The Stacks project

Lemma 33.37.10. Let $A$ be a Noetherian local ring of dimension $1$. Let $L = \prod A_\mathfrak p$ and $I = \bigcap \mathfrak p$ where the product and intersection are over the minimal primes of $A$. Let $f \in L$ be an element of the form $f = i + a$ where $a \in \mathfrak m_ A$ and $i \in IL$. Then some power of $f$ is in the image of $A \to L$.

Proof. Since $A$ is Noetherian we have $I^ t = 0$ for some $t > 0$. Suppose that we know that $f = a + i$ with $i \in I^ kL$. Then $f^ n = a^ n + na^{n - 1}i \bmod I^{k + 1}L$. Hence it suffices to show that $na^{n - 1}i$ is in the image of $I^ k \to I^ kL$ for some $n \gg 0$. To see this, pick a $g \in A$ such that $\mathfrak m_ A = \sqrt{(g)}$ (Algebra, Lemma 10.60.8). Then $L = A_ g$ for example by Algebra, Proposition 10.60.7. On the other hand, there is an $n$ such that $a^ n \in (g)$. Hence we can clear denominators for elements of $L$ by multiplying by a high power of $a$. $\square$

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