Cohomology on the centric orbit category of a fusion system

Let $k$ be a commutative ring with identity. If $G$ is a finite group, $H$ is a subgroup of $G$, and $M$ is a $kG$-module, then we use the usual notation $\mathrm{tr}_{H}^{G}\colon M^{H}\to M^{G}$ for fixed points and the relative trace map. If $p$ is a rational prime which is zero in $k$, then the relative trace is zero in cases where there is an element of $G$ outside $H$ but normalizing $H$ and acting with small nilpotence degree on $M$. We state this when $k=\mathbb{F}_{p}$, the only case we need.

In this paper, if $X$ and $Y$ are two subsets of a group $G$, we write $[X,Y]$ for the subgroup of $G$ generated by the set of commutators $[x,y]=xyx^{-1}y^{-1}$ with $x\in X$ and $y\in Y$. Our iterated commutators are right-associated: set $[X,Y;1]=[X,Y]$, and inductively $[X,Y;i]=[X,[X,Y;i-1]]$ for $i\geq 2$. If $G$ has a left action on some module $V$, the notation $[X,V;i]$ should be interpreted in the semidirect product of $V$ by $G$, in which case $[X,V;i]$ is a subspace of $V$, and $[x,v;i]=(x-1)^{i}v$ for all $x\in G$ and $v\in V$.

The techniques we have generally take advantage of situations in a finite $p$-group in which some subgroup $G$ normalizes another subgroup $P$ and acts with small nilpotence degree on it, usually action which is quadratic (or trivial): $[G,G,P]=1$. Then the following lemma of Miyamoto [Miy81, Lemma 2] provides a bound on the nilpotence degree of the action of $G$ on $H^{j}(P,A)$ that is linear in $j$ when $A$ is finite abelian.

The notation we use for fusion systems follows [AKO11]. We apply morphisms from right to left. Let $\mathcal{F}$ be a saturated fusion system on a finite $p$-group $S$. A subgroup $Q$ of $S$ is fully $\mathcal{F}$-normalized (respectively, fully $\mathcal{F}$-centralized) if $|N_{S}(Q)|\geq|N_{S}(Q^{\prime})|$ (respectively, $|C_{S}(Q)|\geq|C_{S}(Q^{\prime})|$) for each conjugate $Q^{\prime}$ of $Q$ in $\mathcal{F}$, i.e., for each subgroup of the form $\varphi(Q)$ with $\varphi\in\operatorname{Hom}_{\mathcal{F}}(Q,S)$. A subgroup $Q$ of $S$ is $\mathcal{F}$-centric if $C_{S}(Q^{\prime})\leq Q^{\prime}$, i.e. $C_{S}(Q^{\prime})=Z(Q^{\prime})$, for each $\mathcal{F}$-conjugate $Q^{\prime}$ of $Q$. The symbol $\mathcal{F}^{c}$ denotes the set of $\mathcal{F}$-centric subgroups, and also the full subcategory with the same objects. Note that a subgroup of $S$ is $\mathcal{F}$-centric if and only if it is fully centralized and contains its centralizer in $S$ [AKO11, Definition I.3.1]. By one of the axioms for saturation, a fully normalized subgroup $P\leq S$ is also fully centralized and fully automized: $\operatorname{Aut}_{S}(P)$ is a Sylow $p$-subgroup of $\operatorname{Aut}_{\mathcal{F}}(P)$ [AKO11, Proposition I.2.5].

For each pair of subgroups $P,Q\leq S$, $\operatorname{Inn}(Q)$ acts on $\operatorname{Hom}_{\mathcal{F}}(P,Q)$ by left composition. The orbit category $\mathcal{O}(\mathcal{F})$ of $\mathcal{F}$ is the category with the same objects as $\mathcal{F}$ and with morphism sets

the orbits under this action. If $\mathcal{X}$ is a collection of subgroups of $S$ which is closed under $\mathcal{F}$-conjugacy and also closed under passing to overgroups in $S$, then we abuse notation by using $\mathcal{X}$ also for the full subcategory of $\mathcal{F}$ with object set $\mathcal{X}$, and we write $\mathcal{O}(\mathcal{X})$ for the corresponding orbit category. Other than the full orbit category itself, we will only need to work with the centric orbit category, the case $\mathcal{X}=\mathcal{F}^{c}$.

Since the morphisms in a fusion system model conjugation of $p$-subgroups in a finite group, it is natural that there is a notion of Mackey functor for fusion systems. We first want to recall from [DP15, Section 2] the definition of a Mackey functor in this setting in the form that is most useful later. Let $k$ be a commutative ring with identity. Let $M=(M^{*},M_{*})$ be a pair of functors from $\mathcal{O}(\mathcal{F})$ to $k\text{-}\textsf{mod}$ with $M^{*}$ contravariant and $M_{*}$ covariant. Set $\mathrm{r}_{P}^{Q}=M^{*}([\iota_{P}^{Q}])$, $\mathrm{t}_{P}^{Q}=M_{*}([\iota_{P}^{Q}])$, and $\mathrm{iso}([\varphi])=M_{*}([\varphi])$ for each $P\leq Q\leq S$ and each isomorphism $[\varphi]\colon P\to\varphi(P)$ in $\mathcal{O}(\mathcal{F})$. Then $M$ is a Mackey functor for $\mathcal{F}$ if the following conditions hold [DP15, Definition 2.1, Proposition 2.2].

1. (1)

   $M(P)\overset{\mathrm{def}}{=}M^{*}(P)=M_{*}(P)$ for each $P\leq S$,

2. (2)

   (Isomorphism) $M_{*}([\varphi])\overset{\mathrm{def}}{=}\mathrm{iso}(\varphi)=M^{*}([\varphi]^{-1})$ for each isomorphism $[\varphi]$ in $\mathcal{O}(\mathcal{F})$, and

3. (3)

   (Mackey formula) for each $P,Q\leq R\leq S$,

   $$r_{Q}^{R}\circ t_{P}^{R}=\sum_{x\in[Q\backslash R/P]}t_{Q\cap{}^{x}P}^{Q}\circ r_{Q\cap{}^{x}P}^{{}^{x}P}\circ\mathrm{iso}(c_{x}|_{P}).$$

A morphism $M\to N$ of Mackey functors is a family of $k$-module homomorphisms $\eta_{P}\colon M(P)\to N(P)$ such that $\eta=(\eta_{P})$ is both a natural transformation from $M^{*}\to N^{*}$ and a natural transformation $M_{*}\to N_{*}$ simultaneously. A subfunctor of $M$ is a subfunctor of $M^{*}$ which is simultaneously a subfunctor of $M_{*}$, and quotient functors are defined objectwise.

The simple objects in $\operatorname{Mack}_{k}(\mathcal{F})$ are parametrized by pairs $(T,V)$, where $T$ is a subgroup of $S$ taken up to $\mathcal{F}$-conjugacy, and where $V$ is a simple (irreducible) $k\operatorname{Out}_{\mathcal{F}}(T)$-module taken up to isomorphism [DP15, Section 3]. When convenient we view $V$ as a $k\operatorname{Aut}_{\mathcal{F}}(T)$-module via inflation. The corresponding simple Mackey functor $S_{T,V}$ has the property that $S_{T,V}(T)=V$ and $S_{T,V}(Q)=0$ for all subgroups $Q$ such that $T$ is not $\mathcal{F}$-conjugate to a subgroup of $Q$.

Let $T\leq S$ and let $V$ be a simple $k\operatorname{Out}_{\mathcal{F}}(T)$-module. We give the description of the functor $S_{T,V}$ on objects and isomorphisms in $\mathcal{O}(\mathcal{F})$ from p.153 of [DP15], since this will be important for the proof of Theorem 1.1, but interestingly the effect of $S_{T,V}$ on nonisomorphisms is not so important for our argument. For that we refer the interested reader to the description in [DP15]. Our treatment is a little different from (but equivalent to) that in [DP15], since we need to pay somewhat closer attention to precisely how $S_{T,V}(Q)$ decomposes as a direct sum of $k\operatorname{Out}_{\mathcal{F}}(Q)$-modules.

The set $\operatorname{Hom}_{\mathcal{F}}(T,Q)$ is an $\operatorname{Aut}_{\mathcal{F}}(Q)$-$\operatorname{Aut}_{\mathcal{F}}(T)$ biset with action on either side given by composition. The orbits $\operatorname{Hom}_{\mathcal{F}}(T,Q)/\operatorname{Aut}_{\mathcal{F}}(T)$ are in correspondence with the set of subgroups of $Q$ that are $\mathcal{F}$-conjugate to $T$. Likewise the double orbits $\operatorname{Aut}_{\mathcal{F}}(Q)\backslash\operatorname{Hom}_{\mathcal{F}}(T,Q)/\operatorname{Aut}_{\mathcal{F}}(T)$ are in correspondence with the $\operatorname{Aut}_{\mathcal{F}}(Q)$-orbits of such subgroups. Let

be a set of representatives for the double orbits.

For each $\alpha\in\operatorname{Hom}_{\mathcal{F}}(T,Q)$, we temporarily set $U=\alpha(T)$ and denote (formally) by $\alpha\otimes V$ the $k\operatorname{Aut}_{\mathcal{F}}(U)$-module, isomorphic to $V$ as a $k$-module via $\alpha\otimes v\mapsto v$, with action

for each $\varphi\in\operatorname{Aut}_{\mathcal{F}}(U)$ and $v\in V$. Since $T$ acts trivially on $V$, $U$ acts trivially on $\alpha\otimes V$. Set

the image of the relative trace, where here $N_{Q}(U)$ acts on the $\operatorname{Out}_{\mathcal{F}}(U)$-module $\alpha\otimes V$ through the composite

with $UC_{Q}(U)$ acting trivially. Note $W_{\alpha}$ is a $k$-submodule of $\alpha\otimes V$. It has the structure of a $kN_{\operatorname{Aut}_{\mathcal{F}}(Q)}(U)$-module on which $N_{Q}(U)$ acts trivially.

Write $U^{Q}$ for the $Q$-conjugacy class of $U$, and $N_{\operatorname{Aut}_{\mathcal{F}}(Q)}(U^{Q})$ for the stabilizer of this class in $\operatorname{Aut}_{\mathcal{F}}(Q)$. Since $\operatorname{Inn}(Q)$ is a normal subgroup of $\operatorname{Aut}_{\mathcal{F}}(Q)$ that acts transitively on $U^{Q}$,

By construction, the subgroup $N_{\operatorname{Aut}_{\mathcal{F}}(Q)}(U)\cap\operatorname{Inn}(Q)=N_{\operatorname{Inn}(Q)}(U)$ acts trivially on $W_{\alpha}$, and we may regard $W_{\alpha}$ as a module for $N_{\operatorname{Aut}_{\mathcal{F}}(Q)}(U^{Q})$ via the composite

Set now

where $\varphi$ runs over a set of representatives for the left cosets of $N_{\operatorname{Aut}_{\mathcal{F}}(Q)}(U^{Q})$ in $\operatorname{Aut}_{\mathcal{F}}(Q)$. Then $S_{T,V}(Q)_{\alpha}$ is a $k\operatorname{Out}_{\mathcal{F}}(Q)$-module, that is, $Q$ still acts trivially. The value of $S_{T,V}$ on the subgroup $Q$ is then

an $\operatorname{Out}_{\mathcal{F}}(Q)$-invariant direct sum decomposition.

Now let $Q^{\prime}$ be another subgroup of $S$ and $\beta\colon Q\to Q^{\prime}$ an isomorphism in $\mathcal{F}$. The bijection $\operatorname{Hom}(T,Q)\xrightarrow{\beta\circ-}\operatorname{Hom}(T,Q^{\prime})$ determines the $k$-module isomorphism $\alpha\otimes V\cong\beta\alpha\otimes V$ intertwining the actions of $N_{\operatorname{Aut}_{\mathcal{F}}(Q)}(\alpha(T))$ and $N_{\operatorname{Aut}_{\mathcal{F}}(Q^{\prime})}(\beta(\alpha(T)))$ with respect to conjugation by $\beta$. It induces an isomorphism of $k$-modules

intertwining the actions of $N_{\operatorname{Aut}_{\mathcal{F}}(Q)}(\alpha(T)^{Q})$ and $N_{\operatorname{Aut}_{\mathcal{F}}(Q^{\prime})}(\beta(\alpha(T))^{Q^{\prime}})$. The component of $\mathrm{iso}(\beta)$ at $\alpha$ is the corresponding map of induced modules

and $\mathrm{iso}(\beta)$ is the sum of these maps.

Note that if $V$ is a simple $\mathbb{F}_{p}\operatorname{Out}_{\mathcal{F}}(T)$-module, then $S_{T,V}$ takes values in $\mathbb{F}_{p}$-mod. Further, since $V$ is simple, $S_{T,V}$ is a simple functor [DP15].

Let $k$ be a commutative $\mathbb{Z}_{(p)}$-algebra and let $M$ be a contravariant functor from the orbit category of a group or a fusion system to the category of $k$-modules. A common technique for computing the higher limits of $M$ (and especially for showing that such higher limits vanish) is to use a filtration of $M$ each of whose successive quotient functors is atomic, namely a functor which vanishes except on a single conjugacy class of subgroups. This method does not always work as, for example, in the case of the center functor [Oli18]. But the idea is often effective for making reductions even when it doesn’t work directly. And ultimately it is all that is needed for the proof of the main theorem here.

Let $G$ be a finite group and let $A$ be a $\mathbb{Z}_{(p)}G$-module. Define a functor

via $F_{A}(1)=A$ and $F_{A}(P)=0$ when $P\neq 1$. The action of $G=\operatorname{Aut}_{\mathcal{O}_{p}(G)}(1)$ on $A=F_{A}(1)$ is the given one. The higher limits of $F_{A}$ are denoted $\Lambda^{i}(G,A)$ and arise as the higher limits of atomic functors, as was first shown by Jackowski, McClure, and Oliver [AKO11, Proposition 5.20].

Let $P_{1},\dots,P_{n}$ be a set of representatives for the $\mathcal{F}$-conjugacy classes of subgroups of $S$ such that if $i<j$, then $P_{j}$ is not conjugate to a subgroup of $P_{i}$. Then one can make a filtration $0=M_{0}\subset M_{1}\subseteq\cdots\subset M_{n}=M$, in which $M_{j}$ is the functor equal to $M$ on the union of the conjugacy classes $P_{i}$ with $i\leq j$, and $0$ otherwise. Then $(M_{i}/M_{i-1})(P)=M(P)$ if $P$ is conjugate to $P_{i}$, and it is zero otherwise, i.e. the quotient is atomic.

In general, we use the notation $M_{Q}$ for the atomic subquotient functor of $M$ corresponding to the $\mathcal{F}$-conjugacy class of $Q$, namely the functor with values $M_{Q}(P)=M(P)$ if $P$ is $\mathcal{F}$-conjugate to $Q$, and $M_{Q}(P)=0$ otherwise.

The next lemma is proved using long exact sequences on higher limits corresponding to short exact sequences of functors arising out of a filtration of the above type.

The functors $\Lambda^{*}(G,M)$ vanish in many cases. See for example Section III.5 of [AKO11] for many results along these lines. In the next lemma we state two such vanishing results.

Recall that a radical $p$-chain of length $m$ in the finite group $G$ is a sequence $O_{p}(G)=P_{0}<P_{1}<\cdots P_{m}$ such that $P_{i}=O_{p}(N_{G}(P_{1},\dots,P_{i}))$ for each $i=0,\dots,n$, where here $N_{G}(P_{1},\dots,P_{i})$ denotes the intersection of the normalizers in $G$ of the $P_{i}$.

We want to show (Theorem 1.2) that the restriction of the contravariant part of each Mackey composition factor $S_{T,V}$ of $H^{j}(-,\mathbb{F}_{p})$ is acyclic when $j\leq p-2$. For doing this, Díaz and Park show we can restrict attention to composition factors $S_{T,V}$ with $T$ not $\mathcal{F}$-centric.