Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 4 / 4

For the conjugation action, fixed elements have coefficients constant on $G$-conjugacy classes. Therefore $(\mathcal{O}G{)}^G=Z(\mathcal{O}G)$. More generally an $\mathcal{O}$-basis of $(\mathcal{O}G{)}^H$ consists of sums of the $H$-conjugation orbits on $G$.

For $a\in (\mathcal{O}G{)}^H$ define the relative trace, or transfer, byThe expression is independent of the coset representatives and is $G$-fixed. If $z\in Z(\mathcal{O}G)$, then $z{\mathrm{Tr}}_H^G(a)={\mathrm{Tr}}_H^G(za)$; hence its image is a transfer ideal of conjugation-fixed elements in the center of an associative algebra.

Let $P$ be a Sylow $p$-subgroup of $H$. Trace transitivity gives ${\mathrm{Tr}}_P^G={\mathrm{Tr}}_H^G{\mathrm{Tr}}_P^H$, proving inclusion of the $P$-image in the $H$-image. Conversely $[H:P]$ is a unit of $\mathcal{O}$, and for an $H$-fixed element $a$,Thus ${\mathrm{Tr}}_H^G(a)={\mathrm{Tr}}_P^G(a/[H:P])$, proving the reverse inclusion and

Use the basis of $D$-conjugation orbit sums in $kG$. A singleton orbit is exactly an element of $C_G(D)$, so these basis vectors span the subring $k{C}_G(D)$. For a nonsingleton orbit with representative $x$, its stabilizer is the proper subgroup $E=C_D(x)<D$, and its orbit sum is ${\mathrm{Tr}}_E^D(x)$. Thus every nonsingleton basis vector belongs to the sum of proper-subgroup transfer images.

Conversely the coefficient at a $D$-fixed basis element $x\in C_G(D)$ in ${\mathrm{Tr}}_E^D(a)$ is $[D:E]a_x=0$ in characteristic $p$ whenever $E<D$. Hence the two spans intersect trivially, givingFor $u\in (kG{)}^D$, multiplication by a transfer element satisfies $u{\mathrm{Tr}}_E^D(a)={\mathrm{Tr}}_E^D(ua)$ and ${\mathrm{Tr}}_E^D(a)u={\mathrm{Tr}}_E^D(au)$. Therefore the second summand is a two-sided ideal, as required; it need not be nilpotent in this general setting.

Projection onto the subring in part (b), along the two-sided ideal, is the Brauer homomorphism:It is a unital algebra homomorphism because it is the quotient by that ideal followed by the displayed subring identification.

Put $N=N_G(D)$. The projection is $N$-equivariant, since $N$ preserves the centralizer. Its restricted kernel isEquivalently it is spanned by the sums of the $N$-conjugation orbits on $G$ disjoint from $C_G(D)$. Normality of $D$ in $N$ ensures that an $N$-orbit either lies wholly inside that centralizer or wholly outside it.

Write $N=N_G(D)$, $C=C_G(D)$, and use the commutative transfer idealsThe target fixed algebra is commutative, because $C\le N$ makes every $N$-fixed element of $kC$ central in $kC$.

The Brauer morphism and relative trace identity isIndeed let $D$ act on $G/D$ by left multiplication in the trace sum. Its fixed cosets are exactly $gD$ with $g\in N$; every other orbit has size divisible by $p$. After Brauer projection the contributions on each such nonfixed orbit are equal, so they cancel. Fixed cosets give precisely the right-hand side. Since the Brauer projection of $(kG{)}^D$ is all of $kC$, this proves a surjection $I_D\twoheadrightarrow J_D$.

We use the trace criterion for defect groups: a block idempotent $e$ belongs to $I_D$ exactly when its defect group is conjugate into $D$; among these, ${\mathrm{Br}}_D(e)\ne 0$ exactly when its defect group is $D$ up to conjugacy. The criterion can be read from the class-sum basis: if $E=C_D(x)$ thenThis coefficient is nonzero exactly when $E$ is a Sylow $p$-subgroup of $C_G(x)$. Thus $I_D$ is spanned by classes whose class defect is conjugate into $D$, and the kernel of its Brauer projection is the sum of the corresponding proper-subgroup transfer ideals. The block version follows by refinement of the central idempotents in these ideals.

For clarity, the refinement fact needed here is primitive idempotents under a surjection of commutative Artinian ideals: a surjection between ideals of commutative Artinian algebras bijects the primitive idempotents in the module source whose images are nonzero with the primitive idempotents in the target. Decompose the algebras into local factors. An ideal in a local Artinian factor either contains the identity and is the whole factor, or lies in its nilpotent maximal ideal and has no nonzero idempotents. A surviving local factor has a local quotient, so its identity remains primitive. This proves the stated refinement fact without imposing nilpotence on the entire kernel.

The primitive idempotents in $I_D$ are the block idempotents with defect conjugate into $D$, and the surviving ones have defect $D$. Applying refinement to the surjection above givesThis is the Brauer morphism on a defect transfer ideal; the target is the transfer ideal $J_D$, not the entire fixed center of an associative algebra.

If $N_G(D)\le H\le G$, then $N_H(D)=N_G(D)=N$ and $C_H(D)=C_G(D)=C$. Applying part (d) to $G$ and to $H$ therefore gives the same target idealCompose the two bijections. A block idempotent $e$ of $G$ with defect $D$ corresponds to the unique block idempotent $f$ of $H$ with defect $D$ satisfyingThis proves the requested form of Brauer first main theorem, including uniqueness and the equality of the Brauer images.