(injective object)
An object \(I\) of an abelian category \(\mca\) is injective if \(\Hom(\wait, I)\) is an exact contravariant functor from \(\mca\) to abelian groups.
An injective resolution of \(A\) is a complex \(I^\bullet\) of injective objects with a map \(A \to I^0\) such that \[ 0 \to A \to I^0 \to I^1 \to \cdots \] is exact.
If every object of \(\mca\) is isomorphic to a subobject of an injective object, then \(\mca\) has enough injectives.
An object \(I\) is injective if and only if every map \(A \to I\) extends along every monomorphism \(A \injects B\). Over \(\ZZ\) the injective objects are the divisible groups, so \(\QQ\) and \(\QQ/\ZZ\) are injective and no nonzero finitely generated abelian group is.
If \(\mca\) has enough injectives, every object has an injective resolution, so the right derived functors \(R^iF\) of every left exact additive functor \(F\) on \(\mca\) are defined. \(\mods{A}\) has enough injectives for every ring \(A\), and so does \(\mods{\OO_X}\) on any ringed space [@Har10a, Proposition III.2.2]; hence \(H^i(X,\wait)=R^i\Gamma(X,\wait)\) is defined on \(\mods{\OO_X}\). Flasque and other acyclic resolutions also compute the derived functors of global sections.
Dependencies
D-DEFABCATAbelian categoriesusesD-DEFEXACTExactness, and left, right, and exact functorsuses
Backlinks
D-DEFPROJOProjective objects and projective modulesrelated-toD-DEFDERIVDerived functorsuses