I’m reading this pdf on dimension theory in commutative algebra: https://faculty.math.illinois.edu/~r-ash/ComAlg/ComAlg5.pdf
and having trouble in reading the proof of Corollary 5.4.8 which says I can immediately prove this corollary from the theorem above (i.e. Theorem 5.4.7), but I really can’t see how the author proved the corollary with the theorem.
First, in the proof, it says “replace $R$ by $R_P$ and $R/(a)$ by $(R_P)_Q$” but I don’t see why I can replace $R/(a)$ by $(R_P)_Q$ instead of $(R_P)/(a)$.
And also, secondly, I really can’t figure out how the author derived the formula $\text{ht} P -1$with the theorem.
Can anyone help me with this? Can anyone give me some more details? Thanks in advance.
$\textbf{My attempts}$
I could derive $\text{dim} R_P/(a)=\text{coht} (a)=\text{coht} Q$, $\text{ht} P/(a)=\text{dim} {R/(a)}_{P/(a)}$ and also $\text{coht} (a) = \text{coht} Q, \text{ht} (a) = \text{ht} Q$ direclty from the definitions but I don’t know this would complete the proof. Can I use the fact localization and quotient commute?
$\textbf{Theorem and corollary}$
Theorem 5.4.7 : Let $R$ be a Noetherian local ring with maximal ideal $m$ and let $a\in m$ be a non zero divisor. Then, $\text{dim} R/(a)=\text{dim} R-1$.
Corollary 5.4.8 : Let $a$ be a non zero divisor belonging to the prime ideal $P$ of the Noetherian ring $R$. Then $\text{ht} P/(a)=\text{ht} P-1$.
proof : In (5.4.7), replace $R$ by $R_P$ and $R/(a)$ by $(R_P)_Q$, where $Q$ is a minimal prime ideal over $(a)$.