Injective Cogenerator - General Theory

General Theory

In topological language, we try to find covers of unfamiliar objects.

Finding a generator of an abelian category allows one to express every object as a quotient of a direct sum of copies of the generator. Finding a cogenerator allows one to express every object as a subobject of a direct product of copies of the cogenerator. One is often interested in projective generators (even finitely generated projective generators, called progenerators) and minimal injective cogenerators. Both examples above have these extra properties.

The cogenerator Q/Z is quite useful in the study of modules over general rings. If H is a left module over the ring R, one forms the (algebraic) character module H* consisting of all abelian group homomorphisms from H to Q/Z. H* is then a right R-module. Q/Z being a cogenerator says precisely that H* is 0 if and only if H is 0. Even more is true: the * operation takes a homomorphism

f:H → K

to a homomorphism

f*:K* → H*,

and f* is 0 if and only if f is 0. It is thus a faithful contravariant functor from left R-modules to right R-modules.

Every H* is very special in structure: it is pure-injective (also called algebraically compact), which says more or less that solving equations in H* is relatively straightforward. One can often consider a problem after applying the * to simplify matters.

All of this can also be done for continuous modules H: one forms the topological character module of continuous group homomorphisms from H to the circle group R/Z.