Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 6 / a

The divisibility in (i) is the standard dimension test for a projective modular representation. If $P$ is a projective $kG$-module and $S$ is a Sylow p-subgroup of order $p^d$, then ${\mathrm{Res}}_S^{G}P$ is projective over $kS$. The group algebra of a p-group in characteristic p is local, so every finitely generated projective $kS$-module is free. ConsequentlyThis will apply to $P=\overline{W}$ in the implication (v)$\Rightarrow$(i).

Assume (ii), and write the ring decomposition asThe natural column module $W\cong {\mathcal{O}}^n$ is projective over $M_n(\mathcal{O})$ by Morita equivalence; extending it by zero across $B$ makes it projective over $\mathcal{O}G$. After extending scalars to $K$, the action factors through $M_n(K)$ on its natural module $K^n$, which is simple. Therefore

Assume (iii). Reduction of the projective lattice $W$ is projective, because a direct-summand decomposition of a free $\mathcal{O}G$-module remains one after tensoring with $k$. Completeness of $\mathcal{O}$ and idempotent lifting show that $\overline{W}$ is indecomposable: otherwise a nontrivial idempotent of ${\mathrm{End}}_{kG}(\overline{W})$ would lift and split $W$, hence split the simple $KG$-module $M$.

Let $\chi$ be the ordinary irreducible character of $M$. Since $W$ is projective, $\chi$ vanishes on p-singular elements and restricts to the projective character $\Phi$ of $\overline{W}$. ThusAn indecomposable projective module that is not simple has, besides its identity, a nonzero noninvertible endomorphism obtained by projecting onto its simple head and embedding the isomorphic simple socle. Hence its endomorphism algebra has dimension at least two. Therefore $\overline{W}$ is simple as well as projective, proving

Under (ii), tensor the direct-product decomposition with the residue field $k$. Since $W$ is free over $\mathcal{O}$,Henceand the reduced action map is the projection onto the first factor. This proves

Assume (v). If the generic fibre $M=K{\otimes }_{\mathcal{O}}W$ had a nonzero proper $KG$-submodule, intersecting it with $W$ and rescaling to obtain a saturated lattice would give a nonzero proper $kG$-submodule of $\overline{W}$. Thus $M$ is simple.

Because $\overline{W}$ is projective, its restriction to a Sylow p-subgroup $S$ is projective. The local algebra $kS$ has only free finitely generated projectives, so $|S|=p^d$ dividesConsequently