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[Edit : I have changed the formulation of the question. Sorry for the trouble]

Here is a stupid question, maybe trivial.

Let $p$ be a prime number, $q = p^n$ where $n$ is an integer, $R = \mathbb{Z}_q$ the unramified extension of $\mathbb{Z}_p$ ($p$-adic integers) of degree $n$.

Let $G$ be a non trivial group. Let $\delta : G \longrightarrow \mathbb{Z}_p^*$ be a character (write $\delta = \bar{\delta} + p a + \ldots $, where $0 \leq \bar{\delta} \leq p-1$ (abuse of notation here !!)).

Assume that I have a 1-cocycle $\bar{\eta} : G \longrightarrow \mathbb{F}_q$ for $\bar{\delta}$ (reduction mod $p$ of $\delta$), i.e. a map such that $\bar{\eta}(\sigma \tau ) = \bar{\delta} (\sigma) \bar{\eta} (\tau) + \bar{\eta}(\sigma)$ $\forall \sigma, \tau \in G$.

Can I find a lift $\eta : G \longrightarrow R$ of $\bar{\eta}$ (no cocycle condition on $\eta$) such that $$ \delta(\sigma) \eta(\tau) - \eta(\sigma \tau) + \eta(\sigma) = p \bar{a}(\sigma) \bar{\eta}(\tau) \mbox{ mod } p^2 ?$$

The answer is yes when we work over $\mathbb{Z}_p$ ($n=1)$), because any element $\bar{x} \in \mathbb{F}_p$ admits a lift of the form $x = x_0 + p^2 x_2 + p^3 x_3 + \ldots $ with $0 \leq x_i \leq p-1$ (note the gap between $p^0$ and $p^2$).

What happens when $n>1$?

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Maybe I'm not understanding the question, but doesn't it suffice to choose $\eta(\sigma)$ arbitrarily and then just define $\eta(\tau) = A\eta(\sigma) - p \bar{a} \bar{\eta}(\sigma)$? –  David Loeffler Jun 14 '12 at 7:12
    
You're right. This was not exactly the question I had in mind : I will edit it and be more precise. –  user33624 Jun 14 '12 at 13:14

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