Formally, an Abelian category and an object are selected . Then a long exact sequence is called the form
injective resolution of when all are injective.
existence
In the Abelian category, each object is a sub-object of an injective object, i. H. if there is a monomorphism for every object , where injective is, it is also said, have enough injective objects . An important example of such categories is the category of links modules over a ring .
Under these conditions there is also an injective resolution for every object . First there is a monomorphism , then a monomorphism
and then further by induction .
properties
Is
an injective resolution and
an exact sequence, every homomorphism (not necessarily unique) can be converted into a commutative diagram
complete. An important consequence of this property is that every two injective resolutions of an object are of the same homotopy type .