Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 2 / a

Let $g$ be a p-regular element. Its eigenvalues on $M$ are roots of unity of order prime to $p$. If ${\hat{\lambda }}_1,\dots ,{\hat{\lambda }}_d$ are their Teichmuller lifts to characteristic-zero roots of unity, the Brauer character isIt depends only on the conjugacy class of $g$, is additive in short exact sequences, and equals the restriction of an ordinary character whenever the representation lifts.

For linear independence, choose a splitting p-modular system and let $P_i$ be the projective cover of the simple module $S_i$. A projective lattice lifting $P_i$ has an ordinary character ${\Phi }_i$ that vanishes on p-singular elements. Reduction and ordinary character orthogonality givebecause $S_i$ is the head of $P_i$. Pairing a relation ${\sum }_j{c}_j{\chi }_{S_j}=0$ with every ${\Phi }_i$ yields $c_i=0$ for every $i$. HenceThis is the linear-independence part of the Brauer–Nesbitt theorem.