On Galois extensions of geometric fixed point spectra

Abstract: In this article, the cyclotomic Galois action on topological K-theory adjoined with a primitive $n$th root of unity and the famous $GL_1(\mathbf{Z}/n)$- and $GL_2(\mathbf{Z}/n)$-Galois actions on topological modular forms with $Γ_1(n)$- and $Γ(n)$-level structures are unified and generalised. This is done by defining a quotient stack in derived algebraic geometry parametrising constant finite abelian subgroups of $\mathbf{P}$-divisible groups, and studying how these quotient stacks and various notions of torsors interact with Galois extensions formed by taking algebraic and categorical invariants. Taking global sections then yields the titular Galois extensions on $K$-geometric fixed points, recovering and refining the well-known examples above and providing new ones. As an application, the $\infty$-category of perfect modules over a variety of $H$-equivariant ring spectra $R$ are decomposed into simple pullbacks of nonequivariant categories, leading to Mayer--Vietoris sequences for localising invariants of $R$. For example, this occurs for equivariant topological K-theory for all $p$-groups as well as any finite nonabelian simple group of order less than 500.

• The paper constructs faithful Galois extensions from affine flat torsors on injection stacks of P-divisible groups, transferring geometric structure to E∞-rings and 2-rings through global sections.
• The framework recovers cyclotomic extensions in equivariant K-theory and level-structure extensions in topological modular forms, while producing new integral examples such as KO[1/n] to KU[1/n, ζn + ζ̄n].
• The resulting Galois actions can force Tate-term vanishing in KNPR decompositions, yielding pullback descriptions and Mayer–Vietoris sequences for localising invariants including algebraic K-theory, THH, and TC.

This paper develops a systematic framework for producing Galois extensions of $E_\infty$-rings and 2-rings from the geometry of $P$-divisible groups, and applies it to equivariant stable homotopy theory. The central objects are stacks of injections into a $P$-divisible group, which the author shows carry canonical actions of automorphism groups that define affine flat torsors. Taking global sections of these torsors yields faithful Galois extensions on geometric fixed points of tempered cohomology theories, unifying and refining classical examples such as the cyclotomic action on topological $K$-theory and the $\mathrm{GL}_1(\mathbb{Z}/n)$- and $\mathrm{GL}_2(\mathbb{Z}/n)$-actions on level structures in topological modular forms. As an application, the paper produces simplified decompositions of categories of perfect equivariant modules, leading to Mayer–Vietoris sequences for localising invariants.

Torsors in spectral algebraic geometry

The foundational section establishes a dictionary between torsors over spectral stacks and Galois extensions. A $P$-$G$-torsor over a stack $M$ is a map $\varphi\colon Y \to X$ with a $P$0-action such that the quotient stack identifies with $P$1 and the structure map satisfies property $P$2. The author verifies that such torsors automatically satisfy the principality condition — the Čech nerve of $P$3 agrees with the action groupoid — so every torsor is an effective epimorphism, and derives closure properties under base change and descent.

Two results connect this to Galois theory. First, a $P$4-$P$5-Galois extension of $P$6-rings (one where the induced action on $P$7 is free) gives rise to an affine $P$8-torsor on spectra, and conversely any flat affine $P$9-torsor between affines arises this way. Second, the combination of affineness and flatness is shown to be highly restrictive: an affine flat $P$0-torsor over $P$1 pulls back along every affine $P$2 to a map inducing a $P$3-$P$4-Galois extension on $P$5. This rigidity is what makes the later construction work.

The relationship with Rognes' general notion of faithful $P$6-Galois extension is more subtle. The author notes plainly that the converse fails: $P$7 is a faithful $P$8-Galois extension but induces the identity on $P$9, and $K$0 is not an equivalence. Nevertheless, for Landweber exact complex-periodic rings satisfying a faithfulness condition on stabilisers at closed points, the quotient $K$1 is shown to be an affine flat $K$2-torsor over Lurie's moduli stack of oriented formal groups; this gives a geometric model for quotients by finite subgroups of Morava stabiliser groups.

The moduli stack of subgroups

For an oriented $K$3-divisible group $K$4 over a stack $K$5 and a finite abelian group $K$6, the paper studies the substack $K$7 of the homomorphism stack classifying injections from the Pontryagin dual of $K$8, together with its tautological $K$9-action. The key observation is that while the full homomorphism stack fails to give an affine quotient (the zero homomorphism has full stabiliser), restricting to injections removes this obstruction. The main geometric theorem states:

The quotient map $\mathrm{GL}_1(\mathbb{Z}/n)$0 is an affine flat $\mathrm{GL}_1(\mathbb{Z}/n)$1-torsor over $\mathrm{GL}_1(\mathbb{Z}/n)$2.

The proof reduces to the affine case, where freeness of the action on $\mathrm{GL}_1(\mathbb{Z}/n)$3 yields a $\mathrm{GL}_1(\mathbb{Z}/n)$4-Galois extension, and flatness follows from the earlier rigidity criterion. By finiteness of the acting group, the torsor is moreover finite étale.

From any torsor, the paper extracts faithful Galois extensions in several settings: $\mathrm{GL}_1(\mathbb{Z}/n)$5 in commutative algebras over $\mathrm{GL}_1(\mathbb{Z}/n)$6, $\mathrm{GL}_1(\mathbb{Z}/n)$7 in presentably symmetric monoidal stable categories, and — when the base is 0-affine or perfect — their global sections and compact-object refinements as Galois extensions of $\mathrm{GL}_1(\mathbb{Z}/n)$8-rings and 2-rings respectively. Notably, the categorical statements require no affineness or flatness hypotheses; only the passage to global sections needs 0-affineness.

Explicit computations illustrate the scope: for $\mathrm{GL}_1(\mathbb{Z}/n)$9, the subgroup stack satisfies $\mathrm{GL}_2(\mathbb{Z}/n)$0 via étale rigidity, and for the universal oriented elliptic curve, $\mathrm{GL}_2(\mathbb{Z}/n)$1 recovers the moduli stack with $\mathrm{GL}_2(\mathbb{Z}/n)$2-level structure.

Galois extensions on geometric fixed points

The bridge to equivariant homotopy theory uses Lurie's tempered cohomology theories: an oriented $\mathrm{GL}_2(\mathbb{Z}/n)$3-divisible group $\mathrm{GL}_2(\mathbb{Z}/n)$4 over $\mathrm{GL}_2(\mathbb{Z}/n)$5 produces a global $\mathrm{GL}_2(\mathbb{Z}/n)$6-ring $\mathrm{GL}_2(\mathbb{Z}/n)$7 whose $\mathrm{GL}_2(\mathbb{Z}/n)$8-fixed points are the global sections of the homomorphism stack $\mathrm{GL}_2(\mathbb{Z}/n)$9, and whose $P$0-geometric fixed points are the global sections of $P$1. Combining this identification with the torsor theorem yields the main algebraic result: for $P$2 0-affine, the tautological $P$3-action on $P$4 makes

$P$5

a faithful $P$6-Galois extension of $P$7-rings, and similarly for perfect module categories as 2-rings. Restricting along the global Weyl homomorphism $P$8 gives faithful $P$9-Galois extensions whenever the pair $G$0 is Weyl faithful. The result is further refined to a statement about $G$1-ambidextrous global $G$2-rings $G$3, using joint conservativity of geometric fixed point functors and base change for geometric fixed points.

Several restrictions are stated candidly. For nonabelian $G$4, the geometric fixed points of tempered cohomology theories vanish, forcing the abelian hypothesis. The $G$5-action on genuine fixed points is generally not Galois — the null homomorphism has nontrivial stabiliser — hence the focus on geometric fixed points. Any action on $G$6 is trivial, excluding the global sphere. The comparison between these residual actions and the classical Weyl group actions of equivariant stable homotopy theory holds only under additional hypotheses: it always holds for split extensions $G$7, and for torus- and elliptic-curve-based theories where separability of the Galois algebra forces discreteness of the relevant automorphism space. Whether the factorisation through $G$8 holds for arbitrary tempered cohomology theories is left open.

Recovered and new examples

The framework recovers known Galois extensions and produces new ones. For equivariant complex $G$9-theory, $M$0 with the cyclotomic $M$1-action, recovering the familiar extension $M$2. Applying the same machinery to real $M$3-theory — constructed here via an oriented torus over $M$4 — yields the less standard extension $M$5, where the action twists the residual $M$6-linear action with complex conjugation; notably this is not a $M$7-Galois extension.

For equivariant elliptic cohomology $M$8, the case $M$9 produces integral lifts of the classical $\varphi\colon Y \to X$0-Galois extensions $\varphi\colon Y \to X$1 associated with $\varphi\colon Y \to X$2-level structures, without inverting $\varphi\colon Y \to X$3; for instance $\varphi\colon Y \to X$4 computes to $\varphi\colon Y \to X$5, and its homotopy fixed points recover the naïve delocalisation of Mahowald–Rezk's $\varphi\colon Y \to X$6. The case $\varphi\colon Y \to X$7 recovers exactly the $\varphi\colon Y \to X$8-Galois extension $\varphi\colon Y \to X$9 from $P$00-level structures. Mixed cases $P$01 yield almost-integral refinements of twisted $P$02-level extensions. The author notes that integrality of $P$03 in related Katz–Mazur and Tate $P$04-theories is established only indirectly, via maps to splitting algebras, and explicit computations are deferred to future work.

Decompositions of equivariant module categories

The main application concerns KNPR-decompositions, following Krause and Naumann–Pol–Ramzi: for an $P$05-equivariant $P$06-ring $P$07, the category $P$08 admits an iterated pullback description involving homotopy fixed points and Tate constructions $P$09. The paper identifies three sufficient conditions for the Tate term to vanish — trivial Weyl group, invertibility of $P$10 in $P$11, or the Weyl action witnessing a faithful Galois extension — the last being precisely what the main theorem supplies. Since each resulting pullback square is an excision square, every localising invariant $P$12 (nonconnective algebraic $P$13-theory, THH, TC) yields fibre sequences $P$14.

Concrete consequences include: for split groups $P$15 with $P$16, a single pullback decomposition valid for all tempered cohomology theories; for equivariant $P$17-theories, decompositions for all finite $P$18-groups (with trivial Weyl actions when $P$19 is abelian), for $P$20 and $P$21, for $P$22, $P$23, $P$24, $P$25, and finally for $P$26, the last nonabelian simple group of order below 500, where the corner factor identifies explicitly as $P$27 for $P$28. These decompositions fail for some composite-order groups already at $P$29, and for larger symmetric groups such as $P$30 the required Tate vanishing is unavailable since certain Weyl actions are not Galois.

Limitations and open questions

Several hypotheses are load-bearing and their removal is open. The 0-affineness assumption on the base stack $P$31 is needed for global sections to preserve the Galois property; without it, only the relative categorical statements survive. It remains unknown whether the quotient by the full $P$32-action (rather than its restriction through $P$33) defines an affine torsor; partial evidence suggests it may fail, though no counterexample has been found. The factorisation of the canonical $P$34-action through $P$35 is established only for split groups and for torus- and elliptic-curve-based theories, and the analogous question for arbitrary tempered cohomology theories — including higher-height examples from topological automorphic forms — is posed but unresolved. Explicit computations of geometric fixed points for the Katz–Mazur and Tate variants, and for mixed-level elliptic cases beyond the étale regime, are deferred. Finally, the catalogue of groups admitting simple decompositions is ad hoc, governed by case-by-case verification of Tate vanishing rather than a structural classification.

Conclusion

The paper constructs a uniform geometric source of Galois extensions — affine flat torsors of injection stacks attached to oriented $P$36-divisible groups — and shows that global sections produce faithful Galois extensions on geometric fixed points of tempered cohomology theories, simultaneously recovering the cyclotomic and level-structure Galois actions of equivariant $P$37-theory and TMF and yielding new integral refinements. The resulting Tate vanishing phenomena simplify KNPR-decompositions of equivariant perfect module categories across a substantial list of finite groups, converting them into excision squares and hence Mayer–Vietoris sequences for all localising invariants. The framework's dependence on 0-affineness, the unresolved status of the full $P$38-torsor, and the absence of a general criterion for when Weyl actions are Galois delineate the boundaries of the current theory.