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Let $A$ be a commutative ring. Let $P$ be a prime ideal of $A$. Let $I$ be an ideal of $A$ such that $I \subset P$. Let $\bar A = A/I$. Let $\bar P = P/I$. Is $\bar A_{\bar P}$ isomorphic to $A_P/IA_P$?

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up vote 3 down vote accepted

Yes. This is because $A_p/IA_p \cong (A-p)^{-1}(A/I)$, and the latter ring is isomorphic to $(A/I)_{p/I}$ because the image of $A-p$ under the projection $A \rightarrow A/I$ is $A/I - p/I$.

One way to observe the first isomorphism is to note that $A_p/IA_p \cong A/I \otimes_A A_p \cong (A-p)^{-1}(A/I)$ (The latter isomorphism follows from the fact that given an $A$-module $M$, $S^{-1}A \otimes_A M \cong S^{-1}M$).

Later edit at OP's request: (To show the canonical map is an isomorphism) The canonical map $\overline{A}_{\overline{p}} \rightarrow A_p/IA_p$ is clearly surjective. Let $\frac{a+I}{s+I}$ be mapped to $0$. Then $\frac{a}{s} \in IA_p$. Then $ta \in I$ for some $t \in A-p$. So, $ta + I = (t+I)(a+I) = 0$, and from this you show that $\frac{a+I}{s+I} = 0$, proving injectivity.

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I would like to know rather a direct proof. How do you prove that the canonical morphism $\bar A_{\bar P} \rightarrow A_P/IA_P$ is an isomorphism? – Makoto Kato Nov 18 '12 at 6:06
What are you having trouble with? – Rankeya Nov 18 '12 at 6:13
The canonical map is clearly surjective. Let $\frac{a+I}{s+I}$ be mapped to $0$. Then $\frac{a}{s} \in IA_p$. Then $ta \in I$ for some $t \in A-p$. So, $ta + I = 0$ which implies that $a+I/1+I = 0$, and from this you show that $a+I/s+I = 0$. – Rankeya Nov 18 '12 at 6:27
If you write it up as another answer, I will accept it. – Makoto Kato Nov 18 '12 at 6:30
I added it to the answer above. Glad to know it helped. – Rankeya Nov 18 '12 at 6:34

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