A generalization of rejection lemma of Drozd-Kirichenko
The paper provides a new efficient tool for the investigation of (not only) representation-finite orders and artinian algebras. For an order \(\Lambda\) over a complete discrete valuation domain, let \(\text{Ind }\Lambda\) denote the set of isomorphism classes of indecomposable \(\Lambda\)-lattices. The rejection lemma of \textit{Yu. A. Drozd} and \textit{V. V. Kirichenko} [Izv. Akad. Nauk SSSR, Ser. Mat. 36, 328-370 (1972; Zbl 0234.13009)] says that for a bijective (i.e. projective and injective in the category \(\Lambda\)-\(\mathbf{lat}\) of \(\Lambda\)-lattices) \(B\in\text{Ind }\Lambda\) with \(B\not\cong\text{Rad }B\), the set \(\text{Ind }\Lambda\setminus\{B\}\) is of the form \(\text{Ind }\Gamma\) for an overorder \(\Gamma\) of \(\Lambda\). In view of this result, the author studies the sets \({\mathcal S}=\text{Ind }\Lambda\setminus\text{Ind }\Gamma\) with \(\Gamma\supset\Lambda\) which he calls rejectable. Note that the rejection lemma extends to the case \(B\cong\text{Rad }B\) if generalized overorders \(\Gamma\) in the sense of Drozd and Kirichenko are allowed, that is, orders \(\Gamma\) connected with \(\Lambda\) by a ring homomorphism \(\Lambda\to\Gamma\) with torsion cokernel. Then every proper subset \({\mathcal S}\subset\text{Ind }\Lambda\) determines the smallest generalized overorder \(\Gamma\) with \({\mathcal S}\subset\text{Ind }\Lambda \setminus\text{Ind }\Gamma\). The author calls \(\mathcal S\) cofaithful whenever this \(\Gamma\) is an overorder of \(\Lambda\). Now assume that the ambient algebra of \(\Lambda\) is semisimple. Generalizing Auslander's proof of the Brauer-Thrall-I conjecture (= Rojter's theorem), the author shows that a rejectable set \(\mathcal S\) with bounded rational rank for all \(E\in{\mathcal S}\) is necessarily finite. The second starting point of his investigation is an observation on Auslander-Reiten sequences for overorders \(\Gamma\supset\Lambda\) given at the end of the paper, namely: if \(M\hookrightarrow E\twoheadrightarrow N\) is an Auslander-Reiten sequence in \(\Lambda\)-\(\mathbf{lat}\) with \(M\in\Gamma\)-\(\mathbf{lat}\), then the map \(\Gamma E\twoheadrightarrow\Gamma N\) is of the form \(p\oplus 1\colon E'\oplus F\to N'\oplus F\) such that \(M\hookrightarrow E'\twoheadrightarrow N'\) is an Auslander-Reiten sequence in \(\Gamma\)-\(\mathbf{lat}\) unless \(M\in\Gamma\)-\(\mathbf{lat}\) is injective. In order to analyse rationally exact sequences like \(M\to\Gamma E\to\Gamma N\), the author considers arbitrary cochain complexes \({\mathbf A}\colon\dots\to A_0\to A_1\to A_2\to\dots\): If \(A_1=A\oplus T\) with \(T\) indecomposable non-injective and \(\partial_1|_T\) not a split monomorphism (i.e. the complex does not split off a complex of the form \(T@>1 >>T\)) then the right Auslander-Reiten sequence \(\mathbf T\) of \(T\) gives rise to a map \(\varepsilon\colon{\mathbf T}\to{\mathbf A}\) which extends to a triangle \({\mathbf T}\to{\mathbf A}\to{\mathbf C}\to{\mathbf T}[1]\), where the mapping cone \(\mathbf C\) of \(\varepsilon\) splits off a complex of the form \(T@>1 >>T\). The author's method consists in an iterated application of this process. This leads to an implicit characterization of finite rejectable sets \({\mathcal S}=\text{Ind }\Lambda\setminus\text{Ind }\Gamma\) in terms of the Auslander-Reiten quiver of \(\Lambda\), restricted to \({\mathcal S}\). As an application, the list of four element rejectable sets (with their truncated Auslander-Reiten structure) is exhibited.
scientific article; zbMATH DE number 1248134