Question about a definition of Brauer modular character I would just like to have a more comprehensive view of the following abstract definition of the Brauer character (cf: Gregory Karpilovsky).
$G$ denotes a finite group and $F$ an arbitrary field of prime characteristic $p$. All $FG$-modules are assumed to be finitely generated.
Write $G_{p'}$ for the union of all conjugacy classes of $p'$ elements of $G$. A $p$-regular element is the same as a $p'$-element.
Richard Brauer (1901-1977) found a construction with associates to each $FG$-module $V$, a map:
\begin{equation}
\beta_V: \: \: G_{p'} \longrightarrow R 
\end{equation}
where $R$ is a complete discrete valuation ring of characteristic $0$ with residue field $F$.
Let $(F,R,K)$ be a $p$-modular system, $V$ a $FG$-module, $g \in G_{p'}$ and $H=\langle g \rangle$. 
Let $U$ be a projective $RH$-module. If $\chi_U$ is the $R$-character of $H$ afforded by $U$, then $\beta_V(g)=\chi_U(g)$.
I don't understand this abstract definition of the Brauer character $\beta_V$: indeed, the $R$-valued function $\beta_V$ defined on $G_{p'}$ is called the Brauer character of $G$ afforded by $V$. 
In particular, I don't understand why the values of $\beta_V$ lie in $\mathbb{Z}[\epsilon]$, where $\epsilon$ is a primitive $m$-th root of unity and $m$ is the least common multiple of the orders of $p'$-elements of $G$.
Would you have an example to illustrate this ? I thank you in advance for any suggestions that would make this definition clearer.
 A: Example: Consider the 3-modular system $(F,R,K) = (\mathbb{F}_3, \mathbb{Z}_3, \mathbb{Q}_3)$ and the cyclic group $G = \langle g \rangle \cong C_6$ of order six. There is a two dimensional $F G$-Module $V$ given by the representation $g \mapsto \begin{pmatrix} 1&1\\2&0 \end{pmatrix}$. For $h = g^3$, lets try to compute the value $\beta_V(h) = \beta_V \begin{pmatrix} 2&0\\0&2 \end{pmatrix}$ by using the abstract definition.
For that purpose, we have to consider the subgroup $H = \langle h \rangle \cong C_2$ and the restriction $V_{|H}$ as a $F H$-module. Since $|H| = 2$ does not divide the characteristics of $F$, $V_{|H}$ must be projective by Maschke's theorem, and the general theory guarantees that it has a unique projective lift to a $RH$-module $U$. In general, it might not be easy to find $U$, but in this case we easily see that it has to be the module given by the representation $h \mapsto \begin{pmatrix} -1&0\\0&-1 \end{pmatrix}$. To convince yourself that $U$ is projective, note that $U \cong Re \oplus Re$, where $e = \frac{1}{2}(1-h) \in RH$ is idempotent. At this point, the abstract definition gives us $\beta_V(h) = \chi_U(h) = -2.$

Note that the only reason for choosing a projective lift in the definition above is to guarantee that $\beta_V$ is well defined. For computational purposes, we actually may choose any lift of $V_{|H}$ as a $RH$-lattice! More generally, if $H \leq G$ is any subgroup containing some $p'$-element $h$ such that $V_{|H}$ lifts to some $RH$-lattice $U$, we always have $\beta_V(h) = \chi_U(h)$.
To answer your last question: By the ordinary representation theory (in characteristic 0), we know that $\chi_U(h)$ is the sum of n-th roots of unity, where $n$ is the order of $h$. So the same must be true for $\beta_V(h)$.
