Proposition 51.21.7. Let $I$ be an ideal of a Noetherian ring $A$. Let $t \geq 0$ be an upper bound on the number of generators of $I$. There exist $N, c \geq 0$ such that for $n \geq N$ the maps

$A/I^ n \to A/I^{n - c}$

satisfy the equivalent conditions of Lemma 51.20.2 with $e = t$.

Proof. Immediate consequence of Lemmas 51.21.6 and 51.20.2. $\square$

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