×

Tame algebras and generic modules. (English) Zbl 0741.16005

We recall that a finite dimensional algebra \(A\) over an algebraically closed field \(K\) is said to be of tame representation type if for any \(d\in\mathbb{N}\) there are finitely many \(K[t]\)-\(A\)-bimodules \(N_ 1,\dots,N_ r\) which are free of rank \(d\) as left \(K[t]\)-modules, and such that every indecomposable right \(A\)-module of dimension \(d\) is isomorphic to \(K[t]/(t-a)\otimes_{K[t]}N_ j\) for some \(1\leq j\leq r\) and \(a\in K\).
In the paper reviewed a definition of tame representation type for an arbitrary ring \(R\) with an identity element is given. The definition coincides with the above one for finite dimensional \(K\)-algebras. Let \(R\) be a ring. If \(G\) is a right \(R\)-module, then \(G\) can be regarded as a left \(\hbox{End}_ R(G)\)-module and its length as a left \(\hbox{End}_ R(G)\)-module is called the endolength of \(G\). The \(R\)-module \(G\) is said to be generic if \(G\) is indecomposable of finite endolength but of infinite length over \(R\). The ring \(R\) is called generically tame if for each \(d\in \mathbb{N}\) there are only finitely many isomorphism classes of generic \(R\)-modules of endolength \(d\).
It is shown in the paper that if \(R\) is a hereditary noetherian prime ring, or \(R\) is a simple noetherian ring, then \(R\) is generically tame. Moreover, if \(R\) is a generically tame noetherian PI-ring then the Krull dimension of \(R\) is at most one. It is also shown in the paper that if \(R\) is a prime Goldie ring then \(R\) has a unique faithful indecomposable module of finite endolength – the restriction of the simple module \(S\) for the simple artinian quotient ring of \(R\). This module is generic if and only if \(R\) is not simple artinian, and it has endolength equal to the uniform dimension of \(R\).
One of the main results of the paper asserts that for a finite dimensional algebra \(A\) over an algebraically closed field \(K\) the following conditions are equivalent:
(a) \(A\) has tame representation type;
(b) \(A\) is generically tame;
(c) \(\hbox{End}_ A(G)/\hbox{rad End}_ A(G)\cong K(t)\) for every generic \(A\)-module \(G\);
(d) \(\hbox{End}_ A(G)\) is a PI-ring for every generic \(A\)-module \(G\).
If \(A\) is of tame representation type, then for any generic \(A\)-module \(G\) the ring \(\hbox{End}_ A(G)\) is split over its radical \(\hbox{rad} \hbox{End}_ A(G)\), and any two splittings are conjugate. Moreover, for each generic \(A\)-module \(G\) there exists a finitely generated localization \(R_ G\) of \(K[t]\) and an \(R_ G\)-\(A\)-bimodule \(N_ G\) such that the following hold.
(1) As a left \(R_ G\)-module, \(N_ G\) is free of rank equal to the endolength of \(G\). If \(Q_ G\) is the quotient field of \(R_ G\), then \(Q_ G\otimes_{R_ G}N_ G\cong G\).
(2) The functor \((-)\otimes_{R_ G}N_ G\) from \(R_ G\)-modules to \(A\)-modules preserves isomorphism classes, indecomposability and Auslander-Reiten sequences.
(3) For each \(d\in \mathbb{N}\), almost all indecomposable \(A\)-modules of dimension \(d\) arise as \(R_ G/(r)\otimes_{R_ G}N_ G\) for some \(G\) and some \(0\neq r\in R_ G\).
It is also shown that the algebra \(A\) is domestic if and only if \(A\) has only finitely many isomorphism classes of generic modules.
Reviewer: D.Simson (Toruń)

MSC:

16G60 Representation type (finite, tame, wild, etc.) of associative algebras
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
PDFBibTeX XMLCite
Full Text: DOI