Characteristic classes (purities) of short exact sequences in the category of Abelian groups (Q1820246)

If \({\mathcal R}\) is a proper class of short exact sequences of abelian groups and \(\hat {\mathcal R}\) is the class of short exact sequences E with nE\(\in {\mathcal R}\) for some integer \(n\neq 0\), then \(\hat {\mathcal R}\) is a proper class (Th.1). For arbitrary classes \({\mathcal P}\) and \({\mathcal T}\) of modules the symbol i(\({\mathcal P},{\mathcal T})\) denotes the smallest proper class for which all modules from \({\mathcal P}\) are co-projective and all modules from \({\mathcal T}\) are co-injective. If \({\mathcal P}\) and \({\mathcal T}\) are arbitrary classes of abelian groups and \({\mathcal B}\) is the class of all bounded groups, then \(\hat {\i}({\mathcal P},{\mathcal T})=i({\mathcal P}\cup {\mathcal B},{\mathcal T}\cup {\mathcal B})\) (Th.2). The rest of the paper is devoted to the study of the global dimension of proper classes. Thus, gl.dim \(\hat {\mathcal R}\leq \max (2,gl.\dim {\mathcal R})\) for every proper class \({\mathcal R}\) (Th.4) and if \({\mathcal P}\) and \({\mathcal T}\) are arbitrary classes of modules over a hereditary ring, then gl.dim i(\({\mathcal P},{\mathcal T})\leq 2\) (Th.6).