construction of an injective resolution

The categoryMathworldPlanetmath of modules has enough injectivesPlanetmathPlanetmath. Let M be a module, and let I0 be an injective moduleMathworldPlanetmath such that


is exact. Then, let M0 be the image of M in I0, and construct the factor module I0/M0. Then, since the category of modules has enough injectives, we can find a module I1 such that


is exact. ϕ0 induces a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ϕ:I0I1, whose kernel is M0. We thus have an exact sequenceMathworldPlanetmathPlanetmathPlanetmathPlanetmath


One can continue this process to construct injective modules In for any n (the resolution may terminate: Im=0 for some N with all m>N).

Title construction of an injective resolution
Canonical name ConstructionOfAnInjectiveResolution
Date of creation 2013-03-22 17:11:02
Last modified on 2013-03-22 17:11:02
Owner guffin (12505)
Last modified by guffin (12505)
Numerical id 5
Author guffin (12505)
Entry type DerivationPlanetmathPlanetmath
Classification msc 16E05