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Remark 64.18.5. If $\mathcal{F} = \left\{ \mathcal{F}_ n\right\} _{n\geq 1}$ is a $\mathbf{Z}_\ell $-sheaf on $X$ and $\bar x$ is a geometric point then $M_ n = \left\{ \mathcal{F}_{n, \bar x}\right\} $ is an inverse system of finite $\mathbf{Z}/\ell ^ n\mathbf{Z}$-modules such that $M_{n+1}\to M_ n$ is surjective and $M_ n = M_{n+1}/\ell ^ n M_{n+1}$. It follows that

\[ M = \mathop{\mathrm{lim}}\nolimits _ n M_ n = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_{n, \bar x} \]

is a finite $\mathbf{Z}_\ell $-module. This follows from Algebra, Lemmas 10.98.2 and 10.96.12 and the fact that $M/\ell M = M_1$ is finite over $\mathbf{F}_\ell $. Therefore, $M\cong \mathbf{Z}_\ell ^{\oplus r} \oplus \oplus _{i = 1}^ t \mathbf{Z}_\ell /\ell ^{e_ i}\mathbf{Z}_\ell $ for some $r, t\geq 0$, $e_ i\geq 1$. The module $M = \mathcal{F}_{\bar x}$ is called the stalk of $\mathcal{F}$ at $\bar x$.

Comments (2)

Comment #8876 by ZW on

I think one can prove that such is -adically complete, and then use induction to show that is finitely generated by . Maybe this is what is meant by Nakayama.

There are also:

  • 2 comment(s) on Section 64.18: On l-adic sheaves

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