Chromatic support: stabilizer orbits and thick tensor ideals

Source and verification

This article explains family 311 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Height as a complete support invariant

Chromatic homotopy theory studies spectra through homology theories of different heights. Family 311 claims that height completely determines the thick tensor ideals among dualizable $K(n)$-local spectra, at every prime and positive height. The manuscript reaches this topological classification through a rigidity theorem for stabilizer-invariant equations in a deformation ring. (OpenAI, 2026)

The two conclusions should be distinguished. The arithmetic theorem classifies invariant prime and radical ideals; it does not classify all nonradical ideals. The topological theorem concerns dualizable objects, a class larger than the compact objects of the $K(n)$-local category.

🏷️ Preliminaries: localization, duality and support

A spectrum is a stable homotopy object in which suspension is invertible. Its smash product is the analogue of a tensor product, and a cofiber sequence is the analogue of an exact sequence. Morava $K$-theory $K(n)$ detects chromatic height $n$. Localization $L_{K(n)}$ makes maps inducing isomorphisms in this homology theory into equivalences.

Within the localized category, the product and unit are

$$\begin{array}{c}X\otimes Y=L_{K(n)}(X\wedge Y),\\ 1=L_{K(n)}S.\end{array}$$

An object is dualizable if it has a dual with evaluation and coevaluation maps satisfying the usual triangle identities. This is the categorical version of finite-dimensional linear duality. A thick tensor ideal is a class of dualizable objects closed under finite sums, shifts, cofiber sequences, retracts and tensoring with any dualizable object. It is not a class required to be closed under arbitrary infinite sums.

A finite $p$-local spectrum has type $k$ if $K(i{)}_{\ast }F=0$ for $i<k$ and $K(k{)}_{\ast }F\ne 0$, with $K(0)=H\mathbb{Q}$. Choosing such a finite spectrum $F(k)$, define ${\mathcal{D}}_k$ to be the thick tensor ideal generated by $L_{K(n)}F(k)$.

Claimed spectral classification

For every prime $p$ and $n\ge 1$, the dualizable category $\mathcal{D}$ has exactly $n+2$ thick tensor ideals:

$$\begin{array}{c}\mathcal{D}={\mathcal{D}}_0,\\ {\mathcal{D}}_0⊋{\mathcal{D}}_1⊋\cdots ⊋{\mathcal{D}}_{n+1},\\ {\mathcal{D}}_{n+1}=0.\end{array}$$

The ideals are independent of the chosen finite type generators.

To measure an object’s support, use Morava $E$-theory, an even-periodic refinement with coefficient ring

$$E_0=W({\mathbb{F}}_{p^n})[[u_1,\dots ,u_{n-1}]].$$

Here $W({\mathbb{F}}_{p^n})$ is the unramified complete discrete valuation ring with uniformizer $p$ and that residue field. The parameters describe marked deformations of the height-$n$ Honda formal group. A formal group law is a power series describing addition near an identity; height in characteristic $p$ records the first nonzero power in its multiplication-by-$p$ series.

The standard height ideals are

$$\begin{array}{c}{I}_0=0,I_h=(p,u_1,\dots ,u_{h-1}),\\ 1\le h\le n.\end{array}$$

The locus defined by $I_h$ has formal-group height at least $h$; exact height $h<n$ additionally requires $u_h\ne 0$.

For a dualizable $X$, the degree-zero and degree-one groups of completed $E$-homology form a finite $E_0$-module $M_X$. Its support $S(X)$ is the set of prime ideals at which this module does not vanish after localization. This is a closed subset $V(\mathrm{Ann}{M}_X)$ of the coefficient spectrum.

🏷️ The arithmetic theorem and its quantifiers

Let $A_n$ be the endomorphism ring of the height-$n$ Honda group over ${\overline{\mathbb{F}}}_p$. The unit group $A_n^{\times }$ acts on $E_0$ by changing the deformation’s marking. An open subgroup contains all sufficiently small congruence transformations. The extended stabilizer $G_n$ also includes the residue-field Galois action.

The manuscript claims that for every open subgroup $U\subseteq A_n^{\times }$, its invariant proper prime ideals are exactly

$$0,I_1,\dots ,I_n.$$

Its invariant radical ideals are those ideals together with $E_0$ itself. The same radical list holds for the full canonical $G_n$-action. This is the finite-residue-field formulation of Chai’s invariant-ideal assertion, strengthened to open stabilizer subgroups.

The central mechanism is local orbit density in the sense of equations: a nonzero analytic germ on a height locus cannot vanish on the stabilizer orbit sufficiently near its base point. This is not a claim that the orbit is open in an analytic topology.

🏷️ Positive characteristic: coordinates from a Tate lattice

Work over the complete algebraically closed characteristic-$p$ field $C={\mathbb{C}}_p^{♭}$, and let $k={\overline{\mathbb{F}}}_p$ be its coefficient field. Write ${\mathfrak{m}}_C$ for the elements of norm less than one. At an exact-height-$h$ point $x$ with $1\le h<n$, put $r=n-h$.

The finite kernels of the universal deformation form a height-$n$ algebraic $p$-divisible group. At $x$, its connected part has height $h$ and its étale quotient has height $r$. A Tate lattice records compatible systems of $p$-power torsion in that quotient; choose a ${\mathbb{Z}}_p$-basis $z_1,\dots ,z_r$.

A universal-cover comparison regards these fixed vectors as points of a perfected Honda group. Specializing at nearby parameters gives analytic functions

$$\theta (y)=({\Theta }_y(z_i){)}_{i=1}^r,\theta (x)=0.$$

They test whether the fixed vectors remain in the nearby Tate lattice. The manuscript proves that their parameter differential is invertible on the height-at-least-$h$ locus.

The infinitesimal argument explains this assertion’s content. If a parameter tangent vector makes all these derivatives zero, the corresponding deformation over $C[ϵ]/{ϵ}^2$ retains compatible torsion sections. Those sections split the étale quotient. Fixed-height rigidity trivializes the connected factor, with its marking, and the Kodaira–Spencer comparison then forces the parameter vector to vanish. Source and target tangent spaces both have dimension $r$, proving invertibility. The finite-level identities are checked before passage to the whole $p$-divisible group; no uniform convergence neighborhood over all torsion levels is assumed.

🏷️ Frobenius roots and finite analytic jets

Fix $d\in A_n$ and consider the nearby orbit points

$$y_N=(1-p^{N}d)x.$$

For large $N$, these transformations belong to any specified open subgroup. Set $q=p^h$. Integral comparison series with the height-$h$ Honda group turn the transport identity into the exact numerical equation

$$\begin{array}{c}{U}_N=s_N^{q^N},\\ s_N⟶s(d)=({\nu }_x(d{z}_i){)}_{i=1}^r.\end{array}$$

Powers are coordinatewise. The map ${\nu }_x$ is a nonzero natural map on perfectoid test spaces, not just an abstract linear map on one set of valued points.

There is a convergence issue that the manuscript treats explicitly. The coefficients of the change of coordinates are analytic germs, but their domains may shrink as the coefficient index grows. The full coordinate change is therefore used formally. At each $y_N$, one instead chooses an actual integral series matching any fixed finite number of those coefficient germs eventually. Its tails are bounded because the arguments have norm less than one.

Suppose a nonzero analytic germ $P$ vanishes on the orbit. Its expansion in the formal coordinates has coefficients $B_{\ell }$ fixed before choosing $d$. Finite Taylor comparison gives, for every fixed truncation degree $m$,

$$\left|\sum _{|\ell |\le m}{B}_{\ell }{U}_N^{\ell }\right|\le D_m\Vert U_N{\Vert }^{m+1}.$$

The constant and threshold may depend on $m$ and $d$. Taking unique $q^N$-th roots in the perfect field $C$ and then the limit gives

$$\begin{array}{rl}& \underset{N\to \infty }{\mathrm{limsup}}\left|\sum _{|\ell |\le m}{B}_{\ell }^{1/q^N}s(d{)}^{\ell }\right|\\ & \le \Vert s(d){\Vert }^{m+1}.\end{array}$$

A Frobenius-separation lemma converts these estimates into

$$\sum _{\ell }\lambda (B_{\ell })s(d{)}^{\ell }=0$$

for every algebraic $k$-linear functional $\lambda :C\to k$. The lemma works on finite coefficient spans, choosing Frobenius subsequences that fix the finite-field coefficients there; it does not assume that $\lambda$ is continuous or interchange it with an infinite evaluation.

Choose one $\lambda$ nonzero on a fixed nonzero $B_{\ell }$. Because the whole family $B_{\ell }$ was chosen independently of $d$, this produces one common nonzero formal equation over $k$ on the subgroup

$$S_x=\{({\nu }_x(d{z}_i){)}_{i=1}^r:d\in A_n\}.$$

🏷️ Why that formal equation is impossible

The first input is a formal-closure theorem. If a subgroup of a product of height-$h$ Honda groups satisfies a nonzero formal equation over $k$, the ideal of all such equations defines a proper reduced formal subgroup. The proof uses finite truncations, Frobenius separation and separated completion to verify compatibility with the group law.

A classification of the resulting $p$-divisible subgroup up to isogeny then supplies a nonzero row of Honda endomorphisms annihilating it. A common power of $p$ clears denominators, and torsion-freeness of morphism groups turns rational vanishing into actual vanishing. In the present case the conclusion would be

$$\sum _{i=1}^r{a}_i{\nu }_{x}d(z_i)=0(d\in A_n),$$

where not all $a_i$ are zero and the sum uses the target formal-group law.

The second input interprets this natural relation on the Fargues–Fontaine curve. The source uses established comparisons between Honda-group sheaves and sections of vector bundles. The height-$s$ object corresponds to a stable bundle ${\mathcal{E}}_s$ of rank $s$ and degree one; natural maps become bundle maps.

Here is the decisive independence argument. Let $r<n$ sections of ${\mathcal{E}}_n$ be independent over ${\mathbb{Q}}_p$. Saturate their image to a subbundle $\mathcal{F}$ of rank $b\le r$. A generically independent selection of $b$ sections gives a nonzero determinant section, while semistability gives

$$0\le \mathrm{deg}\mathcal{F}\le b/n<1.$$

Degrees are integers, so this degree is zero. A nonzero effective divisor has positive degree; hence the determinant section has no zeros and the chosen sections trivialize $\mathcal{F}$. Since global functions on the curve are ${\mathbb{Q}}_p$, all $r$ original sections then lie in ${\mathbb{Q}}_p^b$. Their independence forces $b=r$. They are therefore independent even over the function field of the curve.

At the generic point the division-algebra action spans the full matrix algebra: the action is unital and injective by central simplicity, and dimensions agree. Extend the vanishing endomorphism identity linearly to that algebra. A matrix can send one selected $z_i$ to a vector on which ${\nu }_x$ is nonzero and send every other $z_j$ to zero. A nonzero $a_i$ is invertible in the target division algebra, so the resulting identity is impossible.

This proves the required absence of formal equations on $S_x$, and hence orbit-germ density. Naturality is indispensable: the bundle argument does not apply to an arbitrary linear map between valued-point groups.

🏷️ From orbit germs to invariant ideals

For an invariant prime containing $p$, choose the largest $h$ with $I_h$ contained in it. If $h=n$, maximality already identifies the prime. Otherwise an auxiliary formal-curve argument produces a point of the quotient where $u_h\ne 0$ and all parameters are small.

Any nonzero extra equation modulo $I_h$ has bounded coefficients. Translation by a small point is an invertible operation on bounded-coefficient series, so the equation remains a nonzero ambient analytic germ at that point. Invariance makes it vanish on the orbit, contradicting density. The quotient point need not come from an injective map of its coordinate ring into $C$; ambient nonvanishing is supplied separately by the translation lemma.

For primes avoiding $p$, the manuscript uses the Gross–Hopkins period map instead. It is locally an analytic isomorphism to projective space, where the split division algebra supplies all tangent directions. Vanishing on sufficiently small ${\mathbb{Z}}_p$-parameter grids forces an analytic germ to be zero by repeated one-variable uniqueness. This argument requires no openness of the orbit in the ${\mathbb{C}}_p$ topology.

Finally an invariant radical ideal has finitely many minimal primes. Their stabilizers are closed and of finite index, hence open. The prime theorem applies to each one, and their intersection is the smallest member of the height chain. This explains why the open-subgroup theorem is useful even when the ultimate target is invariance under the full extended stabilizer.

🏷️ Transferring the height chain to topology

The manuscript imports an established support comparison, valid at every prime:

$$\begin{array}{c}S(X\otimes Y)=S(X)\cap S(Y),\\ X\in ⟨Y{⟩}_{\otimes }⟺S(X)\subseteq S(Y),\\ S(L_{K(n)}F(k))=V(I_k),\\ 0\le k\le n.\end{array}$$

Its proof uses Morava-$E$ descent, tensor-nilpotence detection and the classification of support for perfect $E$-modules. Dualizable local spectra map to perfect modules even when they are not compact in the local spectral category. The direction used here has no lower bound on $p$; prime restrictions in a converse realization theorem are irrelevant to this deduction.

The radical classification makes the possible supports a finite chain:

$$V(I_0)⊋\cdots ⊋V(I_n)⊋\varnothing .$$

Any nonzero thick tensor ideal contains an object whose support is the greatest support occurring in it: the maximum is attained because the possibilities are finite. Support containment then shows that this single object generates the whole ideal. Its support equals that of one finite type generator, giving precisely ${\mathcal{D}}_0,\dots ,{\mathcal{D}}_n$; the zero ideal supplies the final member.

The Balmer spectrum has proper prime thick tensor ideals as its points. Here every proper member of the chain is prime, including zero, since two nonempty coefficient supports both contain $V(I_n)$ and cannot have empty intersection. Thus there are $n+1$ points. Their closures are

$$\overline{\{{\mathcal{D}}_j\}}=\{{\mathcal{D}}_l:j\le l\le n+1\}.$$

The point ${\mathcal{D}}_1$ is generic and ${\mathcal{D}}_{n+1}=0$ is closed. For example at height one, the three ideals are the whole category, the nonzero proper ideal ${\mathcal{D}}_1$, and zero; the spectrum has two points.

The map from coefficient primes to these points sends $\mathfrak{q}$ to ${\mathcal{D}}_{h(\mathfrak{q})+1}$, where $h(\mathfrak{q})$ is the largest index with $I_h\subseteq \mathfrak{q}$. This is a formal-group height stratum, not the Krull height of the prime.

The selected review covers the statements, finite-jet argument, endomorphism-to-bundle contradiction and support-classification deduction. The full deformation and universal-cover comparisons, formal subgroup rigidity, perfectoid sheaf identifications and imported period-map/descent theorems remain substantial inputs rather than independently established results of this note.

🏷️ Key steps and proof outlines

Finite jets replace an unjustified infinite analytic substitution

The orbit argument constructs formal coordinates

$$U_i(X)=\sum _{j\ge 1}{a}_j(X){\theta }_i(X{)}^j,\theta (0)=0,$$

with invertible linear part $L=d{U}_0$. Each coefficient $a_j$ is an analytic germ, but their convergence neighborhoods need not have one common positive radius. The full formal series should therefore not be evaluated as though it were a convergent coordinate change.

At each orbit displacement ${\delta }_N\to 0$, the source instead has integral numerical coefficients $a_{N,j}$ agreeing with any fixed finite list of germs once $N$ is sufficiently large. The resulting numerical series $U_N$ converges because its arguments have norm less than one. For fixed $m$, integrality bounds its tail after degree $m$ by $\Vert \theta ({\delta }_N){\Vert }^{m+1}$. Taylor expansion of the finitely many retained analytic terms gives

$$U_N=J_{m}U({\delta }_N)+O_m(\Vert {\delta }_N{\Vert }^{m+1}),$$

where $J_{m}U$ is the Taylor polynomial through total degree $m$.

At $m=1$, the remainder is eventually smaller than $\Vert L{\delta }_N\Vert$. The ultrametric inequality then gives $\Vert U_N\Vert =\Vert L{\delta }_N\Vert$, hence

$$\Vert {\delta }_N\Vert \le \Vert L^{-1}\Vert \Vert U_N\Vert .$$

Now formally expand a vanishing analytic germ as $P={\sum }_{\ell }{B}_{\ell }{U}^{\ell }$. Truncate to the polynomial $Q_m(Z)={\sum }_{|\ell |\le m}{B}_{\ell }{Z}^{\ell }$. The formal identity says that the actual analytic germ $P-Q_m(J_{m}U)$ vanishes to order $m+1$. Since a fixed polynomial is Lipschitz on a sufficiently small polydisc, the preceding jet estimate and $P({\delta }_N)=0$ imply

$$|Q_m(U_N)|\le D_m\Vert U_N{\Vert }^{m+1}.$$

Every substitution is finite. The constants and the threshold may depend on $m$; no common convergence neighborhood for all coefficients is asserted. These are the numerical estimates to which Frobenius separation is applied. (OpenAI, 2026)

Orbit density classifies equations, not just generic points

Assume the local orbit-density theorem and let $\mathfrak{p}$ be invariant under an open stabilizer subgroup. In positive characteristic, choose the largest $h$ for which $I_h\subseteq \mathfrak{p}$. If $h<n$, the image of $u_h$ in the quotient is nonzero. The source’s formal-curve lemma supplies a small point where $u_h$ remains nonzero, placing it on the exact-height-$h$ locus.

If $\mathfrak{p}/I_h$ contained a nonzero power series, the bounded-translation lemma would make it a nonzero analytic germ at that point. Invariance makes it vanish on the nearby orbit, contradicting orbit density. Therefore $\mathfrak{p}=I_h$. The characteristic-zero case uses the separate period-map orbit theorem and gives $\mathfrak{p}=0$.

To pass to radical ideals, take their finitely many minimal primes. The stabilizer permutes this finite set, so each prime has finite-index stabilizer. The action’s continuity and adic closedness of ideals make this stabilizer closed; its complement is a finite union of closed cosets, hence it is open. The open-subgroup prime theorem applies to each minimal prime. Since the standard primes form a chain, their intersection is just the smallest one. This explains why the theorem must cover open subgroups, even when the final application uses the full stabilizer.

A finite chain of supports makes every ideal singly generated

The established support comparison is the topological interface:

$$X\in ⟨Y{⟩}_{\otimes }⟺S(X)\subseteq S(Y).$$

After the invariant-radical theorem, every object support belongs to the finite chain $V(I_0),\dots ,V(I_n),\varnothing$. In any nonzero thick tensor ideal $\mathcal{J}$, choose an object $Y$ whose support is largest among the supports that actually occur in $\mathcal{J}$. Such a largest support is attained because the list is finite.

Every $X\in \mathcal{J}$ has support inside $S(Y)$ and therefore belongs to $⟨Y{⟩}_{\otimes }$. The reverse inclusion follows from $Y\in \mathcal{J}$. Thus the ideal is singly generated as a conclusion, with no finite-generation assumption on $\mathcal{J}$. A known finite type-$k$ spectrum realizes the same support as $Y$, so comparison in both directions identifies their generated ideals. The classification does not require realizing arbitrary coefficient modules by spectra.

🏷️ Manuscripts and verification scope

• Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra — pinned manuscript and companion source.

No family scope page was present at lean/docs/311.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra.