Relative projective module

For a subgroup $H\le G$, an $RG$-module $M$ is relatively H-projective when every $RG$-epimorphism onto $M$ that splits after restriction to $H$ already splits over $G$. Equivalently, $M$ is a direct summand of

For $\alpha \in {\mathrm{End}}_{RH}(M)$, the relative trace is

Let a finite group $G$ act on an algebra $A$ by conjugation and let $H\le G$. The transfer ideal from $H$ isThe notation records both the source subgroup and the ambient fixed-point algebra.

D. Higman's criterion says that an $RG$-module $M$ is relative projective module for $H$ exactly when there is $\alpha \in {\mathrm{End}}_{RH}(M)$ satisfying

If $[G:H]$ is invertible in $R$, every $RG$-module is relatively H-projective. An $RG$-module is then projective exactly when its restriction to $RH$ is projective.

A vertex of an indecomposable $kG$-module is a subgroup minimal among those for which the module is relatively projective. Vertices exist, form one conjugacy class, and are p-groups when $k$ has characteristic $p$. The vertices of the trivial module are the Sylow p-subgroups.

Every noncyclic finite finite p-group $P$ has a quotient $C_p\times C_p$. Over any field of characteristic $p$, define a square-zero representation with basis $v_0,\dots ,v_r,w_1,\dots ,w_r$ by $x{v}_i=w_i$ for $1\le i\le r$, $x{v}_0=0$, $y{v}_i=w_{i+1}$ for $0\le i<r$, $y{v}_r=0$, and $xW=yW=0$. Here $x=g-1$, $y=h-1$, and $W$ is the span of the $w_i$. An endomorphism induces the same scalar on the top and on $W$; the remaining maps from the top to $W$ form a square-zero ideal. Thus the endomorphism ring is local and the module is indecomposable. Inflating to $P$ and choosing $p\nmid 2r+1$ gives infinitely many distinct dimensions with vertex $P$: the trace of a relative-trace identity from any proper subgroup would force dimension zero in the field. This construction works over finite fields too.

Let $N_G(D)\le H\le G$. Set $\mathcal{X}=\{D\cap {}^{g}D:g\notin H\}$ and $\mathcal{Y}=\{H\cap {}^{g}D:g\notin H\}$. Green correspondence pairs indecomposable modules with vertex $D$ for $G$ and $H$: restriction has a unique vertex-$D$ summand, with the other summands relatively projective for $\mathcal{Y}$; induction has a unique vertex-$D$ summand, with the other summands relatively projective for $\mathcal{X}$. The two assignments are inverse and preserve sources. Its more general form applies to vertices contained in $D$ but not conjugate into any member of $\mathcal{X}$.

Let $N_G(D)\le H\le G$ and let $U$ be indecomposable with vertex $D$ over $H$. The Mackey restriction formula writes ${\mathrm{Res}}_H^G{\mathrm{Ind}}_H^{G}U=U\oplus T$, where the vertices in $T$ lie in $H\cap {}^{g}D$ for $g\notin H$. None of these subgroups contains an $H$-conjugate of $D$: equality would put $g$ in $N_G(D)H=H$. Every summand of the induction with a $G$-conjugate of vertex $D$ contributes a vertex-$D$ summand on restriction. Krull-Schmidt decomposition therefore forces exactly one such summand, with multiplicity one. Choosing sources and using transitivity of induction proves the inverse correspondence and preservation of sources.

The Green correspondent is the unique indecomposable summand of restriction or induction having the selected vertex of an indecomposable module, in the setting of Green correspondence. The two correspondence maps are inverse and preserve sources of an indecomposable module up to the appropriate normalizer conjugacy.

If $N_G(D)\le H\le G$, Brauer correspondence pairs the blocks of a group algebra of $RG$ and $RH$ having defect group of a block $D$. This is Green correspondence for their bimodules, with vertex of an indecomposable module $\Delta D$ in $G\times G$. The normalizer of $\Delta D$ lies in $H\times H$, and the corresponding restriction and induction have unique summands with that vertex.

For an indecomposable module $M$ with vertex of an indecomposable module $D$, a source is an indecomposable $RD$-module $S$ which is a direct summand of ${\mathrm{Res}}_D^{G}M$ and for which $M$ is a direct summand of ${\mathrm{Ind}}_D^{G}S$. It has vertex $D$ as a $D$-module. Sources at a fixed vertex are conjugate by its normalizer.

Over a modular field or a complete discrete valuation coefficient ring, an indecomposable module has trivial source when a source is the rank-one trivial module of its vertex. These are precisely the indecomposable direct summands of permutation modules. A general trivial source module is a direct sum of these indecomposables.

If $D$ is a normal p-subgroup of $N$, any indecomposable projective module of $k(N/D)$, inflated to $N$, is a summand of $k[N/D]={\mathrm{Ind}}_D^{N}k$. It has trivial source and vertex $D$. Its Brauer quotient of a module at $D$ is the entire module, since $D$ acts trivially and all proper-subgroup relative traces multiply by an index divisible by $p$. This prevents its vertex from being smaller than $D$.

For coset permutation modules, Mackey restriction formula and Frobenius reciprocity identify the homomorphism space with a free module having one orbit-sum basis vector per double coset. The same basis works over the residue field, so reduction of homomorphisms is surjective. Applied to endomorphisms of a permutation module, idempotent refinement theorem lifts each modular direct summand to a trivial source lattice. Lifting inverse isomorphisms between reductions shows uniqueness among trivial source lattices.