Example 63.18.4. Let $X=\mathop{\mathrm{Spec}}(\mathbf{C})$ and $\Phi : \mathbf{Z}_\ell \to \mathbf{Z}_\ell $ be multiplication by $\ell $. More precisely,

\[ \Phi = \left\{ \mathbf{Z}/\ell ^ n\mathbf{Z} \xrightarrow {\ell } \mathbf{Z}/\ell ^ n\mathbf{Z}\right\} _{n \geq 1}. \]

To compute the kernel, we consider the inverse system

\[ \ldots \to \mathbf{Z}/\ell \mathbf{Z}\xrightarrow {0} \mathbf{Z}/\ell \mathbf{Z}\xrightarrow {0}\mathbf{Z}/\ell \mathbf{Z}. \]

Since the images are always zero, $\mathop{\mathrm{Ker}}(\Phi )$ is zero as a system.

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