Skip to main content

Postmodern Algebra

Sam Macdonald

Section 4.2 Injective Resolutions

Injective resolutions are analogous to projective resolutions, but now we want to approximate our module \(M\) by injectives.

Definition 4.32. Injective Resolution.

Let \(M\) be an \(R\)-module. An injective resolution of \(M\) is a complex
\begin{equation*} E=0 \longrightarrow E_{0} \longrightarrow E_{1} \longrightarrow E_{2} \longrightarrow \cdots \end{equation*}
with \(\mathrm{H}_{0}(E)=M\) and \(\mathrm{H}_{n}(E)=0\) for all \(n \neq 0\text{.}\) We may abuse notation and instead say that an injective resolution of \(M\) is an exact sequence
\begin{equation*} 0 \longrightarrow M \longrightarrow E_{0} \longrightarrow E_{1} \longrightarrow E_{2} \longrightarrow \cdots . \end{equation*}

Remark 4.33.

This is the first example we have encountered where we have a cocomplex rather than a complex. Its homology should technically be referred to as cohomology, and written with superscripts:
We can construct injective resolutions in a similar fashion to how we constructed projective resolutions.

Proof.

By Theorem 3.32, every \(R\)-module embeds into an injective module. So we start by taking an injective \(R\)-module \(E_{0}\) containing \(M\text{,}\) and look at the cokernel of the inclusion:
\begin{equation*} 0 \longrightarrow M \stackrel{i_{0}}{\longrightarrow} E_{0} \stackrel{\pi_{0}}{\longrightarrow} \text { coker } i_{0} \longrightarrow 0 \text {. } \end{equation*}
Now coker \(i_{0}\) includes in some other injective module \(E_{1}\text{.}\)
Take \(\partial_{0}:=i_{1} \pi_{0}\text{.}\) Since \(i_{1}\) is injective,
\begin{equation*} \operatorname{ker} \partial_{0}=\operatorname{ker}\left(i_{1} \pi_{0}\right)=\operatorname{ker} \pi_{0}=\operatorname{im} i_{0} \end{equation*}
Notice also that coker \(i_{0}=\operatorname{im} \partial_{0}=\operatorname{ker}\left(E_{1} \rightarrow \operatorname{coker} \partial_{0}\right)\text{.}\) So we can now we continue in a similar fashion, by finding an injective module \(E_{2}\) that coker \(\partial_{0}\) embeds into.
By construction and since \(i_{2}\) is injective, \(\operatorname{ker} \partial_{1}=\operatorname{im} \partial_{0}\text{,}\) and our complex is exact at \(E_{1}\text{.}\) The process continues analogously.
We can again define a minimal injective resolution for \(M\) as one where at each step we take the smallest injective module that coker \(i_{n}\) embeds into; this is called the injective hull of \(M\text{.}\) Perhaps unsurprisingly, one can show that the minimal injective resolution of a finitely generated module over a local ring is unique up to isomorphism. The analogues to the betti numbers are called Bass numbers, although now there are some major differences. When we construct a minimal free resolution, we have only to count copies of \(R\) in each homological degree, while there are many different building clocks for injective modules - the injective hulls of \(R / P\text{,}\) where \(P\) ranges over the prime ideals in \(R\text{.}\) So for each homological degree \(i\text{,}\) we get one bass number for each prime ideal \(P\text{.}\)

Example 4.35.

Let’s construct a minimal free resolution for the abelian group \(\mathbb{Z}\text{.}\) We start by including \(\mathbb{Z}\) in \(\mathbb{Q}\text{,}\) and then note that the cokernel \(\mathbb{Q} / \mathbb{Z}\) is actually injective, by Theorem 3.27 and Lemma 3.22. So \(\mathbb{Q} / \mathbb{Z}\) embeds in itself, and our resolution stops there. So the short exact sequence
\begin{equation*} 0 \longrightarrow \mathbb{Z} \longrightarrow \mathbb{Q} \longrightarrow \mathbb{Q} / \mathbb{Z} \longrightarrow 0 \end{equation*}
is in fact a minimal injective resolution for \(\mathbb{Z}\text{.}\)
PrevTopNext