The endomorphism ring of a localized coherent functor
Let \(C\) be a commutative artinian ring and let \(\Lambda\) be an Artin \(C\)-algebra. The category of finitely presented objects in the category of additive functors from mod-\(\Lambda\) to \(Ab\) is denoted by \(Ab(\Lambda)\) (since the author identifies this with the free abelian category over the ring \(\Lambda)\). It is an open question as to what are the possible values for the Krull-Gabriel dimension of \(Ab(\Lambda)\) (\(0\), \(2\), \(\infty\) are known to occur). The author proves that the value 1 cannot occur. For the case that \(\Lambda\) is a finite-dimensional algebra over an algebraically closed field this has been shown by \textit{H. Krause} [Generic modules over Artin algebras, Proc. Lond. Math. Soc., III. Ser. (to appear)]. Herzog's proof involves showing that if \(S_0\) is the Serre subcategory of \(Ab(\Lambda)\) consisting of the finite length functors and if \(A\in Ab(\Lambda)\) then the endomorphism ring of the image of \(A\) in the localisation of \(Ab(\Lambda)\) at \(S_0\) is a locally Artin \(C\)-algebra. Then he finishes by using results on generic modules (in particular, results from \textit{W. W. Crawley-Boevey} [in ``Representations of algebras and related topics'', Lond. Math. Soc. Lect. Note Ser. 168, 127-184 (1992; Zbl 0805.16028)]).
categories of finitely presented objects
scientific article; zbMATH DE number 1019701
scientific article