Invariant Primes in Lubin-Tate Space and Hovey-Strickland at Every Height

Abstract: Let $H_n$ be the one-dimensional Honda formal group of height $n$ over $\mathbf F_{p<sup>n}$ and let [ R_n=W(\mathbf F_{p^n})[[u_1,\ldots,u_{n-1}]],\qquad A_n=R_n/(p). ] We prove, for every height $n$ and every prime $p$, that the prime ideals of $A_n$ stable under an open subgroup of the Morava stabilizer group are exactly the height ideals $(u_1,\ldots,u_j)$. We also prove that a stable prime of $R_n$ avoiding $p$ is zero. The resulting radical-ideal classification implies the Hovey-Strickland classification of thick tensor ideals in the category of dualizable $K(n)$-local spectra via the forward implication of Barthel-Heard-Naumann. The special-fiber argument is local and geometric. After cutting an invariant prime by a one-parameter curve, we construct from the Cartier structure equation a smooth formal quotient $\mathcal{Q}$ and a distinguished subgroup $\mathcal{H}$. The generic fiber of $\mathcal{H}$ is identified with the deformation space of connected-étale extensions. A Cartier obstruction map from the full Honda endomorphism order is compared with evaluation on Tate vectors through completed universal covers. Fargues-Fontaine vector bundles give a period-detection statement. Chai's rigidity theorem then promotes detection by homomorphisms to formal Zariski density after a renormalization of the valuation. A uniform fixed-jet argument transfers this density to Morava-stabilizer orbits. The generic-fiber assertion is proved separately from the Gross-Hopkins period map.

• The paper classifies stable prime ideals in mod-$p$ Lubin-Tate deformation spaces using height filtration and stabilizer group actions, proving that for every open subgroup of the Morava stabilizer group, the stable prime ideals are precisely the ideals generated by the initial coefficients of the p-series
• The classification is applied to the Hovey-Strickland problem, identifying thick tensor ideals of dualizable $K(n)$-local spectra using Cartier normal form techniques and evaluation through completed universal covers
• The renormalized density argument and fixed-jet comparison with stabilizer displacements are crucial for proving the formal density of obstruction classes and the classification of invariant subvarieties of Lubin-Tate Spectrum.

Main results

The paper classifies the prime ideals in the mod-$p$ Lubin–Tate deformation space that are stable under an open subgroup of the Morava stabilizer group. Fix a prime $p$ and a height $n$ Honda formal group $H_n$ over $\mathbb{F}_{p^n}$, and write

$$R_n=W(\mathbb{F}_{p^n})[[u_1,\ldots,u_{n-1}]],\qquad A_n=R_n/(p).$$

For $0\leq j\leq n-1$, let

$J_j=(u_1,\ldots,u_j)\subset A_n,$

with $J_0=(0)$. The principal theorem asserts that, for every open subgroup $U\subseteq S_n=\operatorname{End}(H_n)^\times$, the $p$0-stable prime ideals of $p$1 are precisely

$p$2

The mixed-characteristic statement is complementary. If $p$3 is prime, stable under an open subgroup of $p$4, and $p$5, then

$p$6

Consequently, the proper radical ideals of $p$7 stable under an open stabilizer subgroup are exactly

$p$8

together with $p$9. The result holds uniformly for every height and every prime, including $n$0 and $n$1 (2610.10072).

The paper then applies this invariant-radical classification to the Hovey–Strickland problem. Using the forward implication of Barthel–Heard–Naumann, it concludes that the thick tensor ideals of dualizable $n$2-local spectra are exactly

$n$3

together with the zero ideal $n$4. Here $n$5 is any finite $n$6-local spectrum of chromatic type $n$7. The Balmer spectrum consequently has the expected chromatic chain, with specialization order determined by

$n$8

The conclusion is stronger than a classification of $n$9-stable closed subsets. It identifies all stable prime and radical ideals, and therefore rules out invariant subvarieties that are not generated by the height filtration. The theorem is also independent of the large-prime numerical hypotheses appearing in the converse direction of the Barthel–Heard–Naumann results.

Reduction to a one-parameter degeneration

The special-fiber argument begins by assigning a generic formal height to a prime $H_n$0. If the generic height is $H_n$1, then

$H_n$2

The ideals $H_n$3 are visibly prime and stabilizer invariant: they are generated by the initial coefficients of the $H_n$4-series, and an invertible change of formal coordinate preserves the corresponding height conditions.

The essential task is to exclude a strict containment $H_n$5. Rather than attempting to analyze the entire multivariable formal scheme directly, the paper chooses a test curve

$H_n$6

such that the image of $H_n$7 is nonzero while all deformation parameters lie in the maximal ideal $H_n$8. This produces a valued field $H_n$9 and a connected–étale sequence

$\mathbb{F}_{p^n}$0

where $\mathbb{F}_{p^n}$1 has height $\mathbb{F}_{p^n}$2 and $\mathbb{F}_{p^n}$3 has height $\mathbb{F}_{p^n}$4.

The curve is not required to be invariant under the stabilizer. This point is logically important. Stabilizer invariance is used before restriction to the curve: if $\mathbb{F}_{p^n}$5 and $\mathbb{F}_{p^n}$6, then $\mathbb{F}_{p^n}$7, and both equations may subsequently be evaluated on the chosen curve. The proof therefore does not impose an unwarranted equivariance condition on the test curve.

The reduction converts the invariant-prime problem into a dynamical question about stabilizer displacements along a one-dimensional valued deformation. The remaining argument must show that sufficiently many small stabilizer translates of a point on the curve cannot remain inside a proper formal subvariety.

Cartier normal forms and extension geometry

The central geometric construction uses the covariant $\mathbb{F}_{p^n}$8-typical Cartier module $\mathbb{F}_{p^n}$9 of the formal part of $$R_n=W(\mathbb{F}_{p^n})[[u_1,\ldots,u_{n-1}]],\qquad A_n=R_n/(p).$$0. In a suitable $$R_n=W(\mathbb{F}_{p^n})[[u_1,\ldots,u_{n-1}]],\qquad A_n=R_n/(p).$$1-basis, its structure operator is

$$R_n=W(\mathbb{F}_{p^n})[[u_1,\ldots,u_{n-1}]],\qquad A_n=R_n/(p).$$2

A normal-form theorem decomposes every sufficiently positive Cartier curve uniquely as

$$R_n=W(\mathbb{F}_{p^n})[[u_1,\ldots,u_{n-1}]],\qquad A_n=R_n/(p).$$3

where $$R_n=W(\mathbb{F}_{p^n})[[u_1,\ldots,u_{n-1}]],\qquad A_n=R_n/(p).$$4 and $$R_n=W(\mathbb{F}_{p^n})[[u_1,\ldots,u_{n-1}]],\qquad A_n=R_n/(p).$$5 is supported in the first $$R_n=W(\mathbb{F}_{p^n})[[u_1,\ldots,u_{n-1}]],\qquad A_n=R_n/(p).$$6 Cartier coordinates. This gives a smooth formal group

$$R_n=W(\mathbb{F}_{p^n})[[u_1,\ldots,u_{n-1}]],\qquad A_n=R_n/(p).$$7

of dimension $$R_n=W(\mathbb{F}_{p^n})[[u_1,\ldots,u_{n-1}]],\qquad A_n=R_n/(p).$$8Q$$is induced by Cartier normal-form addition.</p> <p>The subgroup</p> <p>$$R_n=W(\mathbb{F}_{p^n})[[u_1,\ldots,u_{n-1}]],\qquad A_n=R_n/(p).$9

has dimension $0\leq j\leq n-1$0. The Cartier relation implies

$0\leq j\leq n-1$1

and every coordinate of the $0\leq j\leq n-1$2-series of $0\leq j\leq n-1$3 has total degree at least

$0\leq j\leq n-1$4

Thus positive valuations are multiplied by at least $0\leq j\leq n-1$5 under multiplication by $0\leq j\leq n-1$6: $0\leq j\leq n-1$7

The formal group $0\leq j\leq n-1$8 is identified with the connected–étale extension space: $0\leq j\leq n-1$9 After extending scalars to an algebraically closed field, it becomes isomorphic to $J_j=(u_1,\ldots,u_j)\subset A_n,$0, the $J_j=(u_1,\ldots,u_j)\subset A_n,$1-fold product of the height-$J_j=(u_1,\ldots,u_j)\subset A_n,$2 Honda formal group. This identification supplies the correct geometric target for the obstruction classes associated with Honda endomorphisms.

The construction is significant because it converts the deformation-theoretic problem into a linear extension problem. The $J_j=(u_1,\ldots,u_j)\subset A_n,$3 étale Tate directions of $J_j=(u_1,\ldots,u_j)\subset A_n,$4 become $J_j=(u_1,\ldots,u_j)\subset A_n,$5 independent connected extension parameters, while the height-$J_j=(u_1,\ldots,u_j)\subset A_n,$6 Honda factor records the connected part of the degeneration.

Honda obstruction classes

Let

$J_j=(u_1,\ldots,u_j)\subset A_n,$7

be the maximal order in the Honda division algebra. For each $J_j=(u_1,\ldots,u_j)\subset A_n,$8, the paper chooses an integral Cartier lift of the special-fiber endomorphism and defines an obstruction class

$J_j=(u_1,\ldots,u_j)\subset A_n,$9

The class vanishes exactly when $J_0=(0)$0 lifts to an endomorphism of the deformed formal group. It satisfies the additive and $J_0=(0)$1-compatibility relations

$J_0=(0)$2

The latter identity produces the decisive valuation growth: $J_0=(0)$3 whenever the class is nonzero. This growth is both useful and problematic. It produces highly controlled obstruction points, but it makes them collapse too rapidly in the original $J_0=(0)$4-adic topology to support a density argument.

The use of the full Honda order, rather than only the unramified Witt subring, is essential. The later matrix-spanning argument requires the entire rational division algebra

$J_0=(0)$5

which becomes a full matrix algebra after scalar extension.

Evaluation through completed universal covers

After passing to a completed algebraic closure $J_0=(0)$6 of $J_0=(0)$7, choose an identification

$J_0=(0)$8

Every integral Tate vector

$J_0=(0)$9

defines an evaluation homomorphism

$U\subseteq S_n=\operatorname{End}(H_n)^\times$0

If $U\subseteq S_n=\operatorname{End}(H_n)^\times$1 is a basis of the Tate module, the resulting map

$U\subseteq S_n=\operatorname{End}(H_n)^\times$2

is an isomorphism.

The construction must use completed universal covers. Evaluation on the discrete connected torsion sheaf over the reduced valuation ring would lose the relevant points, because the connected Honda group has no nonzero field-valued torsion. The Tate vectors become visible only after passing through nilpotent thickenings and taking the inverse limit. This technical choice is not cosmetic: it is what makes the étale quotient detectable inside the connected extension group.

The paper proves the obstruction–evaluation identity

$U\subseteq S_n=\operatorname{End}(H_n)^\times$3

where $U\subseteq S_n=\operatorname{End}(H_n)^\times$4 is the analytic projection induced by the connected Honda quotient. This formula links the Cartier obstruction classes to the action of the Honda endomorphism algebra on Tate vectors.

The associated Fargues–Fontaine vector-bundle argument gives an exact sequence

$U\subseteq S_n=\operatorname{End}(H_n)^\times$5

The kernel is a semistable degree-zero bundle and hence is trivial. From this, the paper derives the period-detection statement: every nonzero homomorphism

$U\subseteq S_n=\operatorname{End}(H_n)^\times$6

detects at least one obstruction class,

$U\subseteq S_n=\operatorname{End}(H_n)^\times$7

for some $U\subseteq S_n=\operatorname{End}(H_n)^\times$8.

The implication is substantial. It says that the obstruction classes are not contained in the kernel of any nonzero Honda-valued formal homomorphism. The proof depends on extending the action of $U\subseteq S_n=\operatorname{End}(H_n)^\times$9 to a matrix algebra and using its ability to move a nonzero Tate vector independently in arbitrary matrix directions.

Renormalized density

The valuation growth $p$00 prevents ordinary $p$01-adic density. The paper therefore introduces an asymptotic valued field by taking an ultraproduct and renormalizing the valuation of a sequence $p$02 by

$p$03

The resulting completed field contains the original coefficient field with trivial valuation, while sequences of obstruction points

$p$04

have positive valuation and remain analytically accessible.

The main density proposition states that the points $p$05, for $p$06, are formally Zariski dense in $p$07. Suppose a nonzero formal series vanished on every $p$08. The ideal of all such equations is radical and stable under scalar units. Chai’s rigidity theorem then forces each irreducible component of its zero locus to be a smooth $p$09-divisible formal subgroup. Dieudonné–Manin semisimplicity supplies a nonzero homomorphism to $p$10 vanishing on each proper component.

A finite-union argument shows that one of these components contains the obstruction points associated with a full-rank $p$11-subspace of $p$12. Multiplication by a sufficiently large power of $p$13 then forces the corresponding Honda-valued homomorphism to vanish on every obstruction point, contradicting the period-detection theorem. Therefore no nonzero formal equation can vanish on all $p$14.

This is the central density mechanism of the paper. Chai rigidity converts algebraic invariance into subgroup structure; Dieudonné–Manin theory converts subgroup structure into a nonzero annihilating homomorphism; the Fargues–Fontaine argument rules out that annihilator.

From obstruction parameters to stabilizer orbits

The obstruction group $p$15 is auxiliary, whereas the prime ideal lives in the Lubin–Tate coordinates

$p$16

The paper therefore establishes a fixed-jet comparison between obstruction coordinates and actual stabilizer displacements.

For $p$17, define

$p$18

for sufficiently large $p$19, and let

$p$20

Solving the Cartier gauge equation gives a formal map

$p$21

with invertible linear term satisfying

$p$22

In the renormalized field,

$p$23

The fixed-jet qualification is essential. A Cartier curve has infinitely many coordinates, and coordinatewise convergence is insufficient for substituting it into a formal equation. The appendix proves uniform estimates at every finite Taylor order. In particular, truncations stabilize modulo increasingly high powers of the obstruction variables, and the error bounds are independent of the output Cartier coordinate.

This repairs a genuine technical difficulty in the argument. Without uniform control over finite jets, the passage from infinite Cartier data to a formal map $p$24 would not be justified. The proof therefore relies not merely on formal inverse-function reasoning, but on a uniform tail ideal estimate and a contraction argument for the Cartier gauge equation.

Suppose now that $p$25 is a nonzero equation of a hypothetical strict enlargement of $p$26 inside an invariant prime $p$27. Stabilizer invariance gives

$p$28

The fixed-jet comparison transfers these equations to

$p$29

which is nonzero because $p$30 has invertible Jacobian. In the asymptotic field, $p$31 vanishes on every $p$32. This contradicts formal density. Hence no strict enlargement exists, and

$p$33

This completes the special-fiber classification. The argument does not merely show that stabilizer orbits are large in an informal sense; it identifies a formal coordinate system in which the orbit displacements contain a formally dense set.

The mixed-characteristic generic fiber

The classification of primes avoiding $p$34 uses a separate argument and does not depend on the Cartier-density construction. Let $p$35 be a nonzero prime with $p$36. Its rigid generic fiber is nonempty, has dimension at most $p$37, and contains a smooth point.

The Gross–Hopkins period map

$p$38

is étale and stabilizer equivariant. At a smooth point of the generic fiber of $p$39, infinitesimal stabilizer actions yield tangent directions in the image under $p$40. After scalar extension,

$p$41

so these infinitesimal actions span the entire tangent space of projective space.

Because $p$42 is étale, they also span the tangent space of Lubin–Tate space. Stabilizer invariance would therefore force the supposed proper subspace to have full tangent dimension, contradicting the dimension bound. Thus a stable prime avoiding $p$43 must be zero.

The separation of the generic-fiber proof from the special-fiber proof is methodologically useful. The special fiber requires valuation-theoretic density and Cartier geometry, whereas the generic fiber follows from the period map and the full infinitesimal action of the division algebra.

Consequences for radical ideals and tensor ideals

Every stable prime containing $p$44 descends modulo $p$45 to one of the ideals $p$46. Every stable prime avoiding $p$47 is zero. Since the ideals

$p$48

form a chain, the minimal primes over a stable radical ideal cannot be distinct and incomparable. Consequently every proper stable radical ideal is itself one of the $p$49.

The passage to tensor-triangular geometry uses the completed Morava $p$50-homology of dualizable $p$51-local spectra. Its support is a closed invariant subset of $p$52, and the invariant-radical classification forces this support to be one of the chromatic closed subsets defined by $p$53. Barthel–Heard–Naumann then identify the resulting thick tensor ideals with the ideals generated by localized finite spectra of the corresponding chromatic types (2610.10072).

The result applies at all heights and primes. In particular, the numerical condition

$p$54

that appears in the converse realization theorem of Barthel–Heard–Naumann is not used here. The paper proves only the forward classification required to deduce the list of thick tensor ideals from invariant radical ideals.

Limitations and open questions

The proof depends on several deep structural inputs: Zink’s Cartier theory, Chai’s rigidity and extension results, the Fargues–Fontaine classification of vector bundles, Scholze–Weinstein’s fully faithful theory of $p$55-divisible groups, and the Gross–Hopkins period map. The paper’s contribution is the way these ingredients are assembled, particularly the renormalized density argument and the fixed-jet comparison with stabilizer displacements.

The argument also treats open-subgroup stability rather than arbitrary stability under more general subgroups or correspondences. The theorem identifies invariant primes and radical ideals, but it does not provide a classification of all nonradical stable ideals. Nor does the proof establish the converse tensor-triangular realization theorem independently of the hypotheses in the broader Barthel–Heard–Naumann framework.

A further question left open by the proof is whether the obstruction-period density and fixed-jet mechanism admit formulations that avoid the auxiliary ultraproduct-valued field. The present construction is effective for formal density, but its dependence on valuation renormalization means that it is not directly an ordinary rigid-analytic density statement in the original Lubin–Tate space.

Conclusion

The paper proves that the only primes in the special fiber of Lubin–Tate space stable under an open Morava stabilizer subgroup are the height ideals, and that no nonzero stable prime survives in mixed characteristic while avoiding $p$56. The special-fiber proof proceeds through Cartier normal forms, connected–étale extension spaces, Honda obstruction classes, Fargues–Fontaine period detection, Chai rigidity, renormalized formal density, and a uniform fixed-jet comparison with stabilizer orbits. The resulting radical-ideal classification yields the Hovey–Strickland classification of thick tensor ideals in dualizable $p$57-local spectra at every height and every prime (2610.10072).