I'm studying on this notes. I'm finding some difficulties on proposition 12 on page 15. Let me recall what we are trying to prove:
At first we are trying to prove that if inj dim$_{A_p}\;A_p<\infty$ then $\mu_i(p)=0$ if $i
Let $M$ be a finitely generated module on a local ring $A$ and $0\rightarrow M\rightarrow E_0\rightarrow E_1\rightarrow\cdots$ ($d_0:E_0\rightarrow E_1$) a minimal injective resolution. If $f$ is $A$-regular and $M$-regular and if $D=d_0(E_0)$ then we have the following exact sequence:
$0\rightarrow Hom_A(A/fA,D)\rightarrow Hom_A(A/fA,E_1)\rightarrow\cdots$
that is a minimal injective resolution of the $A/fA$-module $Hom_A(A/fA,D)$ that is isomorphic to $M/fM$.
This was the implication i$\Rightarrow$ iv. Any help on this issue?