The notes I follow define a Fermionic coherent state $|c\rangle$ as \begin{equation} \hat{c}|c\rangle=c|c\rangle \end{equation} where $\hat{c}$ is the Fermionic annihilation operator and $c$ is a Grassman number. Then the notes give its explicit form in the Fock occupation space \begin{equation} |c\rangle=e^{\hat{c}^{\dagger}c}|0\rangle \end{equation} Is it really a good form of it? The following calculation seems to contradict the definition : \begin{equation} e^{\hat{c}^{\dagger}c}|0\rangle=|0\rangle+\hat{c}^{\dagger}c|0\rangle=|0\rangle-c\hat{c}^{\dagger}|0\rangle=|0\rangle-c|1\rangle \end{equation} \begin{equation} \hat{c}|c\rangle=\hat{c}\left(|0\rangle-c|1\rangle\right)=0+c\hat{c}|1\rangle=c|0\rangle \end{equation} In the first line, I expanded the exponent using the fact that the square of a Grassman number is 0 or the Pauli exclusion principle (I can't create two fermions in a place). Then I simply acted by the annihilation operator on that state. It doesn't look like an eigenvalue. Is that representation in the Fock space wrong or do I make a mistake somewhere?
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$\newcommand{\ket}[1]{|#1\rangle}$Your definitions and calculation are correct. To prove that $\hat{c}\ket{c}=c\ket{c}$ from where you left you have to use the fact that Grassman numbers verify $c^2=0$: $$\hat{c}\ket{c} = c\ket{0} = c\ket{0} - c^2\ket{1} = c(\ket0 - c\ket1) = c\ket c$$
