Finite presentation and purity in categories
نویسنده
چکیده
For any module M over an associative ring R, let σ[M ] denote the smallest Grothendieck subcatgory of Mod-R containingM . If σ[M ] is locally finitely presented the notions of purity and pure injectivity are defined in σ[M ]. In this paper the relationship between these notions and the corresponding notions defined in Mod-R are investigated, and the connection between the resulting Ziegler spectra is discussed. An example is given of an M such that σ[M ] does not contain any nonzero finitely presented objects. 1 Local finite presentation of categories σ[M ] Given an R-module M , let σ[M ] denote the category subgenerated by M [15]. This is the smallest Grothendieck subcatgory of Mod-R containing M . We say that a category C is locally finitely presented, lfp, if it has a set of finitely presented objects such that every object of C is a direct limit of copies of objects from this set. Recall that the object C of C is finitely presented if the functor (C,−) commutes with direct limits. This can be characterized by the fact that the kernel of any epimorphism X → C in C is finitely generated provided that X is finitely generated. Locally finitely presented abelian categories are Grothendieck [2] and they share many properties with module categories. In general σ[M ] need not be locally finitely presented although it is easy to see (e.g. [1, 1.70]) that it is locally α-presentable for
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