Representation theory of Artin algebras.

*(English)*Zbl 0834.16001
Cambridge Studies in Advanced Mathematics. 36. Cambridge: Cambridge University Press. xiv, 423 p. (1995).

The representation theory of Artin algebras grew up during the last 25 years as an important part of the module theory. It has been developed rapidly and became already an independent field of study having many important and deep applications in module theory, ring theory, integral representation theory, abelian group theory, the theory of quantum groups and in the representation theory of Lie groups and algebras. It has also very important links with many nonalgebraic theories including model theory for modules and algebraic geometry [see W. Geigle and H. Lenzing, J. Algebra 144, 273-343 (1991; Zbl 0748.18007)] and E. Brieskorn [Invent. Math. 4, 336-358 (1968; Zbl 0219.14003)].

An important role in the development of the theory has been played by the Brauer-Thrall conjectures and classical decomposition problems studied by G. Köthe and R. Brauer in the third and the fourth decade of this century [see C. M. Ringel, Lect. Notes Math. 831, 104-136 (1980; Zbl 0444.16019)].

The book by Auslander, Reiten and Smalø is one of the first textbooks on this subject. It provides us with an elementary, systematic, homogeneous and excellent introduction to the representation theory of Artin algebras. The main aim of the book is to give an introduction to the part of the theory built around almost split sequences – a notion introduced by M. Auslander and I. Reiten in 1974-75. However the book develops also a modern technique of linear representations of quivers with relations and their interpretation as modules over factor algebras of path algebras of quivers. There are other important aspects of the theory not covered by this book such as representations of partially ordered sets, vector space categories, bocses, Galois covering technique, integral representation theory, tilting theory, derived categories, homologically finite subcategories, tame and wild algebras and quasi- hereditary algebras. Some of them are presented in the following books: D. J. Benson [“Representations and Cohomology” (1991; Zbl 0718.20001)], C. W. Curtis and I. Reiner [“Methods of Representation Theory” (1990; Zbl 0698.20001)], Yu. Drozd and V. V. Kirichenko [“Finite Dimensional Algebras” with appendix on “Quasi-hereditary Algebras” by V. Dlab (1994; Zbl 0816.16001)], P. Gabriel and A. V. Rojter [“Representations of Finite Dimensional Algebras”, Algebra VIII, Encycl. Math. Sci. 73 (1992)], D. Happel [“Triangulated Categories in the Representation Theory of Finite Dimensional Algebras” (1988; Zbl 0635.16017)], C. U. Jensen and H. Lenzing [“Model Theoretic Algebra” (1989; Zbl 0728.03026)], M. Prest [“Model Theory and Modules” (1988; Zbl 0634.03025)], C. M. Ringel [“Tame Algebras and Integral Quadratic Forms” (1984; Zbl 0546.16013)], D. Simson [“Linear Representations of Partially Ordered Sets and Vector Space Categories” (1992; Zbl 0818.16009)].

Throughout we assume that \(R\) is an associative ring with an identity element. We recall that \(R\) is an Artin algebra if the center \(Z(R)\) is a commutative artinian ring and \(R\) is a finitely generated module over \(Z(R)\). In particular, every finite dimensional algebra \(R\) over a field \(K\) is an Artin algebra. We always suppose that \(R\) is connected, that is, \(R\) has no decomposition into a product of two algebras.

One of the main aims of the representation theory of Artin algebras is to describe the structure of the category \(\text{mod}(R)\) of finitely generated right \(R\)-modules. In particular we are interested in describing and classifying all indecomposable modules \(X\) in \(\text{mod} (R)\) up to isomorphism, their endomorphism (local) algebras \(\text{End}(X)\), the abelian groups \(\text{Hom}_R (X,Y)\) and \(\text{Ext}^1_R (X,Y)\) viewed as End\((Y)\)-End\((X)\)-bimodules for each pair \(X\), \(Y\) of indecomposable modules in \(\text{mod}(R)\).

The theory is developed under the assumption that simple right \(R\)- modules and the indecomposable projective right \(R\)-modules are given up to isomorphism. We are also interested in describing the Jacobson radical \(\text{rad}(\text{mod }R)\) of the category \(\text{mod} (R)\), the radical power sequence \[ \text{rad}(\text{mod }R) \supseteq \text{rad}^2(\text{mod }R) \supseteq \text{rad}^3(\text{mod }R) \supseteq \dots\supseteq \text{rad}^\infty(\text{mod }R) \supseteq (\text{rad}^\infty(\text{mod }R))^2 \supseteq \cdots \] and the quotients \(\text{rad}^j \text{mod} (R)/\text{rad}^{j + 1} \text{mod} (R)\), \(j \geq 1\), where \(\text{rad}^\infty(\text{mod }R) = \bigcap^\infty_{j = 1} \text{rad}^j(\text{mod }R)\). We recall that \(\text{rad}(\text{mod }R)\) is the intersection of all two-sided ideals in \(\text{mod} (R)\), and given two indecomposable modules \(X\) and \(Y\) in \(\text{mod} (R)\) the homomorphism \(f : X \to Y\) belongs to \(\text{rad}(\text{mod }R)\) if and only if \(f\) is not an isomorphism.

It follows that the description of the quotient \(\text{rad}(\text{mod }R)/\text{rad}^2(\text{mod }R)\) leads to a description of the Auslander-Reiten translation quiver \((\Gamma_R, \tau)\) of \(R\), whose vertices are the isomorphism classes \([X]\) of indecomposable modules \(X\) in \(\text{mod} (R)\), and there is an arrow \([X] \to [Y]\) in \(\Gamma_R\) if and only if there is an irreducible homomorphism \(X \to Y\). Here \(\tau = D \circ Tr\) is the Auslander-Reiten translation.

One of the technical aims of the theory is to find a simple construction allowing us to produce new indecomposable modules from given ones and such that it produces all indecomposable modules in \(\text{mod}(R)\) from the simple ones and the indecomposable projective ones, provided \(R\) is of finite representation type, that is, the number of the isomorphism classes of indecomposable modules in \(\text{mod} (R)\) is finite. We are also interested in describing all homomorphisms in \(\text{mod} (R)\) by constructing a minimal generating set of homomorphisms between indecomposable modules in \(\text{mod} (R)\) such that every homomorphism in \(\text{mod} (R)\) is a linear combination of compositions of homomorphisms in this generating set.

The book by Auslander, Reiten and Smalø is a perfect introduction to the theory and it presents elementary useful efficient tools and theoretical results allowing us to solve the problems formulated above, at least for algebras of finite representation type. The main tools are: the Auslander transpose, the Auslander-Reiten translation, irreducible morphisms, the class of short exact sequences (*): \(0 \to X \to Y \to Z \to 0\) in \(\text{mod}(R)\) called almost split sequences and the Auslander-Reiten translation quiver \((\Gamma_R, \tau)\) of \(R\). They are presented in Chapters 4, 5 and 7. We recall that the non-split sequence (*) with indecomposable modules \(X\) and \(Y\) is said to be an almost split sequence if every homomorphism \(g : U \to Z\) in \(\text{mod} (R)\) which is not a splittable epimorphism has a factorization through \(Y \to Z\). The translation \(\tau\) is defined in such a way that \(\tau Z \cong X\) when \(X\) and \(Z\) appear as the end terms of the almost split sequence (*). Criteria for \(R\) to be of finite representation type are given. In particular it is shown in Chapter 6 that a connected Artin algebra \(R\) is of finite representation type if and only if the Auslander-Reiten translation quiver \((\Gamma_R, \tau)\) contains a finite connected component. Elementary facts concerning the shapes of connected components of the Auslander-Reiten translation quiver \((\Gamma_R, \tau)\) are presented in Chapter 7.

The notions of an almost split sequence, irreducible morphism and the Auslander-Reiten translation quiver are introduced and discussed in detail in the book mainly in connection with the radical of the category \(\text{mod} (R)\) and with algebras of finite representation type. In particular the representation theory of Nakayama algebras, selfinjective algebras, group algebras, skew group algebras and hereditary algebras are discussed in detail.

In the case \(R\) is hereditary the idea of applying the preprojective modules, the preinjective modules, a valued quiver technique, an associated quadratic form, Weyl group, root system and the Coxeter transformation is presented in Chapter 8. In particular it is proved that a connected hereditary Artin algebra is of finite representation type if and only if the associated valued quiver is a Dynkin diagram. Although some information on regular components of \((\Gamma_R, \tau)\) are included in the case \(R\) is hereditary the only representation-infinite algebra presented in the book in detail together with its representation theory is the hereditary Kronecker algebra.

In Chapter 9 directing modules, short chains, short cycles, sincere modules and modules determined by their composition factors are investigated.

In Chapter 10 stable equivalence of algebras is investigated. In particular algebras with radical square zero and algebras stably equivalent to symmetric Nakayama algebras are studied in detail. We recall that two algebras \(R\) and \(T\) are said to be stably equivalent if their stable module categories \(\underline {\text{mod}}(R)\) and \(\underline {\text{mod}} (T)\) are equivalent, where \(\underline {\text{mod}} (R)\) means the factor category of \(\text{mod}(R)\) modulo the two- sided ideal in \(\text{mod} (R)\) consisting of all homomorphisms that factorize through projective modules.

The book finishes by Chapter 11 on modules determining morphisms and morphisms determined by modules.

The book contains an extensive bibliography. Each chapter is followed by a set of exercises, and by a brief but very informative set of notes and remarks. The book ends with a set of conjectures and open problems.

The book is a well-written and welcome addition to this subject. It does the field a great service by presenting foundational material in a clear, accessible form. It illustrates very well the role of almost split sequences and the Auslander-Reiten technique in the representation theory of Artin algebras. The first part of the book contains an excellent introduction to the basic aspects of the subject.

It is suitable for any mathematician (including graduate students) wanting a brief introduction to this active field. It would make an excellent text for a topics course in the subject.

An important role in the development of the theory has been played by the Brauer-Thrall conjectures and classical decomposition problems studied by G. Köthe and R. Brauer in the third and the fourth decade of this century [see C. M. Ringel, Lect. Notes Math. 831, 104-136 (1980; Zbl 0444.16019)].

The book by Auslander, Reiten and Smalø is one of the first textbooks on this subject. It provides us with an elementary, systematic, homogeneous and excellent introduction to the representation theory of Artin algebras. The main aim of the book is to give an introduction to the part of the theory built around almost split sequences – a notion introduced by M. Auslander and I. Reiten in 1974-75. However the book develops also a modern technique of linear representations of quivers with relations and their interpretation as modules over factor algebras of path algebras of quivers. There are other important aspects of the theory not covered by this book such as representations of partially ordered sets, vector space categories, bocses, Galois covering technique, integral representation theory, tilting theory, derived categories, homologically finite subcategories, tame and wild algebras and quasi- hereditary algebras. Some of them are presented in the following books: D. J. Benson [“Representations and Cohomology” (1991; Zbl 0718.20001)], C. W. Curtis and I. Reiner [“Methods of Representation Theory” (1990; Zbl 0698.20001)], Yu. Drozd and V. V. Kirichenko [“Finite Dimensional Algebras” with appendix on “Quasi-hereditary Algebras” by V. Dlab (1994; Zbl 0816.16001)], P. Gabriel and A. V. Rojter [“Representations of Finite Dimensional Algebras”, Algebra VIII, Encycl. Math. Sci. 73 (1992)], D. Happel [“Triangulated Categories in the Representation Theory of Finite Dimensional Algebras” (1988; Zbl 0635.16017)], C. U. Jensen and H. Lenzing [“Model Theoretic Algebra” (1989; Zbl 0728.03026)], M. Prest [“Model Theory and Modules” (1988; Zbl 0634.03025)], C. M. Ringel [“Tame Algebras and Integral Quadratic Forms” (1984; Zbl 0546.16013)], D. Simson [“Linear Representations of Partially Ordered Sets and Vector Space Categories” (1992; Zbl 0818.16009)].

Throughout we assume that \(R\) is an associative ring with an identity element. We recall that \(R\) is an Artin algebra if the center \(Z(R)\) is a commutative artinian ring and \(R\) is a finitely generated module over \(Z(R)\). In particular, every finite dimensional algebra \(R\) over a field \(K\) is an Artin algebra. We always suppose that \(R\) is connected, that is, \(R\) has no decomposition into a product of two algebras.

One of the main aims of the representation theory of Artin algebras is to describe the structure of the category \(\text{mod}(R)\) of finitely generated right \(R\)-modules. In particular we are interested in describing and classifying all indecomposable modules \(X\) in \(\text{mod} (R)\) up to isomorphism, their endomorphism (local) algebras \(\text{End}(X)\), the abelian groups \(\text{Hom}_R (X,Y)\) and \(\text{Ext}^1_R (X,Y)\) viewed as End\((Y)\)-End\((X)\)-bimodules for each pair \(X\), \(Y\) of indecomposable modules in \(\text{mod}(R)\).

The theory is developed under the assumption that simple right \(R\)- modules and the indecomposable projective right \(R\)-modules are given up to isomorphism. We are also interested in describing the Jacobson radical \(\text{rad}(\text{mod }R)\) of the category \(\text{mod} (R)\), the radical power sequence \[ \text{rad}(\text{mod }R) \supseteq \text{rad}^2(\text{mod }R) \supseteq \text{rad}^3(\text{mod }R) \supseteq \dots\supseteq \text{rad}^\infty(\text{mod }R) \supseteq (\text{rad}^\infty(\text{mod }R))^2 \supseteq \cdots \] and the quotients \(\text{rad}^j \text{mod} (R)/\text{rad}^{j + 1} \text{mod} (R)\), \(j \geq 1\), where \(\text{rad}^\infty(\text{mod }R) = \bigcap^\infty_{j = 1} \text{rad}^j(\text{mod }R)\). We recall that \(\text{rad}(\text{mod }R)\) is the intersection of all two-sided ideals in \(\text{mod} (R)\), and given two indecomposable modules \(X\) and \(Y\) in \(\text{mod} (R)\) the homomorphism \(f : X \to Y\) belongs to \(\text{rad}(\text{mod }R)\) if and only if \(f\) is not an isomorphism.

It follows that the description of the quotient \(\text{rad}(\text{mod }R)/\text{rad}^2(\text{mod }R)\) leads to a description of the Auslander-Reiten translation quiver \((\Gamma_R, \tau)\) of \(R\), whose vertices are the isomorphism classes \([X]\) of indecomposable modules \(X\) in \(\text{mod} (R)\), and there is an arrow \([X] \to [Y]\) in \(\Gamma_R\) if and only if there is an irreducible homomorphism \(X \to Y\). Here \(\tau = D \circ Tr\) is the Auslander-Reiten translation.

One of the technical aims of the theory is to find a simple construction allowing us to produce new indecomposable modules from given ones and such that it produces all indecomposable modules in \(\text{mod}(R)\) from the simple ones and the indecomposable projective ones, provided \(R\) is of finite representation type, that is, the number of the isomorphism classes of indecomposable modules in \(\text{mod} (R)\) is finite. We are also interested in describing all homomorphisms in \(\text{mod} (R)\) by constructing a minimal generating set of homomorphisms between indecomposable modules in \(\text{mod} (R)\) such that every homomorphism in \(\text{mod} (R)\) is a linear combination of compositions of homomorphisms in this generating set.

The book by Auslander, Reiten and Smalø is a perfect introduction to the theory and it presents elementary useful efficient tools and theoretical results allowing us to solve the problems formulated above, at least for algebras of finite representation type. The main tools are: the Auslander transpose, the Auslander-Reiten translation, irreducible morphisms, the class of short exact sequences (*): \(0 \to X \to Y \to Z \to 0\) in \(\text{mod}(R)\) called almost split sequences and the Auslander-Reiten translation quiver \((\Gamma_R, \tau)\) of \(R\). They are presented in Chapters 4, 5 and 7. We recall that the non-split sequence (*) with indecomposable modules \(X\) and \(Y\) is said to be an almost split sequence if every homomorphism \(g : U \to Z\) in \(\text{mod} (R)\) which is not a splittable epimorphism has a factorization through \(Y \to Z\). The translation \(\tau\) is defined in such a way that \(\tau Z \cong X\) when \(X\) and \(Z\) appear as the end terms of the almost split sequence (*). Criteria for \(R\) to be of finite representation type are given. In particular it is shown in Chapter 6 that a connected Artin algebra \(R\) is of finite representation type if and only if the Auslander-Reiten translation quiver \((\Gamma_R, \tau)\) contains a finite connected component. Elementary facts concerning the shapes of connected components of the Auslander-Reiten translation quiver \((\Gamma_R, \tau)\) are presented in Chapter 7.

The notions of an almost split sequence, irreducible morphism and the Auslander-Reiten translation quiver are introduced and discussed in detail in the book mainly in connection with the radical of the category \(\text{mod} (R)\) and with algebras of finite representation type. In particular the representation theory of Nakayama algebras, selfinjective algebras, group algebras, skew group algebras and hereditary algebras are discussed in detail.

In the case \(R\) is hereditary the idea of applying the preprojective modules, the preinjective modules, a valued quiver technique, an associated quadratic form, Weyl group, root system and the Coxeter transformation is presented in Chapter 8. In particular it is proved that a connected hereditary Artin algebra is of finite representation type if and only if the associated valued quiver is a Dynkin diagram. Although some information on regular components of \((\Gamma_R, \tau)\) are included in the case \(R\) is hereditary the only representation-infinite algebra presented in the book in detail together with its representation theory is the hereditary Kronecker algebra.

In Chapter 9 directing modules, short chains, short cycles, sincere modules and modules determined by their composition factors are investigated.

In Chapter 10 stable equivalence of algebras is investigated. In particular algebras with radical square zero and algebras stably equivalent to symmetric Nakayama algebras are studied in detail. We recall that two algebras \(R\) and \(T\) are said to be stably equivalent if their stable module categories \(\underline {\text{mod}}(R)\) and \(\underline {\text{mod}} (T)\) are equivalent, where \(\underline {\text{mod}} (R)\) means the factor category of \(\text{mod}(R)\) modulo the two- sided ideal in \(\text{mod} (R)\) consisting of all homomorphisms that factorize through projective modules.

The book finishes by Chapter 11 on modules determining morphisms and morphisms determined by modules.

The book contains an extensive bibliography. Each chapter is followed by a set of exercises, and by a brief but very informative set of notes and remarks. The book ends with a set of conjectures and open problems.

The book is a well-written and welcome addition to this subject. It does the field a great service by presenting foundational material in a clear, accessible form. It illustrates very well the role of almost split sequences and the Auslander-Reiten technique in the representation theory of Artin algebras. The first part of the book contains an excellent introduction to the basic aspects of the subject.

It is suitable for any mathematician (including graduate students) wanting a brief introduction to this active field. It would make an excellent text for a topics course in the subject.

Reviewer: D.Simson (Toruń)

##### MSC:

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

16G10 | Representations of associative Artinian rings |

16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |

16G20 | Representations of quivers and partially ordered sets |

16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |

16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |

16D90 | Module categories in associative algebras |