overline{\lim}_{n}\sup_{i}\sum_{k}a_{nk}(i)x_ k.\) The authors prove the following: (1) \(V(Ax)=V(x)\) for all \(x\in m\) and \(A\in W\). (2) In order that \(\phi\) in the algebraic dual of m satisfies \(\phi (x)\leq \overline{\lim}_{p}\sup_{n}(p+1)^{- 1}\sum^{p}_{i=0}x_{n+i}\) and \(\phi (Ax)=\phi (x)\), \(A\in W\) it is necessary and sufficient that \(\phi\) (x)\(\leq V(x)\), \(x\in m\). (3) If W contains a translative method on m, then V generates Banach limits which are invariant under W. (4) W consists of consistent methods for bounded sequences.

Reviewer: T.Leiger

### MSC:

16Gxx | Representation theory of associative rings and algebras |

16P10 | Finite rings and finite-dimensional associative algebras |

16B50 | Category-theoretic methods and results in associative algebras (except as in 16D90) |

18E30 | Derived categories, triangulated categories (MSC2010) |

18-02 | Research exposition (monographs, survey articles) pertaining to category theory |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

16D50 | Injective modules, self-injective associative rings |

16Exx | Homological methods in associative algebras |

16E10 | Homological dimension in associative algebras |