# Injective object

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An object in an Abelian category such that for any monomorphism the mapping is surjective. Every injective subobject of an object is a retract of . A product of injective objects is an injective object. If every object in is isomorphic to a subobject of an injective object in , one says that 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 . If is the category of modules over a Noetherian commutative ring , then the indecomposable injective modules are the injective hulls of the fields of fractions of the quotient rings , where is an arbitrary prime ideal in .

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### Examples.

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

2) The category of right -modules contains enough injective objects (cf. Injective module).

3) The category of sheaves of modules on a ringed topological space contains enough injective objects. Examples of such objects are sheaves all stalks of which are injective -modules. If is a scheme, the converse statement holds for quasi-coherent -modules: Every stalk of an injective sheaf is an injective -module.

How to Cite This Entry:
Injective object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Injective_object&oldid=15500
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article