Quadratic bimodules and quadratic orders

Abstract

In some branches of mathematics, especially the representation theory of Artin algebras and orders, it has been an important problem to study the (geometric) quotient GL n (Λ)\M n (M) for a (Λ, Λ)-module M (g · m := gmg −1 for g ∈ GL n (Λ) and m ∈ M n (M)). An effective formulation of such problems is a bimodule problem [4,9], which is a study of the matrix category Mat(C, M) associated to an additive category C and a (C, C)-module M (Section 2.1). The bimodule problems have nice module-theoretical interpretation by means of prinjective modules [22,29,30]. They have already many useful applications in the representation theory of finite-dimensional algebras, artinian rings, orders, vector space categories, and bocses, see [6,7,9,16,19,25,28,29].In this paper, we introduce the concept of quadratic extensions of rings (Section 1.2), which are closely related to many rings known in representation theory, for example Bass orders [10,17], Bäckström orders [28], Green orders [27], special biserial algebras [31], and clannish algebras [5] (see Sections 1.3, 3.4, and 3.5). Using quadratic extensions, we introduce the concept of quadratic bimodules (Section 3.1), where we can easily formulate self-reproducing systems [21] and clans [5] in terms of quadratic bimodules. Then we study their matrix category Mat(C, M) associated to a quadratic (C, C)-module M, especially we give a description of all indecomposable objects in terms of walks (Sections 3.3, 6.1), which can be regarded as a generalization of Green walks [13] (Section 3.4), strings and bands [5,31] (Section 3.5). As an application, we give a description of finitely generated indecomposable modules over quadratic orders (Section 1.3(2)) in Section 7.3. In forthcoming papers, our results will be applied in construction of large new classes of orders of tame representation type. Some of the results of this paper were presented at the Conference and Workshop Constanta 2000 and were announced in [19] without proof.In a word, our method is an application of the categorical reduction to quadratic bimodules. One advantage of our categorical method is that we do not need base algebraically closed fields. Thus we can apply our results to many important examples like algebras over finite fields, orders over p-adic integers, and so on. One basic tool is a quadratic reduction functor (Section 5.2) which reduces the matrix category of a given quadratic bimodule to that of a certain new quadratic bimodule. The reduction of self-reproducing systems in [21] can be regarded as a typical example of our reduction. Although our reduction is in the same philosophy of the well-known reduction of bocses and arbitrary bimodule problems [3,4,9], it gives us much more information thanks to the fact that quadratic bimodules are closed under quadratic reductions. Another basic tool is a singular functor − : Mat(C, M) → D (Section 2.3), which can be defined quite naturally also for large classes of bimodule problems. It is a crucial result that the singular functor gives a universal composition...