On Hereditary and Bass Orders

Drozd, Ju A; Kiričenko, V V; Roiter, A. V.

doi:10.1070/im1967v001n06abeh000625

Cited by 23 publications

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“…P 8.2.3. Let us recall a famous lemma below due to Drozd and Kirichenko [DK], on which the theory of Bass orders has been constructed [DKR,R,HN1,HN2]. It can be regarded as a special case of both 8.2.1 and 8.2.2.…”

Section: For Anymentioning

confidence: 99%

“…In [I1], a subset S of ind(lat ) is called rejectable if S = ind(lat )−ind(lat ) for some overorder of . A characterization of rejectable subsets with single elements is known as the Rejection Lemma of Drozd and Kirichenko [DK] (see Section 8.2.3), which is a fundamental of the theory of Bass orders [DKR,R,HN1,HN2]. More generally, a characterization of finite rejectable subsets was given in terms of the AuslanderReiten quiver of in [I1].…”

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τ-Categories II: Nakayama Pairs and Rejective Subcategories

Iyama

2005

Algebr Represent Theor

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We study Nakayama pairs in τ -categories, which are pairs of objects connected by a certain diagram of Auslander-Reiten sequences. Using them, we naturally introduce orderlike τ -categories. Then we study rejective subcategories of τ -categories. A typical example of orderlike τ -categories is given by the category of lattices over an order. In this case, its rejective subcategories correspond bijectively to its overrings. MathematicsSubject Classifications (2000): Primary: 16G30; secondary: 16E65, 16G70, 18E05.For a τ -category C, we introduce the concept of Nakayama pairs and rejective subcategories, which have been studied previously for the category of lattices over an order over a complete discrete valuation ring R.A Nakayama pair is a pair (A, B) of objects connected by an invertible ladder (Section 2.1), which was introduced in [I2] as a certain diagram formed by Auslander-Reiten sequences. When C is the category lat (Section 0.1) of lattices over an R-order of finite representation type, (A, B) is a Nakayama pair for any indecomposable projective -lattice B and the corresponding indecomposable relative injective -lattice A = n + (B), where n + := Hom R (Hom ( , ), R) is the Nakayama functor (Section 3.3). We call a τ -category C orderlike if there exists a 'good' bijection n + : ind + 1 C → ind − 1 C such that (n + (B), B) is a Nakayama pair for any B ∈ ind + 1 C (Section 3.1). Then lat is orderlike if the R-order is of finite representation type. In Part I of this paper, we study Nakayama pairs and orderlike τ -categories.One main result Section 3.5 states that under weak assumptions, a completely graded orderlike τ -category C has an order structure, i.e. there exists an order over a formal power series ring k [[x]] such that C is equivalent to the category pr of finitely generated projective -modules. Moreover, in [I3], we will show that such C is equivalent to lat for some k [[x]]-order by using a homological Current address:

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Section: For Anymentioning

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τ-Categories II: Nakayama Pairs and Rejective Subcategories

Iyama

2005

Algebr Represent Theor

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“…(<=) Every local Bass order is sigma-I from [DKR,12.1] and Theorem 2.15 shows that every triad and special quasi-triad are sigma-I. 0 An appropriate remark is that the sigma-I property is left-right symmetric.…”

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Local orders whose lattices are direct sums of ideals

Haefner¹

1990

Trans. Amer. Math. Soc.

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“…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.…”

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“…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.…”

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Quadratic bimodules and quadratic orders

Iyama

2005

Journal of Algebra

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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...

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On Hereditary and Bass Orders

Cited by 23 publications

References 6 publications

τ-Categories II: Nakayama Pairs and Rejective Subcategories

τ-Categories II: Nakayama Pairs and Rejective Subcategories

Local orders whose lattices are direct sums of ideals

Quadratic bimodules and quadratic orders

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