---
title: Reduced Emerton–Gee Stacks
url: https://www.emergentmind.com/topics/reduced-emerton-gee-stacks
type: topic
---

# Reduced Emerton–Gee Stacks

Searching arXiv for the cited Emerton–Gee stack papers to ground the response in current arXiv records.
Reduced Emerton–Gee stacks are the underlying reduced substacks of the special fibers of Emerton–Gee moduli stacks of étale \((\varphi,\Gamma)\)-modules, or more generally of \(L\)-parameters and related structures. They retain the topological and finite-type algebraic geometry of mod \(p\) families—points, irreducible components, dimensions, closure relations, and scheme-theoretic images—while discarding nilpotent thickening data. Across the recent literature, they appear in several complementary forms: as the reduced special fiber of the original Emerton–Gee stack for \(\mathrm{GL}_n\); as explicit quotient stacks in low rank; as loci cut out by crystalline or potentially semistable conditions; and, for tame and classical groups, as reduced moduli stacks of \(L\)-parameters whose irreducible components admit recursive descriptions. A recurring theme is that reduced geometry is sufficiently rigid to encode component combinatorics, Serre-weight labels, and many local-model comparisons, even when the full formal or derived stack carries subtler infinitesimal structure [2304.05317].

## 1. Foundational meaning and formal-algebraic setting

In the original \(\mathrm{GL}_n\) setting, the Emerton–Gee stack is a moduli stack of projective rank \(n\) étale \((\varphi,\Gamma)\)-modules, equivalently a moduli stack of \(p\)-adic Galois representations in families. Its reduced special fiber is the algebraic stack obtained after base change to the residue field and passage to the underlying reduced substack. For rank \(n\), irreducible components of the reduced special fiber are labeled by Serre weights, and this reduced stack is equidimensional of dimension \(\frac{n(n-1)}{2}[K:\mathbf Q_p]\) in the background recalled in the component-function and spectral Satake literature [2306.00141].

For tame groups, the formal-algebraic existence of reduced Emerton–Gee stacks is established in a stronger form. The moduli stack \(X_{{}^{L}G}\) of \(L\)-parameters for a tame group is constructed as a Noetherian formal algebraic stack, and its reduced mod-\(p\) fiber \(X_{{}^{L}G,\mathrm{red}}\) is proved to be an algebraic stack of finite presentation over \(\mathbf F_p\) [2304.05317]. In that framework, the reduced stack is not merely a formal shadow of the \(p\)-adic stack: it is an honest finite-type algebro-geometric object, and the paper develops parabolic, toric, and Herr-complex tools specifically to analyze it.

This formal-algebraic picture extends to further group-valued variants. For general flat algebraic groups \(G/\mathbf Z_p\), the Emerton–Gee stack is defined Tannakianly as the stack of exact tensor functors
\[
\operatorname{Rep}_{\mathbf Z_p}(G)\to {}^{\varphi,\Gamma}(\mathcal O_{\mathcal E,R}),
\]
and is shown to be a formal algebraic stack locally of finite presentation over \(\operatorname{Spf}(\mathbf Z_p)\) [2411.12661]. The paper does not isolate a separately named “reduced Emerton–Gee stack,” but once formal algebraicity is known, the special fiber and its underlying reduced substack exist in the usual sense. A plausible implication is that, for these general-group versions, reduced geometry should again be interpreted as the finite-type mod-\(p\) geometry of the classical formal stack, rather than as a derived or higher-categorical construction.

## 2. Reducedness, classicality, and the derived/classical distinction

A central source of confusion is the distinction between reduction and classical truncation. The derived-stack construction of Laurent \(F\)-crystals on the absolute prismatic site produces a derived stack \(\chi\) whose underlying classical stack is naturally equivalent to the Emerton–Gee stack \(\chi_{\mathrm{EG}}\):
\[
{}^{\rm cl}\chi \simeq \chi_{\rm EG}.
\]
The same work proves that \(\chi\) is classical up to nilcompletion on truncated animated rings, in the sense that it agrees with the étale sheafification of the left Kan extension of its classical truncation after nilcompletion [2309.05066].

That result is about classicality, not reducedness. The paper explicitly distinguishes these operations: classical truncation forgets higher derived directions, whereas reduction kills ordinary nilpotent functions on the resulting classical stack. It therefore does **not** prove that the Emerton–Gee stack is reduced, nor does it compute its nilreduction or irreducible components [2309.05066]. The same distinction persists in the general-group sequel: for connected reductive groups, the derived stack of Laurent \(F\)-crystals with \(G\)-structure is classical after nilcompletion, and for generalized reductive groups the analogous statement holds for a modified Emerton–Gee stack, but again the reduced special fiber is not separately analyzed as a named object [2411.12661].

This distinction is especially important when discussing “reduced Emerton–Gee stacks.” Reducedness is an ordinary scheme- or stack-theoretic property of the special fiber. Classicality results rule out extra derived nilpotents, but they do not show that ordinary nilpotent elements in the classical structure sheaf vanish. Thus the recent derived work clarifies the relation between derived and classical Emerton–Gee stacks, while leaving ordinary reducedness questions to the geometric analyses of the classical reduced special fiber [2309.05066].

## 3. Explicit reduced geometry in rank one

Rank one is the case in which the reduced structure is most completely understood. The rank-one Emerton–Gee stack \(X_1\) of étale \((\varphi,\Gamma)\)-modules with coefficients in \(p\)-adically complete rings admits the explicit description
\[
X_1 \cong \big[\mathcal T/\widehat{\mathbf G}_m\big],
\]
where \(\mathcal T\) is the formal moduli functor of continuous characters \(W_K\to A^\times\), and the \(\widehat{\mathbf G}_m\)-action is trivial. After choosing a geometric Frobenius,
\[
\mathcal T \cong \operatorname{Spf}\mathcal O[[I_K^{\mathrm{ab}]]\times \widehat{\mathbf G}_m.
\]
The classification theorem states that every rank-one étale \((\varphi,\Gamma)\)-module is uniquely of the form
\[
M\cong A(\delta)\otimes_A L
\]
for a unique continuous character \(\delta:W_K\to A^\times\) and a unique invertible \(A\)-module \(L\) up to isomorphism [2206.02888].

On the reduced level, this makes the geometry completely explicit. The reduced character space decomposes as
\[
\coprod_{\delta}\mathbf G_m \xrightarrow{\sim} \mathcal T_{\mathrm{red}},
\]
where \(\delta\) runs over residual inertial characters \(I_K^{\mathrm{ab}}\to \mathbf F^\times\). Consequently,
\[
(X_1)_{\mathrm{red}} \cong \coprod_{\delta} [\mathbf G_m/\mathbf G_m].
\]
Thus the reduced rank-one Emerton–Gee stack is a disjoint union of copies of \([\mathbf G_m/\mathbf G_m]\), indexed by residual inertial characters, equivalently by rank-one Serre weights in the sense used there [2206.02888]. The same method also yields the analogous description for rank-one étale \(\varphi\)-modules without \(\Gamma\)-action.

This rank-one computation is the clearest model for what “reduced Emerton–Gee stack” means concretely: the nilpotent/formal directions are entirely carried by the formal character space, while the reduced structure is the mod-\(p\) union of unramified-twist families. A plausible implication is that higher-rank reduced stacks should likewise be controlled by residual representation-theoretic data, but with far more complicated extension and intersection behavior [2206.02888].

## 4. The \(\mathrm{GL}_2\) case: components, intersections, smooth loci, and irregular crystalline strata

For \(\mathrm{GL}_2\), the reduced Emerton–Gee stack \(\mathcal X_{2,\mathrm{red}}\) is a pure \([K:\mathbf Q_p]\)-dimensional algebraic stack over \(\mathbf F\), and its irreducible components \(\mathcal X_\sigma\) are indexed by non-Steinberg Serre weights \(\sigma\) [2210.05002]. The finite-type points of \(\mathcal X_\sigma\) are exactly the residual Galois representations \(\bar\rho\) for which \(\sigma\in W(\bar\rho)\), so components admit a direct mod-\(p\) Galois-theoretic interpretation.

The incidence geometry of these reduced components is unusually explicit. For non-isomorphic non-Steinberg Serre weights \(\sigma,\tau\), the condition
\[
\dim(\mathcal X_\sigma\cap \mathcal X_\tau)=[K:\mathbf Q_p]-1
\]
characterizes codimension-one intersections, and the paper relates this directly to extension groups of Serre weights. In particular,
\[
\operatorname{Ext}^1_{\mathrm{GL}_2(\mathcal O_K)}(\sigma,\tau)\neq 0
\implies
\dim(\mathcal X_\sigma\cap \mathcal X_\tau)=[K:\mathbf Q_p]-1,
\]
and for weakly regular \(\sigma,\tau\), the converse holds [2210.05002]. The geometry distinguishes type I and type II codimension-one intersections, the latter being exactly the phenomenon where a codimension-one locus lies on three top-dimensional components.

A different line of work identifies many individual reduced components explicitly. For \(K/\mathbf Q_p\) unramified and \(p>2\), many components of the reduced Emerton–Gee stack are shown, via comparison with the Breuil–Kisin image stack \(\mathcal Z\), to be smooth quotient stacks. Under the conditions
\[
\vec b\neq (0,\dots,0),\qquad \vec b\neq (p-2,\dots,p-2),
\]
and excluding any contiguous block \((0,p-2,\dots,p-2,p-1)\) of length at least \(2\), the corresponding component is isomorphic to a quotient of
\[
\GL_2\times \SL_2^{\,f-1}
\]
by
\[
\mathbb G_m^{\,f+1}\times \mathbb G_a^{\,f},
\]
and its global functions are
\[
\Gamma(\mathcal Z(\sigma),\mathcal O)\cong \mathbb F[x,y][y^{-1}].
\]
Through the comparison with \(\mathcal X_{2,\mathrm{red}}\), these become smooth reduced components of the \(\mathrm{GL}_2\) Emerton–Gee stack [2209.09439].

The reduced geometry of lower-dimensional crystalline loci inside \(X_{2,\mathrm{red}}\) is also now understood in the irregular range. For unramified \(K/\mathbf Q_p\), the reduced closed substack
\[
X^{\underline r}_{\mathrm{red}}
\subset X_{2,\mathrm{red}}
\]
cut out by crystalline lifts of Hodge type \(\underline r\) is irreducible whenever \(\underline r\) is \(p\)-bounded and non-Steinberg, that is,
\[
0\le r_{i,1}-r_{i,2}\le p
\quad\forall i,
\]
and not all gaps equal \(p\). Its codimension inside \(X_{2,\mathrm{red}}\) is
\[
\#\{i:r_{i,1}=r_{i,2}\},
\]
the number of irregular embeddings. These loci are identified with scheme-theoretic images of explicit irreducible Breuil–Kisin strata \(Z^\tau(J)\), giving a precise reduced-geometric model for irregular crystalline conditions [2309.13665].

## 5. Higher-rank reduced components and rings of functions

For general \(n\), the reduced special fiber \(\mathcal X_{n,\mathrm{red},\mathbf F}\) decomposes as a union of irreducible components
\[
\mathcal X_{n,\mathrm{red},\mathbf F}=\bigcup_\sigma \mathcal X(\sigma),
\]
indexed by Serre weights \(\sigma\) of \(\mathrm{GL}_n(k)\) [2306.00141]. For sufficiently generic weights—more precisely, for \((3n-1)\)-deep \(\sigma\)—an individual component admits a local-model presentation
\[
\mathcal X(\sigma)\cong [\widetilde{\mathcal C}_\sigma/T^{\mathcal J}],
\]
with \(T^{\mathcal J}\) acting by shifted conjugation on an explicit irreducible monodromy Schubert-type subscheme \(\widetilde{\mathcal C}_\sigma\) of a product of affine flag varieties [2306.00141].

The principal consequence is a precise description of the ring of global functions:
\[
\mathcal O(\mathcal X(\sigma))\cong \mathbf F[x_1,\dots,x_{n-1},x_n^{\pm1}]
\]
for \((3n-1)\)-deep \(\sigma\) [2306.00141]. The construction uses explicit affine-flag charts, torus invariants on dense opens, and an extension theorem for minor functions on affine Schubert varieties via Demazure resolutions. This gives a rigid global invariant of the reduced component, although the paper does not claim that the component itself is isomorphic to \(\mathbf A^{n-1}\times \mathbf G_m\).

A complementary perspective identifies these function rings with mod \(p\) Hecke algebras. For a non-Steinberg Serre weight \(\sigma\), there is a natural injective morphism
\[
\mathcal O(\mathcal C_\sigma)\hookrightarrow \mathcal H(\sigma),
\]
where \(\mathcal C_\sigma\) is the irreducible component of the reduced special fiber corresponding to \(\sigma\), and \(\mathcal H(\sigma)\) is the Hecke algebra
\[
\operatorname{End}_{GL_n(K)}\!\left(\operatorname{c\text{-}Ind}_{GL_n(\mathcal O_K)}^{GL_n(K)}\sigma\right).
\]
If \(\sigma\) is \(((e+1)(n-1)+2)\)-deep, this map is an isomorphism:
\[
\mathcal O(\mathcal C_\sigma)\xrightarrow{\sim}\mathcal H(\sigma).
\]
Via the mod \(p\) Satake isomorphism, this yields
\[
\mathcal O(\mathcal C_\sigma)\cong \mathbb F[y_1,\dots,y_{n-1},y_n^{\pm1}]
\]
for sufficiently generic \(\sigma\) [2402.14011].

This Hecke-theoretic description leads to geometric stratifications of the reduced component. Canonical functions \(f_i\in \mathcal O(\mathcal C_\sigma)\) define open loci
\[
\mathcal C_{\sigma,I}:=\bigcap_{i\in I}\mathcal C_\sigma(f_i),
\]
with a supersingular stratum \(\mathcal C_\sigma^{\mathrm{ss}}=\mathcal C_{\sigma,\varnothing}\). Under a slightly stronger depth condition, these strata carry a parabolic structure, via natural morphisms to reduced components for Levi subgroups [2402.14011]. This suggests that, in higher rank, reduced components are not only indexed combinatorially by Serre weights but also internally organized by parabolic and Satake-theoretic data.

## 6. Tame groups, classical groups, \(GSp_4\), and unitary recursion

For tame groups, the reduced moduli stack \(X_{{}^{L}G,\mathrm{red}}\) is an algebraic stack of finite presentation, and its geometry can be analyzed by parabolic and toric factorization of mod \(p\) \(L\)-parameters [2304.05317]. The sequel constructs formal substacks
\[
X_{K,{}^LG}^{\mathrm{ss},\tau,\underline\lambda}
\]
of potentially semistable \(L\)-parameters with fixed inertial and Hodge type, proves that their mod-\(p\) fibers are equidimensional of dimension
\[
\dim_{\mathbf Q_p}\widehat G/P_{\underline\lambda},
\]
and uses them to study irreducible components of the reduced Emerton–Gee stack for classical groups [2309.05773].

The topological reduction is to maximally non-split Borel loci. The paper proves that the generic points of irreducible components lie in the Borel locus, and that maximal-dimensional irreducible components of a suitable maximally non-split open substack of the Borel stack are in bijection with irreducible components of the reduced Emerton–Gee stack itself [2309.05773]. Under the formal “classical structure” axioms, this yields a recursive description of irreducible components in terms of components for a Levi \(M\cong \mathrm{Res}_{K/F}G\times H_M\) and for the smaller group \(H_M\). Relatively non-Steinberg components correspond to irreducible components of \(X_{F,{}^LM,\mathrm{red}}\), while relatively Steinberg components correspond to irreducible components of \(X_{F,{}^LH_M,\mathrm{red}}\) when \(\dim U/[U,U]\ge 2\) [2309.05773].

For even unitary groups over \(F=\mathbf Q_p\), this recursion terminates in a concrete classification: irreducible components of
\[
X_{F,{}^LU_{2m},\mathrm{red}}
\]
are in natural bijection with parahoric Serre weights [2309.05773]. This is one of the strongest currently available group-theoretic descriptions of reduced Emerton–Gee stacks beyond \(\mathrm{GL}_n\).

A related but distinct tame-group paper gives a group-theoretic description of irreducible components away from Steinberg parts. For regular parahoric Serre weights \(\sigma\), it defines closed substacks \(\mathcal C(\sigma)\subset X_{{}^{L}G,\mathrm{red}}\) as closures of loci of parameters factoring through a unique Borel and having prescribed regular inertial presentation. These are the explicit candidates for irreducible components of the reduced Emerton–Gee stack in the regular non-Steinberg range [2306.02093]. The full exhaustion of all components is conjectural there, but the construction is already sufficient for geometric Breuil–Mézard formulations.

The \(GSp_4\) analogue illustrates how reduced geometry interacts with potentially crystalline deformation theory. The symplectic Emerton–Gee stack \(\mathcal X\) is a Noetherian formal algebraic stack over \(\operatorname{Spf}\mathcal O\), and its underlying reduced mod-\(p\) stack
\[
\mathcal X_{,\mathrm{red}}
\]
is algebraic of finite presentation, equidimensional of dimension
\[
4[K:\mathbf Q_p],
\]
with irreducible components naturally labeled by Serre weights [2304.13879]. In the unramified case, potentially crystalline substacks admit local-model comparisons, and at tame points under genericity assumptions the relevant versal rings are domains, hence the completed local rings are formally unibranch. The underlying reduced special fiber of a potentially crystalline substack is identified as a union of those components \(\mathcal X_\sigma\) indexed by Serre weights predicted by the Jordan–Hölder factors of the corresponding Deligne–Lusztig representation [2304.13879].

Finally, reduced Emerton–Gee stacks also enter the existence of de Rham lifts for classical groups. In that setting, partial lifting problems are reduced to dimension estimates for closed substacks
\[
X_s\subset X_{F,{}^LM,\overline{\mathbf F}_p}
\]
defined by lower bounds on obstruction groups \(H^2(\mathrm{Gal}_F,U^{ab})\). For unitary groups, the necessary dimension estimates are proved by analyzing special closed substacks of the reduced unitary Emerton–Gee stack using Grassmannian geometry, and this yields potentially crystalline lifts of regular Hodge type for all mod \(p\) \(L\)-parameters for \(U_n\) [2309.00761]. This suggests that reduced Emerton–Gee stacks function not only as repositories of component combinatorics, but also as the ambient spaces in which non-abelian lifting problems become tractable geometric inequalities.

In summary, reduced Emerton–Gee stacks form the finite-type mod-\(p\) geometric core of the Emerton–Gee program. In rank one they are completely explicit; for \(\mathrm{GL}_2\) they exhibit rich intersection and smoothness phenomena; in higher rank they carry explicit function theory and Satake-compatible stratifications; and for tame, classical, symplectic, and unitary groups they support recursive and local-model descriptions of irreducible components. What remains less understood is the passage from these reduced stacks back to the full nonreduced or derived geometry: recent classicality results clarify that no extra derived directions are present beyond nilcompletion, but ordinary nilpotent structure and its relation to reduction remain largely separate questions [2309.05066].

Source: https://www.emergentmind.com/topics/reduced-emerton-gee-stacks