Coalgebras, comodules, pseudocompact algebras and tame comodule type

Abstract

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...

Tame comodule type, roiter bocses, and a geometry context for coalgebras

“…Throughout this paper, we freely use the coalgebra representation theory notation and terminology introduced in [2,16,21,22,28]. The reader is referred to [1,8,10,18] for representation theory terminology and notation, and to [3,4,7,9,13] for a background on the representation theory of bocses.…”

“…We recall from [21] and [25] that an arbitrary K-coalgebra C is defined to be of K-wild comodule type (or K-wild, in short) if the category C-comod of finite-dimensional C-comodules is of K-wild representation type [18,21,23] in the sense that there exists an exact K-linear representation imbedding T : modΓ 3 (K) −→ C-comod, where Γ 3 (K) = K K 3 0 K . A K-coalgebra C is defined to be of K-tame comodule type [25] (or K-tame, in short) if the category C-comod of finite-dimensional left C-comodules is of K-tame representation type ( [18], Sec.…”

“…Throughout this paper, we use the terminology and notation introduced in [21,22,28]. We fix a field K. Given a K-coalgebra C, we denote by C-Comod and C-comod the categories of left C-comodules and left C-comodules of finite K-dimension.…”

“…It is clear that lgthM ∈ Z (I C ) if M is of finite K-dimension. We recall from [21] that the map M → lgthM defines a group isomorphism lgth : K 0 (C) − − → Z (I C ) , where K 0 (C) = K 0 (C-comod) is the Grothendieck group of the category C-comod and Z (I C ) is the direct sum of I C copies of Z .…”

Semiartinian Profinite Algebras have Nilpotent Jacobson Radical

“…At the opposite pole, one finds rings whose upper (Jordan) series "terminates" at 0. There are many important situations in both commutative and non-commutative algebra, where category of pseudocompact algebras with continuous morphisms is dual equivalent to that of coalgebras and morphisms of coalgebras; we refer to [S1,DNR,Gb] for details. In general, questions about nil and nilpotence properties of Jacobson radical of a ring are of central interest in ring theory.…”