The Stacks project

Remark 15.100.9. The awkwardness in the statement of Lemma 15.100.8 is partly due to the fact that there are no obvious maps between the modules $\mathop{\mathrm{Ext}}\nolimits ^ i_{A_ n}(M_ n, N_ n)$ for varying $n$. What we may conclude from the lemma is that there exists a $c \geq 0$ such that for $m \gg n \gg 0$ there are (canonical) maps

\[ I^ c\mathop{\mathrm{Ext}}\nolimits ^ i_{A_ n}(M_ m, N_ m)/I^ n\mathop{\mathrm{Ext}}\nolimits ^ i_{A_ n}(M_ m, N_ m) \to \mathop{\mathrm{Ext}}\nolimits ^ i_{A_ n}(M_ n, N_ n)/\mathop{\mathrm{Ext}}\nolimits ^ i_{A_ n}(M_ n, N_ n)[I^ c] \]

whose kernel and cokernel are annihilated by $I^ c$. This is the (weak) sense in which we get a system of modules.

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