Representation embeddings and the Ziegler spectrum
The right Ziegler spectrum \(Zg_R\) of a ring \(R\) is a topological space whose points are the isomorphism classes of indecomposable pure-injective (right) \(R\)-modules. Given rings \(R\) and \(S\), the author proves (Theorem 7) that every representation embedding from the category \(\text{Mod-}S\) of all right \(S\)-modules to the category \(\text{Mod-}R\) of all right \(R\)-modules induces a homeomorphic embedding of \(Zg_S\) as a closed subset of \(Zg_R\) (a representation embedding from \(\text{Mod-}S\) to \(\text{Mod-}R\) is a functor which is of the form \(-\otimes{_SB_R}\) for some bimodule \(_SB_R\) which is a finitely generated progenerator over \(S\) and which preserves indecomposability and reflects isomorphisms). The aforementioned result is proved by using model theory of modules. It is also shown (Theorem 9) that a functor of the form \(-\otimes{_SB_R}\colon\text{Mod-}S\to\text{Mod-}R\) with \(_SB\) finitely generated and which is of finite endolength as a bimodule induces a continuous map from \(Zg_S\) to \(Zg_R\), whenever it sends indecomposable pure-injectives to indecomposable pure-injectives. Finally, the author considers the question: when is the image of a representation embedding an elementary class? The question is reduced, up to Morita equivalence, to the restriction of scalars functor defined by a ring extension \(R\subseteq S\). The answer is positive under the condition that every pure-injective \(S\)-module is discrete as an \(R\)-module (Proposition 13).
scientific article; zbMATH DE number 955125