The Stacks project

Lemma 15.102.1. Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $K \in D(A)$ be pseudo-coherent. Let $a \in \mathbf{Z}$. Assume that for every finite $A$-module $M$ the modules $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M)$ are $I$-power torsion for $i \geq a$. Then for $i \geq a$ and $M$ finite the system $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M/I^ nM)$ is essentially constant with value

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M) = \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M/I^ nM) \]

Proof. Let $M$ be a finite $A$-module. Since $K$ is pseudo-coherent we see that $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M)$ is a finite $A$-module. Thus for $i \geq a$ it is annihilated by $I^ t$ for some $t \geq 0$. By Lemma 15.101.4 we see that the image of $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, I^ nM) \to \mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M)$ is zero for some $n > 0$. The short exact sequence $0 \to I^ nM \to M \to M/I^ n M \to 0$ gives a long exact sequence

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, I^ nM) \to \mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M) \to \mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M/I^ nM) \to \mathop{\mathrm{Ext}}\nolimits ^{i + 1}_ A(K, I^ nM) \]

The systems $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, I^ nM)$ and $\mathop{\mathrm{Ext}}\nolimits ^{i + 1}_ A(K, I^ nM)$ are essentially constant with value $0$ by what we just said (applied to the finite $A$-modules $I^ mM$). A diagram chase shows $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M/I^ nM)$ is essentially constant with value $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M)$. $\square$

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