construction of an injective resolution

The category of modules has enough injectives. Let $M$ be a module, and let $I^{0}$ be an injective module such that

0\longrightarrow M\longrightarrow I^{0}

is exact. Then, let $M_{0}$ be the image of $M$ in $I^{0}$ , and construct the factor module $I^{0}/M^{0}$ . Then, since the category of modules has enough injectives, we can find a module $I^{1}$ such that

0\longrightarrow I^{0}/M^{0}\lx@stackrel{{\scriptstyle\phi_{0}}}{{% \longrightarrow}}I^{1}

is exact. $\phi_{0}$ induces a homomorphism $\phi\!:\!I^{0}\longrightarrow I^{1}$ , whose kernel is $M^{0}$ . We thus have an exact sequence

0\longrightarrow M\longrightarrow I^{0}\longrightarrow I^{1}.

One can continue this process to construct injective modules $I^{n}$ for any $n\in\mathbb{Z}$ (the resolution may terminate: $I^{m}=0$ for some $N\in\mathbb{Z}$ with all $m>N$ ).

Title	construction of an injective resolution
Canonical name	ConstructionOfAnInjectiveResolution
Date of creation	2013-03-22 17:11:02
Last modified on	2013-03-22 17:11:02
Owner	guffin (12505)
Last modified by	guffin (12505)
Numerical id	5
Author	guffin (12505)
Entry type	Derivation
Classification	msc 16E05