We develop a theory of Smith-Treumann localization and relative parity sheaves in the context of Fargues-Scholze's Geometrization of the Local Langlands Correspondence. We then apply this theory to prove some conjectures of Treumann-Venkatesh concerning mod-$\ell$ Local Langlands functoriality between a reductive group $G$ and its fixed subgroup under an order $\ell$ automorphism. As another application, we calculate the Fargues-Scholze parameters of mod-$\ell$ Howe-unramified toral representations.

§ INTRODUCTION

§.§ Modular Langlands functoriality
Let $E$ be a local field of residue characteristic $p$ and $G$ be a reductive group over $E$. For simplicity, we assume for now that $G$ is split; beyond the Introduction, the text always treats general $G$. The Local Langlands Correspondence predicts, vaguely speaking, that the smooth representations of $G(E)$ over a field $k$ are controlled by the Langlands dual group $\chG$ over $k$.

For example, letting $W_E \subset \Gal(E^s/E)$ be the Weil group of $E$, the Local Langlands Correspondence predicts that should be a natural parametrization of irreducible smooth representations of $G(E)$ over $k$ by “$L$-parameters”, which are homomorphisms $W_E \rightarrow \chG(k)$ up to conjugacy. The existence of such a parametrization has surprising implications for representation theory: for example, a homomorphism $\chpsi \co \chH \rightarrow \chG$ of dual groups induces an obvious map

\[
\{\text{$L$-parameters for $\chH$}\} \xrightarrow{\chpsi_*} \{\text{$L$-parameters for $\chG$}\}
\]

and therefore suggests some Langlands functoriality operation from (packets of) irreducible representations of $H(E)$ to (packets of) irreducible representations of $G(E)$. In practice, it is usually difficult to construct such operations directly, or to describe them explicitly in representation-theoretic terms.

In this paper we investigate Langlands functoriality for a specific class of dual homomorphisms $\chpsi$ that was identified by Treumann-Venkatesh in [50]. It considers the situation where the reductive group $H$ arises as the fixed points of an order $\ell$ automorphism $\sigma$ of another reductive group $G$, where $\ell$ is a prime different from $p$. Furthermore, we take $k$ to be a field of characteristic $\ell$. In this situation (and under some hypotheses), Treumann-Venkatesh constructed a dual homomorphism $\chpsi \co \chH \rightarrow \chG$ over $k$, with a specific property in terms of the Satake isomorphism that we will discuss later. We refer to this situation as modular functoriality, because its construction depends on special features of modular arithmetic (and this paper depends in turn on special features of modular representation theory). By contrast, Langlands' original conjectures were focused on the case where $k$ has characteristic zero. We note that in many examples the map $\chpsi$ does not lift to characteristic zero, and the resulting functoriality is truly specific to modular arithmetic.

The story extends to non-split groups. In that case, the dual group $\chG$ should be augmented to the $L$-group $\ld G \cong \wh{G} \rtimes W_E$, where the action of $W_E$ on $\wh{G}$ reflects the twisting of $G$ over $E$ relative to its split form. This generalization is necessary to treat some of the most interesting examples.

A familiar example is cyclic base change, where $G = \Res_{E'/E} H$ for a cyclic $\ell$-extension $E'/E$ and $\sigma$ is a generator of $\Gal(E'/E)$ acting on $G$ in the natural way, so that $G^{\sigma} = H$. This particular example lifts to characteristic $0$, but many do not; several examples are tabulated in the ArXiv version of Treumann-Venkatesh's paper [49]. We also consider here some interesting examples that are ruled out by the hypotheses of [49, 50]: a useful one is where $\sigma$ is conjugation by a strongly regular element of $G$, in which case $H$ is a (not necessarily split!) maximal torus. Note that the work of Treumann-Venkatesh only treats examples where $G$ is simply connected and $H$ is semi-simple.

§.§ Conjectures of Treumann-Venkatesh
We describe some conjectures of Treumann-Venkatesh that will be proved in this paper, up to technical hypothesis that exclude small $\ell$. We again restrict our attention to the case where $G$ and $H$ are split, for simplicity. We let $\sH(G,K)$ be the spherical Hecke algebra of $G$ (with respect to some maximal compact subgroup $K$ stable under $\sigma$) with coefficients in $k$. Then the Satake isomorphism supplies a $k$-algebra isomorphism

\[
\sH(G,K) \cong K_0 (\Rep_k(\chG)).
\]

Under some technical assumptions, Treumann-Venkatesh construct a $k$-algebra homomorphism $\br \co \sH(G, K) \rightarrow \sH(H, K^\sigma)$ which they call the (normalized) Brauer homomorphism. (The construction uses in an essential way the assumption that char $k = \ell = \ord(\sigma)$.) This implies the existence of a commutative diagram

\[
\begin{tikzcd}
\sH(G,K) \ar[r, "\sim","\text{Satake}"'] \ar[d, "\br"]  & K_0 (\Rep_k(\chG)) \ar[d, dashed] \\
\sH(H, K^\sigma) \ar[r, "\sim", "\text{Satake}"']   & K_0(\Rep_k( \chH))
\end{tikzcd}
\]

It is then natural to ask if the dashed map is induced by restriction along a homomorphism $\chpsi \co \chH \rightarrow \chG$. If so, then $\chpsi$ will be called a “$\sigma$-dual homomorphism”. One of the main theorems of Treumann-Venkatesh <cit.> is that if $G$ is simply connected and $H$ is semisimple, then a $\sigma$-dual homomorphism exists, with three possible exceptions if $G$ has type $\rE_6$. In <cit.>, Treumann-Venkatesh pose the open problem of constructing a $\sigma$-dual homomorphism in full generality. Our first theorem addresses this question.

Type     $A_n$     $B_n, D_n$     $C_n$     $G_2, F_4, E_6$     $E_7$     $E_8$

$b(\Phi)$     1     2     $n$     3     19     31

Excluded primes for each root system.

Suppose $\ell >\max\{ b(\chG), b(\chH)\}$ where $b(\chG)$ is the maximum over the bad primes $b(\check{\Phi})$ over all root systems $\check{\Phi}$ of the simple factors of $G$, tabulated in Figure <ref>, and $b(\chH)$ is defined similarly. Then a $\sigma$-dual homomorphism $\chpsi \co \chH \rightarrow \chG$ exists.

Theorem <ref> is not a strict improvement on the work of Treumann-Venkatesh, as our characteristic assumption actually rules out many interesting examples. On the other hand, relaxing the condition that $H$ be semisimple is also interesting in examples; even the case where $H$ is a torus is very useful, as we shall discuss below.

The construction of the $\sigma$-dual homomorphism in [50] is a tour de force: it invokes classification theorems to tabulate all examples of modular functoriality, and analyzes them case-by-case. By contrast, the proof of Theorem <ref> is completely uniform across all cases, and makes no use of classification theorems. The idea is quite natural: instead of contemplating the dual groups, we try to categorify the Brauer homomorphism $\br$ to a Brauer functor $\wcbr$ from the Satake category of $G$ to the Satake category of $H$. After equipping this Brauer functor with a Tannakian structure, we obtain the $\sigma$-dual homomorphism for free from the Geometric Satake equivalence.[A subtle but important point found by Treumann-Venkatesh is that, when taking into account the full $L$-group, a $\sigma$-dual homomorphism may not exist with the “usual” notion of $L$-group due to Langlands. Treumann-Venkatesh suggest in <cit.> that this problem might be repaired by instead using the “$c$-group”, which is the variant of the $L$-group that naturally comes out of the geometric Satake equivalence <cit.>. For non-archimedean local fields there is actually an isomorphism between the $L$-group and the $c$-group (possibly depending on choice of square root of $p$ in $k$), so this distinction is not essential for our purposes, but it seems to support the morality of our approach.]

\[
\begin{tikzcd}
\Sat(\Gr_G; k)  \ar[rr, "\sim","\text{Geom. Satake}"'] \ar[d, "\wcbr"]  & & \Rep_k(\chG) \ar[d, dashed] \\
\Sat(\Gr_H; k)  \ar[rr, "\sim", "\text{Geom. Satake}"']  & & \Rep_k( \chH)
\end{tikzcd}
\]

Although this idea is simple, its implementation is quite involved and will be elaborated upon later; for now we just mention that for the sole purpose of producing the $\sigma$-dual homomorphism, it should be easy to improve the assumption $\ell > \max\{b(\chG), b(\chH)\}$ to “$\ell$ is a good prime for $\chG$ and $\chH$”, which for example holds as long as $\ell > 5$. The extra inefficiency in our Theorem <ref> comes from our desire not to work with the usual Geometric Satake equivalence, but with the version on the $\bdrp$-affine Grassmannians occurring in the work of Fargues-Scholze [22], whose geometric representation theory is not as developed.

We next turn to describe conjectures of Treumann-Venkatesh pertaining to the Local Langlands Correspondence. We write $\Z[\sigma]$ for the group ring of $\tw{\sigma} \cong \Z/\ell\Z$ and let $N := 1 + \sigma + \ldots + \sigma^{\ell-1} \in \Z[\sigma]$. If $\Pi$ is a representation of $G(E) \rtimes \sigma$, then its Tate cohomology groups $\rT^j(\Pi)$, for $j \in \Z/2\Z$, are defined as

\begin{equation}\label{eq: Tate cohomology}
\rT^0 (\Pi) := \frac{\ker(1-\sigma \mid \Pi)}{\Ima(N \mid \Pi)} \hspace{1cm}
\rT^1 (\Pi) := \frac{\ker(N \mid \Pi)}{\Ima(1-\sigma \mid \Pi)}
\end{equation}

and the $G(E)$-action on $\Pi$ induces an $H(E)$-action on each $\rT^{j}(\Pi)$.

Let $\Pi$ be an irreducible smooth representation of $G(E)$ whose isomorphism class is fixed by $\sigma$, so that the $G(E)$-action on $\Pi$ uniquely extends to a $G(E) \rtimes \sigma$-action by <cit.>. Then for each $j \in \Z/2\Z$:

* (Admissibility Conjecture) $\rT^j(\Pi)$ is admissible as a representation of $H(E)$.

* (Functoriality Conjecture) The $L$-parameter of every irreducible $H(E)$-subquotient of $\rT^{j}(\Pi)$ is sent by the $\sigma$-dual homomorphism $\chpsi$ to the $L$-parameter for $\Pi^{(\ell)} := \Pi \otimes_{k, \Frob} \ell$, the Frobenius twist of $\Pi$.

We do not consider Part (1) of Conjecture <ref> in this paper; it is the subject of current work-in-progress. We will focus on Part (2). At the time of its formulation, the Local Langlands Correspondence was constructed only for certain families of groups, so the precise meaning of Conjecture <ref>(2) was left vague for general groups. We may now formulate a precise version for all $G$ using the work of Fargues-Scholze [22]; this is what we will describe next.

For $G = \GL_n$, Vignéras had constructed in [51] the Local Langlands Correspondence over $k$ (using the characteristic zero version due to Harris-Taylor [25]), many years before the work of Fargues-Scholze. This gives a precise meaning to Conjecture <ref>(2) when $G$ and $H$ are both general linear groups, and in the context of cyclic base change, special cases have been proven by other authors, as will be discussed more in <ref>. Even for this case, our results will depend on the work of Fargues-Scholze.

§.§ The Fargues-Scholze correspondence
An output of the work [22] of Fargues-Scholze is a map

\begin{align}\label{eq: LLC}
\left\{ \begin{array}{@{}c@{}}  \text{irreducible admissible representations} \\  \text{$\pi$ of $G(E)$ over $k$}\end{array} \right\}/\sim  & \longrightarrow \left\{ \begin{array}{@{}c@{}}  \text{semi-simple $L$-parameters} \\
\rho_{\pi} \co W_E \rightarrow \ld G(k)  \end{array} \right\}/\sim
\end{align}

that we call the Fargues-Scholze correspondence. We refer to $\rho_\Pi$, which is most naturally regarded as an element of $\rH^1(W_E; \wh{G}(k))$, as the Fargues-Scholze parameter of $\Pi$. The map $\Pi \mapsto \rho_\Pi$ is expected to be the semi-simplification of “the” hypothetical Local Langlands Correspondence. For specific groups including tori and $\GL_n$, the Local Langlands Correspondence had been constructed previously (by class field theory and by Harris-Taylor, respectively), and in these cases it is known that the Fargues-Scholze correspondence is compatible with the previous construction.

For most groups the Fargues-Scholze correspondence is quite mysterious. For example it is expected that (<ref>) is surjective and has finite fibers, but proving this is wide open. For regular supercuspidal representations of quite general groups, Kaletha [29] has prescribed explicit constructions for the $L$-parameters, which are strongly supported by the expected endoscopic character relations; while for general groups almost nothing is known about the Fargues-Scholze parameters of such representations. We remark that in the cases where the Fargues-Scholze correspondence has been explicated, there have been important geometric implications; an example is the work of Koshikawa [30] towards torsion vanishing in the cohomology of Shimura varieties, where the relevant compatibility with the classical LLC is proven in [23].

§.§.§ The functoriality conjecture
We prove the following result concerning modular functoriality in the Fargues-Scholze correspondence, which we take as fulfilling Conjecture <ref>(2), away from small $\ell$.

Assume $\ell >\max\{ b(\chG), b(\chH)\}$. Let $\Pi$ be an irreducible smooth representation of $G(E)$ whose isomorphism class is fixed by $\sigma$. Then for each $j \in \Z/2\Z$ and every $G(E)$-irreducible subquotient $\pi$ of $\rT^{j}(\Pi)$, the $\sigma$-dual homomorphism $\chpsi_* \co \rH^1(W_E; \wh{H}(k)) \rightarrow \rH^1(W_E; \wh{G}(k))$ sends $\rho_{\pi} \mapsto \rho_{\Pi^{(\ell)}}$.

This result appears as Corollary <ref> in the main text. It is a consequence of a more powerful statement, Theorem <ref>, which treats “derived smooth representations” $\Pi \in D^b(\Rep^{\mrm{sm}}_k G(E))$. The derived version is more useful for calculations but less elementary to formulate, so we do not state it here.

Theorem <ref> does not seem to be made any easier by assuming the full categorical conjectures of [22], or the conjectural compatibility with Kaletha's explicit Local Langlands Correspondence, or any other standard conjectures about the Local Langlands Correspondence that we know.
On the other hand, it may not be unreasonable to speculate that Theorem <ref> could be useful for proving some of these other conjectures. For example, as an application of (the derived version of) Theorem <ref>, we calculate explicitly the Fargues-Scholze parameters of certain classes of representations.

§.§.§ Explicit calculation of Fargues-Scholze parameters
The basic idea is that if the $L$-parameter $\rho_{\Pi} \co W_E \rightarrow \ld G(k)$ admits some factorization

\begin{equation}\label{eq: L-param factorization}
\begin{tikzcd}
W_E \ar[rr, "\rho_{\Pi} "] \ar[dr, dashed] &  &  \ld G(k) \\
& \ld H(k) \ar[ur, "\ld \psi"']
\end{tikzcd}
\end{equation}

through an $L$-parameter into $\ld H(k)$ where $H$ is a torus arising as the fixed points of some appropriate $\sigma$, then Theorem <ref> identifies $\rho_\Pi$ explicitly in terms of the $L$-parameter of $\rT^j(\Pi)$, which is computable since the Fargues-Scholze correspondence is completely understood for tori. Note that “most” supercuspidal parameters factor through the $L$-group of some (not necessarily split) torus; for example all of them factor in this way if $p$ is large enough relative to $G$.

We apply this idea to the “Howe-unramified toral supercuspidal representations” considered in work of Chan-Oi [12]. They are generalizations of depth-zero supercuspidals, with an analogous construction but instead using “deeper-level Deligne-Lusztig representations” studied in work of Chan-Ivanov [11]; consequently, they include supercuspidal representations with arbitrarily high depth. The input for the (modular version of the) Chan-Oi construction is an elliptic unramified torus $T \subset G$ and a character $\theta \co T(E) \rightarrow k^\times$.

Under the assumption that $\ell > \max\{b(\chG), b(\chH)\}$, and that $T$ contains a strongly regular element of order $\ell$, we prove (Corollary <ref>) that some irreducible constituent in the Chan-Oi construction has Fargues-Scholze parameter of the form

\begin{equation}\label{eq: intro L-param}
W_E \xrightarrow{\ld \theta} \ld T(k) \xrightarrow{ \ld j} \ld G(k),
\end{equation}

where $\ld j$ is the canonical embedding of the $L$-group of an unramified maximal torus. This is morally in accordance with the prediction with Kaletha's explicit Local Langlands Correspondence for regular supercuspidal representations in [29]; we say “morally” because Kaletha does not consider modular coefficients. Because the precise definition of a Howe-unramified toral representation is complicated, we defer the precise formulation of our result to Theorem <ref>.

We emphasize that the simple appearance of (<ref>) belies the intricacy of Kaletha's recipe, which for example involves twisting the most natural (from a representation-theoretic perspective) guess for the $L$-parameter by a subtle “twisting character”. In the relatively simple example of epipelagic representations of unitary groups, these characters were explicated in [21] and found to be already very complicated there. The geometric construction of Chan-Oi somehow bakes this twisting character into the geometry (see <cit.> for more discussion of this point) which is then reflected in our computation.

§.§ Further results
We mention some further results.

§.§.§ Existence of functorial lifts
The following Theorem guarantees the existence of functorial lifts along any $\sigma$-dual homomorphism.

Assume $\ell > \max\{b(\chG), b(\chH)\}$. Let $\pi$ be an irreducible smooth representation of $H(E)$ over $k$, with Fargues-Scholze parameter $\rho_{\pi} \in \rH^1(W_E; \wh{H}(k))$. Then there exists an irreducible smooth representation $\Pi$ of $G(E)$ over $k$ with Fargues-Scholze parameter $\rho_{\Pi} \cong \chpsi \circ \rho_{\pi}  \in \rH^1(W_E; \wh{G}(k))$.

The more general version (allowing non-split groups) is Theorem <ref>. The more general version allows to treat further examples such as the following.

[Base change]
Taking $G = \Res_{E'/E} H$ for a cyclic $\ell$-extension $E'/E$ and $\sigma$ a generator of $\Gal(E'/E)$ acting in the natural way, Theorem <ref> asserts the existence of base change along $E'/E$. In equal characteristic and for the Genestier-Lafforgue correspondence, this was established in <cit.>.

Take $\sigma$ to be conjugation by a strongly regular $\ell$-torsion element of $G$. Then $H$ is a torus, so every $\rho \in \rH^1(W_E; \wh{T}(k))$ is realized as a Fargues-Scholze parameter. Then Theorem <ref> implies that every $\rho \in \rH^1(W_E; \wh{G}(k))$ that factors through $\ld \psi$ is realized as a Fargues-Scholze parameter. It would be interesting to investigate which $\ld \psi$ can arise, as this might help to show surjectivity of the Fargues-Scholze correspondence over $k$: recall that if $p$ is not too small relative to $G$, then every supercuspidal $L$-parameter factors through the $L$-group of some torus, and we know that the Fargues-Scholze correspondence is surjective for tori.

§.§.§ Functoriality for the Bernstein center
Let $\mf{Z}(G;k)$ be the Bernstein center of $G$ with coefficients in $k$, and similarly for $H$. The Fargues-Scholze correspondence (<ref>) is deduced from the construction of a $k$-algebra homomorphism

\begin{equation}\label{eq: intro FS homomorphism}
\FS_G \co \Exc_k(W_E; \wh{G}) \rightarrow \mf{Z}(G;k)
\end{equation}

where $\Exc_k(W_E; \wh{G})$ is the excursion algebra (over $k$).

Building on ideas of Treumann-Venkatesh, we construct a map of Bernstein centers

\begin{equation}\label{eq: intro TV homomorphism}
\mf{Z}_{\TV} \co \mf{Z}(G;k) \rightarrow  \mf{Z}(H;k)
\end{equation}

which we call the Treumann-Venkatesh homomorphism. We show in Theorem <ref> that if $\ell > \max\{b(\chG), b(\chH)\}$, then there is a commutative diagram

\begin{equation}\label{eq: intro bernstein center functoriality}
\begin{tikzcd}
\Exc_k(W_E, \wh{G})  \ar[r, "\chpsi^*"] \ar[d, "\FS_G"']  & \Exc_k(W_E, \wh{H})  \ar[d, "\FS_H"]\\
\mf{Z}(G)\ar[r, "\mf{Z}_{\TV}"] & \mf{Z}(H)
\end{tikzcd}
\end{equation}

This shows that $\mf{Z}_{\TV}$ realizes functoriality for the Bernstein center. It is in fact the main ingredient to the proof of Theorem <ref>.

§.§ Methods
The proofs of the aforementioned results synthesize recent breakthroughs in several different areas of mathematics, including:

* The work [50, 48] of Treumann-Venkatesh on Smith theory and Langlands functoriality.

* The work [27, 28] of Juteau-Mautner-Williamson on parity sheaves in modular representation theory.

* The work [22] of Fargues-Scholze on the geometrization of the Local Langlands Correspondence.

Let sketch vaguely how these ingredients fit together. In the sketch below we use some abbreviated and simplified notation which does not match that of the main body of the text. We begin by commenting on the proof of Theorem <ref>. As explained earlier, the proof is based on categorifying the Brauer homomorphism of Treumann-Venkatesh to a Brauer functor, from perverse sheaves on the affine Grassmannian $\Gr_G$ for $G$ to perverse sheaves on the affine Grassmannian $\Gr_H$ for $H$, which should be Tannakian (additive, symmetric monoidal, and compatible with the fiber functor).

The construction of the Brauer homomorphism is based on the observation that restrictions of $\sigma$-equivariant functions from $G(E)/K$ to $H(E)/K^\sigma$ has the miraculous property of being compatible with convolution specifically in characteristic $\ell$ (the order of $\sigma$). The naive categorification of this operation would be restriction of sheaves from $\Gr_G$ to $\Gr_H$, but the miracle does not (naively) persist to the level of sheaves. Instead we apply an operation that we call Smith-Treumann localization, which is restriction followed by a certain Verdier quotient. The Smith-Treumann localization functor is then monoidal to some extent, but does not interact well with perversity. However it turns out that it can be made to interact well with parity sheaves in the sense of Juteau-Mautner-Williamson. This allows to lift Smith-Treumann localization to a functor from perverse parity sheaves on $\Gr_G$ to perverse parity sheaves on $\Gr_H$. Then studying the interaction of parity and perversity on affine Grassmannians allows to extend the functor to all perverse sheaves (under conditions on $\ell$).

The preceding construction could have been executed on the “classical” affine Grassmannian as soon as $\ell$ is good for $\chG$ and $\chH$.[Explicitly, this means that $\ell>2$ if $\chG$ has simple factors of type $B,C$ or $D$; $\ell>3$ if $\wh{G}$ has simple factors of type $G_2, F_4, E_6, E_7$; and $\ell>5$ if $\wh{G}$ has simple factors of type $E_8$.] However, at the next step we want to combine it with the constructions of Fargues-Scholze, so we actually need carry everything out on the $\bdrp$-affine Grassmannians and their Beilinson-Drinfeld variants, which are built out of the period rings of $p$-adic Hodge theory. As those objects live in the world of $p$-adic geometry, which behaves very differently from algebraic geometry in several key aspects (for example, there is no good theory of constructible sheaves), there are substantial technical difficulties to overcome in order to develop the appropriate technology in this new setting. We defer discussion of these technical difficulties to the individual sections in which they appear. The formalism that we develop here lays the groundwork for further applications of geometric representation theory to $p$-adic geometry, which we hope to pursue in future work.

(m) [matrix of nodes, row sep=1cm, column sep=4cm]
|[draw, circle, shift=(0.75cm,0)]| \(\Rep_k(\chG)\)     |[draw, circle, shift=(0.75cm,0)]| \(\Rep_k(\chH)\)

|[draw, circle, shift=(0.75cm,0)]| \(\Sat(\Gr_G;k)\)     |[draw, circle, shift=(0.75cm,0)]| \(\Sat(\Gr_H;k)\)

|[draw]| Tate$(\rR\Gamma_c(\Sht_G, -))$     |[draw]|  Tate$(\rR\Gamma_c(\Sht_H, -))$

[->] ($(m-2-1.south) + (0.4cm, 0.1)$) – node[midway, right] $\Sat_G$ ($(m-3-1.south east)!0.5!(m-3-1.north east) + (-0.5,0.1)$);
[->] ($(m-2-2.south)+ (.4cm, 0.1)$) – node[midway, left] $\Sat_H$  ($(m-3-2.south east)!0.5!(m-3-2.north east) +  (-0.5,0.1)$);
[->] (m-1-1.south) – node[midway, right]Geom. Satake (m-2-1.north);
[->] (m-1-2.south) – node[midway, left]Geom. Satake (m-2-2.north);

[->] (m-2-1.east) – node[midway, above] $\wt{\cbr}$ (m-2-2.west);
[->] (m-1-1.east) – node[midway, above] $\chpsi^*$ (m-1-2.west);

[dashed, <->] (m-3-2.west) – node[midway, above] equiv. localization (m-3-1.east);

This cartoon (produced with the aid of ChatGPT after much coaxing) depicts the Brauer functor $\wt{\cbr}$ interfacing with the Tate cohomology of moduli of local shtukas and with the $\sigma$-dual homomorphism $\chpsi$.

Next we turn towards the proof of Theorem <ref>. For this we need to integrate the Brauer functor with the construction of the Fargues-Scholze correspondence for $G$ and for $H$. The Fargues-Scholze correspondence for $G$ can be obtained by constructing excursion operators on the cohomology of moduli spaces $\Sht_G$ of local $G$-shtukas with coefficients in Satake sheaves coming from $\Gr_G$. We need to compare such cohomology groups to the ones obtained from applying the Brauer functor to get Satake sheaves on $\Gr_H$, transporting them to the moduli spaces $\Sht_H$ of local $H$-shtukas, and then taking cohomology. This comparison is mediated by equivariant localization for Tate cohomology. Figure <ref> depicts a cartoon of the strategy. From this comparison we extract relations among certain excursion operators for $\chG$ and for $\chH$, which are ultimately used to establish Theorem <ref>.

§.§ Related work
We note some related work on the problems studied here.

§.§.§ The functoriality conjecture
We focus first on the “functoriality conjecture”, Conjecture <ref>(2). Ronchetti [39] studies Conjecture <ref>(2) in the special case of cyclic base change for $\GL_n$, proving it for depth-zero cuspidal representations “of level zero and minimal-maximal type”.

Dhar-Nadimpalli [18] study Conjecture <ref>(2) in the special case of cyclic base change for $\GL_n$, proving it for generic representations (i.e., those possessing mod-$\ell$ Whittaker models).

The earlier work of the author [20] proved the analogous statement to Theorem <ref> for the Genestier-Lafforgue correspondence (in equal characteristic) in the special case of cyclic base change, and in doing so introduced a primordial form of some ideas developed here. We point out several differences.

* The methods of [20] were based on local-global compatibility, and were thus fundamentally restricted to the function field setting. Here our methods are purely local, and we encompass both mixed characteristic and equal characteristic local fields. To do this we work in the context of $p$-adic geometry, which presents substantial new difficulties.

* The argument of [20] was restricted to the case of cyclic base change, for both local and global reasons. On the local side, the $\sigma$-dual homomorphism is obvious for cyclic base change, and the construction of the Brauer functor in that case exploited some simplifying special features of cyclic base change; the present paper debuts more general and conceptual arguments to treat the local aspects in arbitrary generality (as long as $\ell$ is not too small). We still do not know how to generalize the global arguments even in the function field case, but fortunately they are irrelevant for the new approach.

The author's work [6] with Böckle-Harris-Khare-Thorne explicates Conjecture <ref>(2) for the Genestier-Lafforgue correspondence in the special case of cyclic base change for toral supercuspidals, assuming several conjectures about torsion in the cohomology of deep-level Deligne-Lusztig varieties. The analogous results for the Fargues-Scholze correspondence are subsumed by the $L$-parameter calculations in <ref>.

§.§.§ The $\sigma$-dual homomorphism
The work [40] of Riche-Williamson, which gives a geometric proof of the linkage principle, is not logically related to problems we study here, but has some philosophical similarities. We apply Smith-Treumann localization from $\Gr_G$ to $\Gr_H$ coming from an automorphism of $G$, while Riche-Williamson apply (a slightly different form of) Smith-Treumann localization to a self-embedding of $\Gr_G$ coming from the action of $\mu_\ell \subset \G_m$ via loop rotation.

The paper [34] of Leslie-Lonergan also applies Smith-Treumann localization along this self-embedding, in attempt to give a geometric construction of the Frobenius contraction functor on $\Rep_k(\chG)$. The formalism of localization that they develop is more similar to the one that we use in this paper.

§.§ Organization of the paper
We now indicate the structure of the paper.

In <ref> we collect some notation and abbreviations used commonly throughout the paper.

In <ref> we develop a general formalism of Smith-Treumann localization for diamonds, which refers to a certain type of sheaf restriction operation from a diamond $Y$ to its $\sigma$-fixed points. Then in <ref> we calculate the $\sigma$-fixed points of various diamonds associated to $G$, such as the $\bdrp$-affine Grassmannian and its Beilinson-Drinfeld or twisted variants, as well as moduli spaces of local $G$-shtukas. These calculations are used when applying Smith-Treumann localization to such spaces.

In <ref>, we develop a notion of “relative parity sheaves” on the Beilinson-Drinfeld affine Grassmannians arising in $p$-adic geometry. Then in <ref> we prove that away from small characteristics, (normalized) relative parity sheaves are relative perverse and correspond under the Geometric Satake equivalence to tilting modules; this is analogous to a theorem of Juteau-Mautner-Williamson from [28]. In fact, each of <ref>, <ref>, and <ref> is parallel to some existing theory in algebraic geometry, but there are non-trivial new issues encountered in the setting of $p$-adic geometry, which we try to illuminate at the beginnings of the respective sections.

In <ref>, we construct the Brauer functor and establish its properties, proving Theorem <ref>. Then in <ref> we study the (Tate) cohomology of moduli stacks of shtukas, and integrate it with the Brauer functor in order to prove certain identities of excursion operators. The applications ripen for picking in <ref>, where we combine the preceding sections to prove Theorem <ref>, Theorem <ref>, construct the Treumann-Venkatesh homomorphism and establish the commutative diagram (<ref>).

Finally, in <ref> we study the example where $\sigma$ is conjugation by a strongly regular order-$\ell$ element of an unramified elliptic maximal torus of $G$. We calculate the $\sigma$-dual embedding, the Tate cohomology of deep level Deligne-Lusztig varieties and their compact inductions, and deduce explicit computations of Fargues-Scholze parameters.

The beginnings of individual sections summarize their contents in more detail.

§.§ Acknowledgments
It is a pleasure to acknowledge Brian Conrad, Olivier Dudas, Ian Gleason, Jesper Grodal, David Hansen, Tasho Kaletha, Teruhisa Koshikawa, Gopal Prasad, Simon Riche, Peter Scholze, Sug Woo Shin, Jay Taylor, David Treumann, Akshay Venkatesh, and Geordie Williamson for relevant discussions. Special thanks to Venkatesh for posing the question of finding a uniform construction of the $\sigma$-dual homomorphism to me in 2017, to Scholze for the proof of Lemma <ref>, and to Conrad for help with Lemma <ref>. Parts of this paper were completed at the Hausdorff Institute for Mathematics, funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2047/1 – 390685813. This work was supported by NSF Postdoctoral Fellowship DMS-1902927 and NSF Grant DMS-2302520.

§ NOTATION

§.§ Local fields

We let $E$ be a local field of residue characteristic $p$, $\cO_E$ its ring of integers, $\varpi_E$ a uniformizer, and $\F_q  = \cO_E/\varpi_E$ its residue field.

Let $\brE$ be the completion of the maximal unramified extension of $E$. For a reductive group $G/E$, $B(G)$ denotes the Kottwitz set of $G$, i.e., elements of $G(\brE)$ modulo Frobenius-conjugacy.

Let $E^s$ be a separable closure of $E$ and $W_E \subset \Gal(E^s/E)$ be the Weil group of $E$.

§.§ The group $\Sigma$
Throughout $\ell$ denotes a prime number and $\Sigma$ denotes a finite cyclic group of order $\ell$. The notation $\sigma$ always denotes a generator of $\Sigma$; conversely, if an automorphism $\sigma$ is constructed first then $\Sigma$ always denotes the group generated by $\sigma$. Some constructions (e.g., Tate cohomology) are phrased in terms of $\sigma$, but ultimately all constructions are independent of the choice of $\sigma$.

We denote by $N$ the element $1 + \sigma + \ldots + \sigma^{\ell-1} \in \Z[\Sigma]$.

For an object $Y$ with an action of $\Sigma$, we denote by $Y^\sigma$ or $\Fix(\sigma, Y)$ the $\sigma$-fixed points of $Y$.

§.§ Reductive groups
Throughout $G$ denotes a reductive group over $E$. When the notation $H$ appears, it refers to a reductive arising as the subgroup of $G$ fixed by an action of $\Sigma$. We denote by $\iota \co H \rightarrow G$ the tautological embedding, and we use the same notation for induced maps such as $H(E) \rightarrow G(E)$, $B(H) \rightarrow B(G)$, $\Gr_H \rightarrow \Gr_G$, etc.

For a torus $T$, we denote by $X_*(T)$ and $X^*(T)$ the cocharacter and character groups of $T$, respectively, and by $X_*(T)^+$ and $X^*(T)^+$ the subsets of dominant (co)characters.

We denote by $\Rep^{\sm}_k G(E)$ the category of smooth representations of $G(E)$ over $k$.

§.§ Perfectoid spaces

The notation $(C,C^+)$ will always mean that $C$ is an algebraically closed perfectoid field and $C^+ \subset C$ is an open bounded valuation subring.

Following [22], we denote by $\Perf = \Perf_{\ol{\F}_p}$ the category of perfectoid spaces over $\ol{\F}_p$.

§.§ Coefficients

Throughout $k$ denotes an algebraically closed field of characteristic $\ell  = \# \Sigma	$, where $\ell$ is a prime number different from $p$. We denote by $\OO := W(k)$ the Witt vectors of $k$.

We write $\Frob \co k \rightarrow k$ for the absolute Frobenius automorphism $x \mapsto x^\ell$.

§.§ Sheaves
We will use $\Lambda$ to denote a coefficient ring finite over $\OO$. In particular, $\Lambda$ is $\ell$-adically complete. The six-functor formalism for such sheaves is developed in [42] (the adic case is in 27 of loc. cit.).

We say that a map $f \co Y \rightarrow Y'$ of small v-stacks on $\Perf$ is shriekable if it is compactifiable, representable in locally spatial diamonds, and has locally finite $\dimg$. This is exactly the hypothesis in <cit.> for $Rf_! \co D_{\et}(Y; \Lambda) \rightarrow D_{\et}(Y'; \Lambda)$ to exist, and under the same assumptions its right adjoint $Rf^! \co D_{\et}(Y'; \Lambda) \rightarrow D_{\et}(Y; \Lambda) $ exists.

For a shriekable map $f \co Y \rightarrow Y'$, we denote by $\DD_{Y/Y'}$ the relative Verdier duality functor,

\[
\DD_{Y/Y'}(-) := \cRHom_{ D_{\et}(Y; \Lambda) }(-, f^! \Lambda).
\]

We abuse notation and write $\DD_{Y/Y'} := \DD_{Y/Y'}(\Lambda) = f^! \Lambda$ for the relative dualizing sheaf. We omit $Y'$ from the notation in the case $Y' = \Spd \ol{\F}_p$.

§.§ Categories

For a category $\msf{C}$ with an action of a group $\Gamma$, we let $\msf{C}^{B \Gamma}$ denote the category of $\Gamma$-equivariant objects in $\msf{C}$. This comes equipped with a forgetful functor to $\msf{C}$. If $\msf{C}$ is a triangulated category with a natural stable $\infty$-categorical enhancement, then by $\msf{C}^{B \Gamma}$ we mean the homotopy category of the $\Gamma$-equivariant objects in its stable $\infty$-categorical enhancement.

For an $A$-linear abelian category $\msf{C}$ and a commutative ring homomorphism $A \rightarrow B$, we abbreviate

\[
\msf{C} \otimes_{A} B := \msf{C} \otimes_{A-\Mod} (B-\Mod).
\]

When $A = \OO$ and $B = k$, we write

\[
\FF \co \msf{C} \rightarrow \msf{C} \otimes_{\OO} k
\]

for the tautological base change.

§ SMITH-TREUMANN LOCALIZATION FOR DIAMONDS

In this section we develop for diamonds a form of Smith-Treumann localization, which refers to a type of sheaf-theoretic equivariant localization from a $\Sigma$-equivariant space to its $\Sigma$-fixed points. There is a loose analogy between Smith-Treumann localization and the perhaps more familiar hyperbolic localization for spaces with a $\G_m$-action.

The output of the theory looks similar to that for complex algebraic varieties or schemes (developed in [48] and [40]). However, the details of the proofs are quite different, owing to the different behavior of étale sheaves on adic spaces. For example, the theory relies crucially on finiteness results, which in the case of schemes comes from quasi-compactness (among other things). On the other hand, non-quasicompactness is ubiquitous in $p$-adic geometry. For example, the complement of a closed subspace is usually not quasi-compact, hence the extension-by-zero from such a complement is not constructible; this example already illustrates that we will frequently need to contend with non-constructible sheaves.

The contents of this section are as follows. In <ref> we codify the notion of “extra small” v-stacks. The adjective “extra small”  is a slight strengthening of “small”, and allows to approximate the v-stack by quasicompact subspaces. In <ref>, we define the (large) “Tate category” of a locally spatial diamond, following ideas of Treumann [48] but with modifications as in <cit.> to allow infinite dimensional sheaves. In <ref> we prove technical bounds for the tor-dimension of various sheaf operations, in order to show in that these operations pass to the Tate category, which we do in <ref>. In <ref> we define the “Smith operation”, a kind of localization functor to the $\Sigma$-fixed points, and establish compatibility properties that may be thought of as forms of equivariant localization. Finally, in <ref> we define Tate cohomology and various generalizations of it.

§.§ Extra small v-stacks

Let $\omega_1$ be the first uncountable cardinal. Recall that an $\omega_1$-cofiltered inverse system $I$ is one for which any functor $J \rightarrow I$, with $J$ a countable category, extends to $J^{\triangleleft} \rightarrow I$, where the cone $J^{\triangleleft}$ consists of $J$ plus an extra object with a unique map to each object of $J$.

We say that a v-stack $Y$ is extra small if for any $\omega_1$-cofiltered inverse system $\{S_i = \Spa(R_i, R_i^+)\}_{i \in I}$ of affinoid perfectoid spaces with inverse limit $S = \Spa(R, R^+)$, the natural map

\[
Y(S) \rightarrow \colim_{i \in I} Y(S_i).
\]

is an isomorphism.

The following Lemma justifies the terminology.

If a v-stack $Y$ is extra small, then $Y$ is small.

The proof is contained in the first paragraph of <cit.>.

Informally speaking, all the small v-stacks that have come up in our experience are also extra small. The remark below <cit.> suggests that all “reasonable” v-stacks are extra small.

The last paragraph of the proof of <cit.> shows that $\Bun_G$ is extra small. By minor variations on this argument, the following v-stacks are also extra small.

* The local Hecke stack $\cHck_{G/(\Div^1_X)^I}$ <cit.>, and the global Hecke stack $\Hck_G^I$ <cit.>.

* The Beilinson-Drinfeld Grassmannian $\Gr_{G,(\Div^1_X)^I}$ <cit.>.

* The moduli spaces of shtukas $\Sht_{(G, b, \mu_{\bu}), K}$ <cit.>.

The importance of extra smallness (for us) is to control the cohomological dimension of direct image along open embeddings. For quasi-compact open embeddings, the direct image is actually exact, but we will typically be contending with non-quasicompact open embeddings. Under hypotheses that spaces are extra small, we will be able to approximate open embeddings by quasicompact ones, as articulated in  the following Lemma (pointed out by Peter Scholze).

Let $Y$ be a locally spatial diamond admitting a countable open cover by spatial diamonds $Y_n$. Let $Z \inj Y$ be a closed embedding such that $Z \cap Y_n$ is extra small for each $n$. Then $U := Y \setminus Z$ is a countable union of quasicompact open subspaces.

If $Y_n \setminus Z_n$ is a countable union of quasicompact open subsets for each $n$, then

\[
Y \setminus Z = \bigcup_{n} (Y_n \setminus Z_n)
\]

is also a countable union of quasicompact open subsets. Therefore, renaming $Y_n$ to $Y$ and $Z_n$ to $Z$, we may assume that $Y$ is spatial and $Z$ is extra small, with the goal of showing that $U := Y \setminus Z$ is a countable union of quasicompact open subsets.

Consider the collection $\cZ$ of closed sub-diamonds $Z_i \subset Y$ such that $Z_i$ contains $Z$ and $Y \setminus Z_i$ is a countable union of quasi-compact open subsets. Viewing $\cZ$ as a partially ordered set under inclusion, we claim that $\cZ$ is $\omega_1$-cofiltered. Indeed, if $I$ is countable then we have

\[
Y \setminus \bigcap_{i \in I} Z_i = \bigcup_{i \in I} (Y \setminus Z_i)
\]

is a countable union of countable unions of quasi-compact open subsets, so $\bigcap_{i \in I} Z_i \in \cZ$.

There is an obvious map

\begin{equation}\label{eq: intersection of Z_i}
Z \rightarrow \limit_{Z_i \in \cZ} Z_i,
\end{equation}

which we claim is an isomorphism. Recall that closed sub-diamonds of $Y$ are determined by their underlying topological spaces, as follows from <cit.> (the generalization of this statement to v-stacks is <cit.>), so it suffices to check (<ref>) at the level of topological spaces. Any point of the open subspace $|Y \setminus Z|$ lies in a quasicompact open subset since $Y$ is spatial. This shows the second equality in the sequence of identifications

\[
|\limit_{Z_i \in \cZ} Z_i| = \limit_{Z_i \in \cZ} |Z_i| = |Z|,
\]

verifying the claim.

Since $Y$ is spatial, there exists <cit.> a cover of $Y$ by a strictly totally disconnected space, which is of the form $ \Spa (B,B^+) \surj Y$ by <cit.>. Recall that a closed subspace of a strictly totally disconnected space is again strictly totally disconnected by <cit.>, and then affinoid by <cit.>. Hence for each $Z_i \in \cZ$, the cover pulls backs to a cover $\Spa (A_i, A_i^+) \surj Z_i$ where $\Spa (A_i, A_i^+)$ is strictly totally disconnected. We showed that the system $\{ \Spa(A_i, A_i^+)\}_{Z_i \in \cZ}$ is $\omega_1$-cofiltered. Then, letting $\Spa (A,A^+)  := \varprojlim_{Z_i \in \cZ} \Spa (A_i, A_i^+)$,[So $A = \colim A_i$, which we note is already complete since any Cauchy sequence already lies in some $A_i$ by the $\omega_1$-cofiltered condition.] the composite map

\[
\Spa (A,A^+)  = \limit_{Z_i \in \cZ} \Spa (A_i, A_i^+) \rightarrow \limit_{Z_i \in \cZ} Z_i = Z
\]

factors through some $\Spa (A_i, A_i^+) \rightarrow Z$ since $Z$ is extra small. This gives a retract $Z_i \rightarrow Z$ over $Y$, which must then be an isomorphism since these are closed subdiamonds of $Y$. Hence $Z \in \cZ$, which means by definition that $Y \setminus Z$ is a countable union of quasi-compact open subsets.

Motivated by Lemma <ref>, we make the following definition.

We say that a locally spatial diamond $Y$ is ($\omega_0$-)locally spatial if $Y$ can be written as a (countable) union $\bigcup_n Y_n$ of open spatial sub-diamonds. (Recall that $\omega_0 = \aleph_0$ is the cardinality of the natural numbers.) We say that a closed subdiamond $Z \inj Y$ is $\omega_0$-locally extra small if such a union can be arranged so that $Y_n \cap Z$ is extra small for each $n$.

We say that a morphism of v-stacks $f \co Y' \rightarrow Y$ is ($\omega_0$-)locally spatial if for any spatial diamond $X$ and any map $X \rightarrow Y$, the fibered product $X \times_Y Y'$ is ($\omega_0$-)locally spatial.

Let $i \co Z \inj Y$ be a closed embedding of $\omega_0$-locally spatial diamonds. If $Z \subset Y$ is $\omega_0$-locally extra small, then Lemma <ref> says that the complementary open embedding $j \co U \inj Y$ is $\omega_0$-locally spatial.

We say that a locally spatial diamond $Y$ has $\ell$-cohomological dimension $\leq d$ if its cohomology on étale $\F_\ell$-sheaves has amplitude in $[0, d]$. We say that $Y$ is $\ell$-bounded if has $\ell$-cohomological dimension $\leq d$ for some $d<\infty$.

The following statements are used to bound the $\ell$-cohomological dimension in practice. Recall from <cit.> that the dimension of a locally spectral space is the supremum of the lengths of chains of specializations. By <cit.>, if $Y$ is a spatial diamond of finite dimension, then $Y$ has $\ell$-cohomological dimension $\leq \dim Y + \sup_y \cd_\ell y$ where $y$ runs through maximal points of $Y$.

According to <cit.>, if $f \co Y \rightarrow \Spa (C, C^+)$ is a map of locally spatial diamonds, then $\cd_{\ell} y \leq \dimg f$ for all maximal points $y \in Y$.

Let $A$ be a noetherian ring. Let $Y$ be an $\omega_0$-locally spatial diamond with $\ell$-cohomological dimension $\leq d$. Let $Z \inj Y$ be an $\omega_0$-locally extra small closed subdiamond with (open) complement $j \co U \inj Y$. Then $\rR j_* \co D_{\et}(U; A) \rightarrow D_{\et}(Y;A)$ has cohomological dimension $\leq d + 1$.

The statement is local, so we may immediately reduce to the case that $Y$ is a spatial diamond and $Z \inj Y$ is extra small.

By Lemma <ref>, there is a countable sequence of quasicompact open embeddings $j \co U_n  \inj Y$ such that $U_n \subset U$ and $\colim_n U_n = U$. Hence we have

\[
\rR j_* \cong \Rlim_n \rR j_{n*} j_n^*.
\]

By <cit.>, the (derived) direct image along a quasi-compact open embedding is exact, hence has cohomological dimension zero. This applies to $j_n$ for every $n$. Then, thanks to the countability of the indexing set for the inverse limit, $\Rlim_n A_n$ can be expressed as

\begin{equation}\label{eq: countable Rlim}
\Rlim_n A_n \cong \mrm{cone}(\rR{\prod_n} A_n \rightarrow \rR{\prod_n} A_n)[-1]
\end{equation}

where $\rR\hspace{-.1cm}\prod_n$ is the derived functor of the product, defined as the product of injective resolutions of the terms. Therefore, $\rR\hspace{-.1cm}\prod_n$ has cohomological dimension bounded by the cohomological dimension of $Y$, which by assumption is $d<\infty$. Then from (<ref>) we see that $\Rlim_n$ has cohomological dimension at most $d+1$.

In the situation of Lemma <ref>, if $\cK \in D_{\et}(U; A)$ has finite tor-dimension then $\rR j_* \cK \in D_{\et}(Y; A)$ has finite tor-dimension.

This follows from <cit.>, which applies because $\rR j_*$ has finite cohomological dimension by Lemma <ref> and the fact <cit.> that $Y_{\et}$ has enough points, thanks to $Y$ being locally spatial.

§.§ The Tate category

Let $\Lambda$ be a finite commutative $W(k)$-algebra; we will be most interested in the cases $\Lambda = k$ or $W(k)$. Recall that $\Sigma$ is a cyclic group of order $\ell$ and $\Lambda[\Sigma]$ denotes its group ring.

Let $Y$ be a locally spatial diamond. We define $\Flat^b(Y; \Lambda[\Sigma]) \subset D^b_{\et}(Y; \Lambda[\Sigma])$ to be the full subcategory consisting of complexes with finite tor-dimension <cit.>, i.e., which have tor-amplitude in $[a,b]$ for some $a,b \in \Z$.

We define the (large) Tate category of $Y$ (with respect to $\Lambda$) to be the Verdier quotient category

\[
\Shv(Y; \cT_{\Lambda}) := D^b_{\et}(Y; \Lambda[\Sigma]) / \Flat^b(Y; \Lambda[\Sigma]).
\]

(A “small” variant of the Tate category will appear later in <ref>.) We denote the tautological projection map from $D^b_{\et}(Y; \Lambda[\Sigma])$ to $\Shv(Y; \cT_{\Lambda})$ by

\[
\TT^* \co D^b_{\et}(Y; \Lambda[\Sigma]) \rightarrow \Shv(Y; \cT_{\Lambda}).
\]

We will only be using the large Tate category for $\Lambda =k$, but some arguments are repeated in the small version which also gets used for $\Lambda = W(k)$, so we formulate it in more generality.

For $\Lambda = k$, the following Lemma (pointed out by Jesper Grodal) gives an alternative characterization of the subcategory $\Flat^b(Y; k[\Sigma])$, showing that it is determined by conditions on geometric stalks.

The subcategory $\Flat^b(Y; k[\Sigma] ) \subset D^b_{\et}(Y; k[\Sigma])$ coincides with the full subcategory spanned by objects whose stalks at all geometric points have finite tor-dimension over $k[\Sigma]$.

We apply <cit.> which immediately gives one containment: if $\cK \in D^b_{\et}(Y; k[\Sigma])$ has finite tor-dimension over $k[\Sigma]$, then all its stalks at geometric points have finite tor-dimension over $k[\Sigma]$.

For the other containment, again by <cit.> it suffices to show that if all the geometric stalks of $\cK \in D^b_{\et}(Y; k[\Sigma])$ have finite tor-amplitude, then there is a uniform bound on the tor-amplitude of the stalks. Indeed, since $\cK$ is bounded we may pick a bounded complex representing it, say concentrated in degrees $[a,b]$. It then suffices to show: if $K$ is a complex of $k[\Sigma]$-modules concentrated in degrees $[a,b]$ and has finite tor-dimension, then in fact its tor-amplitude lies in $[a,b]$.

Since $k[\Sigma]$ is Artinian, the properties of being flat and projective over $k[\Sigma]$ coincide by <cit.>. Therefore $\cK$ also has finite projective dimension <cit.>. Furthermore, since $k[\Sigma]$ has finitistic dimension $0$ (by [38], the finitistic dimension of a commutative ring is bounded by the Krull dimension), the projective amplitude of $K$ lies in $[a,b]$. Hence $\cK$ is represented by a complex of projective $k[\Sigma]$-modules supported in degrees $[a,b]$, and therefore has tor-amplitude in $[a,b]$.

Let $\epsilon \co \Lambda[\Sigma] \surj \Lambda$ be the augmentation and $\epsilon^* \co D^b_{\et}(Y; \Lambda) \rightarrow D^b_{\et}(Y; \Lambda[\Sigma])$ be pullback along $\epsilon$. Define $\TT := \TT^* \epsilon^* \co  D^b_{\et}(Y; \Lambda) \rightarrow \Shv(Y; \cT_{\Lambda})$.

The notation $\TT, \TT^*, \epsilon^*$ follows that of Leslie-Lonergan [34].

§.§ Fixed points
For an endomorphism $\sigma$ of a v-stack $Y$, we write $\Fix(\sigma, Y) := Y^\sigma$ for the fibered product

\[
\begin{tikzcd}
Y^\sigma \ar[r] \ar[d] & Y \ar[d, "\Id \times \sigma"] \\
Y \ar[r, "\Delta"] & Y \times Y
\end{tikzcd}
\]

Observe that if $Y$ is a separated locally spatial diamond, then $\Delta$ is a closed embedding, hence $Y^\sigma  \rightarrow Y$ is a closed embedding. We will only be interested in this situation. We denote by $\Sigma = \tw{\sigma}$ the cyclic group of order $\ell$. Recall the equivariant bounded derived category $D_{\et}^b(Y; \Lambda)^{B \Sigma}$, which is the homotopy category of $\cD_{\et}^b(Y; \Lambda)^{B \Sigma}$, the $\infty$-category of functors from $B \Sigma$ to the $\infty$-category $\cD_{\et}^b(Y; \Lambda)$ from <cit.>.

The $\Sigma$-action on $Y$ induces the trivial $\Sigma$-action on $Y^\sigma$, for which we have an equivalence of derived categories

\begin{equation}\label{eq: derived cat for trivial action}
D_{\et}^b(Y^\sigma;\Lambda)^{B \Sigma} \cong D^b_{\et}(Y^\sigma; \Lambda[\Sigma]).
\end{equation}

§.§ Bounding tor-dimension
Here we establish technical results that show the “permanence” of finite tor-dimension under various operations.

Let $Y$ be a locally spatial diamond with a free action of $\Sigma$. Let $q \co Y \rightarrow Y / \Sigma$ denote the quotient. Then for any $\cF \in D^b_{\et}(Y; \Lambda[\Sigma])$, we have $q_* \cF \in \Flat^b(Y/\Sigma; \Lambda[\Sigma])  \subset D^b_{\et}(Y/\Sigma; \Lambda[\Sigma])$.

It suffices to show that for any open subset $U \subset Y/\Sigma$, the $\Lambda[\Sigma]$-module $q_* \cF (U)$ is free. Indeed, since $\Sigma$ acts freely on $Y$ we have

\[
q^{-1}(U) = \coprod_{i \in \Z/\ell \Z}^\ell U_i
\]

where $q$ maps each $U_i$ isomorphically to $U$ and $\sigma$ cyclically permutes $U_i \mapsto U_{i+1}$. Then

\[
q_* \cF (U) \cong  \cF(q^{-1}(U))  \cong \prod_{i=1}^\ell \cF(U_i)
\]

where $\sigma$ acts by sending $(f_i \in \cF(U_i))_{i \in \Z/\ell \Z} \mapsto (\sigma^* (f_i) \in \cF(U_{i-1}))_{i \in \Z/\ell \Z}$, which is visibly free.

Let $Y$ be an $\ell$-bounded, separated, $\omega_0$-locally spatial diamond with an action of $\Sigma$. Assume that $Y^\sigma \inj Y$ is $\omega_0$-locally extra small. Let $U := Y \setminus Y^{\sigma}$ and $j \co U \inj Y$ be its inclusion into $Y$. Then for any $\cF \in D^b_{\et}(Y; \Lambda[\Sigma])$, we have $i^*\rR j_* \cF \in \Flat^b(Y^\sigma; \Lambda[\Sigma])$.

Let $Y/\Sigma$ be the quotient diamond; then $Y/\Sigma$ is also $\ell$-bounded, $\omega_0$-locally spatial, and $Y^\sigma \inj Y/\Sigma$ is $\omega_0$-locally extra small. Let $i \co Y^\sigma \inj Y$ be the inclusion of the $\Sigma$-fixed points. Since the map $q \co Y \rightarrow Y/\Sigma$ is totally ramified over $Y^\sigma$, the composition

\[
Y^\sigma \xrightarrow{i} Y \xrightarrow{q} Y/\Sigma
\]

is a closed embedding, which we denote $\ol{i}$, and the square

\[
\begin{tikzcd}
Y^\sigma \ar[d, equals] \ar[r, "i"] & Y \ar[d,"q"] \\
Y^\sigma \ar[r, "\ol{i}"]  & Y/\Sigma
\end{tikzcd}
\]

is Cartesian. Hence we have an isomorphism

\begin{equation}\label{eq: big tate 2 eq 1}
i^* \rR j_* \cF \cong  \ol{i}^* q_* \rR j_* \cF  \in D^b_{\et}(Y^\sigma; \Lambda[\Sigma]).
\end{equation}

From the commutativity of the diagram

\[
\begin{tikzcd}
U \ar[r, hook, "j"] \ar[d, "q_U"] & Y \ar[d, "q"] \\
U/\Sigma \ar[r, hook, "\ol{j}"] & Y/\Sigma
\end{tikzcd}
\]

we have $\ol{i}^*q_* \rR j_* \cF  \cong \ol{i}^*\rR\ol{j}_* q_{U*} \cF$. Now Lemma <ref> implies that $q_{U*} \cF$ has finite tor-dimension. The extra smallness hypothesis allows us to apply Corollary <ref> to deduce that $\rR\ol{j}_* q_{U*} \cF$ also has finite tor-dimension, hence $\ol{i}^* \rR\ol{j}_* q_{U*} \cF$ also has finite tor-dimension, and then we conclude using (<ref>).

Retain the notation and assumptions from Lemma <ref>. Let $i \co Y^\sigma \hookrightarrow Y$ be the embedding of $\Sigma$-fixed points (assumed to be $\omega_0$-locally extra small). Then for any $\cF \in D^b_{\et}(Y; \Lambda)^{B\Sigma}$, the cone of $i^! \cF \rightarrow i^* \cF$ belongs to $\Flat^b(Y^\sigma; \Lambda[\Sigma])$.

Let $j \co Y \setminus Y^\sigma \inj Y$. Consider the exact triangle $i_* i^! \cF \rightarrow \cF \rightarrow j_* j^* \cF$ in $D_{\et}^b(Y; \Lambda)^{B\Sigma}$. Applying $i^*$ to it yields the exact triangle in $D^b_{\et}(Y^{\sigma}; \Lambda)^{B \Sigma}$,

\[
i^! \cF \rightarrow i^* \cF \rightarrow i^* \rR j_* j^* \cF.
\]

Then Lemma <ref> implies that $\mrm{Cone}(i^! \cF \rightarrow i^* \cF) \cong i^* \rR j_* j^* \cF \in \Flat^b(Y^{\sigma}; \Lambda[\Sigma])$, as desired.

Let $A$ be a finite $\Lambda$-algebra (not necessarily commutative). Let $Y$ and $S$ be locally spatial diamonds, and let $f \co Y \rightarrow S$ be a shriekable morphism.

(1) Then $\rR f_! \co D_{\et}^b(Y; A) \rightarrow D_{\et}^b(S; A)$ has cohomological dimension $\leq 3 \dimg f$.

(2) If furthermore $S$ has $\ell$-cohomological dimension $\leq d$ and $f$ is $\omega_0$-locally spatial, then $\rR f_* \co D_{\et}^b(Y; A) \rightarrow D_{\et}^b(S; A)$ has cohomological dimension $\leq 3 \dimg f + d + 1$.

(1) By the hypothesis that $Y$ and $f$ are locally spatial, we may write $Y \cong \colim_i Y_i$ as a filtered colimit of spatial open subdiamonds $j_i \co Y_i \inj Y$. We have by definition <cit.> that

\[
\rR f_! \cong \colim_i \rR (f|_{Y_i})_! j_i^*.
\]

By <cit.>, each $\rR(f|_{Y_i})_!$ has cohomological dimension at most $3 \dimg f$. Since $j_i^*$ is exact and filtered colimits are exact, we deduce that $Rf_!$ also has cohomological dimension at most $3 \dimg f$.

(2) Similarly to (1), we have

\[
\rR f_* \cong \Rlim_i \rR(f|_{Y_i})_* j_i^*
\]

where now we may arrange the indexing set to be countable, thanks to the hypothesis that $f$ is $\omega_0$-locally spatial. Each $\rR(f|_{Y_i})_* j_i^*$ has cohomological dimension at most $3\dimg f$ by <cit.>. Then we bound the cohomological dimension of the derived inverse limit as in the proof of Lemma <ref>.

Let $A$ be a finite $\Lambda$-algebra (not necessarily commutative). Let $Y$ and $S$ be locally spatial diamonds, and let $f \co Y \rightarrow S$ be shriekable.

(1) Then $\rR f_! \co D^b_{\et}(Y; A) \rightarrow D^b_{\et}(S; A)$ carries $\Flat^b(Y; A)$ to $\Flat^b(S; A)$.

(2) If furthermore $S$ is $\ell$-bounded and $f$ is $\omega_0$-locally spatial, then $\rR f_* $ carries $\Flat^b(Y; A)$ to $\Flat^b(S; A)$.

(1) Suppose $\cF \in \Flat^b(Y; A)$. We need to verify that for any sheaf $M$ (concentrated in degree $0$) in $D^b_{\et}(S; A)$, we have $\rH^{-i}(M \dotimes \rR f_! \cF) = 0$ for $i \gg 0$. By the projection formula, we have

\[
\rH^{-i}(M \dotimes_A \rR f_! \cF) \cong \rH^{-i}(\rR f_! (f^*M \dotimes_A \cF)).
\]

By assumption $f^*M \dotimes_A \cF$ is bounded. Then by Lemma <ref>(1), $\rR f_! (f^*M \dotimes_A \cF)$ is bounded, so $\rH^{-i}(\rR f_! (f^*M \dotimes_A \cF)) = 0$ for $i \gg 0$, as desired.

(2) This follows from <cit.>, which applies because $\rR f_*$ has finite cohomological dimension by Lemma <ref> and the fact <cit.> that $S_{\et}$ has enough points, thanks to $S$ being locally spatial.

Let $Y$ and $S$ be locally spatial diamonds, and let $f \co Y \rightarrow S$ be shriekable. Suppose $\Sigma$ acts trivially on $S$ and freely on $Y$, and $f$ is $\Sigma$-equivariant.

(1) Then $\rR f_! \co D^b_{\et}(Y; \Lambda[\Sigma]) \rightarrow D^b_{\et}(S; \Lambda[\Sigma])$ lands in $\Flat^b(Y; \Lambda[\Sigma])$.

(2) If furthermore $S$ has $\ell$-cohomological dimension $\leq d$ and $f$ is $\omega_0$-locally spatial, then $\rR f_* $ carries $\Flat^b(Y; \Lambda[\Sigma])$ to $\Flat^b(S; \Lambda[\Sigma])$.

By the hypotheses, we may factor $f$ as the composition of $\Sigma$-equivariant maps

\[
Y \xrightarrow{q} Y/\Sigma \xrightarrow{\ol{f}} S.
\]

For (1), apply Lemma <ref> to $q_!$ and Lemma <ref>(1) to $\rR\ol{f}_!$ with $A := \Lambda[\Sigma]$. For (2), apply Lemma <ref> to $q_* = q_!$ and Lemma <ref>(2) to $\rR\ol{f}_*$ with $A:= \Lambda[\Sigma]$.

§.§ Functors on Tate categories

Let $f \co Y \rightarrow S$ denote a $\Sigma$-equivariant map of separated locally spatial diamonds with $\Sigma$-action.

§.§.§ Pullback
Since the pullback functor $f^* \co D^b_{\et}(S^{\sigma}; \Lambda)^{B \Sigma} \rightarrow D^b_{\et}(Y^{\sigma}; \Lambda)^{B \Sigma}$ preserves stalks, it carries $\Flat^b(S^\sigma; \Lambda[\Sigma])$ to $\Flat^b(Y^\sigma; \Lambda[\Sigma])$, hence induces a functor

\[
f^* \co \Shv(S^{\sigma}; \Cal{T}_{\Lambda}) \rightarrow \Shv(Y^{\sigma}; \Cal{T}_{\Lambda}).
\]

§.§.§ Pushforward
Assume furthermore that $f \co Y \rightarrow S$ is shriekable. By Lemma <ref>(1), the functor $\rR f_!\co D^b_{\et}(Y^{\sigma}; \Lambda[\Sigma]) \rightarrow D^b_{\et}(S^{\sigma}; \Lambda[\Sigma])$ carries $\Flat^b(Y^\sigma; \Lambda[\Sigma])$ to $\Flat^b(S^\sigma; \Lambda[\Sigma])$, hence induces a functor

\[
\rR f_! \co \Shv(Y^{\sigma}; \cT_{\Lambda}) \rightarrow \Shv(S^{\sigma}; \cT_{\Lambda}).
\]

If $f$ is furthermore $\omega_0$-locally spatial and $i \co Y^\sigma \inj Y$ is $\omega_0$-locally extra small, then similarly using Lemma <ref>(2) gives a functor

\[
\rR f_! \co \Shv(Y^{\sigma}; \cT_{\Lambda}) \rightarrow \Shv(S^{\sigma}; \cT_{\Lambda}).
\]

§.§ The Smith operation
We define the analogue in $p$-adic geometry of the “Smith operation” $\Psm$ introduced by Treumann in [48].

Let $Y$ be a separated locally spatial diamond with an action of $\Sigma$ and $i \co Y^\sigma \inj Y$ the inclusion of its $\Sigma$-fixed points. The Smith operation is the functor

\begin{equation}\label{eq: Psm for diamonds}
\Psm  := \TT^* \circ i^* \co D_{\et}^b(Y; \Lambda)^{B \Sigma} \rightarrow \Shv(Y^{\sigma}; \Cal{T}_{\Lambda})
\end{equation}

defined as the composition of $i^* \co D_{\et}^b(Y; \Lambda)^{B \Sigma} \rightarrow D_{\et}^b(Y^{\sigma}; \Lambda)^{B \Sigma} \stackrel{\eqref{eq: derived cat for trivial action}}\cong D^b_{\et}(Y^{\sigma}; \Lambda[\Sigma])$ with the projection $\TT^*$ to $\Shv(Y^{\sigma}; \Cal{T}_{\Lambda})$.

The following result may be viewed as a form of equivariant localization for spaces with $\Sigma$-action, comparing the (relative) cohomology of a space with that of its $\Sigma$-fixed points.

Let $Y$ and $S$ be separated locally spatial diamonds, and let $f \co Y \rightarrow S$ be shriekable. Denote $f^{\sigma} := f \circ i \co Y^{\sigma} \rightarrow S^\sigma$ for the induced map on fixed points.

(1) Then the following diagram commutes:

\[
\begin{tikzcd}
D^b_{\et}(Y;\Lambda)^{B \Sigma} \ar[d, "\Psm"] \ar[r, "\rR f_!"]  &  D^b_{\et}(S;\Lambda)^{B \Sigma}   \ar[d, "\Psm"]  \\
\Shv(Y^{\sigma}; \Cal{T}_\Lambda)  \ar[r, "\rR f_!^\sigma
"]  & \Shv(S^{\sigma}; \Cal{T}_\Lambda)
\end{tikzcd}
\]

(2) If $f$ is $\omega_0$-locally spatial and $i \co Y^\sigma \inj Y$ is $\omega_0$-locally extra small, then the following diagram commutes:

\[
\begin{tikzcd}
D^b_{\et}(Y;\Lambda)^{B \Sigma} \ar[d, "\Psm"] \ar[r, "\rR f_*"]  &  D^b_{\et}(S;\Lambda)^{B \Sigma}   \ar[d, "\Psm"]  \\
\Shv(Y^{\sigma}; \Cal{T}_\Lambda)  \ar[r, "\rR f_*^\sigma
"]  & \Shv(S^{\sigma}; \Cal{T}_\Lambda)
\end{tikzcd}
\]

(1) We may replace $S$ by $S^{\sigma}$ and therefore assume that the $\Sigma$-action on $S$ is trivial. Let $\cF \in D_{\et}^b(Y;\Lambda)^{B \Sigma}$. Writing $j \co (Y \setminus Y^\sigma) \inj Y$ for the open complement of $i \co Y^{\sigma} \inj Y$, we have an exact triangle

\[
j_! j^* \cF  \rightarrow  \cF \rightarrow  i_* i^* \cF \in D^b_{\et}(Y; \Lambda)^{B\Sigma}.
\]

By definition $\Sigma$ acts freely on $U$, which by Lemma <ref>(1) implies that $\rR f_! \circ  (j_! j^* \cF) \in \Flat^b(S; \Lambda[\Sigma])$. Hence the cone of $\rR f_!  \cF \rightarrow \rR f^{\sigma}_! (i^*  \cF)$ lies in $\Flat^b(S; \Lambda[\Sigma])$, and therefore becomes $0$ in $\Shv(S; \cT_{\Lambda})$. Therefore

\[
\TT^*(\rR f_! \cF) \cong  \TT^* ( \rR f^{\sigma}_! (i^* \cF)) \cong \rR f_!^\sigma \Psm(\cF) \in \Shv(S; \cT_{\Lambda}),
\]

which exactly expresses the desired commutativity.

(2) The argument is similar to that for (1), but using instead the exact triangle

\[
i_* i^! \cF  \rightarrow  \cF \rightarrow  j_* j^* \cF.
\]

By Lemma <ref>(2), the cone of $\rR f_* i_* i^! \cF  \rightarrow  \rR f_* \cF$ lies in $\Flat^b(S; \Lambda[\Sigma])$, and therefore projects to $0$ in $\Shv(S; \cT_{\Lambda})$. Then using Lemma <ref>, the maps $\rR f_* i_* i^! \cF  \rightarrow  \rR f_* i_* i^* \cF \rightarrow \rR f_* \cF$ project to isomorphisms in $\Shv(S; \cT_{\Lambda})$.

§.§ Tate cohomology

For a $\Lambda[\Sigma]$-module $M$, its Tate cohomology groups were defined in (<ref>). We extend the definition 2-periodically to define $\rT^j(M)$ for all $j \in \Z$.

[Tate cohomology of trivial coefficients]
Equip $k$ with the trivial $\Sigma$-action. Then we have

\[
\rT^j(k)  \cong k \text{ for all } j \in \Z.
\]

Equip $\OO$ with the trivial $\Sigma$-action. Then we have

\[
\rT^j(\OO)  \cong \begin{cases} k & j \equiv 0 \pmod{2}, \\ 0 & j \equiv 1 \pmod{2}. \end{cases}
\]

Let $V$ be a $k$-vector space. Then $V^{\otimes \ell}$ has an action of $\Sigma$, where $\sigma$ acts by cyclic rotation of the factors, and a straightforward calculation shows that

\[
\rT^0(V^{\otimes \ell}) \cong V^{(\ell)} := V \otimes_{k, \Frob_\ell} k,
\]

the Frobenius twist of $V$.

Next we consider generalizations of Tate cohomology to complexes and sheaves.

§.§.§ Tate cohomology of complexes
The Tate cohomology of a complex of $\Lambda[\Sigma]$-modules is defined in <cit.>. We summarize below. The exact sequence of $\Lambda[\Sigma]$-modules

\[
0 \rightarrow \Lambda \rightarrow \Lambda[\Sigma] \xrightarrow{1-\sigma} \Lambda[\Sigma] \rightarrow \Lambda \rightarrow 0
\]

induces a morphism

\begin{equation}\label{eq: lambda shift by 2}
\Lambda \rightarrow \Lambda[2] \in D^b(\Lambda[\Sigma]).
\end{equation}

Note that it becomes an isomorphism in the Tate category, since the middle two terms project to $0$. Given a bounded-below complex of $\Lambda[\Sigma]$-modules $C^{\bu}$, we define its Tate cohomology as

\begin{align*}
\rT^n(C^{\bu}) &= \colim_{j \rightarrow \infty} \Hom_{D^+(\Lambda[\Sigma])}(\Lambda, C^{\bu}[n+2j])
\end{align*}

where the transition maps are those induced by (<ref>). Evidently $\rT^n(C^{\bu})$ is 2-periodic in $n$, and we occasionally the subscript as $n \in \Z/2\Z$. It is clear that this construction descends to the derived category.

Now we state some assertions that may be specific[Our argument uses $\Lambda = k$, but the assertions may be true in greater generality, as far as we know.] to $\Lambda = k$. By <cit.>, a module over $k[\Sigma]$ is flat if and only if it is free. Since Tate cohomology of free $k[\Sigma]$-complexes vanishes (by inspection), this construction further factors through the Tate category, inducing

\[
\rT^n \co \Shv(\pt; \cT_{k}) \rightarrow  \mrm{Vect}_{/k}.
\]

We write $\rT^* := \bigoplus_n \rT^n$ and $\rH^* := \bigoplus_n \rH^n$.

If $C^{\bu} \in  D^b(k[\Sigma])$ is bounded, then the argument of <cit.> gives a natural isomorphism

\[
\rT^i(C^{\bu}) \cong \Hom_{\Shv(\pt; \cT_k)}(k, \TT^* C^{\bu}[i]).
\]

[Tate cohomology of trivial complexes]
If $C^{\bullet} \in D^b(\OO)$, then (cf. Definition <ref> for notation) we have

\[
\rT^*(\epsilon^* C^\bullet) \cong \rH^*(\mrm{Tot}(\ldots \xrightarrow{\ell} C^{\bu} \xrightarrow{0} C^{\bu} \xrightarrow{\ell} C^{\bu}  \xrightarrow{0} \ldots))
\]

so that

\[
\rT^n(\epsilon^* C^\bullet) \cong  \bigoplus_{i \equiv n \pmod{2}} \rH^i(C^{\bu} \dotimes_{\OO} k).
\]

If $C^{\bullet} \in D^b(k)$, then we have

\[
\rT^*(\epsilon^* C^\bullet) \cong \rH^*(C^{\bullet}) \otimes_k \rT^*(k)
\]

where $k$ is equipped with the trivial $\Sigma$-action. By Example <ref>, we therefore have

\[
\rT^n(\epsilon^* C^\bullet) \cong \bigoplus_{i \in \Z} \rH^i(C^{\bullet}) .
\]

§.§.§ Tate cohomology sheaves

Let $S$ be a locally spatial diamond with trivial $\Sigma$-action, so that $\Shv(S; \Cal{T}_\Lambda)$ is defined. Given $\Cal{F} \in D^b_{\et}(S; \Lambda[\Sigma])$, we define Tate cohomology sheaves

\[
\rT^n (\Cal{F}) := \colim_{j \rightarrow \infty} \cHom_{D^b_{\et}(S; \Lambda[\Sigma])}(\Lambda, \cF[n+2j])
\]

where the transition maps are induced by (<ref>). The $\rT^i(\cF)$ are étale sheaves of $T^0(\Lambda)$-modules, where $T^0(\Lambda)$ is the 0th Tate cohomology of the trivial $\Sigma$-module $\Lambda$.

For $\Lambda = k$, as in Remark <ref> we also have the description

\[
\rT^i (\cF) \cong \cHom_{\Shv(S; \cT_k)}(k, \TT^* \cF[i]).
\]

§.§.§ Tate cohomology for a morphism
Assume that $Y$ and $S$ are separated locally spatial diamonds with an action of $\Sigma$, and $f \co Y \rightarrow S$ is a $\Sigma$-equivariant shriekable morphism.

For $\Cal{F} \in D^b_{\et}(Y; \Lambda)^{B \Sigma}$, we have $\rR f_! (\cF) \in D^b_{\et}(S; \Lambda)^{B \Sigma}$. If $S$ has the trivial $\Sigma$-action, then we can form the “relative Tate cohomology” sheaves $\rT^n (\rR f_! \cF)$ on $S$.

Note that if $\Sigma$ acts trivially on $Y$ and on $S$, then the construction $\cF \mapsto \rR f_! (\cF)$ factors over $\Shv(Y; \cT_k)$ by Lemma <ref>. In this situation we will also regard $\rT^i ( \rR f_!)$ as a functor on $\Shv(Y; \cT_k)$.

§.§.§ The long exact sequence for Tate cohomology
Let assumptions be as in <ref>, and suppose furthermore that the $\Sigma$-action on $S$ is trivial. Given a distinguished triangle $\Cal{F}' \rightarrow \Cal{F} \rightarrow \Cal{F}'' \in D^b_{\et}(Y;\Lambda)^{B\Sigma}$, we have a long exact sequence of étale sheaves on $S$,

\[
\begin{tikzcd}[row sep = tiny]
& & \ldots \ar[r] & \rT^{-1} (\rR f_!( \Cal{F}''))  \\
\ar[r] &  \rT^0 (\rR f_!(\Cal{F}')) \ar[r] &  \rT^0 (\rR f_!(\Cal{F})) \ar[r] &  \rT^0 (\rR f_!( \Cal{F}'')) \\
\ar[r] &  \rT^1 (\rR f_!( \Cal{F}')) \ar[r] &  \rT^1 (\rR f_!( \Cal{F} )) \ar[r] &  \rT^1 (\rR f_!(\Cal{F}'')) \\
\ar[r] &  \rT^2 (\rR f_!(\Cal{F}' )) \ar[r]  & \ldots
\end{tikzcd}
\]

§ $\SIGMA$-FIXED POINT CALCULATIONS

Let $G$ be a reductive group over a $E$ and $\sigma$ an order $\ell \neq p$ automorphism of $G$ such that $H := G^\sigma$ is (connected) reductive.

By <cit.> and <cit.>, $H$ is automatically (connected) reductive if $G$ is semisimple simply connected.

In this section we will analyze the fixed points of $\sigma$ on various spaces affiliated with $G$, and relate them to spaces affiliated with $H$, such as:

* The $\bdrp$-affine Grassmannians, as well as their variants such as Beilinson-Drinfeld affine Grassmannians, convolution affine Grassmannians, and “twisted” affine Grassmannians (which are the target of the generalized Grothendieck-Messing period maps).

* The moduli stacks of local shtukas.

These calculations are used later when applying Smith-Treumann localization to such spaces.

§.§ Affine Grassmannians
We begin by formulating the results for affine Grassmannians and their variants.

Let $\CC$ be an algebraically closed field. Let $\Gr_{G, \CC}^{\alg}, \Gr_{H, \CC}^{\alg}$ be the algebraic affine Grassmannians over $\CC$. Then the map $\Gr_{H,\CC}^{\alg} \rightarrow \Fix(\sigma, \Gr_{G,\CC}^{\alg})$ is an isomorphism on underlying reduced ind-schemes.

The proof of Proposition <ref> will be given shortly below, in <ref>. For now we assume it and deduce a few consequences for mixed-characteristic Beilinson-Drinfeld Grassmannians.

Let $S$ be a separated locally spatial diamond over $(\Div_{X}^1)^I$. Then the natural map

\[
\Gr_{H, S/(\Div^1_X)^I} \rightarrow \Fix(\sigma, \Gr_{G, S/(\Div^1_X)^I})
\]

is an isomorphism.

Since $\Gr_{G, S/(\Div^1_X)^I}$ is separated, $\Fix(\sigma, \Gr_{G, S/(\Div^1_X)^I})$ is a closed subdiamond of $\Gr_{G, S/(\Div^1_X)^I}$. The map $
\Gr_{H, S/(\Div^1_X)^I} \rightarrow \Fix(\sigma, \Gr_{G, S/(\Div^1_X)^I})$ is a closed embedding by the argument of <cit.>, hence qcqs, so by <cit.> it is an isomorphism if and only if it induces a bijection on $\Spa(C, C^+)$-points for all $(C,C^+)$. Checking this reduces to the case where $S = \Spa(C,C^+)$, in which case we abbreviate $\Gr_{G, C}:= \Gr_{G, \Spa(C,C^+)/(\Div^1_X)^I}$ and similarly for $H$.

A $\Spa(C,C^+)$-point of $\Gr_{G, C}$ consists of an untilt $C^{\sharp}/E$ and a $(C^{\sharp}, \cO_{C^\sharp})$-point of $\Gr_{G}^{\bdrp}$, the $\bdrp$-affine Grassmannian built using $C^{\sharp}$ as discussed in <cit.>. Any choice of uniformizer $\xi$ for $\bdr(C^{\sharp})$ induces an isomorphism $\bdr(C^{\sharp}) \cong C^{\sharp}((\xi))$, which induces (cf. <cit.>) a commutative diagram

\[
\begin{tikzcd}
\Gr_{H}^{\bdrp}(C^\sharp, \cO_{C^\sharp}) \ar[r, hook] \ar[d, equals] & \Fix(\sigma, \Gr_{G}^{\bdrp})(C^\sharp, \cO_{C^\sharp})  \ar[r, hook] \ar[d, equals] &   \Gr_{G}^{\bdrp}(C^\sharp, \cO_{C^\sharp}) \ar[d, equals] \\
\Gr_{H,C^\sharp}^{\alg}(C^{\sharp}) \ar[r, hook] & \Fix(\sigma, \Gr_{G,C^\sharp}^{\alg})(C^{\sharp}) \ar[r, hook] & \Gr_{G,C^\sharp}^{\alg}(C^{\sharp})
\end{tikzcd}
\]

By Proposition <ref>, the map $\Gr_{H, C^\sharp}^{\alg}(C^{\sharp}) \rightarrow \Fix(\sigma, \Gr_{G, C^\sharp}^{\alg})(C^{\sharp})$ is a bijection, hence so is the parallel map in the top row. Hence the map

\[
\Gr_{H, C}(C,C^+) \rightarrow \Fix(\sigma, \Gr_{G,C})(C,C^+)
\]

is also a bijection, as desired.

The same argument works for variants like $(\Div_{\cY}^1)^I$ instead of $(\Div_X^1)^I$.

We also deduce variants for convolution Grassmannians and the “twisted” Grassmannians $\Gr^{\twi}_G$ from <cit.>.

Let $m  \in \Z_{\geq 1}$.

* Let $\CC/E$ be an algebraically closed field. Let $\wt{\Gr}_{G, \CC}^{\alg, (m)}$ be the $m$-step convolution Grassmannian for $G$, and similarly for $H$. Then the natural map

\[
\wt{\Gr}_{H, \CC}^{\alg, (m)}  \rightarrow \Fix(\sigma, \wt{\Gr}_{G, \CC}^{\alg, (m)} )
\]

is an isomorphism.

* Let $I$ be a finite non-empty set and let $S$ be a separated locally spatial diamond with a map $S \rightarrow (\Div_{X}^1)^I$. Let $\wt{\Gr}_{G, S/(\Div_{X}^1)^I}^{(m)}$ be the $m$-step convolution Beilinson-Drinfeld Grassmannian for $G$, and similarly for $H$. Then the natural map

\[
\wt{\Gr}_{H, S/(\Div_{X}^1)^I}^{(m)} \rightarrow \Fix(\sigma, \wt{\Gr}_{G, S/(\Div_{X}^1)^I}^{(m)})
\]

is an isomorphism.

* Let $I$ be a finite non-empty set. Let $\Gr_{G,I}^{\twi} = \colim_{\mu_{\bu}} \Gr_{G, \prod_{i \in I} \Spd E_i, \leq \mu_{\bu}}^{\twi}$ be the twisted affine Grassmannian (cf. <cit.>). Then the natural map

\[
\Gr_{H, I}^{\twi} \xrightarrow{\sim} \Fix(\sigma, \Gr_{G,I}^{\twi})
\]

is an isomorphism.

(1) The map $\wt{\Gr}_{G}^{\alg, (m)} \rightarrow (\Gr_G^{\alg})^m$ sending $(\cE^0 \dashrightarrow \cE_1 \dashrightarrow \ldots \dashrightarrow \cE_m)$ to $(\cE^0 \dashrightarrow \cE_1, \cE^0 \dashrightarrow \cE_2, \ldots, \cE^0 \dashrightarrow \cE_m)$ is a $\Sigma$-equivariant isomorphism. Under this isomorphism, the statement transports to that of Proposition <ref>.

(2) As in the proof of Proposition <ref>, the statement reduces to the case where $S = \Spa(C, C^+)$. Then as in the proof of Proposition <ref>, the statement follows from the algebraic case over some untilt $C^{\sharp}$ of $C$, which was treated in (1).

(3) As in the proof of Proposition <ref>, the statement reduces to the case where $S = \Spa(C,C^+)$, where it is then a consequence of (2).

§.§ Group-theoretic preliminaries
Recall that a Borus of a reductive group $G$ is a pair $(B,T)$ of a Borel subgroup $B <G$ and a split maximal torus $T < B$.

Suppose $\Sigma$ stabilizes a Borus $(B,T)$ of $G$. Then $(B^\sigma, T^\sigma)$ is a Borus of $H$.

Since $B^{\sigma}$ is clearly a solvable of $H$, in order to show that it is a Borel subgroup of $H$ it suffices by a result of Chevalley <cit.> to show that it is parabolic, i.e., that $G^\sigma/B^\sigma$ is projective.

Suppose for the moment that we have shown $B^\sigma$ is a Borel subgroup of $H$. We will argue that $T^\sigma$ is a maximal torus of $H$. Let $U$ be the unipotent radical of $B$. Applying <cit.>, which implies that the map on $\Sigma$-fixed points of a smooth $\Sigma$-equivariant morphism is again smooth, to $B \rightarrow B/U \cong T$ shows that $B^\sigma \rightarrow (B/U)^\sigma  \cong T^\sigma$ is a smooth map whose non-empty fibers are $U^\sigma$-torsors. Since we also have the section $T^\sigma \inj B^\sigma$, we conclude that the map $B^\sigma/U^\sigma \rightarrow (B/U)^\sigma \cong T^\sigma$ is an isomorphism. This shows that $T^\sigma$ is a maximal torus of $H$ (and in particular is connected).

Now we return to the issue of showing that $H/B^\sigma$ is projective. By assumption $G/B$ is projective, hence its closed subscheme $(G/B)^\sigma$ is projective. We will prove that $H/B^\sigma$ is closed and open in $(G/B)^\sigma$. Evidently $H$ acts by translation on $(G/B)^\sigma$, and it suffices to show that each $H$-orbit in $(G/B)^\sigma$ is open, since then the orbits are finite in number and pairwise disjoint, hence comprise the connected components. For any $g \in G(\ol{E})$ representing a point in $(G/B)^\Sigma$, the orbit map $G^\sigma \rightarrow (G/B)^\sigma$ through $gB$ is a smooth morphism by <cit.> again, hence has open image.

Let $S$ be a seaparated locally spatial diamond over $(\Div_{X}^1)^I$. Suppose that $G$ has a $\Sigma$-stable Borel subgroup $B$. Let $B_H := B^\sigma$, a Borel subgroup of $H$ by Lemma <ref>. Then the natural map

\[
\Gr_{B_H, S/(\Div_{X}^1)^I} \rightarrow \Fix(\sigma, \Gr_{B, S/(\Div_{X}^1)^I})
\]

is an isomorphism.

The map $\Gr_{B, S/(\Div_{X}^1)^I} \rightarrow \Fix(\sigma, \Gr_{B, S/(\Div_{X}^1)^I})$ is a closed embedding by the same argument as <cit.>, hence qcqs, so by <cit.> it suffices to check that it induces a bijection on geometric points. A variation on <cit.> shows that the maps $\Gr_{B_H, S/(\Div_{X}^1)^I} \rightarrow \Gr_{H, S/(\Div_{X}^1)^I}$ and $\Gr_{B, S/(\Div_{X}^1)^I} \rightarrow \Gr_{G, S/(\Div_{X}^1)^I}$ are bijections on geometric points. Hence in the commutative diagram

\[
\begin{tikzcd}
\Gr_{B_H, S/(\Div_{X}^1)^I} \ar[r] \ar[d] &  \Fix(\sigma, \Gr_{B, S/(\Div_{X}^1)^I}) \ar[d] \\
\Gr_{H, S/(\Div_{X}^1)^I} \ar[r] & \Fix(\sigma, \Gr_{G, S/(\Div_{X}^1)^I})
\end{tikzcd}
\]

both vertical maps and the bottom horizontal map induce bijections on geometric points (the latter by Proposition <ref>). Therefore the upper horizontal map also induces a bijection on geometric points, concluding the proof.

§.§ Proof of Proposition <ref>

We write $LG^{\alg}$ for the loop group of $G$ with respect to the loop variable $t$, so $LG^{\alg}(R) = G(R((t)))$. We write $L^+G^{\alg} \subset LG^{\alg}$ for the arc group, with functor of points $L^+G^{\alg}(R) = G(R[[t]])$. The algebraic affine Grassmannian is the fppf quotient $LG^{\alg}/L^+G^{\alg}$, which has a natural ind-scheme structure.

For the purpose of proving Proposition <ref>, we may base change $G$ to the algebraically closed field $\CC$. Since $\sigma$ is a semisimple automorphism of $G$ over an algebraically closed field, a theorem of Steinberg <cit.> implies that $\sigma$ stabilizes a Borus $(B,T)$ of $G$. By Lemma <ref>, $(B^\sigma, T^\sigma)$ is Borus of $H := G^\sigma$.

§.§.§ Iwahori stratification

Let $\Iw^G$ be the Iwahori subgroup of $L^+G$ corresponding to $B$. This induces a stratification by $\Iw^G$-orbits (cf. <cit.>)

\[
\Gr_{G, \CC}^{\alg, \red} = \coprod_{\lambda \in X_*(T)} \Gr_{G, \lambda}^{\alg}.
\]

Furthermore, $\Iw^G$ is stable under $\sigma$ and $\Iw^H := (\Iw^G)^\sigma$ is the Iwahori subgroup of $L^+H$ corresponding to $B^\sigma$, so we have an analogous stratification

\[
\Gr_{H, \CC}^{\alg, \red} = \coprod_{\lambda \in X_*(T^\sigma)} \Gr_{H, \lambda}^{\alg}.
\]

The action of $\sigma$ on $G$ induces an action on $X_*(T)$. We will show that:

* If $\lambda$ is not fixed by $\sigma$, then $\Fix(\sigma, \Gr_{G, \lambda}^{\alg}) = \emptyset$.

* If $\lambda$ is fixed by $\sigma$, then $\Fix(\sigma, \Gr_{G, \lambda}^{\alg})  \xleftarrow{\sim} \Gr_{H, \lambda}^{\alg}$.

The combination of (1) and (2) clearly suffices to prove Proposition <ref>.

We prove (1) now: since we arranged $\sigma$ to preserve $\Iw^G$, it sends $\Gr_{G, \lambda}$ to $\Gr_{G, \sigma(\lambda)}$. As distinct $\Iw^G$-orbits are disjoint, if $\lambda$ is not fixed by $\sigma$ then $\Fix(\sigma, \Gr_{G, \sigma(\lambda)}) = \emptyset$.

Next we turn to (2). For this we analyze the structure of the strata.

§.§.§ Fixed points of strata

For any root $\alpha \in \mf{R}$, let $U_\alpha \subset G$ be the corresponding root subgroup. For an affine root $\alpha + m \hbar$ of $G$, denote by $U_{\alpha + m \hbar}$ the corresponding subgroup of $LG$ (cf. <cit.>), which for any isomorphism $u_{\alpha} \co \G_a \xrightarrow{\sim} U_\alpha$, identifies with the image of $x \mapsto u_\alpha (xt^m)$. Set

\[
\delta_{\alpha} = \begin{cases} 1 & \alpha \in \mf{R}^+ \\ 0 & \text{otherwise},\end{cases}
\]

and for $\lambda \in X_*(T)$ define

\begin{equation}\label{eq: Iwlambda}
\Iwu^{G,\lambda} := \prod_{\alpha \in \mf{R}} \left( \prod_{\delta_{\alpha} \leq m < \tw{\lambda, \alpha}} U_{\alpha+m\hbar}\right),
\end{equation}

where the product may be taken in any order. Then as explained in the proof of <cit.>, the action of $\Iwu^{G,\lambda}$ on $t^\lambda \in \Gr_G(\CC)$ induces an isomorphism

\begin{equation}\label{eq: iwahori orbit}
\Iwu^{G,\lambda} \xrightarrow{\sim} \Gr_{G, \lambda}^{\alg}.
\end{equation}

Suppose $\sigma$ fixes $\lambda$. Then (<ref>) implies that

\[
\Fix(\sigma, \Gr_{G, \lambda}^{\alg})  = (\Iwu^{G,\lambda})^{\sigma} \cdot t^\lambda.
\]

Denote by $\Iwu^{G,\lambda}(\alpha)$ the factor of $\Iwu^{G,\lambda}$ in (<ref>) indexed by $\alpha$. If $\sigma(\alpha) \neq \alpha$, then $\Iwu^{G,\lambda}(\alpha)^\sigma = 0$. On the other hand, if $\sigma \alpha = \alpha$ then (since $\sigma$ was arranged to preserve $\mf{R}^+$) we have $\Iwu^{G,\lambda}(\alpha)^\sigma = \Iwu^{H, \lambda}(\alpha)$, so that

\[
(\Iwu^{G,\lambda})^{\sigma} \cdot t^\lambda \cong \Gr_{H, \lambda}^{\alg}.
\]

This completes the proof of Proposition <ref>.

§.§ Moduli of local shtukas
We now turn to examine the moduli stacks of local shtukas, for which a reference is Scholze's Berkeley lectures, especially <cit.>.

Let $(G,b, \{\mu_i\}_{i \in I})$ be a local shtuka datum (cf. <cit.>, except we allow the equal characteristic case as well): $G$ is a reductive group over $E$, $b \in B(G)$, and $\mu_i$ is a conjugacy class of cocharacters $\G_m \rightarrow G_{E^s}$. For any compact open subgroup $K \subset G(E)$ there is a moduli space of local shtukas with $K$-level structure $\Sht_{(G,b, \{\mu_i\}_{i \in I}), K}$, which is functorial in $K$. There is a moduli description of $\Sht_{(G, b, \{\mu_i\}_{i \in I}),K}$ given (in the mixed characteristic case) in <cit.>.

For $\{\mu_i\}_{i \in I} \leq \{\mu_i'\}_{i \in I}$ in the component-wise Bruhat order, there are closed embeddings
$\Sht_{(G, b, \{\mu_i\}),K} \inj \Sht_{(G, b, \{\mu_i'\}),K}$. It will also be convenient to consider the following variant without any cocharacter “bounds”:

We define $\Sht_{(G, b,I),K} := \varinjlim_{\{\mu_i\}_{i \in I}} \Sht_{(G,b,\{ \mu_i\}_{i \in I}),K}$.

§.§.§ Grothendieck-Messing period map

We recall some facts about the the generalized Grothendieck-Messing period map

\begin{equation}\label{eq: Grothendieck-Messing}
\pi_{\GM} \co \Sht_{(G,b,I),K} \rightarrow \Gr^{\twi}_{G,I}
\end{equation}

from <cit.>. Note that we have not imposed any “bound” above; our (<ref>) comes from the colimit over $\{\mu_i\}_{i \in I}$ of the map from <cit.>.

The image of $\pi_{\GM}$ is an open subset of $\Gr^{\twi, a}_{G,I} \subset \Gr^{\twi}_{G,I}$ called the admissible locus. In terms of points, $\Gr^{\twi}_{G,I}$ parametrizes $G$-torsors $\cP_{\eta}$ on $\cY_{(0, \infty)}$ plus additional data $\varphi_{\cP_{\eta}}$ and $\iota_r$ from <cit.> that we do not need to reference right now. The admissible locus is cut out by the condition that the Newton point $\nu_{\cP_{\eta}}$ and the Kottwitz point $\kappa_{\cP_\eta}$ are both identically zero <cit.>.

For a basic $b \in B(G)$, denote by $\Gr^{\twi, b}_{G,I} $ the subspace where $\nu_{\cP_\eta} = \nu(b)$ and $\kappa_{\cP_{\eta}} = \kappa(b)$. This is an open subspace, and when $b=1_G$ this recovers the admissible locus, i.e., $\Gr^{\twi, 1_G}_{G,I} = \Gr^{\twi, a}_{G,I} \subset \Gr^{\twi}_{G,I}$.

If $b \in B(G)$ is basic, then there is a pure inner twist $G_b$ and a canonical isomorphism

\begin{equation}\label{eq: basic twist}
\tr_b \co  B(G) \xrightarrow{\sim} B(G_b)
\end{equation}

sending $b \in B(G)$ to $1_{G_b} \in B(G_b)$. There is also a compatible isomorphism $\Gr_{G,I}^{\twi} \cong \Gr_{G_b,I}^{\twi}$ as in <cit.>, which takes $\Gr_{G,I}^{\twi, b}$ to $\Gr_{G_b,I}^{\twi,a}$. So we could have formulated the subspaces $\Gr^{\twi, b}_{G,I}$ as admissible loci for pure inner twists $G_b$. However, we prefer to keep the distinction, for psychological reasons if nothing else.

Let $\iota \co B(H) \rightarrow B(G)$ be the map induced by the inclusion $\iota \co H \inj G$. The induced map $\iota \co  \Gr_{H,I}^{\twi} \inj \Gr_{G,I}^{\twi}$ is also a closed embedding (by the proof of <cit.>), etc.

Since $X_*(T_{E^s}^\sigma)_{\Q} \subset X_*(T_{E^s})_{\Q}$, the subset $\iota^{-1}(1_G) \subset B(H)$ consists of basic elements. For each $b' \in \iota^{-1}(1_G)$, there is an embedding $H_{b'} \inj G_{b'} \cong G$. As a variant of <cit.>, this induces an embedding $\Gr_{H_{b'},I}^{\twi} \inj \Gr_{G,I}^{\twi}$, identifying $\Gr_{H_{b'},I}^{\twi,a}$ with $\Gr_{H,I}^{\twi, b'}$ as sub-v-sheaves of $\Gr_{G,I}^{\twi}$.

We have

\[
\Gr^{\twi,a}_{G,I} \cap \Gr^{\twi}_{H,I} = \coprod_{b' \in \iota^{-1}(1_G)} \Gr^{\twi, b'}_{H,I}.
\]

as subdiamonds of $\Gr^{\twi}_{G,I}$.

In Corollary <ref> we have seen that $\Fix(\sigma, \Gr_{G,I}^{\twi}) = \Gr_{H,I}^{\twi}$ (with the obvious embedding). The admissible locus $\Gr^{\twi,a}_{G,I}  \subset \Gr^{\twi}_{G,I}$ is cut out by the condition that $\nu(\cP_{\eta}) = 0$ and $\kappa(\cP_\eta) = 0$. Hence $\Fix(\sigma, \Gr_{G,I}^{\twi,a})$ is the pullback of the simultaneous vanishing loci of the Newton and Kottwitz maps from $\Gr_{G,I}^{\twi}$ to $\Gr_{H,I}^{\twi}$, which is precisely the expression on the RHS.

§.§.§ Fixed points of moduli of local shtukas
We now recall more about the structure of $\Sht_{(G, b, I),K}$ in terms of the Grothendieck-Messing period map. According to <cit.>, $\Gr_{G, I}^{\twi,a}$ carries a $\ul{G(E)}$-torsor $\PP_\eta^G$, and $\pi_{\GM}^G \co \Sht_{(G, b, I),K} \rightarrow \Gr_{G, I}^{\twi,a}$ is the étale cover parametrizing $K$-lattices $\PP^G \subset \PP_{\eta}^G$.

For $b' \in \iota^{-1}(1_G)$, write $\iota^*_{b'} K \subset H_{b'}(E)$ for the pre-image of $K$ under the map $H_{b'} \inj G$ from Remark <ref>. We abbreviate $\iota^* K = \iota^*_{1_H}K = K^\sigma \subset H(E)$.

Suppose $K \subset G(E)$ is a pro-$p$, $\Sigma$-stable open compact subgroup. Then we have

\[
\Fix(\sigma, \Sht_{(G,1_G, I),K}) = \coprod_{b' \in \iota^{-1}(1_G)} \Sht_{(H_{b'}, 1_{H_{b'}}, I), \iota^*_{b'} K}.
\]

as sub-v-sheaves of $\Sht_{(G,1_G, I),K}$ (with embedding on the RHS induced by $H_{b'} \inj G_{b'} \cong G$ of Remark <ref>). In particular, $\Sht_{(H,1_H,I), \iota^* K}$ is an open-closed subdiamond of $\Fix(\sigma, \Sht_{(G,1_G,I),K})$.

We compute using the Grothendieck-Messing period map. In view of Lemma <ref>, it suffices to show that for all $x \in \Gr^{\twi, b'}_{H,I}$, the inclusion $\Sht_{(H_{b'}, 1_{H_{b'}}, I), \iota^*_{b'} K} \rightarrow \Sht_{(G,1_G, I),K}$ sends $(\pi_{\GM}^H)^{-1}(x)$ isomorphically to $\Fix(\sigma, (\pi_{\GM}^G)^{-1}(x))$. By Remark <ref>, we may reduce this statement to the case where $b'=1$. The $\ul{G(E)}$-torsor $\PP_\eta^G$ corresponding to $x$ has an $H$-structure $\PP^H_\eta$ since $x$ lies in the image of $\pi_{\GM}^H$, and $(\pi_{\GM}^H)^{-1}(x)$ parametrizes $\iota^* K$-lattices in $\PP^H_\eta$. The action of $H(E)$ on a fixed $\iota^*K$-lattice $L_0$ induces an isomorphism $(\pi_{\GM}^H)^{-1}(x) \cong H(E)/\iota^*K$.

On the other hand, $\Fix(\sigma, (\pi_{\GM}^G)^{-1}(x))$ parametrizes $K$-lattices in $\PP_\eta^G$ which are stable under $\Sigma$. As $\PP_\eta^G \cong \PP^H_\eta \times^{H(E)} G(E)$, the action of $G(E)$ on $L_0$ induces an isomorphism $\Fix(\sigma, (\pi_{\GM}^G)^{-1}(x)) \cong \Fix(\sigma, G(E)/K)$, with $\sigma$ acting in the natural way. Hence it suffices to see that

\begin{equation}\label{eq: fixed lattices}
H(E)/K^\sigma \xrightarrow{\sim} (G(E)/K)^\sigma.
\end{equation}

The long exact sequence of non-abelian cohomology of $\Sigma$ reads

\[
K^\sigma \inj G(E)^\sigma  \rightarrow  (G(E)/K)^\sigma \rightarrow \rH^1(\Sigma; K) \rightarrow \rH^1(\Sigma; G(E)).
\]

Since $K$ is pro-$p$ and $\Sigma$ has order $\ell \neq p$, we have $\rH^1(\Sigma; K) = 0$. This shows (<ref>), completing the proof.

§ PARITY SHEAF THEORY ON $P$-ADIC AFFINE GRASSMANNIANS

The theory of parity sheaves, originating in work of Juteau-Mautner-Williamson [27], has driven many recent developments in geometric representation theory: see [3, 4, 37, 2, 52] for a sampling of applications. Parity sheaves are defined somewhat similarly to perverse sheaves, but with parity conditions instead of inequalities on cohomological degrees.

In this section we develop a theory of relative parity sheaves on affine Grassmannians arising in $p$-adic geometry. The usage of the adjective “relative” is in the same sense as “relative perversity” of Hansen-Scholze [24]: it means parity along the geometric fibers of a morphism. In practice the morphism is the projection of a Beilinson-Drinfeld type affine Grassmannian to the base. Leslie-Lonergan [34] introduced the “Tate-parity sheaves” as the analogue of parity sheaves in Tate categories, and we also develop a theory of “relative Tate-parity sheaves” on families of affine Grassmannians.

The main results of the structure theory of relative (Tate-)parity sheaves include the calculation of Ext groups, the vanishing of maps between incongruous objects, the Krull-Schmidt property, the rigidity of indecomposable objects, and the existence of relative (Tate-)parity sheaves associated to each stratum. Some of the difficulties are similar to those present in the theory of perverse sheaves on $p$-adic affine Grassmannians: the lack of a definitive dimension theory for adic spaces, and the failure of constructibility for the sheaves of interest. There are also some new technical issues, coming from the fact that the Ext-vanishing properties underlying the structure theory of parity sheaves have an “absolute” nature, which is in tension with the “relative” nature of our definitions. The theory of ULA sheaves developed in [22] ultimately helps us to overcome these difficulties.

We now give an overview of the contents of this section. In <ref> we define relative parity complexes, etc. on $\Gr_{G, S/\Div_X^1}$ for a small v-stack $S$ over $\Div_X^1$ and establish their structure theory (mostly for the case where $S$ is strictly totally disconnected). In <ref> we introduce a “small” version of the Tate category for $\Gr_{G, S/\Div_X^1}$, the adjective “small” referring to that the sheaves are required to have strong finiteness properties, and a “small” version of Smith-Treumann localization. In <ref> we define relative Tate-parity complexes, etc. on $\Gr_{G, S/\Div_X^1}$ and establish their structure theory. In <ref> we define “even maps” and prove that they preserve relative parity, and next in <ref> we apply this to certain Demazure resolutions in order to show that relative parity sheaves exist for all strata, and that relative parity is preserved by convolution. In <ref> we study the interaction of relative parity sheaves with modular reduction $\FF$ and the operation $\TT$. This is used in <ref> to define the “lifting functor” à la Leslie-Lonergan, which will constitute a key step in the construction of the Brauer functor.

§.§ Relative parity sheaves
Let $S$ be a small v-stack with a map $S \rightarrow \Div^1_X$. Then we may form $\Gr_{G,S/\Div^1_X} := \Gr_{G, \Div^1_X} \times_{\Div_X^1} S$ and $\cHck_{G, S/\Div^1_X} := \cHck_{G, \Div^1_X} \times_{\Div_X^1} S$ as in <cit.>.

§.§.§ Schubert stratification
Suppose $S \in \Perf$ is strictly totally disconnected. Then <cit.> applies and we have a presentation

\[
\cHck_{G, S/\Div^1_X} = \varinjlim_{\mu \in X_*(T)^+} \cHck_{G, S/\Div^1_X, \leq \mu}
\]

where $T$ is a split maximal torus in $G_{E^s}$; here $\cHck_{G, S/\Div^1_X, \leq \mu}$ is the subfunctor of $\cHck_{G, S/\Div^1_X} $ parametrizing modifications which are given by some $\mu' \leq \mu$ at each geometric point of $S$. The open subfunctor $\cHck_{G, S/\Div^1_X, \mu} \inj \cHck_{G, S/\Div^1_X, \leq \mu}$ is defined as the complement of $\cHck_{G, S/\Div^1_X, \leq \mu'}$ for $\mu' < \mu$.

The pullback of $\cHck_{G, S/\Div^1_X, \leq \mu}$ (resp. $\cHck_{G, S/\Div^1_X, \mu}$) to $\Gr_{G,S/\Div^1_X}$ is denoted $\Gr_{G,S/\Div^1_X, \leq \mu}$ (resp. $\Gr_{G,S/\Div^1_X, \mu}$). Let $L^+_{S/\Div^1_X} G := L^+_{\Div^1_X}G \times_{\Div_X^1} S$. Then $L^+_{S/\Div^1_X} G$ acts on $\Gr_{G,S/\Div^1_X}$ by left translation, and the orbits are the $\Gr_{G,S/\Div^1_X, \mu}$.

We denote by $i_{\mu}$ the locally closed embedding $i_\mu \co \Gr_{G,S/\Div^1_X, \mu} \inj \Gr_{G,S/\Div^1_X}$. By <cit.>, $\Gr_{G,S/\Div^1_X, \mu}$ has the structure of a fibration over the diamond of the (opposite) partial flag variety $(P_{\mu}^-)^{\di}_S$, with the fibers being iterated extensions of $(\Lie G)^{\di}_{\mu \leq m} \{m\}$ where $(\Lie G)_{\mu \leq m}$ is the subspace on which $\G_m$ acts with weights $\leq m$ via the adjoint action composed with $\mu$. (Here $\{m\}$ is a “Breuil-Kisin twist”.)

For the projection
$\pi \co \Gr_{G,S/\Div^1_X, \mu} \rightarrow S$, each $\rR^n \pi_* \Lambda$ has $\Lambda$-free geometric stalks and vanishes if $n$ is odd.

Both $(P_{\mu}^-)^{\di} $ and $(\Lie G)^{\di}_{\mu \leq m}$ have the property that their cohomology with constant coefficients is supported in even degrees. The preceding description shows that $\Gr_{G,S/\Div^1_X, \mu}$ is an iterated fibration over $(P_{\mu}^-)^{\di}$ with fibers of the form $(\Lie G)^{\di}_{\mu \leq m}$ (note that the Breuil-Kisin twist is trivializable over geometric points), so the result follows from the Serre spectral sequence, which is forced to degenerate by the even-ness.

§.§.§ Categories of sheaves

We will take coefficients in a ring $\Lambda$ which is a complete local PID over $\Z_\ell$, i.e., a field of characteristic $\ell$ or a complete DVR with residue characteristic $\ell$. In our applications of interest, $\Lambda$ will be either $k$ or $\OO := W(k)$.

We define $\DSY{G}{\Lambda}$ to be the full subcategory of $D_{\et}^{\ULA}(\Gr_{G,S/\Div^1_X};\Lambda)^{\bd}$ spanned by the image of $D_{\et}^{\ULA}(\cHck_{G,S/\Div^1_X};\Lambda)^{\bd}$ under $*$-pullback.

As a consequence of <cit.>, the category $\DSY{G}{\Lambda}$ is stable under the operations of Verdier duality, $- \dotimes_\Lambda -$, $\cRHom_\Lambda(-, -)$, $i_! i^*, \rR i_* i^*, i_!  i^!, \rR i_*  i^!$ where $i = i_\mu$ is the inclusion of any stratum. (The analogous statement does not hold in the “multiple legs” situation, unless the map from $S$ factors through the locus where the legs are disjoint.)

Recall that an additive category is called Krull-Schmidt if each object is a finite direct sum of indecomposable objects with local endomorphism rings (hence an object is indecomposable if and only if its endomorphism ring is local). We will show that $\Dula{G}{\Lambda}{1}$ is Krull-Schmidt.

All objects of $\DSY{G}{\Lambda}$ admit a finite filtration by objects of the form $i_{\mu !} \Lambda$, and a finite filtration by objects of the form $\rR i_{\mu *} \Lambda$.

This follows immediately from <cit.>.

For any object $\cK \in \DSY{G}{\Lambda}$, $\End(\cK)$ is finitely generated as a $\Lambda$-module.

Applying Lemma <ref>, it suffices to show that

\[
\Hom(i_{\mu!} \Lambda, \rR i_{\mu'*} \Lambda) \cong \Hom(i^*_{\mu'} i_{\mu !} \Lambda, \Lambda) \text{
is finitely generated as a $\Lambda$-module.}
\]

If $\mu \neq \mu'$, then $i^*_{\mu'} i_{\mu!} \Lambda = 0$. If $\mu' = \mu$, then $i^*_{\mu'} i_{\mu !}  \Lambda \cong \Lambda$, so that $\Hom(i^*_{\mu'} i_{\mu !} \Lambda, \Lambda) \cong \Lambda$.

The category $\DSY{G}{\Lambda}$ is Krull-Schmidt.

The usual t-structure on the category $\DSY{G}{\Lambda}$ is bounded. Therefore, it is Karoubian (i.e., every idempotent splits) by <cit.>. By <cit.>, a Karoubian category such that $\End(\cK)$ is semiperfect (cf. <cit.> for a reference on semiperfect rings) for every $\cK$ is Krull-Schmidt. It therefore suffices to show that $\End(\cK)$ is semiperfect for every $\cK \in \DSY{G}{\Lambda}$. For this we note that any finite $\Lambda$-algebra is semiperfect, and we showed in Lemma <ref> that $\End(\cK)$ is a finite $\Lambda$-module.

§.§.§ Relative parity complexes
A pariversity on $\Gr_{G,S/\Div^1_X}$ is a function $\dagger$ from $X_*(T)^+$, thought of as the set of strata on the geometric fibers over $S$, to $\Z/2\Z$.

[Dimension pariversity]
In this section we fix $\dagger$ to be the “dimension pariversity” $\dagger_G$, defined by

\[
\dagger_G(\lambda) := \tw{2\rho, \lambda} \pmod{2}  \in \Z/2\Z
\]

where $2\rho$ is the sum of the positive roots of $G$.

The definition below is a “relative to $S$” (in a sense parallel to the notion of relative perversity in <cit.>) version of the definition of parity sheaves in <cit.>.

Let $S$ be a small v-stack. For a geometric point $\ol{s} = \Spa(C,C^+) \rightarrow S$ and $\cK \in \DSY{G}{\Lambda}$, we denote by $\cK|_{\ol{s}}$ the $*$-restriction of $\cK$ along $\Gr_{G,\ol{s}/\Div^1_X} \rightarrow \Gr_{G,S/\Div^1_X}$. For each $\mu \in X_*(T)^+$ we let $i_\mu \co \Gr_{G,\ol{s}/\Div^1_X, \mu} \inj \Gr_{G,\ol{s}/\Div^1_X}$ be the locally closed embedding of the corresponding stratum.

* For $? \in \{*,!\}$, we say $\cK \in \DSY{G}{\Lambda}$ is relative $?$-even if for all geometric points $\ol{s} \rightarrow S$ and all $\mu \in X_*(T)^+$ and all $n \in \Z$, the cohomology sheaf $\cH^n(i_\mu^? (\cK|_{\ol{s}}))$ is constant and $\Lambda$-free, and vanishes for $n \not\equiv \dagger(\mu) \pmod{2}$.

* For $? \in \{*,!\}$, we say that $\cK \in \DSY{G}{\Lambda}$ is relative $?$-odd if $\cK[1]$ is relative $?$-even.

* For $? \in \{*,!\}$, we say that $\cK \in \DSY{G}{\Lambda}$ is relative $?$-parity if $\cK$ is either relative $?$-even or relative $?$-odd.

* We say $\Cal{K} \in \DSY{G}{\Lambda}$ is a relative even complex if $\cK$ is relative $*$-even and relative $!$-even. We say that $\cK \in \DSY{G}{\Lambda}$ is a relative odd complex if $\Cal{K}[1]$ is even.

* We say $\cK \in \DSY{G}{\Lambda}$ is a relative parity complex if it is isomorphic to a finite direct sum of relative even and relative odd complexes. The full subcategory of $\DSY{G}{\Lambda}$ spanned by relative parity complexes is denoted $\PSY{G}{\Lambda}{1}$.

The following result is part of <cit.>, but we state it for emphasis and as an opportunity to spell out the proof.

Let $f \co S' \rightarrow S$ be a map of small v-stacks over $\Div_X^1$ and $i_{\mu} \co \Gr_{G, S/\Div^1_X, \mu}  \inj \Gr_{G, S/\Div^1_X}$ be the locally closed immersion of a Schubert cell. Consider the diagram

\[
\begin{tikzcd}
\Gr_{G, S'/\Div^1_X, \mu}  \ar[d, "i'_\mu"] \ar[r, "f_\mu"]  & \Gr_{G, S/\Div^1_X, \mu} \ar[d, "i_\mu"]  \\
\Gr_{G, S'/\Div^1_X}  \ar[r, "\wt{f}"]  \ar[d] & \Gr_{G, S/\Div^1_X}  \ar[d] \\
S' \ar[r, "f"] & S
\end{tikzcd}
\]

where all squares are Cartesian. Then we have a natural isomorphism $f_\mu^*(i_\mu)^! \cong  (i_\mu')^!\wt{f}^*$ of functors $\Dula{G}{\Lambda}{1} \rightarrow D_{(L^+G)}^{\ULA}(\Gr_{G, S'/\Div^1_X, \mu} ; \Lambda)^{\bd}$.

We abbreviate $\DD_{/S}$ (resp. $\DD_{/S'}$) for the relative Verdier duality over $S$ (resp. $S'$). By <cit.>, relative Verdier duality is compatible with base change:

\begin{equation}\label{eq: ula base change 1}
f_\mu^* \DD_{/S} \xrightarrow{\sim} \DD_{/S'} (f_{\mu})^* \quad \text{ and } \quad \wt{f}^* \DD_{/S} \xrightarrow{\sim} \DD_{/S'} \wt{f}^*.
\end{equation}

As relative Verdier duality is involutive on ULA objects <cit.>, (<ref>) implies that

\begin{equation}\label{eq: ula base change 2}
f_{\mu}^* \cong \DD_{/S'} f_{\mu}^* \DD_{/S} \quad \text{ and } \quad \wt{f}^* \cong \DD_{/S'} \wt{f}^* \DD_{/S}.
\end{equation}

Using also that $i_\mu^! \cong \DD_{/S} i_\mu^* \DD_{/S}$ and $(i_\mu')^! \cong \DD_{/S'} (i'_{\mu})^*\DD_{/S'} $ on ULA objects, we have natural isomorphisms

\[
f_\mu^*i_\mu^! \cong f_{\mu}^* \DD_{/S} i_\mu^* \DD_{/S} \stackrel{\eqref{eq: ula base change 2}}\cong \DD_{/S'} f_{\mu}^* i_\mu^* \DD_{/S} \cong \DD_{/S'} (i'_{\mu})^* \wt{f}^* \DD_{/S} \stackrel{\eqref{eq: ula base change 2}}\cong \DD_{/S'} (i'_{\mu})^*\DD_{/S'} \wt{f}^*  \cong  (i_\mu')^!\wt{f}^*.
\]

§.§.§ Structure theory for strictly totally disconnected $S$
Here we prove several structural results about $\PSY{G}{\Lambda}{1}$, parallel to <cit.>, under the assumption that $S$ is strictly totally disconnected. In particular, we have the stratification of $\Gr_{G, S/\Div_X^1}$ by $\Gr_{G, S/\Div_X^1, \mu}$, ranging over $\mu \in X_*(T)^+$, and we let $i_\mu$ be the locally closed embedding $\Gr_{G, S/\Div_X^1, \mu} \inj \Gr_{G, S/\Div_X^1}$. Below all Hom and Ext groups are taken in the category $\DSY{G}{\Lambda}$, which we omit for ease of notation.

Lemma <ref> implies the following connection between relative parity as in Definition <ref>, and an “absolute” notion of parity.

Let $\cK \in \DSY{G}{\Lambda}$. For $? \in \{*, !\}$, if $\cK$ is relative ?-even then for all $\mu \in X_*(T)^+$ and all $n \in \Z$, the cohomology sheaf $\cH^n(i_\mu^? (\cK))$ is constant and $\Lambda$-free, and vanishes for $n \not\equiv \dagger(\mu) \pmod{2}$.

The constancy is by definition of $\DSY{G}{\Lambda}$, and the $\Lambda$-freeness of $*$-restrictions can be checked on stalks so follows from the definition of relative $?$-even, using Lemma <ref>. The fact that $\cH^n(i_\mu^* (\cK))$ vanishes for $n \not\equiv \dagger(\mu) \pmod{2}$ follows immediately from the definition of relative ?-even, while the fact that $\cH^n(i_\mu^!(\cK))$ vanishes for $n \not\equiv \dagger(\mu) \pmod{2}$ can be checked after base change along geometric points $\ol{s}  = \Spa(C,C^+) \rightarrow S$, where it follows from the definition of relative ?-even using Lemma <ref>.

Assume $S$ is strictly totally disconnected. If $\cF \in \DSY{G}{\Lambda}$ is relative $*$-parity and $\cG \in \DSY{G}{\Lambda}$ is relative $!$-parity, then we have a (non-canonical) isomorphism of $\Lambda$-modules

\[
\Ext^{\bu}(\cF, \cG) \cong \bigoplus_{\mu \in X_*(T)^+} \Ext^{\bu}(i_{\mu}^* \cF, i_{\mu}^! \cG)
\]

and both sides are free over $\Lambda$.

The argument is similar to that for <cit.>, but we have to address some issues related to the discrepancy between our relative situation and the “absolute” situation of [27]. Since the statement is compatible with finite direct sums in $\cF$ and $\cG$, and $\DSY{G}{\Lambda}$ is Krull-Schmidt by Lemma <ref>, we may assume that $\cF$ and $\cG$ are indecomposable. Since the statement is compatible with shifts, we may further assume that $\cF$ is relative $*$-even and $\cG$ is relative $!$-even.

We will show by induction, on the number $N$ of strata where $\cF$ is supported, that (i) $\Ext^{\bu}(\cF, \cG)$ is $\Lambda$-free and vanishes in odd degrees and (ii) satisfies the decomposition asserted in the Lemma. If $N=1$, then $\cF \cong i_{\mu !} i_{\mu}^*  \cF$ for some $\mu \in X_*(T)^+$, and we have

\[
\Ext^{\bu}(\cF, \cG) \cong \Ext^{\bu} (i_{\mu!} i_{\mu}^* \cF, \cG) \cong \Ext^{\bu}(i_{\mu}^* \cF, i_{\mu}^!  \cG).
\]

This shows part (ii) of the inductive hypothesis. By Corollary <ref>, $i_{\mu}^* \cF$ and $i_{\mu}^!  \cG$ are both constant complexes on $\Gr_{G,S/\Div^1_X}$ concentrated in even degrees. Then part (i) of the induction hypothesis follows from Lemma <ref>.

If $N>1$, let $i$ be the inclusion of a closed stratum in the support of $\cF$ and consider the excision sequence $j_! j^* \cF \rightarrow \cF \rightarrow i_* i^* \cF$. Applying $\Hom(-, \cG)$, we get a long exact sequence

\begin{align*}
\ldots \rightarrow  \Ext^n(i_*  i^* \cF, \cG) \rightarrow \Ext^n(\cF, \cG) \rightarrow \Ext^n(j_! j^* \cF, \cG)  \rightarrow \ldots
\end{align*}

The induction hypothesis applies to the flanking terms. In particular (i) implies that the long exact sequence splits, and then (i) and (ii) for the flanking terms imply (i) and (ii) for $\Ext^{\bu}(\cF, \cG) $.

Assume $S$ is strictly totally disconnected. If $\cF \in \DSY{G}{\Lambda}$ is relative $*$-even and $\cG\in \DSY{G}{\Lambda}$ is relative $!$-odd, then $\Hom(\cF, \cG) = 0$.

Assume $S$ is strictly totally disconnected. Suppose $\cF, \cG  \in \DSY{G}{\Lambda}$ are indecomposable and either both relative even or both relative odd. If $i_\mu$ is the inclusion of a stratum which is open in the support of both $\cF$ and $\cG$, then the restriction map $\Hom(\cF, \cG) \rightarrow \Hom(i_\mu^* \cF, i_\mu^* \cG)$ is surjective.

It suffices to treat the case where $\cF, \cG$ are both relative even. Let $i$ be in the inclusion of the complementary strata, so that we have an excision triangle of relative $!$-even complexes,

\[
i_* i^! \cG \rightarrow \cG \rightarrow i_{\mu *} i_\mu^*  \cG.
\]

Applying $\Hom(\cF, -)$, we obtain a long exact sequence

\[
\ldots \rightarrow \Hom(\cF, \cG) \rightarrow \Hom(i_\mu^* \cF, i_\mu^* \cG) \rightarrow \Hom(\cF, i_*i^!  \cG[1])  \rightarrow \ldots
\]

and $\Hom( \cF, i_*i^! \cG[1]) = 0$ by Corollary <ref>, because $\cG[1]$ is relative $!$-odd.

Assume $S$ is strictly totally disconnected. Let $\cF  \in \DSY{G}{\Lambda}$ be an indecomposable relative parity complex, and let $j \co U \rightarrow \Gr_{G, S/\Div^1_X}$ be an inclusion of a union of strata open in the support of $\cF$. Then $j^* \cF$ is either zero or indecomposable.

Since $\DSY{G}{\Lambda}$ is Krull-Schmidt by Lemma <ref>, the indecomposability of $\cF$ implies that $\End(\cF)$ is local. As a quotient of a local ring is local, Lemma <ref> implies that $\Hom(j^* \cF, j^* \cF)$ is also local, so $j^* \cF$ is either zero or indecomposable.

Assume $S$ is strictly totally disconnected. Let $\Cal{F}  \in \PSY{G}{\Lambda}{1}$ be an indecomposable relative parity complex. Then $\cF$ enjoys the following properties:

* $\cF$ has support of the form $\Gr_{G,S/\Div^1_X,\leq \mu} $ for some $\mu \in X_*(T)^+$.

* The restriction $i_{\mu}^* \cF$ to the open stratum in $\supp(\cF)$ is a shifted $\Lambda$-free constant sheaf $\Lambda[d]$.

* Any indecomposable relative parity complex $\cG \in \PSY{G}{\Lambda}{1}$ supported on $\Gr_{G,S/\Div^1_X,\leq \mu} $, such that $i_{\mu}^* \cG \cong \Lambda[d]$, is isomorphic to $\cF$.

(1) If $\supp(\cF)$ contains two disjoint open strata $U,U'$ then applying Lemma <ref> to each of $U$ and $U'$ provides a non-local quotient of $\End(\cF)$. But since $\cF$ is indecomposable, the Krull-Schmidt property (Lemma <ref>) implies that $\End(\cF)$ is local.

(2) Follows from Lemma <ref>.

(3) By assumption, there are isomorphisms $\alpha \co i_\mu^* \cF\rightarrow  i_\mu^* \cG$ and $\beta \co i_\mu^* \cG  \rightarrow i_\mu^* \cF$ which are mutual inverses. By Lemma <ref> we can find lifts $\wt{\alpha} \co \cF \rightarrow \cG$ and $\wt{\beta} \co \cG  \rightarrow  \cF$ such that $\wt{\beta} \circ \wt{\alpha} \in \End(\cF)$ does not lie in the unique maximal ideal, hence is invertible. Similarly, $\wt{\alpha} \circ \wt{\beta}  \in \End(\cG)$ is invertible. Therefore, $\wt{\alpha}$ and $\wt{\beta}$ are isomorphisms.

Assume that $S$ is strictly totally disconnected. We say that $\cF \in \PSY{G}{\Lambda}{1}$ is a relative parity sheaf (with respect to $\dagger$) if it is an indecomposable relative parity complex (with respect to $\dagger$). Note that relative parity sheaves are not (necessarily) sheaves.

We write $\cF \cong \cE(\mu, \cL)$ if $\cF$ has support $\Gr_{G, S/\Div_X^1, \leq \mu}$ and $i_\mu^*\cF \cong \cL$. Note that for any particular $\mu$, we are not guaranteed a priori that $\cE(\mu, \cL)$ exists.

§.§ Small Smith-Treumann localization
We develop here a version of the Tate category and Smith-Treumann localization which is “small” in the sense that it consists only of sheaves satisfying strong finiteness conditions. It is more similar to the formalism of [48, 40] but we note that our sheaves are still not “constructible” since the Schubert stratifications are not constructible in $p$-adic geometry (due to failure of quasicompactness). The structure theory of ULA sheaves (relative to $S$) on $\cHck_{G, S/\Div_X^1}$ from [22] is what makes this small version well behaved even in the absence of quasicompactness.

§.§.§ The small Tate category
Let $\perfulaI{H}{\Lambda[\Sigma]}{I} \subset \DulaI{H}{\Lambda}{I}$ be the full subcategory spanned by objects with finite tor-dimension over $\Lambda[\Sigma]$.

We define the small Tate category of $\Gr_{H, S/(\Div^1_X)^I}$ to be the Verdier quotient

\[
\perfulaI{H}{\cT_\Lambda}{I} := \frac{\DulaI{H}{\Lambda[\Sigma]}{I}}{ \perfulaI{H}{\Lambda[\Sigma]}{I}}
\]

and the small Tate category of $\cHck_{H, S/(\Div^1_X)^I}$ to be

\[
\perfulacHckI{H}{\cT_\Lambda}{I} := \frac{\DulacHckI{H}{\Lambda[\Sigma]}{I}}{ \perfulacHckI{H}{\Lambda[\Sigma]}{I}}.
\]

We make analogous definitions for $\Gr_{H, S/(\Div^1_X)^I, \mu}$ and $\Gr_{H, S/(\Div^1_X)^I, \leq \mu}$, etc.

Parallel to Lemma <ref>, we give an alternate characterization of $\perfulaI{H}{\Lambda[\Sigma]}{I}$; note that this one applies without assuming hat $\Lambda = k$ is a field.

The subcategory $\perfulaI{H}{\Lambda[\Sigma]}{I} \subset \DulaI{H}{\Lambda[\Sigma]}{I}$ coincides with the full subcategory spanned by objects whose stalks at all geometric points are perfect complexes over $\Lambda[\Sigma]$.

By <cit.> all geometric stalks of all objects of $\DulaI{H}{\Lambda[\Sigma]}{I}$ are pseudocoherent, and have finite tor-amplitude by <cit.>. Pseudo-coherent plus finite tor-amplitude implies perfect by <cit.>; this shows one containment.

For the other containment, we must show that if $\cK \in \DulaI{H}{\Lambda[\Sigma]}{I}$ has all geometric stalks being perfect complexes over $\Lambda[\Sigma]$, then $\cK$ finite tor-amplitude. By <cit.>, it suffices to exhibit a uniform bound on the tor-amplitude of the stalks. The rest of the proof is as for Lemma <ref>, except we use that for a perfect complex over $\Lambda[\Sigma]$, the properties of being flat and projective coincide by <cit.>.

The fully faithful embedding

\[
\DulaI{H}{\Lambda[\Sigma]}{I} \inj D^b_{\et}(\Gr_{H,S/(\Div_X^1)^I}; \Lambda[\Sigma])
\]

carries $\perfulaI{H}{\Lambda[\Sigma]}{I} $ into $\Flat^b(\Gr_{H,S/(\Div_X^1)^I}; \Lambda[\Sigma])$ and so induces a functor

\begin{equation}\label{eq: perf to shv}
\perfulaI{H}{\cT_\Lambda}{I} \rightarrow \Shv(\Gr_{H,S/(\Div_X^1)^I}; \cT_{\Lambda}).
\end{equation}

which is conservative; similarly for the equivariant version.

We denote by

\[
\TT^* \co \DulaI{H}{\Lambda[\Sigma]}{I} \rightarrow
\perfulacHckI{H}{\cT_\Lambda}{I}.
\]

the tautological projection.

We denote by

\[
\epsilon^* \co  \DulaI{H}{\Lambda}{I} \rightarrow  \DulaI{H}{\Lambda[\Sigma]}{I}
\]

the inflation via the augmentation $\Lambda[\Sigma] \rightarrow \Lambda$.

We write

\[
\TT := \TT^* \varepsilon^* \co \DulaI{H}{\Lambda}{I} \rightarrow \perfulacHckI{H}{\cT_\Lambda}{I}.
\]

These functors are compatible with the ones having the same notations in <ref>, under (<ref>). We use the same notation for the analogous operations for strata, Schubert varieties, and $\cHck_{H, S/(\Div_X^1)^I}$, etc.

§.§.§ The small $\Psm$ operation
We now define the version of the “Smith operation” $\Psm$ from <ref> for small Tate categories. We assume the setup of <ref>: $G$ has a $\Sigma$-action and $H  := H^\sigma$ is reductive, so that $\Fix(\sigma, \Gr_{G,S/(\Div^1_X)^I}) = \Gr_{H,S/(\Div^1_X)^I}$ by Proposition <ref>.

Let $\iota \co  \Gr_{H,S/(\Div^1_X)^I} \inj  \Gr_{G,S/(\Div^1_X)^I}$ be the the inclusion of the $\Sigma$-fixed points. For any $\cF \in D^{\ULA}_{\et}( \Gr_{G,S/(\Div^1_X)^I};\Lambda)^{\bd}$, the restriction $\iota^*\cF \in D^{\ULA}_{\et}( \Gr_{H,S/(\Div^1_X)^I};\Lambda)^{\bd}$ is ULA over $S$.

The condition can be checked v-locally on $S$ by <cit.>. Therefore, we may assume that $G$ has a Borus $(B,T)$. Let $(B_H, T_H) = (B^\sigma, T^\sigma)$ be the corresponding Borus of $H$ (cf. Lemma <ref>). Recall the constant term functors $\CT$ from <cit.>. Applying (a variant of) <cit.>, it suffices to check that $\CT_{B_H}(i^* \cF)$ is ULA over $S$. By proper base change applied to the commutative diagram

\[
\begin{tikzcd}
\Gr_{T, S/(\Div^1_X)^I} & \ar[l] \Gr_{B, S/(\Div^1_X)^I} \ar[r] &  \Gr_{G, S/(\Div^1_X)^I}  \\
\Gr_{T_H,S/(\Div^1_X)^I} \ar[u, hook, "\iota"]  & \ar[l] \Gr_{B_H, S/(\Div^1_X)^I} \ar[u, hook, "\iota"]  \ar[r]  & \Gr_{H, S/(\Div^1_X)^I} \ar[u,hook,  "\iota"]
\end{tikzcd}
\]

all of whose squares are Cartesian (thanks to Proposition <ref> and Proposition <ref>),
we have a natural isomorphism $\CT_{B_H}(\iota^* \cF) \cong \iota^* \CT_B(\cF)$. By (a variant of) <cit.>, $\CT_B(\cF) \in D^{\ULA}_{\et}(\Gr_{T,S/(\Div^1_X)^I};\Lambda)^{\bd}$ is ULA over $S$. Now $\iota \co \Gr_{T_H, S/(\Div^1_X)^I} \inj \Gr_{T, S/(\Div^1_X)^I}$ is an open embedding, hence $\ell$-cohomologically smooth, so $\iota^* \CT_B(\cF) $ is ULA over $S$ by <cit.>.

The (small) Smith operation is the functor

\begin{equation}\label{eq: small Psm for diamonds}
\Psm  := \TT^* \circ \iota^* \co (\DulaI{G}{\Lambda}{I})^{B\Sigma} \rightarrow \perfulaI{H}{\cT_\Lambda}{I},
\end{equation}

which is well-defined by Lemma <ref>.

The functor $\Psm$ has an equivariant version, which we also denote

\[
\Psm \co (\DulacHckI{G}{\Lambda}{I})^{B\Sigma} \rightarrow \perfulacHckI{H}{\cT_\Lambda}{I}
\]

§.§.§ Compatibilities
We now establish some compatibility statements that could be remembered under the slogan, “the Smith operation commutes with all functors”.

Below we let $Y, Y'$ be spaces of the form $\Gr_{G, S/(\Div^1_X)^I}$ or the twisted Grassmannian $\Gr^{\twi}_{G, S/(\Div^1_X)^I}$ or (closures of) strata thereof, and $f \co Y \rightarrow Y'$ be a map induced by a group homomorphism $G \rightarrow G'$. In all such cases, the small Tate categories of $Y^\sigma$ and $(Y')^{\sigma}$ are defined, because of Proposition <ref> and Corollary <ref>. The Smith operation is also defined, and will be denoted

\[
\Psm \co D^{\ULA}_{?}(Y; \Lambda) \rightarrow \Perf^{\ULA}_{?} (Y^\sigma; \cT_\Lambda)
\]

where $?$ refers to the constructible or equivariant conditions.

We allow $G'$ to be the trivial group, in which case $\Gr_{G', S/(\Div_X^1)^I} \cong S$.

Let $f \co Y \rightarrow Y'$ be a $\Sigma$-equivariant morphism of the type in Notation <ref>. It is clear that the pullback functor $f^* \co D^{\ULA}_?((Y')^\sigma; \Lambda)^{B\Sigma} \rightarrow D^{\ULA}_?(Y^\sigma; \Lambda)^{B\Sigma}$ preserves the perfect subcategories, hence descends to a functor

\[
f^* \co \Perf^{\ULA}_?((Y')^\sigma; \cT_{\Lambda})  \rightarrow   \Perf^{\ULA}_?(Y^\sigma; \cT_{\Lambda}).
\]

We have the properties analogous to <ref> in this situation. By relative Verdier duality, we also get

\[
f^! \co \Perf^{\ULA}_?((Y')^\sigma; \cT_{\Lambda})  \rightarrow   \Perf^{\ULA}_?(Y^\sigma; \cT_{\Lambda}).
\]

Below, when we say that diagrams “canonically commute”, we mean that we construct explicit natural isomorphisms.

Let $f \co Y \rightarrow Y'$ be a $\Sigma$-equivariant morphism of the type in Notation <ref>. The following diagrams canonically commute:

\[
\begin{tikzcd}
D^b_{?}(Y;\Lambda)^{B \Sigma} \ar[d, "\Psm"] &  D^b_{?}(Y';\Lambda)^{B \Sigma}   \ar[d, "\Psm"]  \ar[l, "f^*"] \\
\Perf_?(Y^{\sigma}; \Cal{T}_{\Lambda}) & \Perf_?((Y')^{\sigma} ; \Cal{T}_{\Lambda}) \ar[l, "f^*"]
\end{tikzcd} \hspace{1cm}
\begin{tikzcd}
D^b_{?}(Y;\Lambda)^{B \Sigma} \ar[d, "\Psm"] &  D^b_{?}(Y';\Lambda)^{B \Sigma}   \ar[d, "\Psm"]  \ar[l, "f^!"] \\
\Perf_?(Y^{\sigma}; \Cal{T}_{\Lambda}) & \Perf_?((Y')^{\sigma}; \Cal{T}_{\Lambda}) \ar[l, "f^!"]
\end{tikzcd}
\]

Evident from the definitions.

Let $f \co Y \rightarrow Y'$ be a $\sigma$-equivariant morphism of the type in Notation <ref>. By Lemma <ref> and <cit.>, $\rR f_!$ and $\rR f_*$ preserve the perfect subcategories, hence descend to

\[
\rR f_! \co  \Perf_?^{\ULA}(Y^\sigma; \cT_\Lambda) \rightarrow \Perf^{\ULA}_?((Y')^\sigma; \cT_\Lambda)
\]

\[
\rR f_*\co \Perf_?^{\ULA}(Y^\sigma; \cT_\Lambda) \rightarrow  \Perf^{\ULA}_?((Y')^\sigma; \cT_\Lambda)
\]

Let $f \co Y \rightarrow Y'$ be a $\Sigma$-equivariant morphism of the type in Notation <ref>. Then the following diagrams canonically commute:

\[
\begin{tikzcd}
D^{\ULA}_?(Y; \Lambda)^{B\Sigma} \ar[d, "\Psm"] \ar[r, "\rR f_!"]  & D^{\ULA}_?(Y'; \Lambda)^{B\Sigma}  \ar[d, "\Psm"]  \\
\Perf^{\ULA}_?(Y^\sigma; \cT_\Lambda)  \ar[r, "\rR f_!"]  & \Perf^{\ULA}_?((Y')^\sigma; \cT_\Lambda)
\end{tikzcd} \hspace{1cm}
\begin{tikzcd}
D^{\ULA}_?(Y; \Lambda)^{B\Sigma} \ar[d, "\Psm"] \ar[r, "\rR f_*"]  & D^{\ULA}_?(Y'; \Lambda)^{B\Sigma}  \ar[d, "\Psm"]  \\
\Perf^{\ULA}_?(Y^\sigma; \cT_\Lambda)  \ar[r, "\rR f_*"]  & \Perf^{\ULA}_?((Y')^\sigma; \cT_\Lambda)
\end{tikzcd}
\]

We give the argument for the first diagram, the second being similar[Using also that $Y^\sigma \inj Y$ is $\omega_0$-locally extra small (cf. Example <ref>).]. We may base change to the $\Sigma$-fixed locus of $Y'$, and therefore assume that $\Sigma$ acts trivially on $Y'$. Let $i \co Y^\sigma \inj Y$ be the inclusion of the $\Sigma$-fixed points and $j \co U \inj Y$ be the complementary open embedding. For $\cF \in D^{\ULA}_?(Y; \Lambda)^{B\Sigma}$ consider the exact triangle

\[
j_! j^* \cF  \rightarrow  \cF \rightarrow i_* i^* \cF.
\]

Since $\Sigma$ acts freely on $U$, Corollary <ref>(1) implies that $\rR f_! j_! j^*\cF \in \Perf^{\ULA}_?(Y'; \Lambda[\Sigma])$, hence projects to $0$ in $\Perf^{\ULA}_?(Y'; \cT_\Lambda)$. Therefore the map $\rR f_! \cF \rightarrow \rR f_! i_* i^* \cF$ projects to an isomorphism in $\Perf^{\ULA}_?(Y'; \cT_\Lambda)$.

§.§ Relative Tate-parity sheaves
We now develop an analogous theory to <ref> in the setting of the small Tate category, inspired by work of Leslie-Lonergan [34] which does this for the classical affine Grassmannian. We note that it is important here to take integral coefficients in order to have (any hope of) parity vanishing properties, because of Example <ref>.

§.§.§ Krull-Schmidt property
Let $\OO := W(k)$. Note that we take $|I|=1$ below. Recall (cf. <ref>) that we write

\[
\FF \co \DSY{H}{\OO}   \rightarrow \DSY{H}{k}
\]

for the change-of-coefficients functor.

The following Lemma is parallel to <cit.>.

Let $\cF, \cG \in \Dula{H}{\OO}{1}$. Then there is a natural isomorphism

\begin{equation}\label{eq: lem: tate homs of TT}
\Hom_{\perfula{H}{\cT_{\OO}}{1}}(\TT \cF, \TT \cG) \cong \bigoplus_{i \in \Z} \Hom_{\DSY{H}{k}}(\FF \cF, \FF \cG[2i]).
\end{equation}

We have

\[
\cHom_{\perfula{H}{\cT_{\OO}}{1}}(\TT \cF, \TT \cG) \cong \TT \left( \cHom_{\DSY{H}{\OO}} (\cF, \cG) \right).
\]

As in Example <ref>, for any $\cE \in \DSY{H}{\OO}$, we have

\[
\TT \cE \cong  \mrm{Tot}\left(\ldots \cE \xrightarrow{\ell} \cE \xrightarrow{0}  \cE \xrightarrow{\ell}  \cE \xrightarrow{0} \cE  \rightarrow \ldots \right)   \in \perfula{H}{\cT_{\OO}}{1}.
\]

Noting that

\[
\mrm{Tot}\left(\ldots \cE \xrightarrow{\ell} \cE \xrightarrow{0}  \cE \xrightarrow{\ell}  \cE \xrightarrow{0} \cE  \rightarrow \ldots \right) \cong \bigoplus_{i \in \Z} \FF \cE [2i],
\]

the result then follows upon taking cohomology.

The proof of Lemma <ref> did not use the ULA or $L^+H$-constructibility hypotheses.

Any object $\cF \in \Perf^{\ULA}_{(L^+H)}(\Gr_{H, S/\Div^1, \mu}; \cT_{\OO})$ is a finite direct sum of objects of the form $\TT (\OO)$ and $\TT(\OO[1])$.

The same argument as for <cit.> shows that any such $\cF$ is pulled back from the small Tate category of $S$. Then we may conclude by applying <cit.>, with the following remark: In [34] the case $\ell=2$ is excluded because of <cit.> which claims that $\Ext^1_{\F_2}(\F_2, \F_2) \neq 0$, however this is clearly false, so we may also take $\ell=2$.

The category $\perfula{H}{\cT_{\OO}}{1}$ is Krull-Schmidt.

By <cit.>, a Karoubian category such that $\End(\cK)$ is semiperfect for every $\cK$ is Krull-Schmidt. We will check the semiperfect and Karoubian conditions.

First we check that $\End(\cK)$ is semiperfect for any $\cK \in \perfula{H}{\cT_{\OO}}{1}$, by showing that $\End(\cK)$ is a finite $k$-algebra. In fact, it follows from Lemma <ref> and Lemma <ref> that for any $\cK_1, \cK_2 \in \perfula{H}{\cT_{\OO}}{1}$, the whole Ext-algebra $\Ext^{\bu}(\cK_1, \cK_2)$ is finite-dimensional over $k$.

Now it suffices to show that  $\perfula{H}{\cT_{\OO}}{1}$ is Karoubian, i.e., all idempotents are split. According to <cit.>, in a triangulated category an idempotent of a distinguished triangle which splits on any two terms splits on the third. Therefore by devissage, we may reduce to showing that for any indecomposable object in $\Perf^{\ULA}_{(L^+H)}(\Gr_{H, S/\Div^1, \mu}; \cT_{\OO})$, its endomorphism ring is local. By Lemma <ref> such $\cK$ must be isomorphic to $\TT (\OO^{r}[d])$ for some $d$, and then Lemma <ref> computes its endomorphism algebra, which is seen to be local by inspection.

§.§.§ Relative Tate-parity complexes
We define make definitions within the small Tate category analogous to those in <ref>.

Let $\cK \in \perfula{H}{\Cal{T}_{\OO}}{1}$. Fix a pariversity $\dagger \co X_*(T_H) \rightarrow \Z/2\Z$. Below we view Tate cohomology as being indexed by $\Z/2\Z$.

* For $? \in \{*,!\}$, we say $\cK$ is relative $?$-Tate-even if for all geometric points $\ol{s} \rightarrow S$ and all $\mu \in X_*(T_H)^+$,

\[
\rT^{\dagger(\mu)+1}(i_{\mu}^?(\cK|_{\ol{s}})) = 0.
\]

* For $? \in \{*,!\}$, we say $\cK$ is relative $?$-Tate-odd if $\cK[1]$ is relative ?-Tate-even.

* For $? \in \{*,!\}$, we say that $\cK$ is relative $?$-Tate-parity if $\cK$ is either relative $?$-even or relative $?$-odd.

* We say $\cK$ is relative Tate-even (resp. relative Tate-odd) if $\cK$ is both relative $*$-Tate even (resp. odd) and relative $!$-Tate even (resp. odd).

* We say $\cK$ is a relative Tate-parity complex if it is isomorphic within $\perfula{H}{\cT_{\OO}}{1}$ to the direct sum of a relative Tate-even complex and a relative Tate-odd complex. The full subcategory of $\perfula{H}{\cT_{\OO}}{1}$ spanned by relative Tate-parity complexes (with coefficients in $\Cal{T}_{\OO}$) is denoted $\PSY{H}{\Cal{T}_{\OO}}{1}$.

§.§.§ Structure theory for strictly totally disconnected $S$
Here we record several structural results about $\perfula{H}{\cT_{\OO}}{1}$ under the assumption $S$ is strictly totally disconnected. Below all Hom and Ext groups are formed in the category $\perfula{H}{\cT_{\OO}}{1}$; we omit this for ease of notation. By arguments analogous to those in <ref>, we have the following results.

Assume that $S$ is strictly totally disconnected. If $\cF \in \perfula{H}{\cT_{\OO}}{1}$ is relative $*$-Tate-parity and $\cG \in \perfula{H}{\cT_{\OO}}{1}$ is relative $!$-Tate-parity, then we have a (non-canonical) isomorphism of $k$-vector spaces

\[
\Ext^{\bu}(\cF, \cG) \cong \bigoplus_{\mu \in X_*(T_H)^+} \Ext^{\bu}(i_{\mu}^* \cF, i_{\mu}^! \cG)
\]

Assume that $S$ is strictly totally disconnected. If $\cF \in \perfula{H}{\cT_{\OO}}{1}$ is relative $*$-Tate-even and $\cG\in \perfula{H}{\cT_{\OO}}{1}$ is relative $!$-Tate-odd, then $\Hom(\cF, \cG) = 0$.

Assume that $S$ is strictly totally disconnected. If $\cF, \cG  \in \perfula{H}{\cT_{\OO}}{1}$ are indecomposable and either both relative Tate-even or both relative Tate-odd, and $i_\mu$ is the inclusion of a stratum which is open in the support of both $\cF$ and $\cG$, then the restriction map $\Hom(\cF, \cG) \rightarrow \Hom(i_\mu^* \cF, i_\mu^* \cG)$ is surjective.

Assume that $S$ is strictly totally disconnected. Let $\cF  \in \perfula{H}{\cT_{\OO}}{1}$ be an indecomposable Tate-parity complex, and let $j \co U \rightarrow \Gr_{H, S/\Div^1_X}$ be an inclusion of a union of strata open in the support of $\cF$. Then $j^* \cF$ is either $0$ or indecomposable.

Assume that $S$ is strictly totally disconnected. Let $\Cal{F}  \in \perfula{H}{\cT_{\OO}}{1}$ be an indecomposable relative Tate-parity complex. Then $\cF$ enjoys the following properties:

* $\cF$ has support of the form $\Gr_{H,S/\Div^1_X,\leq \mu} $ for some $\mu \in X_*(T_H)^+$.

* The restriction $i_{\mu}^* \cF$ to the open stratum in $\supp(\cF)$ is isomorphic $\TT(\OO[d])$ for some $d$.

* Any indecomposable relative Tate-parity complex $\cG$ supported on $\Gr_{H,S/\Div^1_X,\leq \mu} $, such that $i_{\mu}^* \cG \cong \TT(\OO[d])$, is isomorphic to $\cF$.

Assume that $S$ is strictly totally disconnected. We say that $\cF \in\perfula{H}{\cT_{\OO}}{1}$ is a relative Tate-parity sheaf (with respect to $\dagger$) if it is an indecomposable relative Tate-parity complex (with respect to $\dagger$). We write $\cF \cong \cE_{\rT}(\mu, \cL)$ if $\cF$ has support $\Gr_{H, S/\Div_X^1, \leq \mu}$ and $i_\mu^* \cF \cong \TT(\cL)$.

§.§ Even maps
A stratified v-stack is a v-stack $Y$ plus a decomposition of $Y$ into locally closed strata $i_\mu \co Y_\mu \inj Y$, such that $i_\mu$ is shriekable for all $\mu$. For a stratified small v-stack $Y$, such that the inclusion maps of strata are all shriekable, and a pariversity $\dagger$ (i.e., a function from the set of strata to $\Z/2\Z$), we may define (absolute) parity complexes analogously to Definition <ref> (requiring the restriction to strata to have cohomology sheaves which are $\Lambda$-free local systems). We do not expect it to have good properties in general, but in this section we axiomatize the property that this notion of parity complex will be preserved under “even” maps, which is true in general.

We begin with a simple observation that provides a partial substitute for “Gysin isomorphisms” in $p$-adic geometry (which are not defined, due to the lack of a notion of local complete intersection).

Let $i \co Z \rightarrow Y$ be a shriekable map of stratified Artin v-stacks which are $\ell$-cohomologically smooth of integral $\ell$-dimension (cf. <cit.>). Then $i^!$ is isomorphic to an even shift of $i^*$ on étale local systems on $Y$.

By passing to an étale cover, we reduce to showing that $i^!\Lambda$ is isomorphic to an even shift of $\Lambda$. Writing $\DD_{(-)}$ for the dualizing sheaf and $d, d' \in \Z$ for the $\ell$-dimensions of $Y,Z$ respectively, we have

\[
i^! \Lambda \cong i^! \DD_{Y}[-2d] \cong \DD_{Z}[-2d] \cong \Lambda[2d'-2d].
\]

The following Definition is in imitation of <cit.>.

Let $Y$ and $Y'$ be stratified small v-stacks and $f \co Y \rightarrow Y'$ be shriekable. We say that $f$ is stratified if

* The pre-image of any stratum $Y_\lambda' \inj  Y'$ is a union of strata of $Y$.

* For any stratum $Y_{\mu} \subset Y$ lying over any stratum $Y_{\lambda}' \subset Y'$, the restricted map of strata $f_\lambda^\mu \co Y_\mu \rightarrow Y'_{\lambda}$ is $\ell$-cohomologically smooth with integral $\ell$-dimension.

We further say that $f$ is even if for all $\mu, \lambda$ and any $\Lambda$-free étale local system $\cL$ on $Y_\lambda'$, the complex $\rR^i(f_\lambda^\mu)_* (\cL)$ is $\Lambda$-free and vanishes for odd $i$.

The significance of even maps lies in the fact that their direct image preserves parity, as in the following Proposition.

Let $Y,Y'$ be stratified Artin v-stacks and let $f \co Y \rightarrow Y'$ be an even stratified map (hence shriekable by definition). Suppose that all strata of $Y,Y'$ are $\ell$-cohomologically smooth with integral $\ell$-dimension. Let $\dagger$ be a pariversity on $Y'$, and define the pullback pariversity $\dagger_Y$ on $Y$ via $\dagger_Y(\mu) := \dagger(\lambda)$ if $f$ carries $Y_\mu$ to $Y_\lambda'$.

(1) If $f$ is proper and $\cK \in D_{\et}^b(Y; \Lambda)$ is a $\dagger_Y$-parity complex, then $\rR f_* \cK \in D_{\et}^b(Y'; \Lambda)$ is a $\dagger$-parity complex.

(2) If $f$ is smooth of integral $\ell$-dimension and $\cK \in D_{\et}^b(Y'; \Lambda)$ is a $\dagger$-parity complex, then $f^* \cK \in D_{\et}^b(Y; \Lambda)$ is a $\dagger_Y$-parity complex.

(1) The statement and argument are the same as for <cit.>, so we just sketch it. Using proper base change, we calculate $i_\lambda^* \rR f_! \cK$ and $i_\lambda^! \rR f_* \cK$ by stratifying the fibers by $Y_\mu$, and calculating the cohomology of fibers in terms of cohomology of strata using excision sequences. This equips $i_\lambda^* \rR f_! \cK$ with a filtration whose associated graded is a direct sum of pieces of the form $ \rR(f_\lambda^\mu)_! i_\mu^* \cK$, so its cohomology sheaves are $\Lambda$-free and vanish in the correct parity of degrees by the definition of an even map. The story is similar for $i_\lambda^! \rR f_*$.

(2) It is immediate from the definitions that $f^* \cK$ is $*$-parity with respect to $\dagger_Y$. Consider the commutative diagram

\[
\begin{tikzcd}
Y_\mu \ar[dr] \ar[rr, bend left, "i_\mu"]  \ar[r, "i^\mu_\lambda"] & f^{-1}(Y_\lambda') \ar[d, "f_\lambda"] \ar[r, "i_\lambda'"] & Y \ar[d, "f"] \\
& Y_\lambda' \ar[r,"i_\lambda"]  & Y'
\end{tikzcd}
\]

where the right square is Cartesian. Suppose $\cK \in  D_{\et}^b(Y'; \Lambda)$ is $\dagger$-parity. Since $f$ is smooth of integral $\ell$-dimension, $(i_\lambda')^! f^* \cK \cong f_\lambda^* i_\lambda^!$ has locally constant $\Lambda$-free cohomology sheaves, which can only be non-zero in degrees congruent to $\dagger(\lambda)$ mod 2. By Lemma <ref>, the same holds for $(i^\mu_\lambda)^!(i_\lambda')^! f^* \cK  \cong i_\mu^! f^* \cK$, so $f^* \cK$ is also $!$-parity with respect to $\dagger_Y$.

§.§ Demazure resolutions