The Gabriel-Roiter filtration of the Ziegler spectrum. (Q2851205)

The article under review gives a connection between two concepts in the theory of Artin algebras: the Ziegler spectrum and the Gabriel-Roiter measure. The authors study maps from the category \(\text{Mod\,}A\) of modules over an Artin algebra \(A\), to partially ordered sets. For a module \(X\), let \(\text{sub\,}X\) be the full subcategory of finitely presented \(A\)-modules consisting of submodules of finite direct sums of copies of \(X\). Such subcategories are partially ordered by inclusion. They show that the map sending \(X\) to \(\text{sub\,}X\) is universal in the sense that any map (satisfying some natural conditions) from \(\text{Mod\,}A\) to a partially ordered set factors through it.NEWLINENEWLINE The Gabriel-Roiter measure is a map \(\mu\) from \(\text{Mod\,}A\) to the set of subsets of \(\mathbb N\) (lexicographically ordered), and satisfies the natural conditions above. For each \(I\subseteq\mathbb N\), they define \(\text{Zg\,}I\) as the subset of the Ziegler spectrum of \(A\) consisting of modules \(X\) such that \(\mu(X)\leq I\). Then it is shown that these are closed subsets which give a filtration of the Ziegler spectrum.NEWLINENEWLINE In the last section, they give a couple of applications. One is a compactness property for the collection of submodule closed additive subcategories of the category of finitely presented \(A\)-modules. This is then used to give an alternative proof of a recent theorem of \textit{C. M. Ringel} [Bull. Sci. Math. 136, No. 7, 820-830 (2012; Zbl 1272.16008)].