Injective object

An object $ I $ in an Abelian category $ C $ such that for any monomorphism $ \alpha : A ^ \prime \rightarrow A $ the mapping

$$ \mathop{\rm Hom} _ {C} ( A , I ) \rightarrow \mathop{\rm Hom} _ {C} ( A ^ \prime , I ) ,\ \ \textrm{ where } \ \phi \mapsto \phi {\circ \alpha } , $$

is surjective. Every injective subobject $ I $ of an object $ A $ is a retract of $ A $. A product of injective objects is an injective object. If every object in $ C $ is isomorphic to a subobject of an injective object in $ C $, one says that $ C $ is a category with enough injective objects (e.g., a Grothendieck category has this property). In such categories an object is injective if and only if it is a direct summand of any object containing it. For the objects of such categories one can construct resolutions consisting of injective objects (injective resolutions). This makes it possible to develop homological algebra in these categories.

In locally Noetherian categories (cf. Topologized category) a direct sum of injective objects is an injective object, and each injective object is isomorphic to a direct sum of indecomposable injective objects; this representation is moreover unique [3]. If $ C $ is the category of modules over a Noetherian commutative ring $ \Lambda $, then the indecomposable injective modules are the injective hulls of the fields of fractions of the quotient rings $ \Lambda / \mathfrak p $, where $ \mathfrak p $ is an arbitrary prime ideal in $ \Lambda $[4].

Examples.

1) The category of Abelian groups has enough injective objects. These objects are the complete (divisible) groups.

2) The category $ C _ \Lambda $ of right $ \Lambda $- modules contains enough injective objects (cf. Injective module).

3) The category of sheaves of modules on a ringed topological space $ ( X , {\mathcal O} _ {X} ) $ contains enough injective objects. Examples of such objects are sheaves $ F $ all stalks $ F _ {x} $ of which are injective $ {\mathcal O} _ {X ,x } $- modules. If $ ( X , {\mathcal O} _ {X} ) $ is a scheme, the converse statement holds for quasi-coherent $ {\mathcal O} _ {X , x } $- modules: Every stalk of an injective sheaf is an injective $ {\mathcal O} _ {X , x } $- module.

References

[1]	I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968)
[2]	A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohôku Math. J. , 9 (1957) pp. 119–221
[3]	P. Gabriel, "Des catégories abéliennes" Bull. Soc. Math. France , 90 (1962) pp. 323–448
[4]	E. Matlis, "Injective modules over Noetherian rings" Pacific. J. Math. , 8 (1958) pp. 511–528

Comments

Injective objects can be studied (and are frequently of importance) in non-Abelian categories. For example, Sikorski's theorem [a1] characterizes complete Boolean algebras as the injective objects in the category of Boolean algebras and Boolean homomorphisms (and the MacNeille completion construction (cf. Completion, MacNeille (of a partially ordered set)) provides injective hulls in this category). In a topos an object is injective if and only if it occurs as a retract of some power-object, and injective objects are used in the study of the associated sheaf functor (cf. [a2]).

References

[a1]	R. Sikorski, "A theorem on extensions of homomorphisms" Ann. Soc. Polon. Math. , 21 (1948) pp. 332–335
[a2]	P.J. Freyd, "Aspects of topoi" Bull. Austral. Math. Soc. , 7 (1972) pp. 1–76