Ziegler partial morphisms in additive exact categories

Manuel Cortés-Izurdiaga,Ashish K Srivastava,Pedro A Guil Asensio,Berke Kalebog̃Az

doi:10.1142/s1664360720500125

Abstract

We develop a general theory of partial morphisms in additive exact categories which extends the model theoretic notion introduced by Ziegler in the particular case of pure-exact sequences in the category of modules over a ring. We relate partial morphisms with (co-)phantom morphisms and injective approximations and study the existence of such approximations in these exact categories.

Highlights

We introduce and develop a general theory of partial morphisms in arbitrary additive exact categories, in the sense of Quillen
Partial morphisms were introduced by Ziegler in [22] in his study of Model Theory of Modules, in order to prove the existence of pure-injective envelopes
Recall that a short exact sequence of right modules is called pure if it remains exact upon tensoring by any left module

Summary

Introduction

We introduce and develop a general theory of partial morphisms in arbitrary additive exact categories, in the sense of Quillen. Partial morphisms were introduced by Ziegler in [22] in his study of Model Theory of Modules, in order to prove the existence of pure-injective envelopes. The key idea is that, under quite general assumptions, finding preenvelopes in an exact category with respect to a class X of objects is equivalent to show that there exists enough EX -injectives, where EX is the exact structure consisting of all conflations A → B → C which are Hom(−, X)-exact for every X ∈ X Applying these arguments to Theorem 4.4, we deduce Corollary 5.4, a result which recovers [17, Theorem 2.13(4)]. It is well known that the injective hull u is an injective envelope too, in the sense that any endomorphism f : E → E such that f u = f is an automorphism We compare these notions of small injective extensions with that defined in terms of partial morphisms and prove, in Theorem 3.10, that for nice categories, all of them are equivalent. In Corollary 5.9 we prove that every object in abelian finitely accessible additive category has a pureinjective hull

Preliminaries

Partial Morphisms

Existence of hulls and envelopes

Applications