Abelian categories over additive ones

Definable additive categories: purity and model theory

“…Such a proof is given in [54, 2.10] (for rings in [56, 1.2]). Also see Adelman's direct construction [2]. We will see later that Ab(A) is the category of pp-imaginaries for left A-modules (and that is the opposite of the category of pp-imaginaries for right A-modules).…”

“…Algebraic compactness refers only to solutions: an algebraically compact object may well not realise every pp-type. A simple example is the 2-adic integers Z (2) , regarded as an abelian group (or as a module over itself or over the localisation of Z at 2; it makes no difference). This is algebraically compact but the pp-type which describes an element divisible by every power of 2 (and which is realised in the elementary extension obtained by adding on a copy of Q as a direct summand) is not realised in Z (2) (though 0 is a solution, it is not a realisation).…”

Embedding categories with exactness into their abelianization

“…On the other hand % does not since the inclusion into '• § must be exact. with F exact (Adelman (1973) 2.1). Hence to show that FE'^FE -> FE" has zero homology, it suffices to show that JE'^> JE -»/£" does.…”

“…Let si be an abelian category and if a Serre class in si (Adelman (1973) 3. Determination of "S by projectives NOTATION 3.1.…”

“…The left adjoint to the above inclusion is constructed in Adelman (1973). We denote the image of (d,%) under the left adjoint by %(si) and the unit of the adjunction by J&.…”