On the Krull-Gabriel dimension of an algebra
Let \(A\) be a \(k\)-algebra where \(k\) is an algebraically closed field. Let \(\mathcal C\) denote the category of finitely presented additive functors from \(\text{mod-}A\), the category of finite-dimensional \(A\)-modules, to the category of Abelian groups. Form the quotient category \({\mathcal C}/{\mathcal C}_0\) of \(\mathcal C\) by the subcategory, \({\mathcal C}_0\), of functors of finite length. Repeat the process and define the Krull-Gabriel dimension of \(A\), \(\text{KG}(A)\), to be \(n\) if the trivial category is reached after \(n\) steps [\textit{W. Geigle}, Manuscr. Math. 54, 83-106 (1985; Zbl 0593.16022)]. By \textit{M. Auslander} [in Representations of algebras, Lect. Notes Math. 944, 105-179 (1980; Zbl 0486.16027)] \(\text{KG}(A)=0\) exactly if \(A\) is of finite representation type and, by \textit{H. Krause} [Proc. Lond. Math. Soc., III. Ser. 76, No. 2, 276-306 (1998; Zbl 0908.16016)] and \textit{I. Herzog} [J. Algebra 191, No. 1, 416-426 (1997; Zbl 0873.16009)] (for general Artin algebras), \(\text{KG}(A)\) is never 1 for such algebras. Examples with \(\text{KG}(A)\) equal to 2 and to \(\infty\) were known but it was an open problem whether any other values were possible for Artin algebras. The author proves that for every integer \(n\geq 2\) there is a finite-dimensional algebra \(A\) with \(\text{KG}(A)=n\). The proofs depend on the methods developed in [\textit{J. Schröer}, Hammocks for string algebras, SFB 343 preprint no. 97-010, Universität Bielefeld (1997)]. An alternative route, via computing the Cantor-Bendixson rank of the Ziegler spectrum of \(A\), is taken in [\textit{K. Burke} and \textit{M. Prest}, The Ziegler and Zariski spectra of some domestic string algebras, Algebr. Represent. Theory (to appear)]. There are still open problems concerning the Krull-Gabriel dimension of finite-dimensional algebras. Does the condition \(\text{KG}(A)<\infty\) characterize algebras \(A\) of domestic representation type? Is the condition \(\text{KG}(A)<\infty\) equivalent to \(\text{KG}(A)<\omega\)?
categories of finitely presented additive functors
categories of finite-dimensional modules
scientific article; zbMATH DE number 1419359