# 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 ConstructionOfAnInjectiveResolution 2013-03-22 17:11:02 2013-03-22 17:11:02 guffin (12505) guffin (12505) 5 guffin (12505) Derivation msc 16E05