Daniel Simson

Pure semisimple categories and rings of finite representation type

Professor M. Auslander has pointed out to me that the proof of [4, Theorem 2.31 is not correct. In the present note we prove some result weaker than 14, Theorem 2.31. Unfortunately we are not able to decide if [4, Theorem 2.31 is true, although we can prove it in several particular cases (see [5,6]). We keep the terminology and notation of [4]. We recall that a Grothendieck category A is called pure semisimple if every object in A is a coproduct of Noetherian objects. A ring R is right pure semisimple if the category Mod-R of all right R modules is pure semisimple. In [4, Theorem 2.31 we claim that any pure semisimple category A which has only finitely many isomorphism types of simple objects is equivalent to the category of all modules over a ring of finite representation type. We have proved in [4] that this statement is equivalent to the following one:

Posets of finite prinjective type and a class of orders

Abstract The main aim of this paper is to give a characterisation of finite posets J having only finitely many isomorphism classes of indecomposable socle projective K -linear representations over a given field K , or equivalently, finite posets J having only finitely many indecomposable canonical forms of partitioned matrices of the shape (2.12) (with coefficients in K ) with respect to the J -elementary transformations (E 1 ) and (E 2 ) defined in Section 2. The characterisation is given in Theorem 3.1 in terms of the Tits quadratic form associated to J , in terms of a class of algebraic varieties with an algebraic group action, and by presenting a critical list of 114 minimal posets having infinitely many isomorphism classes of indecomposable socle projective representations. An application of posets of finite prinjective type to the study of indecomposable lattices over a class of orders is given.

Partial Coxeter functors and right pure semisimple hereditary rings

An important problem in the representation theory of artinian rings is to obtain a charcterization of rings of finite representation type (see [2, 7, 16, 201). We recall that a ring R is of finite representation type if R is both left and right artinian and there is only a finite number of isomorphism classes of indecomposable finitely generated R-modules. The main tools for studying the representation theory of hereditary artinian rings are partial Coxeter functors and Coxeter functors (see [4, 8, 10, 141). They are successfuly applied in the investigation of hereditary rings of finite representation type [8, 9-12, 141. In the present paper we study partial Coxeter functors for hereditary artinian rings and we use them as a tool for studying right pure semisimple hereditary rings (which are close to rings of finite representation type). We recall that for any ring R the right pure global dimension r.P.gl.dim R and the left pure global dimension l.P.gl.dim R are defined [19-21, 261. There is an interesting problem how to characterize rings R with r.P.gl.dim R = 0 which are exactly those right artinian rings R all of whose right R-modules are algebraically compact [ 19, 20, 261 or, equivalently, all of whose right R-modules are direct sums of modules of finite length. A ring R with r.P.gl.dim R = 0 is called right pure semisimple [25-271. It is well known that any ring of finite representation type is both left and right pure semisimple 12, 20, 23, 251 and the converse implication also holds true [2, 15, 201. However, it is still an open question if a right pure semisimple ring R is of finite representation type or, equivalently, if r.P.gl.dim R = 0 implies l.P.gl.dim R = 0 (see [29]). A positive solution of this problem is given by Auslander [3] for artin algebras. He also discusses the problem for arbitrary left artinian rings (see also 1181). We recall that a 195 0021.8693/81/070195~24

LOCALLY DYNKIN QUIVERS AND HEREDITARY COALGEBRAS WHOSE LEFT COMODULES ARE DIRECT SUMS OF FINITE DIMENSIONAL COMODULES

02.00/0

On right pure semisimple hereditary rings and an Artin problem

ABSTRACT Let C be an indecomposable hereditary K-coalgebra, where K is an algebraically closed field. We prove that every left C-comodule is a direct sum of finite dimensional C-comodules if and only if C is comodule Morita equivalent (see [19]) with a path K-coalgebra , where Q is a pure semisimple locally Dynkin quiver, that is, Q is either a finite quiver whose underlying graph is any of the Dynkin diagrams , , , , , , , or Q is any of the infinite quivers , , , with , shown in Sec. 2. In particular, we get in Corollaries 2.5 and 2.6 a K-coalgebra analogue of Gabriels theorem [11] characterising representation-finite hereditary K-algebras (see also [[6], Sec. VIII.5]). It is shown in Sec. 3 that if , then the Auslander-Reiten quiver of the category of finite dimensional left comodules has at most four connected components, and is connected if and only if Q has no sink vertices and .

Socle reductions and socle projective modules

The pure semisimplicity conjecture (pssR) stated below is studied in the paper mainly for hereditary rings R. One of our main results is Theorem 3.6 containing various conditions which are equivalent to the conjecture (pssR) for hereditary rings R. It follows from our main results together with recent results of Herzog [16] that in order to prove (pssR) for any R it is sufficient (and necessary) to construct an indecomposable module of infinite length over any hereditary ring R of the form (0 GFFMFG), where F, G are division rings and FMG is a simple F-G-bimodule such that dim MG is finite and dimf M is infinite (see Corollary 5.1). Moreover, the existence of a counterexample R to the pure semisimplicity conjecture is equivalent to a generalized Artin problem for division rings (see 4.3–4.6), which is much more difficult than the Artin problem for division ring extensions solved by Cohn in [5] and by Schofield in [20]. It may frighten people of finding an easy solution to the pure semisimplicity problem. On the other hand, it is concluded in Section 5 that studying generalized Artin problems can help solve the pure semisimplicity conjecture.

A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots

A useful and simple idea in determining the representation type of an artinian ring A is to relate its module category mod(A) with a module category of another ring A’ whose representation type is known or it is easier to study. This was successfully used by several authors. Let us mention the Coxeter functors connection [9, 10,3,38], stable equivalences [2, 231, the connection between radical square zero rings and hereditary rings [ 11, 221, connections between representations of algebras and representations of partially ordered sets [ 19, 20, 24, 253, or more generally with representations of BOWS [29], and the covering technique [ 131. An idea close to the one above is to find some subcategory 9 of mod(A ) whose representations type is known and to look at the factor category mod(A)/[W] of mod(A) modulo the two-sided ideal (I&?] in mod(A) generated by maps in SK If mod(A)/[SY] Zmod(A’)/[8] and the representation type of A’ is known then we get a lot of information about mod(A) [2,22,23]. On the other hand there are several situations when mod(~)/[~] is equivalent to a nice subcategory 55’ in some mod(R) and there are constructive methods for studying indecomposable objects in V. It follows from [6-S, 31, 32, 34, 361 that the matrix problems technique and the differentiation procedure [ 19, 201 lead to this scheme. In connection with these ideas the notion of a right peak ring R is introduced and socle projective modules mod,,(R) over R are studied in [31,32f. It is shown there that several factor categories of mod(A), where A is an artinian ring, are of the form mod,,(R), where R is a right peak ring. In particular, if A = ( t “y.‘), where F is a division PI-ring and T is a

Tame prinjective type and Tits form of two-peak posets I☆

Following our paper [Linear Algebra Appl. 433(2010), 699–717], we present a framework and computational tools for the Coxeter spectral classification of finite posets . One of the main motivations for the study is an application of matrix representations of posets in representation theory explained by Drozd [Funct. Anal. Appl. 8(1974), 219–225]. We are mainly interested in a Coxeter spectral classification of posets such that the symmetric Gram matrix is positive semidefinite, where is the incidence matrix of . Following the idea of Drozd mentioned earlier, we associate to its Coxeter matrix , its Coxeter spectrum , a Coxeter polynomial , and a Coxeter number. In case is positive semi-definite, we also associate to a reduced Coxeter number , and the defect homomorphism . In this case, the Coxeter spectrum is a subset of the unit circle and consists of roots of unity. In case is positive semi-definite of corank one, we relate the Coxeter spectral properties of the posets with the Coxeter spectral properties of a simply laced Euclidean diagram associated with . Our aim of the Coxeter spectral analysis of such posets is to answer the question when the Coxeter type of determines its incidence matrix (and, hence, the poset ) uniquely, up to a -congruency. In connection with this question, we also discuss the problem studied by Horn and Sergeichuk [Linear Algebra Appl. 389(2004), 347–353], if for any -invertible matrix , there is such that and is the identity matrix.

Toroidal Algorithms for Mesh Geometries of Root Orbits of the Dynkin Diagram

Abstract The main aim of this paper is to give a simple criterion for a finite poset I with two maximal elements to have the category I -spr of socle projective representations of tame representation type. Our main result is Theorem 1 which asserts that for any upper chain reducible poset I with two maximal elements (see Definition 8) the category I -spr is of tame representation type if and only if the Tits quadratic form q I : Q I → Q (1.1) of I is weakly non-negative, or equivalently, if and only if I does not contain as a peak subposet any of the one-peak posets N 1 ∗ ,…, N 6 ∗ of Nazarova presented in Theorem 1 or any of the 41 two-peak posets listed in Table 1.

\mathbb{D}_4

By applying symbolic and numerical computation and the spectral Coxeter analysis technique of matrix morsifications introduced in our previous paper [Fund. Inform. 1242013], we present a complete algorithmic classification of the rational morsifications and their mesh geometries of root orbits for the Dynkin diagram