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The Setup:
Let $m\geq 1$ be an integer, $\mathbb{F}$ be a finite field of characteristic $p$ and $W(\mathbb{F})$ the ring of Witt-vectors with residue field $\mathbb{F}$ and $\sigma:W(\mathbb{F})\xrightarrow{\sim} W(\mathbb{F})$ a lift of the Frobenius $\bar{\sigma}:\mathbb{F}\xrightarrow{\sim}...