$\mathcal{Q}:{\operatorname{Ho}}({\mathcal{H}})\to{\operatorname{Ho}}({\mathcal{H}})/{\mathcal{S}}$
factors through ${\mathcal{T}}_{{\mathcal{M}}^{\prime}}$. Certainly,
${\mathcal{S}}$ contains ${\mathcal{S}}^{G}_{\mathcal{M}}$ by induction and
hence $\mathcal{Q}$ factors through some
$\mathcal{Q}^{\prime}:{\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}\to{\operatorname{Ho}}({\mathcal{H}})/{\mathcal{S}}$.
But 3.15(ii) then ensures that $\mathcal{Q}^{\prime}$ kills the essential
image of $(I^{G}_{M_{n+1}})_{\mathcal{M}}$, and hence it factors through
${\mathcal{T}}_{{\mathcal{M}}^{\prime}}$ as claimed.

For (ii), we assume $n>1$ and fix $1\leq i\leq n$. We denote by
$\eta^{G}_{\mathcal{M}}$ (resp. $\eta^{G}_{{\mathcal{M}}_{i}}$, resp.
$\eta^{{\mathcal{M}}_{i}}_{\mathcal{M}}$) the unit of the adjunction
$\pi^{G}_{{\mathcal{M}}}\dashv r^{G}_{{\mathcal{M}}}$ (resp.
$\pi^{G}_{{\mathcal{M}}_{i}}\dashv r^{G}_{{\mathcal{M}}_{i}}$, resp.
$\pi^{{\mathcal{M}}_{i}}_{\mathcal{M}}\dashv
r^{{\mathcal{M}}_{i}}_{\mathcal{M}}$). Then by the recollement in (i) we have
two distinguished triangles

$\displaystyle
i^{G}_{{\mathcal{M}}_{i}}\mathcal{R}^{G}_{{\mathcal{M}}_{i}}({\mathfrak{m}})\longrightarrow$
$\displaystyle{\mathfrak{m}}\stackrel{{\scriptstyle\eta^{G}_{{\mathcal{M}}_{i}}}}{{\longrightarrow}}r^{G}_{{\mathcal{M}}_{i}}\pi^{G}_{{\mathcal{M}}_{i}}({\mathfrak{m}})\longrightarrow\Sigma
i^{G}_{{\mathcal{M}}_{i}}\mathcal{R}^{G}_{{\mathcal{M}}_{i}}({\mathfrak{m}})\quad\text{and}$
$\displaystyle
i^{G}_{{\mathcal{M}}}\mathcal{R}^{G}_{{\mathcal{M}}}({\mathfrak{m}})\longrightarrow$
$\displaystyle{\mathfrak{m}}\stackrel{{\scriptstyle\eta^{G}_{\mathcal{M}}}}{{\longrightarrow}}r^{G}_{{\mathcal{M}}}\pi^{G}_{{\mathcal{M}}}({\mathfrak{m}})\longrightarrow\Sigma
i^{G}_{{\mathcal{M}}}\mathcal{R}^{G}_{{\mathcal{M}}}({\mathfrak{m}})$

for any ${\mathfrak{m}}\in{\operatorname{Ho}}({\mathcal{H}})$. Recall that
$\pi^{G}_{{\mathcal{M}}}=\pi^{{\mathcal{M}}_{i}}_{\mathcal{M}}\circ\pi^{G}_{{\mathcal{M}}_{i}}$
and $r^{G}_{{\mathcal{M}}}=r^{G}_{{\mathcal{M}}_{i}}\circ
r^{{\mathcal{M}}_{i}}_{\mathcal{M}}$. By considering the recollement (3.5) for
the collection of Levi subgroups ${\mathcal{M}}_{i}$ and $L=M_{i}$ instead,
and applying $r^{G}_{{\mathcal{M}}_{i}}$ to the analogous triangle for
$\overline{{\mathfrak{m}}}:=\pi^{G}_{{\mathcal{M}}_{i}}({\mathfrak{m}})$, we
get

$\mathfrak{n}\longrightarrow
r^{G}_{{\mathcal{M}}_{i}}(\overline{{\mathfrak{m}}})\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}r^{G}_{{\mathcal{M}}}\pi^{G}_{{\mathcal{M}}}({\mathfrak{m}})\longrightarrow\Sigma\mathfrak{n}$

where
$\mathfrak{n}:=r^{G}_{{\mathcal{M}}_{i}}(I^{G}_{M_{i}})_{{\mathcal{M}}_{i}}(\mathcal{R}^{G}_{M_{i}})_{{\mathcal{M}}_{i}}(\overline{{\mathfrak{m}}})$
and
$\varphi=r^{G}_{{\mathcal{M}}_{i}}(\eta^{{\mathcal{M}}_{i}}_{\mathcal{M}})_{\overline{{\mathfrak{m}}}}$.
Since we have
$\eta^{G}_{\mathcal{M}}=\varphi\circ\eta^{G}_{{\mathcal{M}}_{i}}$, we may
apply the octahedral axiom to obtain a distinguished triangle

(3.6)
$i^{G}_{{\mathcal{M}}_{i}}\mathcal{R}^{G}_{{\mathcal{M}}_{i}}({\mathfrak{m}})\longrightarrow
i^{G}_{{\mathcal{M}}}\mathcal{R}^{G}_{{\mathcal{M}}}({\mathfrak{m}})\longrightarrow\mathfrak{n}\longrightarrow\Sigma
i^{G}_{{\mathcal{M}}_{i}}\mathcal{R}^{G}_{{\mathcal{M}}_{i}}({\mathfrak{m}}),$

c.f. [5, 0145]. Since $\mathfrak{n}\cong
r^{G}_{{\mathcal{M}}_{i}}\pi^{G}_{{\mathcal{M}}_{i}}{\operatorname{Ho}}(I^{G}_{M_{i}})\mathbf{R}\mathcal{R}^{G}_{M_{i}}({\mathfrak{m}})$,
c.f. 3.15(i), this is the desired triangle. The construction for the left
adjoints is completely dual to this.

Finally, for (iii) we only prove the formula for the essential image of
$r^{G}_{\mathcal{M}}$, the proof for $\ell^{G}_{\mathcal{M}}$ being completely
analogous. We argue again by induction on $n$, the case $n=1$ being immediate
from 3.12. Assume the equality holds for some $n\geq 1$ and any collection of
$n$ standard Levi subgroups. Again, let $M_{1},\ldots,M_{n+1}$ be distinct
standard Levi subgroups and write ${\mathcal{M}}=\\{M_{1},\ldots,M_{n}\\}$ and
${\mathcal{M}}^{\prime}={\mathcal{M}}\cup\\{M_{n+1}\\}$. Note first that
${\operatorname{Im}}(r^{G}_{{\mathcal{M}}^{\prime}})={\operatorname{Ker}}(\mathcal{R}^{G}_{{\mathcal{M}}^{\prime}})$
and
${\operatorname{Im}}(r^{G}_{\mathcal{M}})={\operatorname{Ker}}(\mathcal{R}^{G}_{\mathcal{M}})$.
So by the induction hypothesis, one inclusion is automatic from the triangle
(3.6) applied to ${\mathcal{M}}^{\prime}$ and $i=n+1$. Assuming now that
${\mathfrak{m}}\in{\operatorname{Ker}}(\mathcal{R}^{G}_{{\mathcal{M}}^{\prime}})$,
the triangle (3.6) then implies that
$i^{G}_{\mathcal{M}}\mathcal{R}^{G}_{\mathcal{M}}({\mathfrak{m}})$ lies in
${\operatorname{Im}}(r^{G}_{\mathcal{M}})={\operatorname{Ker}}(\mathcal{R}^{G}_{\mathcal{M}})$.
But since $i^{G}_{\mathcal{M}}$ is fully faithful we may apply 2.4 to obtain

$0=\mathcal{R}^{G}_{\mathcal{M}}i^{G}_{\mathcal{M}}\mathcal{R}^{G}_{\mathcal{M}}({\mathfrak{m}})\cong\mathcal{R}^{G}_{\mathcal{M}}({\mathfrak{m}})$

and the induction hypothesis gives
${\mathfrak{m}}\in\bigcap_{j=1}^{n}{\operatorname{Ker}}(\mathbf{R}\mathcal{R}^{G}_{M_{j}})$.
Replacing ${\mathcal{M}}$ by ${\mathcal{M}}^{\prime}\setminus\\{M_{n}\\}$ and
applying (3.6) to ${\mathcal{M}}^{\prime}$ and $i=n$ instead, the above
argument gives that ${\mathfrak{m}}\in\bigcap_{j\neq
n}{\operatorname{Ker}}(\mathbf{R}\mathcal{R}^{G}_{M_{j}})$ as well and hence
we obtain
${\mathfrak{m}}\in\bigcap_{j=1}^{n+1}{\operatorname{Ker}}(\mathbf{R}\mathcal{R}^{G}_{M_{j}})$
as claimed. ∎

###  3.3. Supersingular modules and the derived adjoints

We must check that that the recollement obtained in 3.13(i) is non-trivial,
i.e. that the quotient categories
${\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}$ are non-zero. This will
follow by considering supersingular modules. In this subsection, we assume
that $k$ is algebraically closed.

######  Proposition 3.17.

Suppose that either $G$ does not have root system of type
$A_{1}\times\cdots\times A_{1}$ (possibly empty product) or that $r_{ss}>0$
and $q>2$. Then there exists a simple supersingular ${\mathcal{H}}$-module of
infinite projective dimension.

###### Proof.

If $G$ does not have root system of type $A_{1}\times\cdots\times A_{1}$ then
any simple supersingular ${\mathcal{H}}$-module has infinite projective
dimension by [20, Theorem 7.7]. So assume that $G$ has type
$A_{1}\times\cdots\times A_{1}$ (non-empty product) and that $q>2$. By [20,
Theorem 7.7] again, our claim amounts to showing that there exists a
$k$-valued character $\xi$ of the group $T(\mathbb{F}_{q})$ such that
$S_{\xi}\neq S$ (cf. (2.3)). But since $p$ does not divide the order of
$T(\mathbb{F}_{q})$ and as $T(\mathbb{F}_{q})$ is abelian, the intersection of
the kernels of all the non-trivial $k$-valued characters of
$T(\mathbb{F}_{q})$ is trivial. Let $\alpha$ be a simple root and
$s=s_{\alpha}\in S$, and note that the group $T_{s}(\mathbb{F}_{q})$ is non-
trivial by our assumption that $q>2$. Then there must exist a character $\xi$
with $\xi\mid_{T_{s}(\mathbb{F}_{q})}\neq 1$ and so $S_{\xi}\neq S$. ∎

######  Proposition 3.18.

Suppose that $M$ is a proper Levi subgroup of $G$ and let
${\mathfrak{m}}\in{\operatorname{Mod}}({\mathcal{H}})$.

  1. (i)

If ${\mathfrak{m}}$ is supersingular, then
${\operatorname{Ho}}({\mathcal{L}^{G}_{M}})({\mathfrak{m}})=0$ in
${\operatorname{Ho}}({\mathcal{H}}_{M})$.

  2. (ii)

If ${\mathfrak{m}}$ is uniformly supersingular, then
$\mathbf{R}{\mathcal{R}^{G}_{M}}({\mathfrak{m}})=0$ in
${\operatorname{Ho}}({\mathcal{H}}_{M})$.

###### Proof.

These were proved in [4, Proposition 4.27] (see also [2, Proposition 5.18]) at
the abelian level. Therefore (i) follows immediately. For (ii), let
$\lambda_{0}^{+}$ be as in 2.1 and let
$\mathcal{O}^{+}=\widetilde{W}\cdot\lambda_{0}^{+}$. Write
$z:=z_{\mathcal{O}^{+}}$ and fix $n\geq 1$ such that $z^{n}{\mathfrak{m}}=0$.
First, take an arbitrary
${\mathfrak{m}}^{\prime}\in{\operatorname{Mod}}({\mathcal{H}})$. Since the
element $z\in{\mathcal{H}}$ is central, multiplication by $z^{n}$ is
${\mathcal{H}}$-linear on ${\mathfrak{m}}^{\prime}$ and hence induces an
${\mathcal{H}}_{M}$-linear endomorphism of
${\mathcal{R}^{G}_{M}}({\mathfrak{m}}^{\prime})={\operatorname{Hom}}_{{\mathcal{H}}_{M}^{+}}({\mathcal{H}}_{M},{\mathfrak{m}}^{\prime})$
given by $f\mapsto z^{n}f$. We claim that this is in fact an automorphism of
${\mathcal{R}^{G}_{M}}({\mathfrak{m}}^{\prime})$.

To see this, we use the fact that
$T_{\lambda_{0}^{+}}^{n+1}=z^{n}T_{\lambda_{0}^{+}}$ (c.f. [2, Lemma 5.17] and
[30, Lemma 2.18]). Now let
$f\in{\mathcal{R}^{G}_{M}}({\mathfrak{m}}^{\prime})$. Suppose first that
$z^{n}f=0$. Then for all $h\in{\mathcal{H}}_{M}$, we have

$0=T_{\lambda_{0}^{+}}z^{n}f(h)=T_{\lambda_{0}^{+}}^{n+1}f(h)=f((T^{M}_{\lambda_{0}^{+}})^{n+1}h)$

by definition of the map $j_{M}^{+}$ (c.f. (2.4)). Since
$T^{M}_{\lambda_{0}^{+}}$ is invertible in ${\mathcal{H}}_{M}$, this means
that $f=0$. To get surjectivity, define
$g:{\mathcal{H}}_{M}\to{\mathfrak{m}}^{\prime}$ by
$g(h)=T_{\lambda_{0}^{+}}f((T^{M}_{\lambda_{0}^{+}})^{-n-1})h)$. This is
${\mathcal{H}}_{M^{+}}$-linear because $T^{M}_{\lambda_{0}^{+}}$ is central in
${\mathcal{H}}_{M}$, and we have

$\displaystyle z^{n}g(h)$
$\displaystyle=T_{\lambda_{0}^{+}}z^{n}f((T^{M}_{\lambda_{0}^{+}})^{-n-1})h)$
$\displaystyle=T_{\lambda_{0}^{+}}^{n+1}f((T^{M}_{\lambda_{0}^{+}})^{-n-1})h)$
$\displaystyle=f(h)$

for all $h\in{\mathcal{H}}_{M}$.

We now specialise to the case where
${\mathfrak{m}}^{\prime}=Q_{f}{\mathfrak{m}}$ is any Gorenstein injective
replacement of ${\mathfrak{m}}$, which by definition comes equipped with an
${\mathcal{H}}$-linear weak equivalence $i_{{\mathfrak{m}}}:{\mathfrak{m}}\to
Q_{f}{\mathfrak{m}}$. Since we may view the commutative square

${{\mathfrak{m}}}$${{\mathfrak{m}}}$${Q_{f}{\mathfrak{m}}}$${Q_{f}{\mathfrak{m}}}$$\scriptstyle{\cdot
z^{n}}$$\scriptstyle{i_{{\mathfrak{m}}}}$$\scriptstyle{i_{{\mathfrak{m}}}}$$\scriptstyle{\cdot
z^{n}}$

as a square in ${\operatorname{Ho}}({\mathcal{H}})$, we deduce that
multiplication by $z^{n}$ induces the zero endomorphism of
$Q_{f}{\mathfrak{m}}$ in the homotopy category. Since $Q_{f}{\mathfrak{m}}$ is
Gorenstein injective, this means by 2.14 that there is a factorisation

${Q_{f}{\mathfrak{m}}}$${Q_{f}{\mathfrak{m}}}$${\mathfrak{e}}$$\scriptstyle{\cdot
z^{n}}$

where $\mathfrak{e}\in{\operatorname{Mod}}({\mathcal{H}})$ is an injective
module. By now applying ${\mathcal{R}^{G}_{M}}$ to that commuting triangle, we
deduce from the bijectivity of
${\mathcal{R}^{G}_{M}}(Q_{f}{\mathfrak{m}})\stackrel{{\scriptstyle\cdot
z^{n}}}{{\longrightarrow}}{\mathcal{R}^{G}_{M}}(Q_{f}{\mathfrak{m}})$ that
${\mathcal{R}^{G}_{M}}(Q_{f}{\mathfrak{m}})$ is isomorphic to a direct summand
of ${\mathcal{R}^{G}_{M}}(\mathfrak{e})$. But since ${\mathcal{R}^{G}_{M}}$ is
the right adjoint of an exact functor, the latter module is injective and thus
so is ${\mathcal{R}^{G}_{M}}(Q_{f}{\mathfrak{m}})$. The result now follows
immediately because
$\mathbf{R}{\mathcal{R}^{G}_{M}}({\mathfrak{m}})\cong{\mathcal{R}^{G}_{M}}(Q_{f}{\mathfrak{m}})$
in ${\operatorname{Ho}}({\mathcal{H}}_{M})$. ∎

######  Remark 3.19.

Similarly to the above proof, take $\lambda_{0}^{\pm}$ as in 2.1 and let
$\mathcal{O}^{\pm}=\widetilde{W}\cdot\lambda_{0}^{\pm}$. Then part (i) of the
above result holds more generally whenever every element of ${\mathfrak{m}}$
is killed by a power of $z_{\mathcal{O}^{-}}$, and part (ii) holds whenever
$z_{\mathcal{O}^{+}}$ acts nilpotently on ${\mathfrak{m}}$. This follows from
the above proof and because the abelian analogue of 3.18 holds in that higher
level of generality, c.f. the proof of [4, Proposition 4.27].

######  Corollary 3.20.

Suppose that either $G$ does not have root system of type
$A_{1}\times\cdots\times A_{1}$ (possibly empty product) or that $r_{ss}>0$
and $q>2$. Then for any collection ${\mathcal{M}}$ of proper standard Levi
subgroups, the quotient category
${\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}$ is non-zero. In particular,
for any proper standard Levi subgroup $M$ of $G$, we have that $I^{G}_{M}$ is
not a Quillen equivalence.

###### Proof.

By 3.17, 3.18 and 3.13(iii) there is a simple module ${\mathfrak{m}}$ which is
non-zero in ${\operatorname{Ho}}({\mathcal{H}})$ and which lies in both
${\operatorname{Ker}}(\mathcal{L}^{G}_{\mathcal{M}})$ and
${\operatorname{Ker}}(\mathcal{R}^{G}_{\mathcal{M}})$. By the recollement in
3.13(i), ${\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}$ is equivalent to
both of these via the functors $\ell^{G}_{\mathcal{M}}$ and
$r^{G}_{\mathcal{M}}$, so we are done. ∎

As another corollary, we deduce that uniformly supersingular modules are Ext-
orthogonal to parabolically induced modules in large enough degree without
assuming any finiteness condition on these modules:

######  Corollary 3.21.

Suppose ${\mathfrak{m}}\in{\operatorname{Mod}}({\mathcal{H}})$ is uniformly
supersingular and let $\mathfrak{n}\in{\operatorname{Mod}}({\mathcal{H}}_{M})$
be any module. Then

${\operatorname{Ext}}_{{\mathcal{H}}}^{i}({\operatorname{Ind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}(\mathfrak{n}),{\mathfrak{m}})=0$

for all $i>r_{ss}+r_{Z}+1$.

###### Proof.

Let $Q_{f}{\mathfrak{m}}$ be the fibrant replacement of ${\mathfrak{m}}$. We
have by the proof of 3.18(ii) that
${\mathcal{R}^{G}_{M}}(Q_{f}{\mathfrak{m}})$ is injective, and so by
3.7(ii)-(iii) we have

${\operatorname{Ext}}_{{\mathcal{H}}}^{i}(I^{G}_{M}(\mathfrak{n}),Q_{f}{\mathfrak{m}})\cong{\operatorname{Ext}}_{{\mathcal{H}}_{M}}^{i}(\mathfrak{n},{\mathcal{R}^{G}_{M}}(Q_{f}{\mathfrak{m}}))=0$

for any $i>0$. Moreover, by definition there is a short exact sequence
$0\to{\mathfrak{m}}\to Q_{f}{\mathfrak{m}}\to\mathfrak{e}\to 0$ with
$\mathfrak{e}$ of finite injective dimension, bounded above by
$d=r_{ss}+r_{Z}$. The long exact sequence on ${\operatorname{Ext}}$ and the
vanishing above then gives us that

${\operatorname{Ext}}_{{\mathcal{H}}}^{i-1}(I^{G}_{M}(\mathfrak{n}),\mathfrak{e})\cong{\operatorname{Ext}}_{{\mathcal{H}}}^{i}(I^{G}_{M}(\mathfrak{n}),{\mathfrak{m}})$

for all $i>1$. The bound on the injective dimension of $\mathfrak{e}$ gives
the result. ∎

######  Remark 3.22.

We note that the proofs of 3.18 and 3.21 remain valid without the assumption
that $k$ is algebraically closed.

Next, we compute the image of all simple modules under the adjoints of
parabolic induction at the homotopy level. For that we need the corresponding
formula at the level of the module categories, due to Abe ([2]).

######  Theorem 3.23.

Let $(P,\mathfrak{n},Q)$ be a standard supersingular triple for
${\mathcal{H}}$. Then for $P^{\prime}=M^{\prime}N^{\prime}$ another standard
parabolic subgroup of $G$, we have

$\mathcal{L}^{G}_{M^{\prime}}(I_{\mathcal{H}}(P,\mathfrak{n},Q))\cong\begin{cases}I_{{\mathcal{H}}_{M^{\prime}}}(P\cap
M^{\prime},\mathfrak{n},Q\cap M^{\prime})&\text{if $P\subseteq P^{\prime}$ and
$\Pi(\mathfrak{n})\subseteq\Pi_{Q}\cup\Pi_{P^{\prime}}$,}\\\
0&\text{otherwise,}\end{cases}$

and

$\mathcal{R}^{G}_{M^{\prime}}(I_{\mathcal{H}}(P,\mathfrak{n},Q))\cong\begin{cases}I_{{\mathcal{H}}_{M^{\prime}}}(P\cap
M^{\prime},\mathfrak{n},Q\cap M^{\prime})&\text{if $Q\subseteq
P^{\prime}$,}\\\ 0&\text{otherwise.}\end{cases}$

###### Proof.

The analogous statement was proved in [2, Theorem 5.20] for the adjoints of
parabolic coinduction and the simple modules presented in the form
$CI_{\mathcal{H}}(P,\mathfrak{n},Q)$. We just check that this still holds for
the adjoints of parabolic induction and the simple modules in the form
$I_{\mathcal{H}}(P,\mathfrak{n},Q)$. Write $\sigma$ for the
${\mathcal{H}}_{M^{\prime}}$-module
$\mathcal{L}^{G}_{M^{\prime}}(I_{\mathcal{H}}(P,\mathfrak{n},Q))$. First, we
have by (2.7) and loc. cit. that

$\displaystyle\sigma$
$\displaystyle\cong\iota_{0}^{w_{0}(M^{\prime})}{\operatorname{Loc}}_{w_{0}(M^{\prime})}(CI_{\mathcal{H}}(w_{0}(P),\iota_{0}^{M}\mathfrak{n},w_{0}(Q)))$
$\displaystyle\cong\begin{cases}\iota_{0}^{w_{0}(M^{\prime})}CI_{{\mathcal{H}}_{M^{\prime}}}(w_{0}(P\cap
M^{\prime}),\iota_{0}^{M}\mathfrak{n},w_{0}(Q\cap M^{\prime}))&\text{if
$P\subseteq P^{\prime}$ and
$\Pi(\mathfrak{n})\subseteq\Pi_{Q}\cup\Pi_{P^{\prime}}$,}\\\
0&\text{otherwise.}\end{cases}$

So we may assume that $P\subseteq P^{\prime}$ and
$\Pi(\mathfrak{n})\subseteq\Pi_{Q}\cup\Pi_{P^{\prime}}$. Next, for any
standard parabolic $Q\cap M^{\prime}\subseteq Q^{\prime}\subseteq(P\cap
M^{\prime})(\mathfrak{n})$ of $M^{\prime}$ with Levi subgroup $L^{\prime}$ we
have
$e_{{\mathcal{H}}_{w_{0}(L^{\prime})}}(\iota_{0}^{M}\mathfrak{n})\cong\iota_{0}^{L^{\prime}}e_{{\mathcal{H}}_{L^{\prime}}}(\mathfrak{n})$,
c.f. [4, Lemma 4.22]. From this and the above we get

$\displaystyle\iota_{0}^{w_{0}M^{\prime}}{\operatorname{Coind}}_{{\mathcal{H}}_{w_{0}L^{\prime}}}^{{\mathcal{H}}_{w_{0}M^{\prime}}}\left(e_{{\mathcal{H}}_{w_{0}L^{\prime}}}(\iota_{0}^{M}\mathfrak{n})\right)$
$\displaystyle\cong\iota_{0}^{w_{0}M^{\prime}}{\operatorname{Coind}}_{{\mathcal{H}}_{w_{0}L^{\prime}}}^{{\mathcal{H}}_{w_{0}M^{\prime}}}\left(\iota_{0}^{L^{\prime}}e_{{\mathcal{H}}_{L^{\prime}}}(\mathfrak{n})\right)$
$\displaystyle\cong\iota_{0}^{w_{0}M^{\prime}}{\operatorname{Ind}}_{{\mathcal{H}}_{w_{0,M^{\prime}}w_{0}L^{\prime}}}^{{\mathcal{H}}_{w_{0}M^{\prime}}}\left(\iota_{w_{0}M^{\prime}}^{w_{0}L^{\prime}}\iota_{0}^{L^{\prime}}e_{{\mathcal{H}}_{L^{\prime}}}(\mathfrak{n})\right)\quad\text{by
\lx@cref{creftypecap~refnum}{ind_coind_thm}}$
$\displaystyle\cong{\operatorname{Ind}}_{{\mathcal{H}}_{L^{\prime}}}^{{\mathcal{H}}_{M^{\prime}}}\left(\iota_{0,w_{0}M^{\prime}}^{w_{0}L^{\prime}}\iota_{w_{0}M^{\prime}}^{w_{0}L^{\prime}}\iota_{0}^{L^{\prime}}e_{{\mathcal{H}}_{L^{\prime}}}(\mathfrak{n})\right)\quad\text{by
\cite[cite]{[\@@bibref{}{Vign5}{}{}, Lemma 4.9]}}$
$\displaystyle\cong{\operatorname{Ind}}_{{\mathcal{H}}_{L^{\prime}}}^{{\mathcal{H}}_{M^{\prime}}}\left(\iota_{0}^{w_{0}L^{\prime}}\iota_{0}^{L^{\prime}}e_{{\mathcal{H}}_{L^{\prime}}}(\mathfrak{n})\right)\quad\text{by
\eqref{transitivity_twists}}$
$\displaystyle\cong{\operatorname{Ind}}_{{\mathcal{H}}_{L^{\prime}}}^{{\mathcal{H}}_{M^{\prime}}}\left(e_{{\mathcal{H}}_{L^{\prime}}}(\mathfrak{n})\right).$

By exactness of $\iota_{0}^{w_{0}(M^{\prime})}$ and from the definitions of
both $I_{{\mathcal{H}}_{M^{\prime}}}(P\cap M^{\prime},\mathfrak{n},Q\cap
M^{\prime})$ and $CI_{{\mathcal{H}}_{M^{\prime}}}(w_{0}(P\cap
M^{\prime}),\iota_{0}^{M}\mathfrak{n},w_{0}(Q\cap M^{\prime}))$, it follows
that we have an isomorphism

$I_{{\mathcal{H}}_{M^{\prime}}}(P\cap M^{\prime},\mathfrak{n},Q\cap
M^{\prime})\cong\iota_{0}^{w_{0}(M^{\prime})}CI_{{\mathcal{H}}_{M^{\prime}}}(w_{0}(P\cap
M^{\prime}),\iota_{0}^{M}\mathfrak{n},w_{0}(Q\cap M^{\prime}))$

as required. The proof for the right adjoint is completely analogous. ∎

We now show that the analogue of the above holds at the homotopy level as
well. For this, we need a couple of preliminary results.

######  Lemma 3.24.

Let ${\mathfrak{m}}$ be a finite length ${\mathcal{H}}$-module. Then
${\mathfrak{m}}$ is ${\mathcal{R}^{G}_{M}}$-acyclic for any standard Levi
subgroup $M$ of $G$.

###### Proof.

Fix a standard Levi subgroup $M$. It was shown in [3, Proposition 3.6(2)] that
for any ${\mathcal{H}}_{M}$-module $\mathfrak{n}$ there are isomorphisms

${\operatorname{Ext}}^{j}_{\mathcal{H}}({\operatorname{Coind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}(\mathfrak{n}),{\mathfrak{m}})\cong{\operatorname{Ext}}^{j}_{{\mathcal{H}}_{M}}(\mathfrak{n},({\mathcal{R}^{G}_{M}})^{\prime}({\mathfrak{m}}))$

for all $j\geq 0$, where $({\mathcal{R}^{G}_{M}})^{\prime}$ denotes the right
adjoint of parabolic coinduction. By applying 3.7(iii), we deduce that finite
length modules are acyclic for $({\mathcal{R}^{G}_{M}})^{\prime}$. Since
$({\mathcal{R}^{G}_{M}})^{\prime}\cong\iota_{0}^{w_{0}(M)}\circ\mathcal{R}^{G}_{w_{0}(M)}$,
c.f. 2.3(i), and $\iota_{0}^{w_{0}(M)}$ is an exact equivalence, we get that
finite length modules are acyclic for $\mathcal{R}^{G}_{w_{0}(M)}$. As $M$ was
arbitrary we are done. ∎

######  Remark 3.25.

One deduces immediately from 3.24, [2, Proposition 5.18] and 3.7(iii) that
supersingular modules of finite length are Ext-orthogonal to all parabolically
induced modules in all positive degrees, strengthening 3.21 in that case.

######  Lemma 3.26.

Let $M$ be a standard Levi subgroup of $G$ and let
${\mathfrak{m}}\in{\operatorname{Mod}}({\mathcal{H}})$ be
${\mathcal{R}^{G}_{M}}$-acyclic. Then there is an isomorphism
$\mathbf{R}{\mathcal{R}^{G}_{M}}({\mathfrak{m}})\cong{\mathcal{R}^{G}_{M}}(m)$
in ${\operatorname{Ho}}({\mathcal{H}}_{M})$.

###### Proof.

Taking a fibrant replacement of ${\mathfrak{m}}$ in the Gorenstein injective
model structure, we get a short exact sequence

$0\to{\mathfrak{m}}\to Q_{f}{\mathfrak{m}}\to\mathfrak{e}\to 0$

where $\mathfrak{e}$ has finite injective dimension. As both ${\mathfrak{m}}$
and $Q_{f}{\mathfrak{m}}$ are ${\mathcal{R}^{G}_{M}}$-acyclic, c.f. 3.7(ii),
so is $\mathfrak{e}$. By applying 3.7(iii), we then get isomorphisms

${\operatorname{Ext}}^{j}_{\mathcal{H}}(I^{G}_{M}(\mathfrak{n}),\mathfrak{e})\cong{\operatorname{Ext}}^{j}_{{\mathcal{H}}_{M}}(\mathfrak{n},{\mathcal{R}^{G}_{M}}(\mathfrak{e}))$

for all $\mathfrak{n}\in{\operatorname{Mod}}({\mathcal{H}}_{M})$ and all
$j\geq 0$. It then follows that ${\mathcal{R}^{G}_{M}}(\mathfrak{e})$ also has
finite injective dimension. By acyclicity of ${\mathfrak{m}}$, the sequence

$0\to{\mathcal{R}^{G}_{M}}({\mathfrak{m}})\to{\mathcal{R}^{G}_{M}}(Q_{f}{\mathfrak{m}})\to{\mathcal{R}^{G}_{M}}(\mathfrak{e})\to
0$

is exact. The above therefore implies that the map
${\mathcal{R}^{G}_{M}}({\mathfrak{m}})\to{\mathcal{R}^{G}_{M}}(Q_{f}{\mathfrak{m}})$
is a weak equivalence. As
$\mathbf{R}{\mathcal{R}^{G}_{M}}({\mathfrak{m}})\cong{\mathcal{R}^{G}_{M}}(Q_{f}{\mathfrak{m}})$
in ${\operatorname{Ho}}({\mathcal{H}}_{M})$, we are done. ∎

######  Proposition 3.27.

Let $(P,\mathfrak{n},Q)$ be a standard supersingular triple for
${\mathcal{H}}$. Then for $P^{\prime}=M^{\prime}N^{\prime}$ another standard
parabolic subgroup of $G$, we have

${\operatorname{Ho}}(\mathcal{L}^{G}_{M^{\prime}})(I_{\mathcal{H}}(P,\mathfrak{n},Q))\cong\begin{cases}I_{{\mathcal{H}}_{M^{\prime}}}(P\cap
M^{\prime},\mathfrak{n},Q\cap M^{\prime})&\text{if $P\subseteq P^{\prime}$ and
$\Pi(\mathfrak{n})\subseteq\Pi_{Q}\cup\Pi_{P^{\prime}}$,}\\\
0&\text{otherwise,}\end{cases}$

and

$\mathbf{R}\mathcal{R}^{G}_{M^{\prime}}(I_{\mathcal{H}}(P,\mathfrak{n},Q))\cong\begin{cases}I_{{\mathcal{H}}_{M^{\prime}}}(P\cap
M^{\prime},\mathfrak{n},Q\cap M^{\prime})&\text{if $Q\subseteq
P^{\prime}$,}\\\ 0&\text{otherwise.}\end{cases}$

###### Proof.

The statement for the left adjoints follows immediately from 3.23. The
statement for the right adjoints follows from 3.24, 3.26 and 3.23 because the
simple modules are finite dimensional, c.f. [26, Lemma 6.9]. ∎

Using the above, we give a new proof of one direction of [20, Theorem 8.5]
without any assumptions on the prime $p$ and without using any Bruhat-Tits
theory. This result is not new, in that the proof of the equivalence in loc.
cit. only uses their assumption on $p$ in one of the two direction, although
they never state this explicitly. Using their restriction on $p$, we can also
prove that Abe’s classification of simple modules in terms of standard simple
supersingular triples lifts to the homotopy level.

######  Corollary 3.28.

Let ${\mathfrak{m}}=I_{\mathcal{H}}(P,\mathfrak{n},Q)$ and
${\mathfrak{m}}^{\prime}=I_{\mathcal{H}}(P^{\prime},\mathfrak{n}^{\prime},Q^{\prime})$
be two simple ${\mathcal{H}}$-modules and write $P=MN$ for the Levi
decomposition of $P$.

  1. (i)

Assume ${\mathfrak{m}}$ has finite projective dimension. Then
$\mathfrak{n}\in{\operatorname{Mod}}({\mathcal{H}}_{M})$ has finite projective
dimension.

  2. (ii)

Assume that $p\nmid\left|\Omega_{\text{tor}}\right|$ and that
${\operatorname{pd}}_{{\mathcal{H}}}({\mathfrak{m}})={\operatorname{pd}}_{{\mathcal{H}}}({\mathfrak{m}}^{\prime})=\infty$.
If ${\mathfrak{m}}\cong{\mathfrak{m}}^{\prime}$ in
${\operatorname{Ho}}({\mathcal{H}})$ then $P=P^{\prime}$, $Q=Q^{\prime}$ and
$\mathfrak{n}\cong\mathfrak{n}^{\prime}$ in
${\operatorname{Ho}}({\mathcal{H}}_{M})$.

###### Proof.

(i) Let $L$ denote the Levi subgroup of $Q$. Then we have
$0=\mathbf{R}\mathcal{R}^{G}_{L}(I_{\mathcal{H}}(P,\mathfrak{n},Q))\cong
I_{{\mathcal{H}}_{L}}(P\cap L,\mathfrak{n},L)$ by 3.27, so that we may assume
that $Q=G$. Applying the left adjoint instead, we now have

$\mathfrak{n}\cong
I_{{\mathcal{H}}_{M}}(M,\mathfrak{n},M)\cong{\operatorname{Ho}}({\mathcal{L}^{G}_{M}})(I_{\mathcal{H}}(P,\mathfrak{n},G))=0$

in ${\operatorname{Ho}}({\mathcal{H}}_{M})$, as required.

(ii) First suppose that $Q\neq Q^{\prime}$. Then we may assume that
$Q\nsubseteq Q^{\prime}$. Write $L^{\prime}$ for the Levi subgroup of
$Q^{\prime}$. Then applying $\mathbf{R}\mathcal{R}^{G}_{M^{\prime}}$ to
${\mathfrak{m}}\cong{\mathfrak{m}}^{\prime}$, we obtain $0\cong
I_{{\mathcal{H}}_{L^{\prime}}}(P^{\prime}\cap
L^{\prime},\mathfrak{n}^{\prime},L^{\prime})$ by 3.27, contradicting our
assumption that ${\mathfrak{m}}^{\prime}$ has infinite projective dimension by
[20, Theorem 8.5]. Thus $Q=Q^{\prime}$, and applying
$\mathbf{R}\mathcal{R}^{G}_{L^{\prime}}$ again we may assume that
$Q=Q^{\prime}=G$ as above. Similarly now, if $P\neq P^{\prime}$ then without
loss of generality we have $P\subsetneqq P^{\prime}$ and if we write
$M^{\prime}$ for the Levi subgroup of $P^{\prime}$ then we obtain a
contradiction from 3.27 by applying
${\operatorname{Ho}}(\mathcal{L}^{G}_{M^{\prime}})$. ∎

Going back to our recollement, we write ‘all’ instead of ‘${\mathcal{M}}$’ in
the subscripts of all the functors and categories featuring in 3.13(i) when
the collection ${\mathcal{M}}$ of standard Levi subgroups considered is that
of _all_ proper standard Levi subgroups. We now characterise precisely which
simple modules of infinite projective dimension lie in
${\operatorname{Ho}}({\mathcal{H}})_{\text{all}}$, viewed as a full
subcategory of ${\operatorname{Ho}}({\mathcal{H}})$ via either
$\ell^{G}_{\text{all}}$ or $r^{G}_{\text{all}}$.

######  Corollary 3.29.

Suppose that $p\nmid\left|\Omega_{\text{tor}}\right|$, and let
$\mathfrak{s}=I_{\mathcal{H}}(P,\mathfrak{n},Q)\in{\operatorname{Mod}}({\mathcal{H}})$
be a simple module of infinite projective dimension. Then we have:

  1. (i)

$\mathfrak{s}\in{\operatorname{Ker}}(\mathcal{L}^{G}_{\text{\emph{all}}})$ if
and only if $\Pi=\Pi(\mathfrak{n})$ and $P=Q$; and

  2. (ii)

$\mathfrak{s}\in{\operatorname{Ker}}(\mathcal{R}^{G}_{\text{\emph{all}}})$ if
and only if $Q=P(\mathfrak{n})=G$.

In particular,
$\mathfrak{s}\in{\operatorname{Ker}}(\mathcal{L}^{G}_{\text{\emph{all}}})\cap{\operatorname{Ker}}(\mathcal{R}^{G}_{\text{\emph{all}}})$
if and only if it is supersingular.

###### Proof.

First observe that for any standard parabolic subgroup
$P^{\prime}=M^{\prime}N^{\prime}$ of $G$, we have that
$I_{{\mathcal{H}}_{M^{\prime}}}(P,\mathfrak{n},Q\cap P^{\prime})$ and
$I_{{\mathcal{H}}_{M^{\prime}}}(P,\mathfrak{n},Q)$ also have infinite
projective dimension, whenever these modules are well-defined. Indeed, this
follows from [20, Theorem 8.5] by our assumption on $p$ and because
$\mathfrak{s}$ is assumed to have infinite projective dimension. The statement
of (ii) then follows from 3.27, and for (i) we get

(3.7)
$\mathfrak{s}\in{\operatorname{Ker}}(\mathcal{L}^{G}_{\text{all}})\iff\text{
for all }P\subseteq P^{\prime}\subsetneqq
G,\;\Pi(\mathfrak{n})\nsubseteq\Pi_{Q}\cup\Pi_{P^{\prime}}.$

So we have to check that the right hand side is equivalent to
$\Pi=\Pi(\mathfrak{n})$ and $P=Q$. One direction is clear. For the other, note
that if $\Pi\neq\Pi(\mathfrak{n})$ then $P^{\prime}=P(\mathfrak{n})$ makes the
right hand side of (3.7) fail. Moreover, if $\Pi=\Pi(\mathfrak{n})$ and $P\neq
Q$ then note that when $Q=G$ the right hand side of (3.7) fails for all
choices of $P^{\prime}$. So assume that $\Pi=\Pi(\mathfrak{n})$ and
$P\subsetneqq Q\subsetneqq G$, in which case we let $P^{\prime}$ be the
standard parabolic corresponding to
$\Pi_{P}\cup(\Pi\setminus\Pi_{Q})\subsetneqq\Pi$. Once again $P^{\prime}$
makes the right hand side of (3.7) fail. This completes the proof of (i). ∎

######  Remark 3.30.

The corresponding result at the abelian level, without any assumption on the
projective dimension, is also true with the same proof, using 3.23 instead of
its homotopy variant. Hence a simple module, or any finite length module more
generally, is supersingular if and only if it is killed by the functors
$\mathcal{L}^{G}_{M}$ and $\mathcal{R}^{G}_{M}$ for every proper standard Levi
subgroup $M$.

Assuming that the root system of $G$ is irreducible, we can extend the above
to all modules of finite length.

######  Proposition 3.31.

Suppose that $p\nmid\left|\Omega_{\text{tor}}\right|$. When the root system of
$G$ is irreducible, the following are equivalent for an ${\mathcal{H}}$-module
${\mathfrak{m}}$ of finite length:

  1. (i)

${\mathfrak{m}}\in{\operatorname{Ker}}(\mathcal{L}^{G}_{\text{\emph{all}}})$;

  2. (ii)

${\mathfrak{m}}\in{\operatorname{Ker}}(\mathcal{R}^{G}_{\text{\emph{all}}})$;
and

  3. (iii)

the inclusion ${\mathfrak{m}}_{\text{ss}}\subseteq{\mathfrak{m}}$ is a weak
equivalence, where ${\mathfrak{m}}_{\text{ss}}$ denotes the largest
supersingular submodule of ${\mathfrak{m}}$.

###### Proof.

From 3.18, both (iii)$\Rightarrow$(i) and (iii)$\Rightarrow$(ii) are clear.
For the converse, fix an ${\mathcal{H}}$-module ${\mathfrak{m}}$ of finite
length and let $M_{1},\ldots,M_{n}$ denote all the proper Levi subgroups of
$G$. Let $\epsilon\in\\{-,+\\}$. For each $1\leq i\leq n$, we let
$\lambda_{i}^{\epsilon}\in\widetilde{\Lambda}$ be chosen as in 2.1 for the
Levi subgroup $M_{i}$ and we write
$\mathcal{O}^{\epsilon}_{i}=\widetilde{W}\cdot\lambda_{i}^{\epsilon}$ and
$z_{i}^{\epsilon}:=z_{\mathcal{O}^{\epsilon}_{i}}$. Then let
${\mathfrak{m}}_{i}^{\epsilon}=\\{x\in{\mathfrak{m}}\mid(z^{\epsilon}_{i})^{l}x=0\text{
for some $l>0$}\\}$ and ${\mathfrak{m}}^{\epsilon}_{\leq i}:=\cap_{j\leq
i}{\mathfrak{m}}^{\epsilon}_{j}=({\mathfrak{m}}^{\epsilon}_{\leq i-1})_{i}$.
Note that $z^{\epsilon}_{i}$ then acts invertibly on the quotient
${\mathfrak{m}}/{\mathfrak{m}}^{\epsilon}_{i}$ for any $i$, so that any simple
subquotient of ${\mathfrak{m}}/{\mathfrak{m}}^{\epsilon}_{i}$ is not
supersingular. Hence we see iteratively that any simple subquotient of
${\mathfrak{m}}/{\mathfrak{m}}^{+}_{\leq n}$ is not supersingular and
similarly that any simple subquotient of ${\mathfrak{m}}^{+}_{\leq
n}/({\mathfrak{m}}^{+}_{\leq n})^{-}_{\leq n}$ is not supersingular. But we
have that ${\mathfrak{m}}_{\text{ss}}=({\mathfrak{m}}^{+}_{\leq n})^{-}_{\leq
n}$ by 3.30 and 3.19. Thus we see that every simple subquotient of
${\mathfrak{m}}_{\text{non-ss}}:={\mathfrak{m}}/{\mathfrak{m}}_{\text{ss}}$ is
not supersingular. We will show that either (i) or (ii) implies that
${\mathfrak{m}}_{\text{non-ss}}$ has finite projective dimension, and thus
that (iii) holds by [16, Lemma 5.8].

Suppose first that $\mathfrak{s}=I_{\mathcal{H}}(P,\mathfrak{n},Q)$ is a
simple subquotient of ${\mathfrak{m}}_{\text{non-ss}}$. If
$P(\mathfrak{n})\neq G$ then $\mathfrak{s}$ is in the essential image of
$I^{G}_{M(\mathfrak{n})}$. If instead $P(\mathfrak{n})=G$ then $P=B$ since
$P\neq G$ and the root system is irreducible, and thus $\mathfrak{s}$ has
finite projective dimension by [20, Theorem 8.5] and the fact that
${\mathcal{H}}_{T}$ has finite global dimension (c.f. 3.6). Either way, it
follows that $\mathfrak{s}\in{\mathcal{S}}^{G}_{\text{all}}$ and hence that
${\mathfrak{m}}_{\text{non-ss}}\in{\mathcal{S}}^{G}_{\text{all}}$.

Next, by applying the functors $\mathcal{L}^{G}_{\text{all}}$ and
$\mathcal{R}^{G}_{\text{all}}$ to the distinguished triangle
${\mathfrak{m}}_{\text{ss}}\to{\mathfrak{m}}\to{\mathfrak{m}}_{\text{non-
ss}}\to\Sigma{\mathfrak{m}}_{\text{ss}}$, we obtain from the above and 3.18
that

(3.8)
$\mathcal{R}^{G}_{\text{all}}({\mathfrak{m}})\cong\mathcal{R}^{G}_{\text{all}}({\mathfrak{m}}_{\text{non-
ss}})\text{ and
}\mathcal{L}^{G}_{\text{all}}({\mathfrak{m}})\cong\mathcal{L}^{G}_{\text{all}}({\mathfrak{m}}_{\text{non-
ss}}).$

But since ${\mathfrak{m}}_{\text{non-ss}}\in{\mathcal{S}}^{G}_{\text{all}}$ we
see from the recollement in 3.13 that

(3.9) ${\mathfrak{m}}_{\text{non-
ss}}\in{\operatorname{Ker}}(\mathcal{L}^{G}_{\text{all}})\iff{\mathfrak{m}}_{\text{non-
ss}}\in{\operatorname{Ker}}(\mathcal{R}^{G}_{\text{all}})\iff{\mathfrak{m}}_{\text{non-
ss}}=0,$

where the latter equality is in ${\operatorname{Ho}}({\mathcal{H}})$. Thus,
combining (3.8) and (3.9), we obtain that either one of (i) or (ii) implies
that ${\mathfrak{m}}_{\text{non-ss}}$ has finite projective dimension as
claimed. ∎

When the hypotheses of 3.31 are satisfied we tentatively call the quotient
category ${\operatorname{Ho}}({\mathcal{H}})_{\text{all}}$ the _supersingular
homotopy category_ of ${\mathcal{H}}$. We will see later that this
triangulated category is in fact the homotopy category of a model structure on
${\operatorname{Mod}}({\mathcal{H}})$.

######  Remark 3.32.

If we write ${\mathcal{T}}$ for
${\operatorname{Ker}}(\mathcal{L}^{G}_{\text{all}})$, then 3.31 tells us that
the modules of finite length in ${\mathcal{T}}$ are, up to weak equivalence,
the supersingular ones whenever the root system of $G$ is irreducible and
$p\nmid\left|\Omega_{\text{tor}}\right|$. Moreover we have
${\mathcal{T}}^{\ddagger}={\mathcal{S}}^{G}_{\text{all}}$ and
${\mathcal{T}}={{}^{\ddagger}}{\mathcal{S}}^{G}_{\text{all}}$. This is similar
to the torsion pair constructed at the abelian level in [27] where they showed
that when $\mathbb{G}=\text{SL}_{2}$, $p>2$ and
$\mathfrak{F}\neq\mathbb{Q}_{p}$, then the supersingular modules are in a
torsion pair with a certain class of modules which they call ‘non-
supersingular’ and which contains all parabolically induced modules, c.f.
Theorems 3.26 & 3.27 in loc. cit. Of course, our Hom-orthogonality boils down
to Ext-orthogonality between suitable replacements at the abelian level, so
that the two questions are not immediately related. But we hope to study
elsewhere the relationship between the Ollivier-Schneider torsion pair and the
adjoints of parabolic induction in a more general setting.

##  4\. Bousfield localisations from recollements

###  4.1. Constructing model structures from a recollement

It is natural to ask whether the various quotient categories
${\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}$ that were constructed in
the last section are actually the homotopy category of some model structure on
${\operatorname{Mod}}({\mathcal{H}})$. To answer this we work in as general a
framework as we can.

######  Setup 4.1.

We let ${\mathcal{A}}$ be a Grothendieck abelian category equipped with a
cofibrantly generated, abelian model structure
$({\mathcal{C}}_{\mathcal{A}},{\mathcal{F}}_{\mathcal{A}},{\mathcal{W}}_{\mathcal{A}})$.
We will work in one of the following two situations:

  1. (i)

the model structure on ${\mathcal{A}}$ is injective, i.e.
${\mathcal{C}}_{\mathcal{A}}={\mathcal{A}}$; or

  2. (ii)

the model structure on ${\mathcal{A}}$ is projective, i.e.
${\mathcal{F}}_{\mathcal{A}}={\mathcal{A}}$.

Any projective/injective abelian model structure on an abelian category
${\mathcal{A}}$ is automatically hereditary and thus stable, and we let
${\mathcal{S}}$ be a thick subcategory of
${\operatorname{Ho}}({\mathcal{A}})$. Then we assume that in either situation
the inclusion
$i:{\mathcal{S}}\stackrel{{\scriptstyle\subseteq}}{{\longrightarrow}}{\operatorname{Ho}}({\mathcal{A}})$
is part of a recollement

(4.1)
$\textstyle{{\mathcal{S}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{{\operatorname{Ho}}({\mathcal{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{\pi_{\mathcal{S}}}}$$\scriptstyle{\mathbf{L}{\mathsf{L}}}$$\scriptstyle{\mathbf{R}{\mathsf{R}}}$$\textstyle{{\operatorname{Ho}}({\mathcal{A}})/{\mathcal{S}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ell_{\mathcal{S}}}$$\scriptstyle{r_{\mathcal{S}}}$

where the functors $\mathbf{L}{\mathsf{L}}$ and $\mathbf{R}{\mathsf{R}}$ are
not a priori assumed to be the total derived functor of some Quillen functors.

We recall that the class of trivially fibrant objects in an injective abelian
model category equals the class $\mathrm{Inj}({\mathcal{A}})$ of injective
objects, and dually the class of trivially cofibrant objects in a projective
abelian model category equals the class $\mathrm{Proj}({\mathcal{A}})$
projective objects (c.f. [7, Corollary 1.1.9]).

Our aim is to localise the model structure on ${\mathcal{A}}$ at
${\mathcal{S}}$ in a suitable way, or more precisely to construct an abelian
model structure on ${\mathcal{A}}$ whose homotopy category is the quotient
${\operatorname{Ho}}({\mathcal{A}})/{\mathcal{S}}$. Our main result is:

######  Theorem 4.2.

Let ${\mathcal{A}}$ be a Grothendieck abelian category equipped with a
cofibrantly generated, abelian model structure which is either injective or
projective. Suppose that ${\mathcal{S}}$ is a thick subcategory of
${\operatorname{Ho}}({\mathcal{A}})$ satisfying 4.1. Let
${\mathcal{W}}_{\mathcal{S}}$, resp. ${\mathcal{F}}_{\mathcal{S}}$, resp.
${\mathcal{C}}_{\mathcal{S}}$, denote the class of objects of ${\mathcal{A}}$
whose image in ${\operatorname{Ho}}({\mathcal{A}})$ lies in ${\mathcal{S}}$,
resp. ${\operatorname{Ker}}(\mathbf{R}{\mathsf{R}})$, resp.
${\operatorname{Ker}}(\mathbf{L}{\mathsf{L}})$.

  1. (i)

If the model structure on ${\mathcal{A}}$ is injective then
${\mathcal{A}}^{i}_{\mathcal{S}}=({\mathcal{A}},{\mathcal{F}}_{\mathcal{A}}\cap{\mathcal{F}}_{\mathcal{S}},{\mathcal{W}}_{\mathcal{S}})$
defines another cofibrantly generated, injective, abelian model structure on
${\mathcal{A}}$ such that there is a canonical equivalence
${\operatorname{Ho}}({\mathcal{A}}^{i}_{\mathcal{S}})\cong{\operatorname{Ho}}({\mathcal{A}})/{\mathcal{S}}$,
the adjunction
${\operatorname{id}}:{\mathcal{A}}\rightleftarrows{\mathcal{A}}^{i}_{\mathcal{S}}:{\operatorname{id}}$
is Quillen and the induced total derived adjunction is naturally isomorphic to
the adjoint pair $(\pi_{\mathcal{S}},r_{\mathcal{S}})$.

  2. (ii)

Dually, if the model structure on ${\mathcal{A}}$ is projective then
${\mathcal{A}}^{p}_{\mathcal{S}}=({\mathcal{C}}_{\mathcal{A}}\cap{\mathcal{C}}_{\mathcal{S}},{\mathcal{A}},{\mathcal{W}}_{\mathcal{S}})$
defines another cofibrantly generated, projective, abelian model structure on
${\mathcal{A}}$ such that there is a canonical equivalence
${\operatorname{Ho}}({\mathcal{A}}^{p}_{\mathcal{S}})\cong{\operatorname{Ho}}({\mathcal{A}})/{\mathcal{S}}$,
the adjunction
${\operatorname{id}}:{\mathcal{A}}^{p}_{\mathcal{S}}\rightleftarrows{\mathcal{A}}:{\operatorname{id}}$
is Quillen and the induced total derived adjunction is naturally isomorphic to
the adjoint pair $(\ell_{\mathcal{S}},\pi_{\mathcal{S}})$.

######  Remark 4.3.

We note that Becker had a similar aim when he constructed his singular
projective/injective model structures on categories of modules over a DG-
algebra in [7], see in particular Propositions 1.4.2 & 1.4.6 in loc. cit.
However his situation was more specific, namely he was working in the setting
where he was trying to localise an abelian model structure at another model
structure on the same underlying abelian category, in such a way that the
homotopy category of one model structure embedded fully faithfully as a thick
subcategory of the homotopy category of the other. Our situation is therefore
more general since we do not assume that ${\mathcal{S}}$ is the homotopy
category of some other injective/projective model structure on
${\mathcal{A}}$, although we have to assume that our model structure is
cofibrantly generated, that we have a recollement at the homotopy level and
that the underlying category is Grothendieck abelian, which Becker didn’t
assume. In fact, the above result will allow us to apply Becker’s
constructions, see 4.7, and our construction in 4.8 generalises Proposition
2.2.1 in loc. cit., which was only concerned with modules over a DG-algebra.

To prove the theorem, we need to check that the conditions in 2.7 are
satisfied. First we have:

######  Lemma 4.4.

The classes of objects ${\mathcal{C}}_{\mathcal{S}}$,
${\mathcal{F}}_{\mathcal{S}}$ and ${\mathcal{W}}_{\mathcal{S}}$ defined in 4.2
are thick.

###### Proof.

Since short exact sequences are distinguished triangles, the result follows
immediately from the fact that ${\mathcal{S}}$,
${\operatorname{Ker}}(\mathbf{R}{\mathsf{R}})$, and
${\operatorname{Ker}}(\mathbf{L}{\mathsf{L}})$ are thick subcategories of the
triangulated category ${\operatorname{Ho}}({\mathcal{A}})$. ∎

Next, we characterise the trivially fibrant/cofibrant objects in the proposed
new model structures.

######  Lemma 4.5.

Suppose that we’re in 4.1. When ${\mathcal{A}}$ is injective then
${\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{F}}_{\mathcal{S}}={\mathcal{W}}_{\mathcal{A}}$,
so that
${\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{F}}_{\mathcal{A}}\cap{\mathcal{F}}_{\mathcal{S}}={\mathcal{W}}_{\mathcal{A}}\cap{\mathcal{F}}_{\mathcal{A}}=\mathrm{Inj}({\mathcal{A}})$.
Dually, when ${\mathcal{A}}$ is projective then
${\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{C}}_{\mathcal{S}}={\mathcal{W}}_{\mathcal{A}}$,
so that
${\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{C}}_{\mathcal{A}}\cap{\mathcal{C}}_{\mathcal{S}}={\mathcal{W}}_{\mathcal{A}}\cap{\mathcal{C}}_{\mathcal{A}}=\mathrm{Proj}({\mathcal{A}})$.

###### Proof.

Again we only prove the injective case, the other one being dual. Note that
${\mathcal{W}}_{\mathcal{A}}\cap{\mathcal{F}}_{\mathcal{A}}=\mathrm{Inj}({\mathcal{A}})$
is immediate from the injectivity of the model structure on ${\mathcal{A}}$,
so that the first equality immediately implies the second. For the first
equality, we have
${\mathcal{W}}_{\mathcal{A}}\subseteq{\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{F}}_{\mathcal{S}}$
by definition of the right hand side. Conversely, assume that
$X\in{\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{F}}_{\mathcal{S}}$. Then $X\cong
i(Y)$ in ${\operatorname{Ho}}({\mathcal{A}})$, for some $Y\in{\mathcal{S}}$,
and $X\in{\operatorname{Ker}}(\mathbf{R}{\mathsf{R}})$. But then we obtain
from 2.4 that $Y=0$ in ${\operatorname{Ho}}({\mathcal{A}})$, and hence that
$X\in{\mathcal{W}}_{\mathcal{A}}$ as required. ∎

Let ${\mathcal{A}}$ be a Grothendieck abelian category. Recall that a
cotorsion pair $({\mathcal{D}},{\mathcal{E}})$ on ${\mathcal{A}}$ is said to
be _cogenerated_ by a set $\mathcal{G}\subseteq{\mathcal{D}}$ if
${\mathcal{E}}=\mathcal{G}^{\perp}$. Furthermore, by definition
${\mathcal{A}}$ has a generating set $\mathscr{S}$ (or equivalently a
generator), which is a set of objects of ${\mathcal{A}}$ such that every
object of ${\mathcal{A}}$ is a quotient of a direct sum of objects in
$\mathscr{S}$. A cotorsion pair on ${\mathcal{A}}$ is called _small_ if it is
cogenerated by a set and if ${\mathcal{D}}$ contains a generating set for
${\mathcal{A}}$. We note that small cotorsion pairs are automatically
functorially complete and that an abelian model structure
$({\mathcal{C}},{\mathcal{F}},{\mathcal{W}})$ on ${\mathcal{A}}$ is
cofibrantly generated if and only if both
$({\mathcal{C}},{\mathcal{F}}\cap{\mathcal{W}})$ and
$({\mathcal{C}}\cap{\mathcal{W}},{\mathcal{F}})$ are small, c.f. [16,
Corollary 6.6] and [7, Proposition 1.2.7].

We will also need a notion of generating set for a triangulated category
${\mathcal{T}}$. A set of objects $\mathcal{G}$ of ${\mathcal{T}}$ will be
called a _set of weak generators_ if an object $X\in{\mathcal{T}}$ is zero if
and only if ${\operatorname{Hom}}_{\mathcal{T}}(G,\Sigma^{n}X)=0$ for all
$n\in{\mathbb{Z}}$. The relevance of this notion to us is the result [15,
Theorem 7.3.1], which in particular implies that the homotopy category
${\operatorname{Ho}}({\mathcal{A}})$ of a hereditary, abelian, cofibrantly
generated model category ${\mathcal{A}}$ has a set of weak generators given by
the cokernels of the generating cofibrations. The proof of loc. cit. requires
some input from the homotopy theory of simplicial sets, so we deemed it
worthwhile to provide a more elementary proof in the case where we’re
considering the module category of a Gorenstein ring. In fact we obtain a
stronger result than weak generation.

######  Proposition 4.6.

If $S$ is a Gorenstein ring, then a module
${\mathfrak{m}}\in{\operatorname{Mod}}(S)$ has infinite projective dimension
if and only if there exists a left ideal $J$ such that
$[S/J,{\mathfrak{m}}]_{S}\neq 0$. In particular, the cyclic modules
$\\{S/J\mid J\text{ a left ideal of }S\\}$ form a set of weak generators for
${\operatorname{Ho}}(S)$.

###### Proof.

We view ${\operatorname{Ho}}(S)$ as the homotopy category of the Gorenstein
injective model structure. Suppose that
${\mathfrak{m}}\in{\operatorname{Mod}}(S)$ has the property that
$[S/J,{\mathfrak{m}}]_{S}=0$ for all left ideals $J$. Without loss of
generality, we may take ${\mathfrak{m}}$ to be Gorenstein injective. We first
claim that if $\mathfrak{n}\in{\operatorname{Mod}}(S)$ is finitely generated
then $[\mathfrak{n},{\mathfrak{m}}]_{S}=0$. This follows by induction on the
number $l$ of generators of $\mathfrak{n}$. Indeed, $\mathfrak{n}$ fits into a
short exact sequence
$0\to\mathfrak{n}^{\prime}\to\mathfrak{n}\to\mathfrak{n}^{\prime\prime}\to 0$
where $\mathfrak{n}^{\prime}$ is generated by $l-1$ elements and
$\mathfrak{n}^{\prime\prime}$ is cyclic, and we may view this exact sequence
as a distinguished triangle in ${\operatorname{Ho}}(S)$ and apply the
cohomological functor $[-,{\mathfrak{m}}]_{S}$ to get the claim.

We now claim that ${\mathfrak{m}}$ is an injective module. Indeed, if $J$ is
any left ideal of $S$ and $f:J\to{\mathfrak{m}}$ is any map, then we must show
that $f$ extends to $S$. But the Noetherianity of $S$ implies that $J$ is
finitely generated and thus, by the above, $f$ must be zero in
${\operatorname{Ho}}(S)$. This implies that $f$ factors through a map
$g:J\to\mathfrak{e}$ for some injective module $\mathfrak{e}$ by 2.14. Since
$\mathfrak{e}$ is injective it follows that $g$ extends to $S$, and thus so
does $f$ as required. ∎

###### Proof of 4.2.

We first note that 4.5 implies that
$({\mathcal{A}},{\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{F}}_{\mathcal{A}}\cap{\mathcal{F}}_{\mathcal{S}})=({\mathcal{A}},\mathrm{Inj}({\mathcal{A}}))$
in case (i) and
$({\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{C}}_{\mathcal{A}}\cap{\mathcal{C}}_{\mathcal{S}},{\mathcal{A}})=(\mathrm{Proj}({\mathcal{A}}),{\mathcal{A}})$
in case (ii) are small cotorsion pairs, c.f. [7, Example 1.2.6]. Since
${\mathcal{W}}_{\mathcal{S}}$ is thick by 4.4, we are left to deal with the
remaining two pairs.

(i) We need to construct a set $\mathcal{G}$ of objects of ${\mathcal{A}}$
such that
${\mathcal{F}}_{\mathcal{A}}\cap{\mathcal{F}}_{\mathcal{S}}=\mathcal{G}^{\perp}$
and ${\mathcal{W}}_{\mathcal{S}}={{}^{\perp}}(\mathcal{G}^{\perp})$. By
assumption on ${\mathcal{A}}$, we have
${\mathcal{F}}_{\mathcal{A}}=\mathcal{G}_{1}^{\perp}$ for some set
$\mathcal{G}_{1}$ of objects in ${\mathcal{W}}_{\mathcal{A}}$. By enlarging
$\mathcal{G}_{1}$ if necessary, we may assume that it contains a generating
set for ${\mathcal{A}}$, since
$({\mathcal{W}}_{\mathcal{A}},{\mathcal{F}}_{\mathcal{A}})$ is small. We now
let $\mathscr{S}$ be a set of weak generators for
${\operatorname{Ho}}({\mathcal{A}})$ and, for each $S\in\mathscr{S}$ and each
$n\in{\mathbb{Z}}$, we choose
$X_{n,S}\in{\mathcal{A}}={\mathcal{C}}_{\mathcal{A}}$ such that
$X_{n,S}\cong\Sigma^{-n}i\mathbf{L}{\mathsf{L}}(S)$ in
${\operatorname{Ho}}({\mathcal{A}})$ and let $\mathcal{G}_{2}=\\{X_{n,S}\mid
S\in\mathscr{S}\text{ and }n\in{\mathbb{Z}}\\}$. Then we claim that
$\mathcal{G}=\mathcal{G}_{1}\cup\mathcal{G}_{2}$ satisfies our requirements.
To see this, first note that if $Y\in\mathcal{G}^{\perp}$ then in particular
$Y\in\mathcal{G}_{1}^{\perp}={\mathcal{F}}_{\mathcal{A}}$. So in order to
obtain
${\mathcal{F}}_{\mathcal{A}}\cap{\mathcal{F}}_{\mathcal{S}}=\mathcal{G}^{\perp}$
it suffices to show that for $Y\in{\mathcal{F}}_{\mathcal{A}}$, we have
$Y\in\mathcal{G}_{2}^{\perp}$ if and only if
$Y\in{\mathcal{F}}_{\mathcal{S}}$. Indeed, by 2.16 we have

$\displaystyle Y\in\mathcal{G}_{2}^{\perp}$
$\displaystyle\iff{\operatorname{Hom}}_{{\operatorname{Ho}}({\mathcal{A}})}(\Sigma^{-n}i\mathbf{L}{\mathsf{L}}(S),\Sigma
Y)=0\text{ for all }n\in{\mathbb{Z}}\text{ and }S\in\mathscr{S}$
$\displaystyle\iff{\operatorname{Hom}}_{{\operatorname{Ho}}({\mathcal{A}})}(S,\Sigma^{n+1}i\mathbf{R}{\mathsf{R}}(Y))=0\text{
for all }n\in{\mathbb{Z}}\text{ and }S\in\mathscr{S}$ $\displaystyle\iff
i\mathbf{R}{\mathsf{R}}(Y)=0\iff Y\in{\mathcal{F}}_{\mathcal{S}}$

as required. Next, note that every object of ${\mathcal{F}}_{\mathcal{S}}$ is
isomorphic in ${\operatorname{Ho}}({\mathcal{A}})$ to an object of
${\mathcal{F}}_{\mathcal{A}}\cap{\mathcal{F}}_{\mathcal{S}}$. Thus we see by
applying 2.16 again that an object $X\in{\mathcal{A}}$ lies in
${}^{\perp}({\mathcal{F}}_{\mathcal{A}}\cap{\mathcal{F}}_{\mathcal{S}})$ if
and only if the image of $X$ in ${\operatorname{Ho}}({\mathcal{A}})$ lies in
${}^{\ddagger}{\operatorname{Ker}}(\mathbf{R}{\mathsf{R}})={\mathcal{S}}$.

This shows that
$({\mathcal{W}}_{\mathcal{S}},{\mathcal{F}}_{\mathcal{A}}\cap{\mathcal{F}}_{\mathcal{S}})$
is a small cotorsion pair, and thus we have that
${\mathcal{A}}^{i}_{\mathcal{S}}$ defines an injective, cofibrantly generated,
abelian model structure as required. By 4.5 we have that the trivial
fibrations in ${\mathcal{A}}$ and in ${\mathcal{A}}^{i}_{\mathcal{S}}$ are the
same, and thus
${\operatorname{id}}:{\mathcal{A}}^{i}_{\mathcal{S}}\to{\mathcal{A}}$ is right
Quillen. Furthermore, since the trivialy fibrant objects in both model
structures are the same we have that
$\mathbf{R}{\operatorname{id}}:{\operatorname{Ho}}({\mathcal{A}}^{i}_{\mathcal{S}})\to{\operatorname{Ho}}({\mathcal{A}})$
identifies, after restricting to the cofibrant-fibrant objects, with the
canonical embedding
$({\mathcal{F}}_{\mathcal{S}}\cap{\mathcal{F}}_{\mathcal{A}})/\mathrm{Inj}({\mathcal{A}})\hookrightarrow{\mathcal{F}}_{\mathcal{A}}/\mathrm{Inj}({\mathcal{A}})$
with image
${\operatorname{Ker}}(\mathbf{R}{\mathsf{R}})\simeq{\operatorname{Ho}}({\mathcal{A}})/{\mathcal{S}}$,
where the latter equivalence is induced by $r_{\mathcal{S}}$. Thus, after
these identifications, the left adjoint of $\mathbf{R}{\operatorname{id}}$ is
naturally isomorphic to $\pi_{\mathcal{S}}$.

(ii) As above, we let $\mathscr{S}$ be a set of weak generators for
${\operatorname{Ho}}({\mathcal{A}})$. Then for each $S\in\mathscr{S}$ and each
$n\in{\mathbb{Z}}$, we choose $X_{n,S}\in{\mathcal{C}}_{\mathcal{A}}$ such
that $X_{n,S}\cong\Sigma^{-n}\ell_{\mathsf{G}}\pi_{\mathsf{G}}(S)$ in
${\operatorname{Ho}}({\mathcal{A}})$ and let $\mathcal{G}=\\{X_{n,S}\mid
S\in\mathscr{S}\text{ and }n\in{\mathbb{Z}}\\}$. Since the model structure on
${\mathcal{A}}$ is projective, there is a projective generator $Z$ for
${\mathcal{A}}$ and we may apply [16, Corollary 6.9] to see that $\mathcal{G}$
cogenerates a small cotorsion pair $({\mathcal{D}},{\mathcal{E}})$, where
${\mathcal{D}}$ is the closure of $\mathcal{G}\cup\\{Z\\}$ under transfinite
compositions and direct summands. As ${\mathcal{C}}_{\mathcal{A}}$ is closed
under those (c.f. Lemma 6.2 in loc. cit.), it follows that
${\mathcal{D}}\subseteq{\mathcal{C}}_{\mathcal{A}}$. We first show that
${\mathcal{E}}=\mathcal{G}^{\perp}={\mathcal{W}}_{\mathcal{S}}$. This is
similar to the argument in (i). Since every object of $\mathcal{G}$ is
cofibrant, we may apply 2.16 to see that

$\displaystyle Y\in\mathcal{G}^{\perp}$
$\displaystyle\iff{\operatorname{Hom}}_{{\operatorname{Ho}}({\mathcal{A}})}(S,\Sigma^{n+1}r_{\mathsf{G}}\pi_{\mathsf{G}}(Y))=0\text{
for all }n\in{\mathbb{Z}}\text{ and }S\in\mathscr{S}$ $\displaystyle\iff
r_{\mathsf{G}}\pi_{\mathsf{G}}(Y)=0\iff Y\in{\mathcal{W}}_{\mathcal{S}}$

as required, where the last equivalence holds because
${\operatorname{Ker}}(\pi_{\mathsf{G}})={\mathcal{S}}$. Next, since we have
${\mathcal{D}}\subseteq{\mathcal{C}}_{\mathcal{A}}$ it suffices to show that a
cofibrant $X\in{\mathcal{C}}_{\mathcal{A}}$ lies in
${}^{\perp}{\mathcal{W}}_{\mathcal{S}}$ if and only if it lies in
${\mathcal{C}}_{\mathcal{S}}$, in order to obtain that
${\mathcal{D}}={\mathcal{C}}_{\mathcal{A}}\cap{\mathcal{C}}_{\mathcal{S}}$.
This is now analogous to (i): we have by 2.16 that
$X\in{{}^{\perp}}{\mathcal{W}}_{\mathcal{S}}$ if and only if the image of $X$
in ${\operatorname{Ho}}({\mathcal{A}})$ lies in
${}^{\ddagger}{\mathcal{S}}={\operatorname{Ker}}(\mathbf{L}{\mathsf{L}})$,
i.e. if and only if $X\in{\mathcal{C}}_{\mathcal{S}}$, as required. This
concludes the proof that the abelian model structure
${\mathcal{A}}^{i}_{\mathcal{S}}$ exists and is cofibrantly generated. The
last part is completely dual to (i). ∎

The construction of these model structures now give us the input necessary to
apply the techniques of Becker in [7]. We then obtain the following result.

######  Corollary 4.7.

We keep the same assumptions as in 4.2.

  1. (i)

If the model structure on ${\mathcal{A}}$ is injective, then
${\mathcal{A}}_{i}^{\mathcal{S}}=({\mathcal{W}}_{\mathcal{S}},{\mathcal{F}}_{\mathcal{A}},{\mathcal{F}}_{\mathcal{S}})$
defines a hereditary, cofibrantly generated, abelian model structure on
${\mathcal{A}}$ whose homotopy category is canonically equivalent to
${\mathcal{S}}$. Furthermore, the adjunction
${\operatorname{id}}:{\mathcal{A}}_{i}^{\mathcal{S}}\rightleftarrows{\mathcal{A}}:{\operatorname{id}}$
is Quillen and its total derived adjunction is naturally isomorphic to the
adjoint pair $(i,\mathbf{R}{\mathsf{R}})$.

  2. (ii)

If the model structure on ${\mathcal{A}}$ is projective, then
${\mathcal{A}}_{p}^{\mathcal{S}}=({\mathcal{C}}_{\mathcal{A}},{\mathcal{W}}_{\mathcal{S}},{\mathcal{C}}_{\mathcal{S}})$
defines a hereditary, cofibrantly generated, abelian model structure on
${\mathcal{A}}$ whose homotopy category is canonically equivalent to
${\mathcal{S}}$. Furthermore, the adjunction
${\operatorname{id}}:{\mathcal{A}}\rightleftarrows{\mathcal{A}}_{p}^{\mathcal{S}}:{\operatorname{id}}$
is Quillen and its total derived adjunction is naturally isomorphic to the
adjoint pair $(\mathbf{L}{\mathsf{L}},i)$.

###### Proof.

Everything follow immediately from 4.2 and [7, Propositions 1.4.2 & 1.4.6],
except the cofibrant generation for which we need to check that the associated
cotorsion pairs are small. In the injective case we have
$({\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{F}}_{\mathcal{S}},{\mathcal{F}}_{\mathcal{A}})=({\mathcal{W}}_{\mathcal{A}},{\mathcal{F}}_{\mathcal{A}})$
and in the projective case we have
$({\mathcal{C}}_{\mathcal{A}},{\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{C}}_{\mathcal{S}})=({\mathcal{C}}_{\mathcal{A}},{\mathcal{W}}_{\mathcal{A}})$
, c.f. 4.5, and so it follows from that these are small. Furthermore, the
proof 4.2 shows that
$({\mathcal{W}}_{\mathcal{S}},{\mathcal{F}}_{\mathcal{A}}\cap{\mathcal{F}}_{\mathcal{S}})$,
in the injective case, and
$({\mathcal{C}}_{\mathcal{A}}\cap{\mathcal{C}}_{\mathcal{S}},{\mathcal{W}}_{\mathcal{S}})$,
in the projective case, are small as well. ∎

We would like to be able to recover the entire recollement in one go, while
the above only gives us one half of it in either situation. Furthermore, the
model structures constructed in 4.7 are no longer injective/projective. We now
see how to fix this by adding one extra hypothesis.

######  Proposition 4.8.

Suppose that ${\mathcal{A}}$ is a Grothendieck abelian category equipped with
both an injective model structure
$({\mathcal{A}},{\mathcal{F}}_{\mathcal{A}},W_{\mathcal{A}})$ and a projective
model structure
$({\mathcal{C}}_{\mathcal{A}},{\mathcal{A}},{\mathcal{W}}_{\mathcal{A}})$ with
the same classes of trivial objects, and let
${\operatorname{Ho}}({\mathcal{A}})$ denote their common homotopy category.
Assume that ${\mathcal{S}}$ is a thick subcategory of
${\operatorname{Ho}}({\mathcal{A}})$ satisfying the requirements of 4.1. Let
${\mathcal{W}}_{\mathcal{S}}$, ${\mathcal{F}}_{\mathcal{S}}$ and
${\mathcal{C}}_{\mathcal{S}}$ be as in 4.2.

  1. (i)

${{}^{\ddagger}}{\mathcal{A}}^{i}_{\mathcal{S}}=({\mathcal{A}},{\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{F}}_{\mathcal{A}},{\mathcal{C}}_{\mathcal{S}})$
defines an injective, cofibrantly generated, abelian model structure on
${\mathcal{A}}$. Furthermore, the identity
${\operatorname{id}}:{{}^{\ddagger}}{\mathcal{A}}^{i}_{\mathcal{S}}\to{\mathcal{A}}_{i}^{\mathcal{S}}$
is a right Quillen equivalence, the adjunction
${\operatorname{id}}:{\mathcal{A}}\rightleftarrows{{}^{\ddagger}}{\mathcal{A}}^{i}_{\mathcal{S}}:{\operatorname{id}}$
is Quillen (for the injective model structure on ${\mathcal{A}}$) and its
total derived adjunction is naturally isomorphic to the adjoint pair
$(\mathbf{L}{\mathsf{L}},i)$.

  2. (ii)

${{}^{\ddagger}}{\mathcal{A}}^{p}_{\mathcal{S}}=({\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{C}}_{\mathcal{A}},{\mathcal{A}},{\mathcal{F}}_{\mathcal{S}})$
defines a projective, cofibrantly generated, abelian model structure on
${\mathcal{A}}$. Furthermore, the identity
${\operatorname{id}}:{{}^{\ddagger}}{\mathcal{A}}^{p}_{\mathcal{S}}\to{\mathcal{A}}_{p}^{\mathcal{S}}$
is a left Quillen equivalence, the adjunction
${\operatorname{id}}:{{}^{\ddagger}}{\mathcal{A}}^{p}_{\mathcal{S}}\rightleftarrows{\mathcal{A}}:{\operatorname{id}}$
is Quillen (for the projective model structure on ${\mathcal{A}}$) and its
total derived adjunction is naturally isomorphic to the adjoint pair
$(i,\mathbf{R}{\mathsf{R}})$.

Finally, the identity
${\operatorname{id}}:{{}^{\ddagger}}{\mathcal{A}}^{i}_{\mathcal{S}}\to{{}^{\ddagger}}{\mathcal{A}}^{p}_{\mathcal{S}}$
is a right Quillen equivalence.

###### Proof.

(i) By 4.5, the pair
$({\mathcal{A}},{\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{F}}_{\mathcal{A}}\cap{\mathcal{C}}_{\mathcal{S}})=({\mathcal{A}},\mathrm{Inj}({\mathcal{A}}))$
is a small cotorsion pair. Furthermore, by assumption and by 4.7(ii) we have
that both ${\mathcal{W}}_{\mathcal{S}}$ an ${\mathcal{F}}_{\mathcal{A}}$ are
of the form $\mathcal{G}^{\perp}$, hence so is their intersection. Thus we
only need to show that
${\mathcal{C}}_{\mathcal{S}}={{}^{\perp}}({\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{F}}_{\mathcal{A}})$
to get a small cotorsion pair. Since every object of
${\mathcal{W}}_{\mathcal{S}}$ is isomorphic in
${\operatorname{Ho}}({\mathcal{A}})$ to an object of of
${\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{F}}_{\mathcal{A}}$ and since those
are fibrant in ${\mathcal{A}}$, we may apply 2.16 to deduce that an object
lies in
${{}^{\perp}}({\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{F}}_{\mathcal{A}})$ if
and only if its image in ${\operatorname{Ho}}({\mathcal{A}})$ lies in
${}^{\ddagger}{\mathcal{S}}={\operatorname{Ker}}(\mathbf{L}{\mathsf{L}})$ as
required.

The functors
${\operatorname{id}}:{{}^{\ddagger}}{\mathcal{A}}^{i}_{\mathcal{S}}\to{\mathcal{A}}_{i}^{\mathcal{S}}$
and
${\operatorname{id}}:{{}^{\ddagger}}{\mathcal{A}}^{i}_{\mathcal{S}}\to{\mathcal{A}}$
both clearly preserve fibrant and trivially fibrant objects so are right
Quillen. In fact, since the trivially fibrant objects in
${{}^{\ddagger}}{\mathcal{A}}^{i}_{\mathcal{S}}$,
${\mathcal{A}}_{i}^{\mathcal{S}}$ and ${\mathcal{A}}$ are the same one gets,
after restricting to cofibrant-fibrant objects, that
$\mathbf{R}{\operatorname{id}}:{\operatorname{Ho}}({{}^{\ddagger}}{\mathcal{A}}^{i}_{\mathcal{S}})\to{\operatorname{Ho}}({\mathcal{A}}_{i}^{\mathcal{S}})$
identifies with the identity functor on
${\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{F}}_{\mathcal{A}}/\mathrm{Inj({\mathcal{A}})}$
and
$\mathbf{R}{\operatorname{id}}:{\operatorname{Ho}}({{}^{\ddagger}}{\mathcal{A}}^{i}_{\mathcal{S}})\to{\operatorname{Ho}}({\mathcal{A}})$
identifies with the canonical inclusion
${\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{F}}_{\mathcal{A}}/\mathrm{Inj({\mathcal{A}})}\hookrightarrow{\mathcal{F}}_{\mathcal{A}}/\mathrm{Inj({\mathcal{A}})}$,
as required.

(ii) Again, by 4.5 the pair
$(W_{\mathcal{S}}\cap{\mathcal{C}}_{\mathcal{A}}\cap{\mathcal{F}}_{\mathcal{S}},{\mathcal{A}})=(\mathrm{Proj}({\mathcal{A}}),{\mathcal{A}})$
is a small cotorsion pair. Furthermore, by 4.7(i) there exists a set
$\mathcal{G}\subseteq{\mathcal{W}}_{\mathcal{S}}$ such that
$\mathcal{G}^{\perp}={\mathcal{F}}_{\mathcal{A}}\cap{\mathcal{F}}_{\mathcal{S}}$.
Now define a set
$\mathcal{G}^{\prime}\subseteq{\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{C}}_{\mathcal{A}}$
by letting the objects of $\mathcal{G}^{\prime}$ be a choice of cofibrant
replacements in ${\mathcal{A}}$ for the objects of $\mathcal{G}$. Since these
are cofibrant, we may apply 2.16 again to deduce that
$(\mathcal{G}^{\prime})^{\perp}={\mathcal{F}}_{\mathcal{S}}$. Furthermore, the
pair $({{}^{\perp}}{\mathcal{F}}_{\mathcal{S}},{\mathcal{F}}_{\mathcal{S}})$
is now a small cotorsion pair, such that
${{}^{\perp}}{\mathcal{F}}_{\mathcal{S}}$ is the closure of
$\mathcal{G}^{\prime}\cup\\{Z\\}$ under transfinite compositions and direct
summands by [16, Lemma 6.9], where $Z$ is a choice of projective generator for
${\mathcal{A}}$. Since both ${\mathcal{W}}_{\mathcal{S}}$ and
${\mathcal{C}}_{\mathcal{A}}$ are closed under those by Lemma 6.2 in loc. cit.
and 4.7(i), we have
${{}^{\perp}}{\mathcal{F}}_{\mathcal{S}}\subseteq{\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{C}}_{\mathcal{A}}$.
To show equality it therefore suffices to show that an object of
${\mathcal{C}}_{\mathcal{A}}$ lies in
${{}^{\perp}}{\mathcal{F}}_{\mathcal{S}}$ if and only if it lies in
${\mathcal{W}}_{\mathcal{S}}$. This is proved via the usual argument using
2.16. The statements on the Quillen functors are proved dually to (i).

Finally,
${\operatorname{id}}:{{}^{\ddagger}}{\mathcal{A}}^{i}_{\mathcal{S}}\to{{}^{\ddagger}}{\mathcal{A}}^{p}_{\mathcal{S}}$
will by definition be a right Quillen equivalence if, for
$X\in{\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{C}}_{\mathcal{A}}$ and
$Y\in{\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{F}}_{\mathcal{A}}$, a map
$f:X\to Y$ is a weak equivalence in
${{}^{\ddagger}}{\mathcal{A}}^{p}_{\mathcal{S}}$ if and only if it is a weak
equivalence in ${{}^{\ddagger}}{\mathcal{A}}^{i}_{\mathcal{S}}$. Note that the
trivial cofibrations in ${{}^{\ddagger}}{\mathcal{A}}^{p}_{\mathcal{S}}$ are
precisely the monomorphisms with projective cokernel, and so are still weak
equivalences in ${{}^{\ddagger}}{\mathcal{A}}^{i}_{\mathcal{S}}$. Thus, by
factorising $f$ in ${{}^{\ddagger}}{\mathcal{A}}^{p}_{\mathcal{S}}$ as a
trivial cofibration followed by a fibration we may assume that $f$ is a
fibration. Then it is a weak equivalence in
${{}^{\ddagger}}{\mathcal{A}}^{p}_{\mathcal{S}}$ if and only if its kernel
lies in ${\mathcal{F}}_{\mathcal{S}}$. But since both $X$ and $Y$ lie in
${\mathcal{W}}_{\mathcal{S}}$ by assumption and ${\mathcal{W}}_{\mathcal{S}}$
is thick, this happens if and only if
$\ker(f)\in{\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{F}}_{\mathcal{S}}={\mathcal{W}}_{\mathcal{A}}$,
c.f. 4.5. Similarly, $f$ is a weak equivalence in
${{}^{\ddagger}}{\mathcal{A}}^{i}_{\mathcal{S}}$ if and only if
$\ker(f)\in{\mathcal{W}}_{\mathcal{S}}\cap{\mathcal{C}}_{\mathcal{S}}={\mathcal{W}}_{\mathcal{A}}$.
∎

By applying Becker’s construction [7, Propositions 1.4.2 & 1.4.6] to
${{}^{\ddagger}}{\mathcal{A}}^{i}_{\mathcal{S}}$ and
${{}^{\ddagger}}{\mathcal{A}}^{p}_{\mathcal{S}}$, we obtain the last piece of
the recollement.

######  Corollary 4.9.

Keep the same assumptions as in 4.8. Then the triple
$({\mathcal{A}}_{i}^{\mathcal{S}})^{\ddagger}=({\mathcal{C}}_{\mathcal{S}},{\mathcal{F}}_{\mathcal{A}},{\mathcal{W}}_{\mathcal{S}})$
defines a hereditary, abelian model structure on ${\mathcal{A}}$ whose
homotopy category is canonically equivalent to
${\operatorname{Ho}}({\mathcal{A}})/{\mathcal{S}}$, and the adjunction
${\operatorname{id}}:({\mathcal{A}}_{i}^{\mathcal{S}})^{\ddagger}\rightleftarrows{\mathcal{A}}:{\operatorname{id}}$
is Quillen (for the injective model structure on ${\mathcal{A}}$) with total
derived adjunction naturally isomorphic to
$(\ell_{\mathcal{S}},\pi_{\mathcal{S}})$. Dually, the triple
$({\mathcal{A}}_{p}^{\mathcal{S}})^{\ddagger}=({\mathcal{C}}_{\mathcal{A}},{\mathcal{F}}_{\mathcal{S}},{\mathcal{W}}_{\mathcal{S}})$
defines a hereditary, abelian model structure on ${\mathcal{A}}$ whose
homotopy category is canonically equivalent to
${\operatorname{Ho}}({\mathcal{A}})/{\mathcal{S}}$, and the adjunction
${\operatorname{id}}:{\mathcal{A}}\rightleftarrows({\mathcal{A}}_{p}^{\mathcal{S}})^{\ddagger}:{\operatorname{id}}$
is Quillen (for the projective model structure on ${\mathcal{A}}$) with total
derived adjunction naturally isomorphic to
$(\pi_{\mathcal{S}},r_{\mathcal{S}})$.

Hence, sticking with the projective model structures, 4.2(ii), 4.7(ii),
4.8(ii) and 4.9 together say that if we are in 4.1 in such a way that both an
injective and a projective model structure exist in ${\mathcal{A}}$ with the
same class of trivial objects, then by using only the projective model
structure we may construct four new abelian model structures on
${\mathcal{A}}$ such that the four adjunctions from the recollement (4.1) are
obtained as total derived functors of the various identity adjunctions between
our new model structures and the projective model structure on
${\mathcal{A}}$.

In particular, by considering
${\mathcal{A}}={\operatorname{Mod}}({\mathcal{H}})$ with its Gorenstein
projective and injective model structures and by letting
${\mathcal{M}}=\\{M_{1},\ldots,M_{n}\\}$ be some fixed collection standard
Levi subgroups of $G$, the above tells us that the recollement in 3.13(i) may
be reconstructed from various model structures on
${\operatorname{Mod}}({\mathcal{H}})$ by only using the Gorenstein projective
model structure. If we define

$\displaystyle{\mathcal{W}}^{G}_{\mathcal{M}}$
$\displaystyle:=\left\\{{\mathfrak{m}}\in{\operatorname{Mod}}({\mathcal{H}})\mid{\mathfrak{m}}\in{\mathcal{S}}^{G}_{\mathcal{M}}\right\\},$
$\displaystyle{\mathcal{C}}^{G}_{\mathcal{M}}:$
$\displaystyle=\left\\{{\mathfrak{m}}\in{\operatorname{Mod}}({\mathcal{H}})\mid{\mathfrak{m}}\in{\operatorname{Ker}}(\mathcal{L}^{G}_{\mathcal{M}})\right\\}$
$\displaystyle=\left\\{{\mathfrak{m}}\in{\operatorname{Mod}}({\mathcal{H}})\mid{\operatorname{pd}}_{{\mathcal{H}}_{M_{i}}}(\mathcal{L}^{G}_{M_{i}}({\mathfrak{m}}))<\infty\text{
for all }1\leq i\leq n\right\\},\quad\text{and}$
$\displaystyle{\mathcal{F}}^{G}_{\mathcal{M}}:$
$\displaystyle=\left\\{{\mathfrak{m}}\in{\operatorname{Mod}}({\mathcal{H}})\mid{\mathfrak{m}}\in{\operatorname{Ker}}(\mathcal{R}^{G}_{\mathcal{M}})\right\\}$
$\displaystyle=\left\\{{\mathfrak{m}}\in{\operatorname{Mod}}({\mathcal{H}})\mid{\mathfrak{m}}\in\bigcap_{1\leq
i\leq n}{\operatorname{Ker}}(\mathbf{R}\mathcal{R}^{G}_{M_{i}})\right\\},$

where the second equalities for ${\mathcal{C}}^{G}_{\mathcal{M}}$ and
${\mathcal{F}}^{G}_{\mathcal{M}}$ both follow from 3.13(iii), then in
particular 4.2(ii) and 4.8(ii) say in this case that there are two cofibrantly
generated, projective abelian model structures
${{}^{\ddagger}}{\operatorname{Mod}}({\mathcal{H}})^{G}_{\mathcal{M}}$ and
${\operatorname{Mod}}({\mathcal{H}})^{G}_{\mathcal{M}}$ with cofibrant objects
${\mathcal{C}}^{G}_{\mathcal{M}}\cap\mathrm{GProj}({\mathcal{H}})$ and
${\mathcal{W}}^{G}_{\mathcal{M}}\cap\mathrm{GProj}({\mathcal{H}})$,
respectively, and trivial objects ${\mathcal{W}}^{G}_{\mathcal{M}}$ and
${\mathcal{F}}^{G}_{\mathcal{M}}$, respectively, so that
${\operatorname{Ho}}({\operatorname{Mod}}({\mathcal{H}})^{G}_{\mathcal{M}})\simeq{\mathcal{S}}^{G}_{\mathcal{M}}$
and
${\operatorname{Ho}}({{}^{\ddagger}}{\operatorname{Mod}}({\mathcal{H}})^{G}_{\mathcal{M}})\simeq{\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}$.

######  Remark 4.10.

Consider the case $n=1$ and write $M=M_{1}$. Then since
$I^{G}_{M}:{\operatorname{Mod}}({\mathcal{H}}_{M})^{\text{GP}}\to{{}^{\ddagger}}{\operatorname{Mod}}({\mathcal{H}})^{G}_{M}$
is a right Quillen equivalence by construction and since
${\operatorname{id}}:{\operatorname{Mod}}({\mathcal{H}})^{G}_{M}\to{\operatorname{Mod}}({\mathcal{H}})^{\text{GP}}$
is _left_ Quillen, it follows formally that ${\operatorname{Ho}}(I^{G}_{M})$
has a right adjoint. While we knew this already, this provides a way of seeing
this without ever appealing to the Gorenstein injective model structure.

Recall next that there is an adjunction

$F_{M}:{\operatorname{Mod}}({\mathcal{H}}_{M})\rightleftarrows{\operatorname{Rep}}_{k}^{\infty}(M):U_{M}$

where
$F_{M}({\mathfrak{m}}):={\mathbb{X}}_{M}\otimes_{{\mathcal{H}}_{M}}{\mathfrak{m}}$
and $U_{M}(V):={\operatorname{Hom}}_{M}({\mathbb{X}}_{M},V)\cong V^{I_{M}}$
for any ${\mathfrak{m}}\in{\operatorname{Mod}}({\mathcal{H}}_{M})$ and
$V\in{\operatorname{Rep}}_{k}^{\infty}(M)$. When $M=G$, we drop the subscript
and write $F$ and $U$ for the corresponding functors. On the smooth
representations side, the parabolic induction functor
${\operatorname{Ind}}_{P}^{G}:{\operatorname{Rep}}_{k}^{\infty}(M)\to{\operatorname{Rep}}_{k}^{\infty}(G)$
is defined by first inflating smooth representations of $M\cong P/N$ to $P$
and then applying the smooth induction functor. This functor has both a left
and right adjoint (c.f. [32, Proposition 4.3]), and its left adjoint can be
described explicitly as the Jacquet functor $V\mapsto J_{P}^{G}(V):=V_{N}$
where $V_{N}$ denotes the $N$-coinvariants of $V$. This construction is
compatible with the parabolic induction for ${\mathcal{H}}$-modules by a
result of Ollivier-Vignéras.

######  Theorem 4.11 ([28, Theorem 1]).

Let $M$ be a standard Levi subgroup of $G$. Then we have a natural isomorphism
of functors
$U\circ{\operatorname{Ind}}_{P}^{G}\cong{\operatorname{Ind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}\circ
U_{M}$.

We note in passing that the above immediately implies the analogue of 3.4(i)
on the smooth representations side.

######  Corollary 4.12.

The $I$-Gorenstein projective model structure on
${\operatorname{Rep}}_{k}^{\infty}(M)$ is the right transfer of the
$I$-Gorenstein projective model structure on
${\operatorname{Rep}}_{k}^{\infty}(G)$ along ${\operatorname{Ind}}_{P}^{G}$.

###### Proof.

We need to show that for a morphism $f$ in
${\operatorname{Rep}}_{k}^{\infty}(M)$, $f$ is a fibration (resp. weak
equivalence) if and only if ${\operatorname{Ind}}_{P}^{G}(f)$ is a fibration
(resp. weak equivalence). But these notions are by definition completely
characterised by passing to pro-$p$ Iwahori invariants. So the result follows
immediately from 4.11 and 3.4(i). ∎

We finish by right transferring our new model structures on
${\operatorname{Mod}}({\mathcal{H}})$ to
${\operatorname{Rep}}_{k}^{\infty}(G)$.

######  Proposition 4.13.

For any collection ${\mathcal{M}}$ standard Levi subgroups of $G$, the right
transfer of the model structures
${\operatorname{Mod}}({\mathcal{H}})^{G}_{\mathcal{M}}$ and
${{}^{\ddagger}}{\operatorname{Mod}}({\mathcal{H}})^{G}_{\mathcal{M}}$ to
${\operatorname{Rep}}_{k}^{\infty}(G)$ along $U$ both exist.

###### Proof.

Since
${\operatorname{id}}:{\operatorname{Mod}}({\mathcal{H}})^{\text{GP}}\to{\operatorname{Mod}}({\mathcal{H}})^{G}_{\mathcal{M}}$
and
${\operatorname{id}}:{\operatorname{Mod}}({\mathcal{H}})^{\text{GP}}\to{{}^{\ddagger}}{\operatorname{Mod}}({\mathcal{H}})^{G}_{\mathcal{M}}$
are right Quillen, they in particular preserve path objects. Hence path
objects in ${\operatorname{Mod}}({\mathcal{H}})^{G}_{\mathcal{M}}$ and
${{}^{\ddagger}}{\operatorname{Mod}}({\mathcal{H}})^{G}_{\mathcal{M}}$ can be
computed exactly as in the Gorenstein projective model structure. Therefore,
the arguments of [10, Proposition 4.10] apply mutatis mutandis to yield the
result. ∎

It is unclear to us at this stage whether the two other model structures
constructed in 4.7(ii) and 4.9, applied to
${\operatorname{Mod}}({\mathcal{H}})$, can be right transferred to
${\operatorname{Rep}}_{k}^{\infty}(G)$ as well. We note that these model
structures are not projective and thus the standard techniques used in [10] in
order to establish the existence of right transfers now require one to
construct a fibrant replacement functor as well as path objects for these
right transfers. An in-depth study of this question, of the right transfers
obtained in 4.13 and of the derived adjunction
$(\mathbf{L}J_{P}^{G},\mathbf{R}{\operatorname{Ind}}_{P}^{G})$ will be done
elsewhere.

###  4.2. Bousfield localisation

We finish this section by explaining how the projective model structures
constructed in the previous subsection are Bousfield localisations of the
projective model structure on ${\mathcal{A}}$. Our approach here is very
similar to [7, §1.5], and in fact Proposition 1.5.3 in loc. cit. shows that
this is automatically true for the model structures constructed in 4.7 and
4.9. We therefore focus on the model structures constructed in 4.2(ii) and
4.8(ii). First we recall the notion of Bousfield localisation from [14]. We
only state the definition in the context of pointed model categories to make
the presentation simpler, but this notion can be defined for all model
categories.

######  Definition 4.14.

Let $\mathscr{M}$ be a pointed model category and $\mathrm{S}$ be a class of
morphisms in $\mathscr{M}$.

  * •

An object $X\in\mathscr{M}$ is called an _$\mathrm{S}$ -colocal object_ if it
is cofibrant and for every morphism $f:A\to B$ in $\mathrm{S}$, the induced
map
${\operatorname{Hom}}_{{\operatorname{Ho}}(\mathscr{M})}(\Sigma^{n}X,A)\to{\operatorname{Hom}}_{{\operatorname{Ho}}(\mathscr{M})}(\Sigma^{n}X,B)$
is a bijection for all $n\geq 0$.

  * •

A morphism $f:A\to B$ in $\mathscr{M}$ is called an _$\mathrm{S}$ -colocal
equivalence_ if for every $\mathrm{S}$-colocal object $X$, the induced map
${\operatorname{Hom}}_{{\operatorname{Ho}}(\mathscr{M})}(\Sigma^{n}X,A)\to{\operatorname{Hom}}_{{\operatorname{Ho}}(\mathscr{M})}(\Sigma^{n}X,B)$
is a bijection for all $n\geq 0$.

The notions of _$\mathrm{S}$ -local object_ and _$\mathrm{S}$ -local
equivalence_ are defined dually to the above.

We note that (co)local objects and equivalences are usually defined in terms
of homotopy function complexes in the literature (c.f. [14, Definition
3.1.4]). The above definition is equivalent to that because for pointed model
categories the homotopy groups of these function complexes are precisely the
${\operatorname{Hom}}$ sets we considered above, see [15, Lemma 6.1.2]. With
this notion at hand, we have the following definition of Bousfield
localisation.

######  Definition 4.15 ([14, Definition 3.3.1]).

Let $\mathscr{M}$ be a pointed model category and $\mathrm{S}$ be a class of
morphisms in $\mathscr{M}$. The _right Bousfield localisation_ of
$\mathscr{M}$ with respect to $\mathrm{S}$ is, if it exists, the model
structure $\mathrm{R}_{\mathrm{S}}\mathscr{M}$ on the same underlying category
as $\mathscr{M}$, with fibrations the same as in $\mathscr{M}$, weak
equivalences the $\mathrm{S}$-colocal equivalences, and cofibrations the maps
satisfying the left lifting property with respect to the trivial fibrations.
The notion of _left Bousfield localisation_ of $\mathscr{M}$ with respect to
$\mathrm{S}$ is defined dually.

######  Lemma 4.16.

Let ${\mathcal{A}}$ be a hereditary abelian model category, $X$ an object of
${\mathcal{A}}$, and $f$ a morphism in ${\mathcal{A}}$.

  1. (i)

In the arrow category of ${\operatorname{Ho}}({\mathcal{A}})$, $f$ is
isomorphic to a fibration between fibrant objects.

  2. (ii)

If $f$ is a fibration between fibrant objects, then $X$ is $f$-colocal if and
only if

${\operatorname{Hom}}_{{\operatorname{Ho}}({\mathcal{A}})}(\Sigma^{n}X,\ker(f))=0$

for all $n\geq 0$.

###### Proof.

These statements are proved in steps (1)-(3) of the proof of [7, Proposition
1.5.3] for injective model structure, but works equally well in general. ∎

######  Proposition 4.17.

Let ${\mathcal{A}}$ be an abelian category equipped with a projective abelian
model structure $({\mathcal{C}},{\mathcal{A}},{\mathcal{W}})$, and let
${\mathcal{S}}$ be a thick subcategory of
${\operatorname{Ho}}({\mathcal{A}})$. Let ${\mathcal{W}}_{\mathcal{S}}$ denote
the class of objects of ${\mathcal{A}}$ whose image in
${\operatorname{Ho}}({\mathcal{A}})$ lies in ${\mathcal{S}}$.

  1. (i)

Suppose that the inclusion
${\mathcal{S}}\subset{\operatorname{Ho}}({\mathcal{A}})$ has a left adjoint
$\mathbf{L}{\mathsf{L}}$. Let ${\mathcal{C}}_{\mathcal{S}}$ denote the class
of objects of ${\mathcal{A}}$ whose image in
${\operatorname{Ho}}({\mathcal{A}})$ lies in
${\operatorname{Ker}}(\mathbf{L}{\mathsf{L}})$, and assume that
$({\mathcal{C}}_{\mathcal{S}}\cap{\mathcal{C}},{\mathcal{A}},{\mathcal{W}}_{\mathcal{S}})$
defines an abelian model structure on ${\mathcal{A}}$. Then this model
structure is the right Bousfield localisation of ${\mathcal{A}}$ with respect
to $\mathrm{S}:=\\{0\to X\mid X\in{\mathcal{W}}_{\mathcal{S}}\\}$; and

  2. (ii)

Suppose that the inclusion
${\mathcal{S}}\subset{\operatorname{Ho}}({\mathcal{A}})$ has a right adjoint
$\mathbf{R}{\mathsf{R}}$. Let ${\mathcal{F}}_{\mathcal{S}}$ denote the class
of objects of ${\mathcal{A}}$ whose image in
${\operatorname{Ho}}({\mathcal{A}})$ lies in
${\operatorname{Ker}}(\mathbf{R}{\mathsf{R}})$, and assume that
$({\mathcal{C}}\cap{\mathcal{W}}_{\mathcal{S}},{\mathcal{A}},{\mathcal{F}}_{\mathcal{S}})$
defines an abelian model structure on ${\mathcal{A}}$. Then this model
structure is the right Bousfield localisation of ${\mathcal{A}}$ with respect
to $\mathrm{S}^{\prime}:=\\{0\to X\mid X\in{\mathcal{F}}_{\mathcal{S}}\\}$.

###### Proof.

We only prove (i), the proof of (ii) being entirely analogous. By the dual of
[21, Proposition 4.9.1], we have
${}^{\ddagger}{\mathcal{S}}={\operatorname{Ker}}(\mathbf{L}{\mathsf{L}})$ and
${\mathcal{S}}={\operatorname{Ker}}(\mathbf{L}{\mathsf{L}})^{\ddagger}$. By
definition of $\mathrm{S}$, the class of $\mathrm{S}$-colocal objects is that
of those cofibrant objects $X$ whose image in
${\operatorname{Ho}}({\mathcal{A}})$ lies in
${}^{\ddagger}{\mathcal{S}}={\operatorname{Ker}}(\mathbf{L}{\mathsf{L}})$,
i.e. the class of $\mathrm{S}$-colocal objects is
${\mathcal{C}}_{\mathcal{S}}\cap{\mathcal{C}}$. Then 4.16 implies that an
$\mathrm{S}$-colocal equivalence is a morphism which, in the arrow category of
${\operatorname{Ho}}({\mathcal{A}})$, is isomorphic to a epimorphism with
kernel in ${\mathcal{W}}_{\mathcal{S}}$.

Now assume that
$({\mathcal{C}}_{\mathcal{S}}\cap{\mathcal{C}},{\mathcal{A}},{\mathcal{W}}_{\mathcal{S}})$
defines an abelian model structure, which we denote by
${\mathcal{A}}_{\mathcal{S}}$. Then we must show that the weak equivalences in
${\mathcal{A}}_{\mathcal{S}}$ are the $\mathrm{S}$-colocal equivalences. For
this, note first that
${\operatorname{id}}:{\mathcal{A}}\to{\mathcal{A}}_{\mathcal{S}}$ is right
Quillen since ${\mathcal{W}}\subseteq{\mathcal{W}}_{\mathcal{S}}$, and we get
a functor
${\operatorname{Ho}}({\operatorname{id}}):{\operatorname{Ho}}({\mathcal{A}})\to{\operatorname{Ho}}({\mathcal{A}}_{\mathcal{S}})$
by 2.8. Hence one gets that an $\mathrm{S}$-colocal equivalence is also
isomorphic to a epimorphism with kernel in ${\mathcal{W}}_{\mathcal{S}}$ in
the arrow category of ${\operatorname{Ho}}({\mathcal{A}}_{\mathcal{S}})$, and
hence is a weak equivalence in ${\mathcal{A}}_{\mathcal{S}}$. For the
converse, suppose that $f$ is a weak equivalence in
${\mathcal{A}}_{\mathcal{S}}$. Note first that
${\mathcal{C}}_{\mathcal{S}}\cap{\mathcal{C}}\cap{\mathcal{W}}_{\mathcal{S}}=\mathrm{Proj}({\mathcal{A}})={\mathcal{C}}\cap{\mathcal{W}}$,
so that the trivial cofibrations in ${\mathcal{A}}$ and in
${\mathcal{A}}_{\mathcal{S}}$ coincide. By factorising $f$ in
${\mathcal{A}}_{\mathcal{S}}$ as a composite of a trivial cofibration and a
trivial fibration, we therefore see that $f$ is isomorphic in the arrow
category of ${\operatorname{Ho}}({\mathcal{A}})$ to a trivial fibration in
${\mathcal{A}}_{\mathcal{S}}$, i.e. an epimorphism with kernel in
${\mathcal{W}}_{\mathcal{S}}$, as required. ∎

######  Corollary 4.18.

The projective model structures constructed in 4.2 and 4.8 are right Bousfield
localisations.

Of course the above has completely dual analogues. In particular, the
injective model structures constructed in 4.2 and 4.8 are left Bousfield
localisations.

##  5\. Simple supersingular Hecke modules in the homotopy category

For the remainder of this paper, we assume that $k$ is algebraically closed.
Our main goal is to determine the isomorphism classes of simple
${\mathcal{H}}$-modules in ${\operatorname{Ho}}({\mathcal{H}})$. By 3.28(ii),
we are reduced to investigating this question for the simple supersingular
modules.

###  5.1. The affine Hecke algebra and simple supersingular
${\mathcal{H}}$-modules

We first recall the definition of the affine Hecke algebra
${\mathcal{H}_{\operatorname{aff}}}$, the classification of simple characters
over it, and the classification of simple supersingular
${\mathcal{H}}$-modules.

Fix a set $\\{\hat{s}\mid s\in
S\\}\subseteq\widetilde{W_{\operatorname{aff}}}$ of lifts of the simple affine
reflections defined as in [26, §4.8]. Then
${\mathcal{H}_{\operatorname{aff}}}$ is generated as a $k$-algebra by
$\\{T_{t}\mid t\in T(\mathbb{F}_{q})\\}$ and $\\{T_{\hat{s}}\mid s\in S\\}$.
In addition, each irreducible component $\Phi^{\prime}$ of $\Phi$ gives rise
to a subset $S^{\prime}\subseteq S$ and a subalgebra of
${\mathcal{H}_{\operatorname{aff}}}$ generated by $k[T(\mathbb{F}_{q})]$ and
$\\{T_{\hat{s}}\mid s\in S^{\prime}\\}$. We will refer to these subalgebras as
the irreducible components of ${\mathcal{H}_{\operatorname{aff}}}$. We will
also need a notion of irreducible components of the finite Hecke algebra
${\mathcal{H}}_{F}$ associated to a face $F\subseteq\overline{C}$. First,
since $S$ is in bijection with the nodes of the affine Dynkin diagram of $G$,
we may associate to the set $S_{F}$ the subgraph of the affine Dynkin diagram
spanned by the corresponding nodes. Each connected component of that subgraph
corresponds under the aforementioned bijection to a subset $S_{F^{\prime}}$ of
$S_{F}$, itself corresponding to a face $F^{\prime}$ such that
$F\subseteq\overline{F^{\prime}}$. We will call the subalgebras
${\mathcal{H}}_{F^{\prime}}$ of ${\mathcal{H}}_{F}$ arising from these
connected components the irreducible components of ${\mathcal{H}}_{F}$.

Next, recall from [20, Remark 4.1] that there is an Ollivier-Schneider
resolution of ${\mathcal{H}_{\operatorname{aff}}}$ given as follows. For each
$0\leq i\leq d=r_{ss}$, write $\mathscr{F}_{i}$ for the set of $i$-dimensional
faces $F$ of $\mathscr{X}$ which are contained in $\overline{C}$. Note that by
[19, Lemma 1.2] this is a set of representatives for the
$G_{\operatorname{aff}}$-orbits of the $i$-dimensional faces in $\mathscr{X}$.
Then there an exact sequence of
$({\mathcal{H}_{\operatorname{aff}}},{\mathcal{H}_{\operatorname{aff}}})$-bimodules
of the form

(5.1)
$0\longrightarrow\bigoplus_{F\in\mathscr{F}_{d}}{\mathcal{H}_{\operatorname{aff}}}(\epsilon_{F})\otimes_{{\mathcal{H}}_{F}}{\mathcal{H}_{\operatorname{aff}}}\longrightarrow\cdots\longrightarrow\bigoplus_{F\in\mathscr{F}_{0}}{\mathcal{H}_{\operatorname{aff}}}(\epsilon_{F})\otimes_{{\mathcal{H}}_{F}}{\mathcal{H}_{\operatorname{aff}}}\longrightarrow{\mathcal{H}_{\operatorname{aff}}}\longrightarrow
0.$

Here, for a face $F\subseteq\overline{C}$, the twist of an
${\mathcal{H}}_{F}$-module by the orientation character $\epsilon_{F}$ is the
one defined in [26, §3.3.1]. This gives a free resolution of
${\mathcal{H}_{\operatorname{aff}}}$ as a left and as a right
${\mathcal{H}_{\operatorname{aff}}}$-module, and the methods of loc. cit. then
show that ${\mathcal{H}_{\operatorname{aff}}}$ is $n$-Gorenstein for $n\leq
d$. Thus the category
${\operatorname{Mod}}({\mathcal{H}_{\operatorname{aff}}})$ has a Gorenstein
projective model structure and the homotopy category
${\operatorname{Ho}}({\mathcal{H}_{\operatorname{aff}}})$ is well-defined.
Furthermore, we obtain by tensoring with (5.1) an exact resolution

(5.2)
$0\longrightarrow\bigoplus_{F\in\mathscr{F}_{d}}{\mathcal{H}_{\operatorname{aff}}}(\epsilon_{F})\otimes_{{\mathcal{H}}_{F}}{\mathfrak{m}}\longrightarrow\cdots\longrightarrow\bigoplus_{F\in\mathscr{F}_{0}}{\mathcal{H}_{\operatorname{aff}}}(\epsilon_{F})\otimes_{{\mathcal{H}}_{F}}{\mathfrak{m}}\longrightarrow{\mathfrak{m}}\longrightarrow
0$

for any
${\mathfrak{m}}\in{\operatorname{Mod}}({\mathcal{H}_{\operatorname{aff}}})$.

Recall from [33, Proposition 2.2] that the characters of
${\mathcal{H}_{\operatorname{aff}}}$ are parametrized by pairs $(J,\xi)$ where
$\xi$ is a character of $k[T(\mathbb{F}_{q})]$ and $J\subseteq S_{\xi}$ where
$S_{\xi}$ is as in (2.3). The character $\chi$ corresponding to such a pair is
defined by

$\displaystyle\chi(T_{t})$ $\displaystyle=\xi(t)\quad\text{for all $t\in
T(\mathbb{F}_{q})$}$ $\displaystyle\chi(T_{\hat{s}})$
$\displaystyle=\begin{cases}0&\text{if $s\notin J$}\\\ -1&\text{if $s\in
J$}\end{cases}.$

We will sometimes write $\chi=(J,\xi)$ to mean that $\chi$ is defined as
above. The character $\chi$ is called a _twisted trivial character_ , resp.
_twisted sign character_ , if $S_{\xi}=S$ and $J=\emptyset$, resp. $J=S$.
There is a notion of supersingularity for characters of
${\mathcal{H}_{\operatorname{aff}}}$, and we have that a character $\chi$ is
supersingular if and only if its restriction to each irreducible component of
${\mathcal{H}_{\operatorname{aff}}}$ does not equal the restriction of a
twisted sign or twisted trivial character (c.f. Theorem 6.15 in loc. cit). We
will use this result as a definition of supersingularity for characters of
${\mathcal{H}_{\operatorname{aff}}}$.

Next, the group $\widetilde{\Omega}$ acts on the characters of
${\mathcal{H}_{\operatorname{aff}}}$ as follows. Given a character $\chi$ of
${\mathcal{H}_{\operatorname{aff}}}$ and
$\tilde{\omega}\in\widetilde{\Omega}$, we let
$\chi^{\tilde{\omega}}(h):=\chi(T_{\tilde{\omega}^{-1}}hT_{\tilde{\omega}})$
for $h\in{\mathcal{H}_{\operatorname{aff}}}$. We denote by
$\widetilde{\Omega}_{\chi}$ the stabiliser of $\chi$ under this action. This
is a subgroup of finite index in $\widetilde{\Omega}$, containing
$T(\mathbb{F}_{q})$. We also let ${\mathcal{H}}_{\chi}$ denote the subalgebra
of ${\mathcal{H}}$ generated by ${\mathcal{H}_{\operatorname{aff}}}$ and
$\\{T_{\tilde{\omega}}\mid\tilde{\omega}\in\widetilde{\Omega}_{\chi}\\}$. This
is isomorphic to
${\mathcal{H}_{\operatorname{aff}}}\otimes_{k[T(\mathbb{F}_{q})]}k[\widetilde{\Omega}_{\chi}]$
as a
$({\mathcal{H}_{\operatorname{aff}}},k[\widetilde{\Omega}_{\chi}])$-bimodule,
and as an algebra is isomorphic to the twisted tensor product given by

$(T_{\tilde{w}}\otimes\tilde{\omega})\cdot(T_{\tilde{w}^{\prime}}\otimes\tilde{\omega}^{\prime})=T_{\tilde{w}}T_{\tilde{\omega}^{\prime}\tilde{w}^{\prime}\tilde{\omega}^{-1}}\otimes\tilde{\omega}\tilde{\omega}^{\prime}$

for $\tilde{w},\tilde{w}^{\prime}\in\widetilde{W_{\operatorname{aff}}}$ and
$\tilde{\omega},\tilde{\omega}^{\prime}\in\widetilde{\Omega}_{\chi}$. For
future use, we record here the following:

######  Lemma 5.1.

For any character $\chi$ of ${\mathcal{H}_{\operatorname{aff}}}$, the algebra
${\mathcal{H}}_{\chi}$ is Gorenstein.

###### Proof.

By the braid relations, we have that ${\mathcal{H}}$ is free over
${\mathcal{H}}_{\chi}$. Thus, the functor
${\mathcal{H}}\otimes_{{\mathcal{H}}_{\chi}}(-)$ is exact and preserves
projectives. Since ${\mathcal{H}}$ is $d$-Gorenstein it follows that for each
$\mathfrak{n}\in{\operatorname{Mod}}({\mathcal{H}}_{\chi})$ and for any $i>d$
we have

${\operatorname{Ext}}^{i}_{{\mathcal{H}}_{\chi}}(\mathfrak{n},{\mathcal{H}}\mid_{{\mathcal{H}}_{\chi}})\cong{\operatorname{Ext}}^{i}_{\mathcal{H}}({\mathcal{H}}\otimes_{{\mathcal{H}}_{\chi}}\mathfrak{n},{\mathcal{H}})=0.$

Thus the injective dimension of ${\mathcal{H}}\mid_{{\mathcal{H}}_{\chi}}$ is
bounded above by $d$. Since ${\mathcal{H}}_{\chi}$ is a direct summand of
${\mathcal{H}}\mid_{{\mathcal{H}}_{\chi}}$, it follows that it has finite
selfinjective dimension as a left module over itself. An analogous argument
shows the corresponding statement for right modules.

It remains to show that ${\mathcal{H}}_{\chi}$ is (left and right) Noetherian.
Write $\Omega_{\chi}$ for the image of $\widetilde{\Omega}_{\chi}$ in
$\Omega$. This group decomposes as
$\Omega_{\chi}=\Omega_{\chi}^{\text{free}}\times\Omega_{\chi}^{\text{tor}}$
where $\Omega_{\chi}^{\text{free}}$ is a free abelian group on some generators
$\omega_{1},\ldots,\omega_{r_{Z}}$ and $\Omega_{\chi}^{\text{tor}}$ is finite.
If we let ${\mathcal{H}}_{\chi}^{\text{free}}$ denote the subalgebra of
${\mathcal{H}}_{\chi}$ generated by ${\mathcal{H}_{\operatorname{aff}}}$ and
all $T_{\hat{\omega}_{i}^{\pm 1}}$, then the braid relations show that
${\mathcal{H}}_{\chi}^{\text{free}}$ is an iterated skew Laurent polynomial
algebra over ${\mathcal{H}_{\operatorname{aff}}}$, and hence is Noetherian
because ${\mathcal{H}_{\operatorname{aff}}}$ is Noetherian (c.f. [33, Theorem
5.1] and [22, Theorem 1.4.5]). Since ${\mathcal{H}}_{\chi}$ is finite over
${\mathcal{H}}_{\chi}^{\text{free}}$, it must be Noetherian as well. ∎

We now suppose that $\chi=(J,\xi)$ is a supersingular character of
${\mathcal{H}_{\operatorname{aff}}}$. Given a finite dimensional, irreducible
representation $(V,\rho)$ of $\widetilde{\Omega}_{\chi}$ on which
$T(\mathbb{F}_{q})$ acts by the character $\xi$, we let $\chi\otimes V$ be the
${\mathcal{H}}_{\chi}$-module whose underlying vector space is $V$ and on
which ${\mathcal{H}_{\operatorname{aff}}}$ acts by $\chi$ and
$\widetilde{\Omega}_{\chi}$ acts via $\rho$. Then every simple supersingular
${\mathcal{H}}$-module is isomorphic to one of the form
${\mathcal{H}}\otimes_{{\mathcal{H}}_{\chi}}(\chi\otimes V)$, and furthermore
two pairs $(\chi,V)$ and $(\chi^{\prime},V^{\prime})$ give rise to isomorphic
simple supersingular ${\mathcal{H}}$-modules if and only if the two pairs are
$\widetilde{\Omega}$-conjugate (c.f. [33, Theorem 6.18]). Here we say that
$(\chi,V)$ and $(\chi^{\prime},V^{\prime})$ are $\widetilde{\Omega}$-conjugate
if $\chi^{\prime}=\chi^{\tilde{\omega}}$ for some
$\tilde{\omega}\in\widetilde{\Omega}$ and $V^{\prime}\cong
V^{\tilde{\omega}}$, where $V^{\tilde{\omega}}$ is the representation of
$\widetilde{\Omega}_{\chi^{\tilde{\omega}}}=\tilde{\omega}\widetilde{\Omega}_{\chi}\tilde{\omega}^{-1}$
with underlying vector space $V$ obtained by letting
$\tilde{\omega}\tilde{\omega}^{\prime}\tilde{\omega}^{-1}\cdot
v:=\tilde{\omega}^{\prime}v$ for all $v\in V$ and
$\tilde{\omega}^{\prime}\in\widetilde{\Omega}_{\chi}$.

######  Remark 5.2.

Suppose that $\tilde{\omega},\tilde{\omega}^{\prime}\in\widetilde{\Omega}$.
Since $\Omega$ is abelian, we have that
$\tilde{\omega}\tilde{\omega}^{\prime}\tilde{\omega}^{-1}=t\tilde{\omega}^{\prime}$
for some $t\in T(\mathbb{F}_{q})$. In particular, we deduce that every
subgroup of $\widetilde{\Omega}$ which contains $T(\mathbb{F}_{q})$ is
automatically normal. This applies in particular to the stabiliser subgroups
$\widetilde{\Omega}_{\chi}$ as above, so that we have
$\widetilde{\Omega}_{\chi}=\widetilde{\Omega}_{\chi^{\tilde{\omega}}}$ and
${\mathcal{H}}_{\chi}={\mathcal{H}}_{\chi^{\tilde{\omega}}}$ for all
$\tilde{\omega}\in\widetilde{\Omega}$.

We finally state a theorem of Koziol, which describes when simple
supersingular modules are trivial in the Gorenstein projective model
structure. The statement in loc. cit. doesn’t actually contain (ii) as below,
but their proof actually shows the statement as written here.

######  Theorem 5.3 ([20, Theorem 7.7]).

Suppose that
${\mathfrak{m}}:={\mathcal{H}}\otimes_{{\mathcal{H}}_{\chi}}(\chi\otimes V)$
is a simple supersingular ${\mathcal{H}}$-module as above, and let $\chi$ be
parametrized by $(J,\xi)$. Then the following are equivalent:

  1. (i)

${\operatorname{pd}}_{{\mathcal{H}}}({\mathfrak{m}})<\infty$;

  2. (ii)

${\operatorname{pd}}_{{\mathcal{H}_{\operatorname{aff}}}}(\chi)<\infty$; and

  3. (iii)

$\Phi$ is of type $A_{1}\times\cdots A_{1}$ (possibly empty product) and
$S_{\xi}=S$.

###  5.2. Supersingular characters in
${\operatorname{Ho}}({\mathcal{H}_{\operatorname{aff}}})$

Our first aim is to understand morphisms between supersingular characters in
the homotopy category
${\operatorname{Ho}}({\mathcal{H}_{\operatorname{aff}}})$, assuming that
$G\neq T$. Fix a face $F\subseteq\overline{C}$. Recall that
${\mathcal{H}}_{F}$ is a Frobenius algebra and in particular selfinjective
(c.f. [10, Example 4.1(ii)]), and thus the Gorenstein projective model
structure on ${\operatorname{Mod}}({\mathcal{H}}_{F})$ has all objects
cofibrant and fibrant. In particular, we see from 2.14 that

(5.3)
${\operatorname{Hom}}_{{\operatorname{Ho}}({\mathcal{H}}_{F})}({\mathfrak{m}},\mathfrak{n})\cong{\operatorname{Hom}}_{{\mathcal{H}}_{F}}({\mathfrak{m}},\mathfrak{n})/{\sim}$

where two morphisms $f$, $g$ are equivalent under $\sim$ if and only if $f-g$
factors through a projective ${\mathcal{H}}_{F}$-module. We collect the
various facts we’ll need about the interactions between the Gorenstein
projective model structures on the categories
${\operatorname{Mod}}({\mathcal{H}}_{F})$, $F\subseteq\overline{C}$, and
${\operatorname{Mod}}({\mathcal{H}_{\operatorname{aff}}})$ below:

######  Lemma 5.4.

  1. (i)

If ${\mathfrak{m}},\mathfrak{n}\in{\operatorname{Mod}}({\mathcal{H}}_{F})$ are
simple then

$[{\mathfrak{m}},\mathfrak{n}]_{{\mathcal{H}}_{F}}\cong\begin{cases}0&\text{if
${\mathfrak{m}}$ is projective}\\\
{\operatorname{Hom}}_{{\mathcal{H}}_{F}}({\mathfrak{m}},\mathfrak{n})&\text{otherwise}.\end{cases}$

  2. (ii)

The functors
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}}$
and ${\operatorname{Res}}^{{\mathcal{H}}_{F^{\prime}}}_{{\mathcal{H}}_{F}}$
(for $F^{\prime}\subseteq\overline{F}\subseteq\overline{C}$) are right Quillen
and preserve weak equivalences.

  3. (iii)

The ‘diagonal’ functor

$\Delta:=({\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}})_{F\subseteq\overline{C}}:{\operatorname{Mod}}({\mathcal{H}_{\operatorname{aff}}})\to\prod_{F\subseteq\overline{C}}{\operatorname{Mod}}({\mathcal{H}}_{F})$

is right Quillen. Moreover, the Gorenstein projective model structure on
${\operatorname{Mod}}({\mathcal{H}_{\operatorname{aff}}})$ is the right
transfer of the product model structure on
$\prod_{F\subseteq\overline{C}}{\operatorname{Mod}}({\mathcal{H}}_{F})$ (i.e.
with all classes of morphisms defined termwise) along $\Delta$.

###### Proof.

For (i), we have that
${\operatorname{Hom}}_{{\operatorname{Ho}}({\mathcal{H}}_{F})}({\mathfrak{m}},\mathfrak{n})=0={\operatorname{Hom}}_{{\mathcal{H}}_{F}}({\mathfrak{m}},\mathfrak{n})$
if the simple modules ${\mathfrak{m}}$ and $\mathfrak{n}$ are not isomorphic
(c.f. (5.3)). If on the other hand ${\mathfrak{m}}\cong\mathfrak{n}$, then by
Schur’s lemma we have that
${\operatorname{Hom}}_{{\mathcal{H}}_{F}}({\mathfrak{m}},{\mathfrak{m}})\cong
k$ is one-dimensional and hence either its quotient
${\operatorname{Hom}}_{{\operatorname{Ho}}({\mathcal{H}}_{F})}({\mathfrak{m}},{\mathfrak{m}})$
is zero or the homotopy relation is trivial. But if
${\operatorname{Hom}}_{{\operatorname{Ho}}({\mathcal{H}}_{F})}({\mathfrak{m}},{\mathfrak{m}})=0$
then ${\operatorname{id}}:{\mathfrak{m}}\to{\mathfrak{m}}$ factors through a
projective, from which we deduce that ${\mathfrak{m}}$ is projective. We thus
get our desired isomorphism.

Part (ii) is immediate from 2.13. For (iii), we obtain as a consequence of
(ii) that $\Delta$ preserves fibrations and weak equivalences as those are
defined termwise in the target category. Since
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}}$
is also faithful for each $F\subseteq\overline{C}$, we in fact get that
$\Delta$ reflects fibrations. We are therefore left to show that $\Delta$ has
a left adjoint and reflects weak equivalences. For the former, the left
adjoint is given by
$({\mathfrak{m}}_{F})_{F\subseteq\overline{C}}\mapsto\bigoplus_{F\subseteq\overline{C}}{\mathcal{H}_{\operatorname{aff}}}\otimes_{{\mathcal{H}}_{F}}{\mathfrak{m}}_{F}$.
For the latter, by 2.8 it suffices to show that $\Delta$ reflects trivial
objects, i.e. that if
${\mathfrak{m}}\in{\operatorname{Mod}}({\mathcal{H}_{\operatorname{aff}}})$ is
such that
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}}({\mathfrak{m}})$
is projective for all $F\subseteq\overline{C}$ then
${\operatorname{pd}}_{{\mathcal{H}_{\operatorname{aff}}}}({\mathfrak{m}})<\infty$.
But this is the reverse implication shown in [20, Lemma 4.2]. ∎

5.4(i) has the following immediate consequence which will be helpful later.

######  Corollary 5.5.

If ${\mathfrak{m}},\mathfrak{n}\in{\operatorname{Mod}}({\mathcal{H}}_{F})$ are
semisimple such that each simple summand of ${\mathfrak{m}}$ is non-
projective, then
$[{\mathfrak{m}},\mathfrak{n}]_{{\mathcal{H}}_{F}}={\operatorname{Hom}}_{{\mathcal{H}}_{F}}({\mathfrak{m}},\mathfrak{n})$,
i.e. the homotopy relation on
${\operatorname{Hom}}_{{\mathcal{H}}_{F}}({\mathfrak{m}},\mathfrak{n})$ is
trivial.

A priori, in order to determine the morphism sets between two supersingular
characters $\chi\neq\chi^{\prime}$ in
${\operatorname{Ho}}({\mathcal{H}_{\operatorname{aff}}})$ we would be required
to compute some explicit Gorenstein projective/injective replacements of these
characters. However, we could not come up with any techniques to do this.
Instead, we will compute the image of these morphisms under the functor
${\operatorname{Ho}}(\Delta)$. Our aim is to show the following:

######  Proposition 5.6.

Let $\chi\neq\chi^{\prime}$ be two supersingular characters of
${\mathcal{H}_{\operatorname{aff}}}$ of infinite projective dimension, given
by the pairs $(J,\xi)$ and $(J^{\prime},\xi^{\prime})$ respectively. Let
$\Phi=\sqcup_{i=1}^{r}\Phi_{i}$ be the decomposition of the root system $\Phi$
into irreducible components.

  1. (i)

Suppose that $\Phi_{1}$ is of rank two and $\Phi_{i}$ is of rank 1 for all
$i\geq 2$, $\xi=\xi^{\prime}$, $S=S_{\xi}$ and $J\cap S_{i}=J^{\prime}\cap
S_{i}$ for all $i\geq 2$. Writing $S_{1}=\\{s,s^{\prime},s^{\prime\prime}\\}$,
assume further that, up to permuting the elements of $S_{1}$ and $\chi$ and
$\chi^{\prime}$, we have

$\displaystyle(\chi(T_{\hat{s}}),\chi(T_{\hat{s^{\prime}}}),\chi(T_{\hat{s^{\prime\prime}}}))$
$\displaystyle=(-1,-1,0),$
$\displaystyle(\chi^{\prime}(T_{\hat{s}}),\chi^{\prime}(T_{\hat{s^{\prime}}}),\chi^{\prime}(T_{\hat{s^{\prime\prime}}}))$
$\displaystyle=(0,-1,0).$

Then
${\operatorname{Ho}}(\Delta)([\chi,\chi^{\prime}]_{\mathcal{H}_{\operatorname{aff}}})$
has dimension 1 over $k$ and doesn’t contain any isomorphisms.

  2. (ii)

In every other case
${\operatorname{Ho}}(\Delta)([\chi,\chi^{\prime}]_{\mathcal{H}_{\operatorname{aff}}})=0$.

######  Remark 5.7.

Later we will consider simple supersingular modules for the whole pro-$p$
Iwahori-Hecke algebra ${\mathcal{H}}$, for which the action of
$\widetilde{\Omega}$ on supersingular characters comes into play. Suppose that
$\omega\in\Omega$ and fix a lift $\hat{\omega}\in\widetilde{\Omega}$.
Recalling that the group $\Omega$ normalises $S$, if $\chi$ is any character
of ${\mathcal{H}_{\operatorname{aff}}}$ corresponding to a pair $(J,\xi)$ then
$\chi^{\hat{\omega}}$ corresponds to the pair
$(J^{\hat{\omega}},\xi^{\hat{\omega}})$ where $J^{\hat{\omega}}=\omega
J\omega^{-1}$ and $\xi^{\hat{\omega}}(t)=\xi(\hat{\omega}^{-1}t\hat{\omega})$
($t\in T(\mathbb{F}_{q})$). It follows in particular that
$\left|J^{\hat{\omega}}\right|=\left|J\right|$. We therefore see that the two
supersingular characters of ${\mathcal{H}_{\operatorname{aff}}}$ from 5.6(i)
are not conjugate under the $\widetilde{\Omega}$-action.

We start preparing for the proof. Suppose that $\chi$ and $\chi^{\prime}$ are
as given in 5.6. Given any
$f\in[\chi,\chi^{\prime}]_{\mathcal{H}_{\operatorname{aff}}}$ we obtain a
collection
$(f_{F})\in\prod_{F\subseteq\overline{C}}{\operatorname{Hom}}_{{\mathcal{H}}_{F}}(\chi,\chi^{\prime})$
by letting each $f_{F}$ be a choice of representative of
${\operatorname{Ho}}(\Delta)(f)_{F}={\operatorname{Ho}}({\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}})(f)$,
and this satisfies the property that
${\operatorname{Res}}^{{\mathcal{H}}_{F^{\prime}}}_{{\mathcal{H}}_{F}}(f_{F^{\prime}})\sim
f_{F}$ whenever $F^{\prime}\subseteq\overline{F}$. Note that either $f_{F}\sim
0$ or otherwise the choice of $f_{F}$ must be unique, c.f. 5.4(i). The
strategy for our proof will be to show more generally that any such collection
of morphisms must in fact be zero, up to homotopy, unless we are in the
specific circumstances of part (i) of the Proposition. We therefore first need
a method for determining whether some $f_{F}$ arising as above is homotopic to
zero. To that end, we have the following elementary but key result.

######  Lemma 5.8.

Let $\chi\neq\chi^{\prime}$ be two characters of
${\mathcal{H}_{\operatorname{aff}}}$, and suppose that we are given
$(f_{F})\in\prod_{F\subseteq\overline{C}}{\operatorname{Hom}}_{{\mathcal{H}}_{F}}(\chi,\chi^{\prime})$
such that
${\operatorname{Res}}^{{\mathcal{H}}_{F^{\prime}}}_{{\mathcal{H}}_{F}}(f_{F^{\prime}})\sim
f_{F}$ whenever $F^{\prime}\subseteq\overline{F}$.

  1. (i)

If $F^{\prime}\subseteq\overline{F}$ and $f_{F^{\prime}}\sim 0$ then
$f_{F}\sim 0$.

  2. (ii)

If $F^{\prime}\subseteq\overline{F}$ and $f_{F}\sim 0$ then either
$f_{F^{\prime}}\sim 0$ or
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}}(\chi)$
is projective.

  3. (iii)

If we have three faces $F_{1},F_{2},F_{3}$ with
$F_{i}\subseteq\overline{F_{3}}$ for $i=1,2$ such that
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F_{3}}}(\chi)$
is not projective and $f_{F_{2}}\sim 0$ then $f_{F_{1}}\sim 0$.

###### Proof.

Part (i) is immediate from 5.4(ii) because right Quillen functors preserve
right homotopies. For part (ii), assume that $f_{F}\sim 0$ and
$f_{F^{\prime}}\not\sim 0$. Then by 5.4(i) we must have
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F^{\prime}}}(\chi)={\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F^{\prime}}}(\chi^{\prime})$
and the map $f_{F^{\prime}}=\lambda{\operatorname{id}}_{\chi}$ for some
$\lambda\in k^{\times}$. Up to rescaling we may assume
$f_{F^{\prime}}={\operatorname{id}}_{\chi}$. Then we have
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}}({\operatorname{id}}_{\chi})\sim
f_{F}\sim 0$ from which we deduce that
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}}(\chi)$
is projective. For (iii), the condition that $f_{F_{2}}\sim 0$ implies by (i)
that $f_{F_{3}}\sim 0$. Since
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F_{3}}}(\chi)$
is not projective it then follows from (ii) that $f_{F_{1}}\sim 0$ as
required. ∎

In order to apply 5.8, we need some criterion to establish when the
restriction of a character of ${\mathcal{H}_{\operatorname{aff}}}$ to some
${\mathcal{H}}_{F}$ is not projective. For this, we have the result below
which we gathered from the arguments of Koziol in [20, §7].

######  Lemma 5.9.

Let $F\subseteq\overline{C}$ be a face, and let $\chi=(J,\xi)$ be a character
of ${\mathcal{H}_{\operatorname{aff}}}$.

  1. (i)

If $S_{F}\cap S_{\xi}\neq S_{F}$ then
${\operatorname{Res}}^{\mathcal{H}_{\operatorname{aff}}}_{{\mathcal{H}}_{F}}(\chi)$
is not projective.

  2. (ii)

Assuming instead that $S_{F}\subseteq S_{\xi}$, then
${\operatorname{Res}}^{\mathcal{H}_{\operatorname{aff}}}_{{\mathcal{H}}_{F}}(\chi)$
is not projective if and only if there exists an irreducible component
${\mathcal{H}}_{F^{\prime}}$ of ${\mathcal{H}}_{F}$ such that
${\operatorname{Res}}^{\mathcal{H}_{\operatorname{aff}}}_{{\mathcal{H}}_{F^{\prime}}}(\chi)$
is not equal to a twisted sign or twisted trivial character. Equivalently,
this holds if and only if there exists $s,s^{\prime}\in S_{F}$ corresponding
to adjacent nodes in the affine Dynkin diagram such that
$\chi(T_{\hat{s}})\neq\chi(T_{\hat{s^{\prime}}})$.

###### Proof.

To prove non-projectivity, it suffices to find
$F\subseteq\overline{F^{\prime\prime}}\subseteq\overline{C}$ such that
${\operatorname{Res}}^{\mathcal{H}_{\operatorname{aff}}}_{{\mathcal{H}}_{F^{\prime\prime}}}(\chi)$
is not projective, because ${\mathcal{H}}_{F}$ is free over
${\mathcal{H}}_{F^{\prime\prime}}$. For (i), we pick $F^{\prime\prime}$ to be
given by $S_{F^{\prime\prime}}=\\{s\\}$, where $s\in S_{F}\setminus S_{\xi}$.
That
${\operatorname{Res}}^{\mathcal{H}_{\operatorname{aff}}}_{{\mathcal{H}}_{F^{\prime\prime}}}(\chi)$
is not projective is then the content of the proof of [20, Lemma 7.5]. For
(ii), we consider the idempotent

$e_{\xi}=\left|T(\mathbb{F}_{q})\right|^{-1}\sum_{t\in
T(\mathbb{F}_{q})}\xi(t)T_{t^{-1}}\in k[T(\mathbb{F}_{q})]$

corresponding to $\xi$, and note that $e_{\xi}$ is central in
${\mathcal{H}}_{F}$ since $S_{F}\subseteq S_{\xi}$ by assumption. It follows
that the character
${\operatorname{Res}}^{\mathcal{H}_{\operatorname{aff}}}_{{\mathcal{H}}_{F}}(\chi)$
factors through the product of blocks $e_{\xi}{\mathcal{H}}_{F}$. This algebra
is the $0$-Hecke algebra associated to the finite subgroup $W_{F}\leq W$
generated by $S_{F}$, and is the tensor product of the $0$-Hecke algebras
$e_{\xi}{\mathcal{H}}_{F^{\prime}}$ corresponding to the irreducible
components of ${\mathcal{H}}_{F}$. But it is shown in [24, Lemma 4.2 & Theorem
5.2] that the trivial and sign are the only projective characters over the
0-Hecke algebra of an irreducible root system. Thus, if the restriction of
$\chi$ to each irreducible component of ${\mathcal{H}}_{F}$ is a twisted sign
or trivial character we obtain that
${\operatorname{Res}}^{\mathcal{H}_{\operatorname{aff}}}_{e_{\xi}{\mathcal{H}}_{F}}(\chi)$
is projective, and then that
${\operatorname{Res}}^{\mathcal{H}_{\operatorname{aff}}}_{{\mathcal{H}}_{F}}(\chi)$
is projective as well since $e_{\xi}{\mathcal{H}}_{F}$ is a product of blocks
of ${\mathcal{H}}_{F}$.

Conversely, suppose that there exists $s,s^{\prime}\in S_{F}$ adjacent in the
affine Dynkin diagram with $\chi(T_{\hat{s}})\neq\chi(T_{\hat{s^{\prime}}})$.
Then set $S_{F^{\prime\prime}}=\\{s,s^{\prime}\\}$, and once again we have
that
${\operatorname{Res}}^{\mathcal{H}_{\operatorname{aff}}}_{{\mathcal{H}}_{F^{\prime\prime}}}(\chi)$
factors through the product of blocks
$e_{\xi}{\mathcal{H}}_{F^{\prime\prime}}$. The latter is isomorphic to a
0-Hecke algebra of type $A_{2},$ $B_{2}$ or $G_{2}$, and hence it follows from
loc. cit. that
${\operatorname{Res}}^{\mathcal{H}_{\operatorname{aff}}}_{e_{\xi}{\mathcal{H}}_{F^{\prime\prime}}}(\chi)$
is not projective, since it is neither a sign nor a trivial character by the
condition that $\chi(T_{\hat{s}})\neq\chi(T_{\hat{s^{\prime}}})$. As
$e_{\xi}{\mathcal{H}}_{F^{\prime\prime}}$ is a product of blocks of
${\mathcal{H}}_{F^{\prime\prime}}$, we are done. ∎

We now make a couple of preliminary observations. Suppose that we have a
collection of morphisms as in 5.8 and fix a face $F\subseteq\overline{C}$. If
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}}(\chi)$
is projective then $f_{F}\sim 0$. Similarly, if there is a face
$F^{\prime}\subseteq\overline{F}$ such that
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F^{\prime}}}(\chi)\neq{\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F^{\prime}}}(\chi^{\prime})$
then $f_{F}\sim 0$ by 5.4(i) and 5.8(i). We therefore only need to consider a
face $F$ such that
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}}(\chi)$
is not projective and the restrictions of $\chi$ and $\chi^{\prime}$ to
${\mathcal{H}}_{F^{\prime}}$ agree for all $F^{\prime}\subseteq\overline{F}$.
In particular, note that the condition
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}}(\chi)={\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}}(\chi^{\prime})$
implies that $\xi=\xi^{\prime}$ since
$k[T(\mathbb{F}_{q})]={\mathcal{H}}_{C}\subseteq{\mathcal{H}}_{F}$.

Assume from now on that we are in the above situation. Then the set

$S(\chi,\chi^{\prime}):=\\{s\in
S\mid\chi(T_{\hat{s}})\neq\chi^{\prime}(T_{\hat{s}})\\}$

must be non-empty since $\chi\neq\chi^{\prime}$ and $\xi=\xi^{\prime}$, and we
denote its elements by $s_{1},\ldots,s_{n}$. For each $1\leq j\leq n$, we then
write $S_{i_{j}}$ for the irreducible component containing $s_{j}$. Observe
further that our assumptions on $F$, $\chi$ and $\chi^{\prime}$ imply that for
each $j$ there is no face $F^{\prime}\subseteq\overline{F}$ with
$S_{F^{\prime}}=S_{F}\cup\\{s_{j}\\}$. In other words, we must have $S_{F}\cap
S_{i_{j}}=S_{i_{j}}\setminus\\{s_{j}\\}$ for every $1\leq j\leq n$. In
particular, the corresponding components $\Phi_{i_{1}},\ldots,\Phi_{i_{n}}$
are all distinct.

######  Lemma 5.10.

Suppose $\chi\neq\chi^{\prime}$ are supersingular characters of
${\mathcal{H}_{\operatorname{aff}}}$ and that
$\underline{f}:=(f_{F})\in\prod_{F\subseteq\overline{C}}{\operatorname{Hom}}_{{\mathcal{H}}_{F}}(\chi,\chi^{\prime})$
is as in 5.8. Assume $F\subseteq\overline{C}$ is a face such that
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}}(\chi)$
is not projective and
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F^{\prime}}}(\chi)={\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F^{\prime}}}(\chi^{\prime})$
whenever $F^{\prime}\subseteq\overline{F}$. With the above notation, we have:

  1. (i)

for each $1\leq j\leq n$, the component $\Phi_{i_{j}}$ is not of type $A_{1}$;
and

  2. (ii)

if $S_{F}\not\subseteq S_{\xi}$ then $f_{F}\sim 0$.

###### Proof.

(i) As explained above we must have $\xi=\xi^{\prime}$ and $S_{F}\cap
S_{i_{j}}=S_{i_{j}}\setminus\\{s_{j}\\}$ for all $1\leq j\leq n$. Suppose
$\Phi_{i_{j}}$ is of type $A_{1}$ for some $j$. Then $S_{i_{j}}$ consists of
two simple affine reflection $s_{j}$ and $s^{\prime}_{j}$, say, with the same
associated coroot. Since
$\chi(T_{\hat{s_{j}}})\neq\chi^{\prime}(T_{\hat{s_{j}}})$ we must have
$s_{j}\in S_{\xi}$ and so $S_{\xi}\cap S_{i_{j}}=S_{i_{j}}$ by definition of
$S_{\xi}$. Without loss of generality, we assume $\chi(T_{\hat{s_{j}}})=-1$
and $\chi^{\prime}(T_{\hat{s_{j}}})=0$. Then by supersingularity of $\chi$ and
$\chi^{\prime}$ we must have $\chi(T_{\hat{s^{\prime}_{j}}})=0$ and
$\chi^{\prime}(T_{\hat{s^{\prime}_{j}}})=-1$. But we also have that
$s^{\prime}_{j}\in S_{F}$ so that
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}}(\chi)\neq{\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}}(\chi^{\prime})$,
contradicting our assumptions.

(ii) Let $s^{\prime}\in S_{F}\setminus S_{\xi}$ and note as above that we
necessarily have $s^{\prime}\neq s_{j}$ for any $1\leq j\leq n$. We can now
define $F_{1},F_{2}\subseteq\overline{F_{3}}$ by $F=F_{1}$,
$S_{F_{2}}=\\{s_{j},s^{\prime}\\}$ and $S_{F_{3}}=\\{s^{\prime}\\}$. Note that
such a face $F_{2}$ exists because of (i). 5.9(i) then tells us that
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F_{3}}}(\chi)$
is not projective. So 5.8(iii) gives us that $f_{F}\sim 0$ as required. ∎

######  Lemma 5.11.

Suppose that $\chi$, $\chi^{\prime}$, $\underline{f}$ and
$F\subseteq\overline{C}$ are given as in 5.10, and assume further that
$S_{F}\subseteq S_{\xi}$.

  1. (i)

There are $s^{\prime},s^{\prime\prime}\in S_{F}$, adjacent in the affine
Dynkin diagram of the root system $\Phi$, such that
$\chi(T_{\hat{s^{\prime}}})\neq\chi(T_{\hat{s^{\prime\prime}}})$.

  2. (ii)

Assume that one of the following holds:

    1. (a)

$s^{\prime},s^{\prime\prime}\notin S_{i_{j}}$ for any $1\leq j\leq n$;

    2. (b)

$\left|S_{i_{j}}\right|>3$ for some $1\leq j\leq n$; or

    3. (c)

$n=\left|S(\chi,\chi^{\prime})\right|>1$.

Then $f_{F}\sim 0$.

###### Proof.

The existence of $s^{\prime},s^{\prime\prime}\in S_{F}$ as in (i) follows from
5.9(ii) because
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}}(\chi)$
is not projective by assumption. For (ii), we let
$F_{1},F_{2}\subseteq\overline{F_{3}}$ with $F=F_{1}$,
$S_{F_{2}}=\\{s_{j},s^{\prime},s^{\prime\prime}\\}$ and
$S_{F_{3}}=\\{s^{\prime},s^{\prime\prime}\\}$, for a suitably chosen $1\leq
j\leq n$ so that such an $F_{2}$ exists. In case (a), any choice of $j$ works.
In case (b), we pick $j$ such that $\left|S_{i_{j}}\right|>3$. In case (c), we
pick $j$ such that $s^{\prime},s^{\prime\prime}\notin S_{i_{j}}$. Either way,
as
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F_{2}}}(\chi)\neq{\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F_{2}}}(\chi^{\prime})$,
it follows from 5.4(i) and 5.8(iii) that we only need to show that
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F_{3}}}(\chi)$
is not projective, which itself holds by 5.9(ii). ∎

By 5.11(ii) we may simplify our notation a bit. We write
$\Phi=\sqcup_{i=0}^{r}\Phi_{i}$ as before and assume that
$S_{1}=\\{s,s^{\prime},s^{\prime\prime}\\}\subseteq S_{\xi}$, where
$\chi(T_{\hat{s^{\prime}}})\neq\chi(T_{\hat{s^{\prime\prime}}})$. Furthermore,
we assume that $s\in S_{1}$ is the unique simple affine reflection for which
$\chi(T_{\hat{s}})\neq\chi^{\prime}(T_{\hat{s}})$.

######  Lemma 5.12.

Let $\chi$, $\chi^{\prime}$, $\underline{f}$ be given as in 5.10, and suppose
that $\Phi$, $\chi$ and $\chi^{\prime}$ satisfy the assumptions above. Let
$F\subseteq\overline{C}$ be any face with $S_{F}\cap
S_{1}=\\{s^{\prime},s^{\prime\prime}\\}$.

  1. (i)

If $S_{\xi}\neq S$ then $f_{F}\sim 0$.

  2. (ii)

If $r>1$ and $\Phi_{i}$ is not of type $A_{1}$ for some $i\geq 2$, then
$f_{F}\sim 0$.

###### Proof.

For (i), pick $t\in S\setminus S_{\xi}$ and note that by assumption $t\notin
S_{1}$. Set $F_{1},F_{2}\subseteq\overline{F_{3}}$ by $F_{1}=F$,
$S_{F_{2}}=\\{t,s^{\prime},s^{\prime\prime}\\}$ and
$S_{F_{3}}=\\{s^{\prime},s^{\prime\prime}\\}$. By 5.9(ii),
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F_{3}}}(\chi)$
is not projective so that by applying 5.8(iii) we are reduced to showing that
$f_{F_{2}}\sim 0$.

Now let $F_{2}\subseteq\overline{F^{\prime}}$ be given by
$S_{F^{\prime}}=\\{t\\}$. We have that
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F^{\prime\prime}}}(\chi)$
is not projective for any $F^{\prime\prime}\subseteq\overline{F^{\prime}}$ by
5.9(i). Next, define
$F^{\prime}_{1},F^{\prime}_{2}\subseteq\overline{F^{\prime}_{3}}$ by
$F^{\prime}_{1}=F_{2}$, $S_{F^{\prime}_{2}}=\\{t,s,s^{\prime}\\}$ and
$S_{F^{\prime}_{3}}=\\{t,s^{\prime}\\}$. As
$F^{\prime}_{3}\subseteq\overline{F^{\prime}}$, the above gives that
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F^{\prime}_{3}}}(\chi)$
is not projective. Since
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F^{\prime}_{2}}}(\chi)\neq{\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F^{\prime}_{2}}}(\chi^{\prime})$,
5.4(i) and another application of 5.8(iii) imply
$f_{F_{2}}=f_{F^{\prime}_{1}}\sim 0$ as required.

For (ii), we may therefore assume that $S_{\xi}=S$. Suppose that $\Phi_{i}$ is
not of type $A_{1}$ for some $i\geq 2$. Since $\chi$ is supersingular we may
find $t^{\prime},t^{\prime\prime}\in S_{i}$, adjacent in the affine Dynkin
diagram, such that
$\chi(T_{\hat{t^{\prime}}})\neq\chi(T_{\hat{t^{\prime\prime}}})$. Again
appealing to 5.9(ii), we have that
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F_{3}}}(\chi)$
is not projective where $S_{F_{3}}=\\{t^{\prime},t^{\prime\prime}\\}$. Define
$F_{1},F_{2}\subseteq\overline{F_{3}}$ by
$S_{F_{1}}=\\{s^{\prime},s^{\prime\prime},t^{\prime},t^{\prime\prime}\\}$ and
$S_{F_{2}}=\\{s,t^{\prime},t^{\prime\prime}\\}$. Once again 5.8(iii) gives
that $f_{F_{1}}\sim 0$ and thus $f_{\tilde{F}}\sim 0$ by 5.8(i), where
$S_{\tilde{F}}=\\{s^{\prime},s^{\prime\prime}\\}$. We already saw that
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{\tilde{F}}}(\chi)$
is not projective and thus 5.8(ii) implies that $f_{F}\sim 0$. ∎

Finally, in order to deal with part (i) of 5.6 and for later use we will need
the following general fact.

######  Lemma 5.13.

Suppose that $R_{1},R_{2}$ are two Gorenstein $k$-algebras such that the
tensor product algebra $R=R_{1}\otimes_{k}R_{2}$ is also Gorenstein. Let
${\mathfrak{m}}_{1},{\mathfrak{m}}_{1}^{\prime}\in{\operatorname{Mod}}(R_{1})$
and let ${\mathfrak{m}}_{2}\in{\operatorname{Mod}}(R_{2})$ with
${\operatorname{pd}}_{R_{2}}({\mathfrak{m}}_{2})<\infty$. If
${\mathfrak{m}}_{1}\cong{\mathfrak{m}}_{1}^{\prime}$ in
${\operatorname{Ho}}(R_{1})$, then
${\mathfrak{m}}_{1}\otimes_{k}{\mathfrak{m}}_{2}\cong{\mathfrak{m}}^{\prime}_{1}\otimes_{k}{\mathfrak{m}}_{2}$
in ${\operatorname{Ho}}(R)$.

###### Proof.

We first show that if $f:{\mathfrak{m}}\to\mathfrak{n}$ is a weak equivalence
in ${\operatorname{Mod}}(R_{1})$ then
$f\otimes{\operatorname{id}}:{\mathfrak{m}}\otimes_{k}{\mathfrak{m}}_{2}\to\mathfrak{n}\otimes_{k}{\mathfrak{m}}_{2}$
is a weak equivalence in ${\operatorname{Mod}}(R)$. Since weak equivalences
are composites of trivial cofibrations and trivial fibrations and are closed
under composition, we are reduced to showing this when $f$ is either a trivial
fibration or a trivial cofibration. Since tensoring over $k$ is exact and by
applying [16, Lemma 5.8], this is in turn equivalent to showing that if
$\mathfrak{e}\in{\operatorname{Mod}}(R_{1})$ has finite projective dimension
over $R_{1}$ then $\mathfrak{e}\otimes_{k}{\mathfrak{m}}_{2}$ has finite
projective dimension over $R$. But since
${\operatorname{pd}}_{R_{2}}({\mathfrak{m}}_{2})<\infty$ by assumption, this
is clear.

Now let $q_{c}:Q_{c}{\mathfrak{m}}_{1}\to{\mathfrak{m}}_{1}$ be a Gorenstein
projective replacement in ${\operatorname{Mod}}(R_{1})$. By the above, we then
have that
$q_{c}\otimes{\operatorname{id}}:Q_{c}{\mathfrak{m}}_{1}\otimes_{k}{\mathfrak{m}}_{2}\to{\mathfrak{m}}_{1}\otimes_{k}{\mathfrak{m}}_{2}$
is a weak equivalence in ${\operatorname{Mod}}(R)$. Further, the assumption
that ${\mathfrak{m}}_{1}\cong{\mathfrak{m}}_{1}^{\prime}$ in
${\operatorname{Ho}}(R_{1})$ implies the existence of a weak equivalence
$g:Q_{c}{\mathfrak{m}}_{1}\to{\mathfrak{m}}^{\prime}_{1}$ in
${\operatorname{Mod}}(R_{1})$. Applying the above again shows that
$g\otimes{\operatorname{id}}:Q_{c}{\mathfrak{m}}_{1}\otimes_{k}{\mathfrak{m}}_{2}\to{\mathfrak{m}}^{\prime}_{1}\otimes_{k}{\mathfrak{m}}_{2}$
is a weak equivalence in ${\operatorname{Mod}}(R)$ as well. Putting the two
together yields an isomorphism
$(g\otimes{\operatorname{id}})\circ(q_{c}\otimes{\operatorname{id}})^{-1}:{\mathfrak{m}}_{1}\otimes_{k}{\mathfrak{m}}_{2}\to{\mathfrak{m}}^{\prime}_{1}\otimes_{k}{\mathfrak{m}}_{2}$
in ${\operatorname{Ho}}(R)$ as required. ∎

###### Proof of 5.6.

Given a map map $f:\chi\to\chi^{\prime}$ in
${\operatorname{Ho}}({\mathcal{H}_{\operatorname{aff}}})$, set $f_{F}$ to be
any lift of
${\operatorname{Ho}}(\Delta)(f)_{F}={\operatorname{Ho}}({\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}})(f)$
to ${\operatorname{Hom}}_{{\mathcal{H}}_{F}}(\chi,\chi^{\prime})$. 5.10, 5.11
and 5.12 imply that if we’re not in the precise situation of (i), then
$f_{F}\sim 0$ for all $F\subseteq\overline{C}$. This shows (ii). In the
setting of (i), we still have that $f_{F^{\prime}}\sim 0$ for all faces
$F^{\prime}$ for which $S_{F^{\prime}}\cap
S_{1}\neq\\{s^{\prime},s^{\prime\prime}\\}$. If on the other hand
$S_{F^{\prime}}\cap S_{1}=\\{s^{\prime},s^{\prime\prime}\\}$ then
$F^{\prime}\subseteq\overline{F_{1}}$, where
$S_{F_{1}}=\\{s^{\prime},s^{\prime\prime}\\}$, so that
${\operatorname{Res}}^{{\mathcal{H}}_{F^{\prime}}}_{{\mathcal{H}}_{F_{1}}}(f_{F^{\prime}})\sim
f_{F_{1}}$. As
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F_{1}}}(\chi)={\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F_{1}}}(\chi^{\prime})$
is not projective, c.f. 5.9(ii), it follows from 5.4(i) that
${\operatorname{Res}}^{{\mathcal{H}}_{F^{\prime}}}_{{\mathcal{H}}_{F_{1}}}(f_{F^{\prime}})=f_{F_{1}}$
and thus that ${\operatorname{Ho}}(\Delta)(f)$ is uniquely determined by
$f_{F_{1}}$. This gives that
$\dim_{k}{\operatorname{Ho}}(\Delta)([\chi,\chi^{\prime}]_{\mathcal{H}_{\operatorname{aff}}})\leq
1$ by 5.4(i) again. Moreover, we see that ${\operatorname{Ho}}(\Delta)(f)$ is
never an isomorphism. Indeed, if $F^{\prime\prime}\subseteq\overline{C}$ is
defined by $S_{F^{\prime\prime}}=\\{s,s^{\prime}\\}$ then
$f_{F^{\prime\prime}}\sim 0$ is not a weak equivalence since 5.9(ii) again
shows that
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F^{\prime\prime}}}(\chi^{\prime})$
is not projective.

We are left to show that there exists a map $f:\chi\to\chi^{\prime}$ in
${\operatorname{Ho}}({\mathcal{H}_{\operatorname{aff}}})$ such that
$f_{F_{1}}={\operatorname{id}}$. Suppose first that $r=1$ and so that
$\Phi=\Phi_{1}$ is irreducible of rank 2. Then note that
${\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F}}(\chi)$
is projective for any face $F$ of positive dimension because $S=S_{\xi}$, c.f.
5.9(ii). Thus we deduce from the complex (5.2) applied to $\chi$ that the
augmentation map

$\varepsilon:C_{0}:=\bigoplus_{F\in\mathscr{F}_{0}}{\mathcal{H}_{\operatorname{aff}}}\otimes_{{\mathcal{H}}_{F}}\chi\to\chi$

is a trivial fibration. Note that $C_{0}$ is in fact a cofibrant replacement
of $\chi$ by 5.4(ii) and that $F_{1}\in\mathscr{F}_{0}$. Now, the map
${\operatorname{id}}:{\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F_{1}}}(\chi)\to{\operatorname{Res}}^{{\mathcal{H}_{\operatorname{aff}}}}_{{\mathcal{H}}_{F_{1}}}(\chi^{\prime})$
has an adjoint ${\mathcal{H}_{\operatorname{aff}}}$-linear morphism
$g:{\mathcal{H}_{\operatorname{aff}}}\otimes_{{\mathcal{H}}_{F_{1}}}\chi\to\chi^{\prime}$.
Extending by zero, this gives a map $f:\chi\cong C_{0}\to\chi^{\prime}$ in
${\operatorname{Ho}}({\mathcal{H}_{\operatorname{aff}}})$ which does satisfy
$f_{F}={\operatorname{id}}$. Indeed, this follows because the map
$\chi\to{\mathcal{H}_{\operatorname{aff}}}\otimes_{{\mathcal{H}}_{F_{1}}}\chi\subseteq
C_{0}$, $x\mapsto 1\otimes x$, is an ${\mathcal{H}}_{F}$-linear splitting of
$\varepsilon$ and thus is equal to its inverse in
${\operatorname{Ho}}({\mathcal{H}}_{F})$, and therefore $f_{F}$ must be the
adjoint morphism of $g$ which is the identity by construction.

In general, first note that $\chi$ and $\chi^{\prime}$ factor through the
algebra
${\mathcal{H}}^{0}_{\operatorname{aff}}:=e_{\xi}{\mathcal{H}_{\operatorname{aff}}}$
which is an affine 0-Hecke algebra of type $\Phi_{1}\times
A_{1}\times\cdots\times A_{1}$ (with $r$ factors), i.e. it has generators
$e_{\xi}T_{s_{\alpha}}$ for each $s_{\alpha}\in S$ which satisfy the braid
relations and such that $(e_{\xi}T_{s_{\alpha}})^{2}=-e_{\xi}T_{s_{\alpha}}$.
From the algebra decomposition
${\mathcal{H}_{\operatorname{aff}}}={\mathcal{H}}^{0}_{\operatorname{aff}}\oplus(1-e_{\xi}){\mathcal{H}_{\operatorname{aff}}}$
we get that ${\mathcal{H}}^{0}_{\operatorname{aff}}$ is also a Gorenstein
algebra and that a Gorenstein projective module, resp. module of finite
projective dimension, over ${\mathcal{H}}^{0}_{\operatorname{aff}}$ is still
Gorenstein projective, resp. of finite projective dimension as an
${\mathcal{H}_{\operatorname{aff}}}$-module. Thus any cofibrant replacement in
${\operatorname{Mod}}({\mathcal{H}}^{0}_{\operatorname{aff}})$ is also a
cofibrant replacement in
${\operatorname{Mod}}({\mathcal{H}_{\operatorname{aff}}})$. Furthermore, we
have a decomposition
${\mathcal{H}}^{0}_{\operatorname{aff}}={\mathcal{H}}_{1}\otimes_{k}{\mathcal{H}}_{2}$
where ${\mathcal{H}}_{1}$ is an affine 0-Hecke algebra of type $\Phi_{1}$ and
${\mathcal{H}}_{2}$ is an affine 0-Hecke algebra of type
$A_{1}\times\cdots\times A_{1}$ ($r-1$ times). We correspondingly have
decompositions $\chi=\chi_{1}\otimes_{k}\chi_{2}$,
$\chi^{\prime}=\chi^{\prime}_{1}\otimes_{k}\chi^{\prime}_{2}$ where
$\chi_{i},\chi^{\prime}_{i}$ is a character of ${\mathcal{H}}_{i}$ ($i=1,2$)
and $\chi_{2}=\chi^{\prime}_{2}$ by assumption, and the sets
$J,J^{\prime}\subseteq S$ also decompose as $J=J_{1}\sqcup J_{2}$ and
$J^{\prime}=J^{\prime}_{1}\sqcup J^{\prime}_{2}$ with
$J_{2}=J^{\prime}_{2}\subseteq S\setminus S_{1}$ and
$J_{1},J^{\prime}_{1}\subseteq S_{1}$.

Let $G_{1}$ and $G_{2}$ be the $\mathfrak{F}$-rational points of two split
semisimple, simply connected groups of type $\Phi_{1}$ and
$A_{1}\times\cdots\times A_{1}$ respectively, with corresponding pro-$p$
Iwahori (affine) Hecke algebras ${\mathcal{H}}_{\operatorname{aff}}^{1}$ and
${\mathcal{H}}_{\operatorname{aff}}^{2}$. Then we see that
${\mathcal{H}}_{i}\cong e_{\mathbf{1}}{\mathcal{H}}_{\operatorname{aff}}^{i}$,
say, and that $\chi_{i}$ and $\chi^{\prime}_{i}$ come from two supersingular
character of ${\mathcal{H}}_{\operatorname{aff}}^{i}$ by taking the pairs
$(J_{i},\mathbf{1})$ and $(J^{\prime}_{i},\mathbf{1})$ ($i=0,1$). Choose a
cofibrant replacement $q_{c}:Q_{c}\chi_{1}\to\chi_{1}$ in
${\operatorname{Mod}}({\mathcal{H}}_{1})$. The argument for $r=1$ and 2.14
then show that there is a map $g:Q_{c}\chi_{1}\to\chi^{\prime}_{1}$ such that
$g\circ q_{c}^{-1}$ goes to the identity on $\chi_{1}$ under restriction of
scalars to $e_{\xi}{\mathcal{H}}_{F_{1}}$. But since
$\chi_{2}=\chi^{\prime}_{2}$ has finite projective dimension over
${\mathcal{H}}_{2}$ (c.f. 5.3), it follows from 5.13 that
$q=(q_{c}\otimes{\operatorname{id}}):Q_{c}\chi_{1}\otimes_{k}\chi_{2}\to\chi_{1}\otimes_{k}\chi_{2}=\chi$
is a weak equivalence in
${\operatorname{Mod}}({\mathcal{H}}^{0}_{\operatorname{aff}})$. Hence
$g\otimes
1:Q_{c}\chi_{1}\otimes_{k}\chi_{2}\to\chi^{\prime}_{1}\otimes_{k}\chi^{\prime}_{2}=\chi^{\prime}$
gives in ${\operatorname{Ho}}({\mathcal{H}_{\operatorname{aff}}})$ a map
$f=(g\otimes 1)\circ q^{-1}$ such that $f_{F_{1}}={\operatorname{id}}$, as
required. ∎

######  Corollary 5.14.

Suppose that $\chi\neq\chi^{\prime}$ are two supersingular characters of
${\mathcal{H}_{\operatorname{aff}}}$ of infinite projective dimension. Then
they are not isomorphic in
${\operatorname{Ho}}({\mathcal{H}_{\operatorname{aff}}})$.

###### Proof.

Any isomorphism $f:\chi\to\chi^{\prime}$ in
${\operatorname{Ho}}({\mathcal{H}_{\operatorname{aff}}})$ would induce an
isomorphism $(f_{F})={\operatorname{Ho}}(\Delta(f))$ in
$\prod_{F\subseteq\overline{C}}{\operatorname{Ho}}({\mathcal{H}}_{F})$. But it
follows immediately from 5.6 that this cannot occur. ∎

######  Remark 5.15.

In case (i) of 5.6, when $r=1$, the above proof actually tells us that

$\displaystyle[\chi,\chi^{\prime}]_{\mathcal{H}_{\operatorname{aff}}}$