1
$\begingroup$

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 and 1 if $i=ht(p)$ ($\mu_i(p)$ is the Bass number $\mu_i(A,p)$). Here is how the proof begins: we argue by inductionon the Krull dimension of $A_p$. If it is 0 we are good. Suppose it's greater than 0, let $f$ be an $A_p$-regular element then $A/fA$ has finite injective dimension on itself. This is what I don't understand, why inj dim$_{A/fA}(A/fA)<\infty$? The notes claim that they are using the following property:

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?

  • 0
    It seems that you want to study the Gorentein rings. I suggest you to use the original Bass' paper which is very well written.2012-10-08

1 Answers 1

1

In fact, one uses a change of rings theorem for injective dimension. In the book of I. Kaplansky, Commutative Rings, this is Theorem 205 and it is called "Second theorem on injective change of rings". A slightly different proof of this theorem is given by the proof of Proposition 6 on page 9 of your notes and this says, in particular, the following: if $f$ is $A$-regular and $M$-regular, then $\mathrm{injdim}_AM<\infty$ implies $\mathrm{injdim}_{A/fA}M/fM<\infty$.

  • 0
    In my opinion you must reduce the proof to the local case. It is the most natural thing to do and Bass does it. I also think that in your notes the removing of $\mathfrak{p}$ from $A_{\mathfrak{p}}$ at a moment is a typo.2012-10-09