Nagao decomposition for restriction to a centralizer

Since $H\le N_G(D)$, the group $D$ is normal in $H$. Hence $D\le O_p(H)$, and F6 shows that every defect group $Q$ of every block $c$ of $kH$ contains $D$. It follows that $C_G(Q)\le C_G(D)\le H\mathrm{.}$ Thus F5 defines $c^G$ for every block $c$ of $kH$.

We record the block corner that controls an error component. Write $b$ for the idempotent of the global block $B$, and let ${\pi }_H:kG\to kH$ delete coefficients outside $H$. For a block $c$ of $kH$, the maps $kHc\hookrightarrow kG,a⟼c{\pi }_H(a):kG⟶kHc$ split the identity $H$-double-coset copy of the block bimodule $kHc$. The corner of left multiplication by $b$ on this copy is left multiplication by $c{\pi }_H(b)$. If that corner were a unit in ${\mathrm{End}}_{H\times H}(kHc)=Z(kHc)$, normalizing the second map by its inverse would split $kHc$ from the restriction of the global block $kGb$. By F5 this would imply $c^G=B$. Therefore, when $c^G\mathrm{\ne }B$, the element $c{\pi }_H(b)$ is a nonunit of the local algebra $Z(kHc)$ and hence is nilpotent.

The coefficients of the central element $\widehat{B}$ are constant on $G$-conjugacy classes, hence on $H$-conjugacy classes. The complement of $H$ in $G$ is $H$-conjugation invariant, so for suitable coefficients $a_x\in \mathcal{O}$ and representatives $x$ of its finitely many $H$-classes, $\widehat{B}-{\pi }_H(\widehat{B})={\sum }_{x\in (G\setminus H)\mathrm{/}H\text{-conj}}{a}_x{\mathrm{Tr}}_{C_H(x)}^H(x)\mathrm{.}$ Here the trace is for the conjugation action: its summands are exactly the elements of the $H$-class of $x$. Moreover, $D\le ̸C_H(x),$ because the opposite containment would put $x$ in $C_G(D)\le H$.

Let $\widehat{c}$ denote the lift of $c$. The lifts are pairwise orthogonal and sum to $1$: their products and the difference between their sum and $1$ are central idempotents reducing respectively to $0$ and $0$, so F1's uniqueness forces those idempotents to vanish. Consequently $M={⨁}_c\widehat{c}M\mathrm{.}$ Define $M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}={⨁}_{c:c^G=B}\widehat{c}M,M_{\mathrm{e}\mathrm{r}\mathrm{r}}={⨁}_{c:c^G\mathrm{\ne }B}\widehat{c}M\mathrm{.}$ F2 decomposes each component into indecomposables, and the first asserted property follows directly from the definition.

Let $U$ be an indecomposable summand of $\widehat{c}M$ for a block $c$ with $c^G\mathrm{\ne }B$, and choose $H$-linear split maps $i:U\to M$ and $r:M\to U$. Put $E={\mathrm{End}}_{\mathcal{O}H}(U)$, which is local by F2. Integral coefficient truncation ${\pi }_H:\mathcal{O}G\to \mathcal{O}H$ commutes with reduction. Thus step 1.2 shows that the reduction of $\widehat{c}{\pi }_H(\widehat{B})$ is nilpotent. Some power of this element therefore belongs to $\mathfrak{m}\mathcal{O}H$, where $\mathfrak{m}$ is the maximal ideal of $\mathcal{O}$. Since $\widehat{c}$ acts as the identity on $U$ and the central element ${\pi }_H(\widehat{B})$ commutes with the $H$-projection $ir$, it follows that $s=r{\pi }_H(\widehat{B})i\in E$ has a power in $\mathfrak{m}E$. Such a power cannot be a unit, so $s$ is a nonunit and belongs to $J(E)$.

The equality $\widehat{B}M=M$ and the $H$-linearity of $i,r$ now give inside $E$ $1_U=s+{\sum }_{x\in (G\setminus H)\mathrm{/}H\text{-conj}}{\mathrm{Tr}}_{C_H(x)}^H\left(a_{x}rxi\right)\mathrm{.}$ Indeed, $a_{x}rxi$ is $C_H(x)$-linear, and taking the $H$-linear corner commutes with each finite relative trace. The first term lies in $J(E)$ by step 2.2. If every displayed trace term were a nonunit, their finite sum would also lie in the maximal ideal $J(E)$, contradicting the equality. Hence one trace term is a unit. For $\beta \in E$ and any $C_H(x)$-endomorphism $\alpha$ of $U$, one has $\beta {\mathrm{Tr}}_{C_H(x)}^H(\alpha )={\mathrm{Tr}}_{C_H(x)}^H(\beta \alpha )$ and ${\mathrm{Tr}}_{C_H(x)}^H(\alpha )\beta ={\mathrm{Tr}}_{C_H(x)}^H(\alpha \beta )$. Thus the image of this relative trace is a two-sided ideal of $E$; since it contains a unit, it contains $1_U$. Higman's criterion makes $U$ relatively $C_H(x)$-projective for the corresponding $x$.

Let $P$ be any vertex of $U$. F3 gives $P\le h{C}_H(x)h^{-1}$ for some $h\in H$. Were $D\le P$, normality of $D$ in $H$ would imply $D=h^{-1}Dh\le C_H(x),$ contrary to step 1.3. Thus $D\le ̸P$, proving the second property. If $D=1$, the hypotheses force $H=G$, so the outside-class sum is empty and all error components are zero; if $M=0$, both conclusions are vacuous. These also cover all boundary cases without an empty-sum inference.