# Why is Nagao's theorem the “Module theoretic version of Brauer's second main theorem”?

Let $$G$$ be a finite group, $$p\in\mathbb{P}$$ a prime, $$\mathbb{F}$$ an algebraically closed field of characteristic $$p$$, and $$D\leq G$$ a $$p$$-subgroup.

Brauer second main theorem states

If $$\chi\in Irr(\mathbb{C}G)$$ is an ordinary character in the $$p$$-block $$B\in Bl(\mathbb{F}G)$$, $$u\in G_p$$ a $$p$$-element, and $$\phi\in IBr_p(C_G(u))$$ an irreducible Brauer character in the $$p$$-block $$b\in Bl(\mathbb{F}C_G(u))$$, then $$d_{\phi,\chi}^u \neq 0 \implies b^G = B$$

Nagao's theorem states

If $$e\in Z(\mathbb{F}G)$$ is an idempotent, $$M \in \mathbb{F}G\mathsf{-mod}$$ a module with $$eM=M$$, $$H$$ a subgroup with $$C_G(D) \leq H \leq N_G(D)$$, and $$Br_D: Z(\mathbb{F}G) \to Z(\mathbb{F}H)$$ the Brauer homomorphism, then the restriction of $$M$$ to $$H$$ decomposes as $$Res_H^G(M) = Br_D(e)M \oplus M'$$ withsome $$\mathbb{F}H$$-module $$M'$$ whose indecomposable summands all are relatively projective w.r.t. to some $$Q\leq H$$ with $$D\not\leq Q$$.

I have seen in several places, for example in Benson's book, that Nagao's theorem is considered to be "the module theoretic version of Brauer's second main theorem". I struggle to see the similarity. I assume Nagao's theorem (at least) implies the 2nd main theorem, but I am unable to prove it right now. Perhaps I'm missing some crucial insight about the connection between the generalised decomposition numbers and the Brauer homomorphism.

To see the connection, it is easiest to work over a local ring $$R$$ of characteristic zero with residue field $$R/J(R) \cong \mathbb{F}$$ (there are some technicalities I am omitting here for the sake of brevity).

The key point is that if $$B$$ is a $$p$$-block with defect group $$P$$ (I avoid $$D$$ for the name of a defect group since you have used it more generally), and $$x$$ is an element of $$P$$, then Nagao's lemma as you state it (but lifted to to the version over $$R$$) applies to an $$RG$$-module affording irreducible character $$\chi \in B$$ with $$D = \langle x \rangle$$ and with $$H = C_{G}(D)$$, $$e = 1_{B}$$ and with $$M$$ an $$RG$$-module affording $$\chi.$$

The block summands of $${\rm Br}_{D}(e)RH$$ are just ( the lifts of) the Brauer correspondent blocks of $$H$$ for $$B$$.

Nagao's theorem (together with Mackey decomposition) tells us that all indecomposable summands of $$N = (e - {\rm Br}_{D}(e))M$$ (viewed as $$RD$$-module) have vertex strictly less than $$D$$. Then Green's indecomposability theorem tells us that each such indecomposable summand is induced from $$\langle x^{p} \rangle$$ (using the fact that $$D$$ is cyclic).

Now take a $$p$$-regular element $$y$$ of $$H$$. Then the primitive idempotents of $$R\langle y \rangle$$ (there are $$|\langle y \rangle |$$ of these as $$y$$ has order prime to $$p$$) give a decomposition of $$N$$ as a sum of indecomposable $$RD$$-modules such that $$y$$ acts as a scalar on each summand. Each of these still has vertex strictly less than $$D$$,so $$x$$ has trace zero on each of them. Since $$y$$ acts a scalar on each summand, we see that $$xy$$ acts with trace zero on $$N$$.

Hence we see that only the Brauer correspondent blocks of $$B$$ for $$H = C_{G}(x)$$ need to be considered when calculating $$\chi(xy).$$ Since $$y$$ was an arbitrary $$p$$-regular element of $$H = C_{G}(x),$$ we do obtain fairly easily the usual statement of Brauer's second main theorem from this.

Thus it is true that (with a little effort) one can deduce Brauer's second main theorem from Nagao's theorem and "standard" theory, and it is reasonable to consider Nagao's theorem as a (strict) generalization of Brauer's second main theorem.