Lemma 15.100.7. A morphism (c, \varphi _ n) of the category of Remark 15.100.6 is an isomorphism if and only if there exists a c' \geq 0 such that \mathop{\mathrm{Ker}}(\varphi _ n) and \mathop{\mathrm{Coker}}(\varphi _ n) are I^{c'}-torsion for all n \gg 0.
Proof. We may and do assume c' \geq c and that the \mathop{\mathrm{Ker}}(\varphi _ n) and \mathop{\mathrm{Coker}}(\varphi _ n) are I^{c'}-torsion for all n. For n \geq c' and x \in I^{c'}E'_ n we can choose y \in I^ cE_ n with x = \varphi _ n(y) \bmod E'_ n[I^ c] as \mathop{\mathrm{Coker}}(\varphi _ n) is annihilated by I^{c'}. Set \psi _ n(x) equal to the class of y in E_ n/E_ n[I^{c'}]. For a different choice y' \in I^ cE_ n with x = \varphi _ n(y') \bmod E'_ n[I^ c] the difference y - y' maps to zero in E'_ n/E'_ n[I^ c] and hence is annihilated by I^{c'} in I^ cE_ n. Thus the maps \psi _ n : I^{c'}E'_ n \to E_ n/E_ n[I^{c'}] are well defined. We omit the verification that (c', \psi _ n) is the inverse of (c, \varphi _ n) in the category. \square
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