This first part of a two-volume set offers a modern account of the representation theory of finite dimensional associative algebras over an algebraically closed field. The authors present this topic from the perspective of linear representations of finite-oriented graphs (quivers) and homological algebra. The self-contained treatment constitutes an elementary, up-to-date introduction to the subject using, on the one hand, quiver-theoretical techniques and, on the other, tilting theory and integral quadratic forms. Key features include many illustrative examples, plus a large number of end-of-chapter exercises. The detailed proofs make this work suitable both for courses and seminars, and for self-study. The volume will be of great interest to graduate students beginning research in the representation theory of algebras and to mathematicians from other fields.

We develop a technique for the study of K-coalgebras and their representation types by applying a quiver technique and topologically pseudocompact modules over pseudocompact K-algebras in the sense of Gabriel [17], [19]. A definition of tame comodule type and wild comodule type for K-coalgebras over an algebraically closed field K is introduced. Tame and wild coalgebras are studied by means of their finitedimensional subcoalgebras. A weak version of the tame-wild dichotomy theorem of Drozd [13] is proved for a class of K-coalgebras. By applying [17] and [19] it is shown that for any length K-category A there exists a basic K-coalgebra C and an equivalence of categories A ∼ = C-comod. This allows us to define tame representation type and wild representation type for any abelian length K-category. Hereditary coalgebras and path coalgebras KQ of quivers Q are investigated. Tame path coalgebras KQ are completely described in Theorem 9.4 and the following Kcoalgebra analogue of Gabriel's theorem [18] is established in Theorem 9.3. An indecomposable basic hereditary K-coalgebra C is left pure semisimple (that is, every left C-comodule is a direct sum of finite-dimensional C-comodules) if and only if the quiver C Q * opposite to the Gabriel quiver C Q of C is a pure semisimple locally Dynkin quiver (see Section 9) and C is isomorphic to the path K-coalgebra K(C Q). Open questions are formulated in Section 10. 104 D. SIMSON The reader is referred to [11], [28] and [56] for the coalgebra and comodule terminology, to [6], [15], [17] and [34] for the category theory terminology, and to [2], [21] and [44] for the representation theory terminology. Given a ring R with an identity element we denote by J(R) the Jacobson radical of R. We recall that an artinian ring R is said to be connected if R is not decomposable in a product of rings; and R is said to be basic if R/J(R) ∼ = F 1 ×. .. × F m , where F 1 ,. .. , F m are division rings. We denote by Mod(R) the category of all right R-modules and by mod(R) the full subcategory of Mod(R) formed by finitely generated R-modules. If R is a K-algebra, we denote by Mod lf (R) the full subcategory of Mod(R) formed by the locally finite-dimensional R-modules, that is, the modules that are directed unions of finite-dimensional right R-submodules [23]. Given a right R-module M we denote by soc M the socle of M , that is, the sum of all simple R-submodules of M. The concepts of tame comodule type, wild comodule type and the main results of this paper were presented during ICRA-IX at Beijing Normal University in August 2000 (see [48]). 2. Linear topological rings and modules. For convenience of the reader we collect from [3], [16], [17], [25] and [33] some facts on linear topological rings and modules, and on pseudocompact K-algebras and their pseudocompact modules (see also [59]). By a topological ring we mean a ring R equipped with a topology such that addition and multiplication are continuous. A topological ring is said to be right linear topological if R has a basis (of neighborhoods of...

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A composition factors matrix C F is studied for any basic Hom-computable K-coalgebra C over an arbitrary field K, in connection with a Cartan matrix C F of C. Left Euler K-coalgebras C are defined. They are studied by means of an Euler integral bilinear form b C :the Euler characteristic χ C (M, N ) of Euler pairs of C-comodules M and N , and an Euler defect ∂ C : K 0 (C) × K 0 (C) → Z of C. It is shown that b C (lgth M, lgth N) = χ C (M, N ) + ∂ C (M, N ), for all M, N in C-comod, and ∂ C = 0, if all simple C-comodules are of finite injective dimension. A diagrammatic characterisation of representationdirected hereditary Hom-computable coalgebras is given.

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