Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra

Proof. Fix m. If all the coefficients through degree m vanish, the corresponding assertion is immediate. Otherwise choose a k-basis \beta_1,\ldots,\beta_a for their span and write

There is a subsequence of positive integers N on which q^N-power Frobenius fixes all the c_{\ell j}: finitely many elements of \overline{\Fp} lie in a common finite field. On this subsequence,

Put V=\max_j|v_j|. If V>0, choose j_0 with |v_{j_0}|=V, set w_j=v_j/v_{j_0}, and let \rho_j\in k be the residue of w_j\in\OO_C. At least one \rho_j is nonzero, and

Pass to a further subsequence on which q^N-power Frobenius also fixes all the \rho_j. The independence of the \beta_j gives \sum_j\beta_j\rho_j\ne0, whence

On the other hand,

since |\beta_j|^{1/q^N}\to1 for every j. The ultrametric inequality therefore implies

along this subsequence. Equation (2.1) gives V\leq\|s\|^{m+1}; this inequality also holds if V=0.

For an arbitrary k-linear functional \lambda, we now have the finite-sum estimate

Here |\lambda(\beta_j)|\leq1, because these values lie in k. The series with coefficients \lambda(b_\ell)\in k converges at s, and the last bound tends to zero as m\to\infty. At s=0, Equation (2.1) for m=0 forces b_0=0, giving the same conclusion.□

Proof. Every f\in R has a unique finite decomposition

An ordinary k-derivation kills all the pth powers, so this identity shows that it is determined by its values on the T_i, through the usual formal partial derivatives. Those partial derivatives themselves are derivations. Hence the ordinary module \Omega^1_{R/k} is free on dT_1,\ldots,dT_r, and the quotient presentation gives

In particular, this ordinary differential module is finite over B.

Consider the completed product ring

and its diagonal ideal J=(Y_i-X_i)_i. Writing Y=X+Z modulo (Z)^2 gives

Thus J/J^2 has exactly the presentation in Equation (2.2). This computation makes no reducedness assumption on the completed product. The group-difference automorphism

carries the diagonal to the identity section in the second factor. The conormal module of that section is B\otimes_k(\mathfrak m_B/\mathfrak m_B^2): it is obtained by retaining just the linear terms in the second coordinates. We conclude that

Let \mathfrak p be a minimal prime of B and set D=B/\mathfrak p, K=\Frac(D), and d=\dim D. Since B is reduced, B_{\mathfrak p} is the field K. Localization of ordinary differentials in Equation (2.3) gives \dim_K\Omega^1_{K/k}=e.

Choose a system of parameters in the complete local domain D. Completeness and the finite-dimensional quotient by those parameters give a finite map k[[x_1,\ldots,x_d]]\to D. It is injective by dimension, since a nonzero ideal in the source would lower its dimension. Thus K is a finite extension of L=\Frac(k[[x_1,\ldots,x_d]]). Monomials with exponents below p form a basis over Frobenius, so [L:L^p]=p^d. This degree is unchanged by the finite extension: Frobenius identifies the extensions K/L and K^p/L^p, and comparison of the two towers over L^p gives

This argument includes inseparable extensions. Because k is perfect, the p-basis description of field differentials now gives \dim_K\Omega^1_{K/k}=d. Indeed, a p-basis of K over K^p has d elements, and their differentials form a K-basis of \Omega^1_{K/k}. Therefore d=e.

Every component has dimension e, so \dim B=e. This equals the embedding dimension of B, making B regular. Lifting a basis of \mathfrak m_B/\mathfrak m_B^2 gives a surjection k[[X_1,\ldots,X_e]]\to B by completeness. Equality of dimensions makes its kernel zero, proving the assertion.□

Proof. Put q=p^h, and let I\ne0 be the defining ideal of \mathcal H in R=k[[T_1,\ldots,T_r]]. By Lemma 2.3, its coordinate ring is a power-series ring B of some dimension e<r. If e=0, then B=k and \mathcal H is the origin; a coordinate projection proves the assertion. Assume henceforth that e>0.

Let \sigma:R\to R raise every coefficient to its qth power and fix the variables. For P\in I, stability under multiplication by p gives

Radicality implies \sigma^{-1}I\subseteq I. Applying \sigma repeatedly produces the ascending chain

It stabilizes because R is Noetherian, and applying an inverse iterate of \sigma to a stationary pair gives I=\sigma I. Consequently \sigma descends to an automorphism of B.

The coordinate pullback of [p] on \mathcal H is the h-fold absolute Frobenius of B, composed with \sigma^{-1}. This h-fold Frobenius on B\cong k[[X_1,\ldots,X_e]] is finite free, with basis given by monomials whose exponents are less than q. Thus [p] on \mathcal H is finite flat of degree q^e=p^{he}. The kernels \mathcal H[p^j] are finite flat of order p^{hej}. The map induced by [p] from level j+1 to level j is a pullback of [p]:\mathcal H\to\mathcal H, hence is faithfully flat. These kernels form a p-divisible subgroup \mathcal H[p^\infty] of \Gamma_h^r[p^\infty], of height he<hr.

They also recover the formal topology. If \overline T_i are the images of the ambient coordinates in B and

then

The first inclusion follows because any monomial of the indicated total degree has an exponent at least q^j.

In the semisimple isogeny category over k, the ambient group is a sum of r copies of the simple object associated to \Gamma_h. The subgroup has smaller height. Projecting a nonzero complementary summand onto one simple factor gives a nonzero rational morphism

whose restriction to \mathcal H[p^\infty] is zero. Choose one common power of p clearing its denominator, and let \varphi be the resulting actual morphism of p-divisible groups. Its restriction is actually zero. To see this, Hom groups of p-divisible groups are p-torsion-free: if p^a f=0, then f\circ[p]^a=0, and [p]^a on the source is an epimorphism of fppf sheaves. A rationally zero restriction is killed by one power of p in its Hom group and hence vanishes.

The compatible coordinate maps at finite levels recover an actual formal homomorphism \Gamma_h^r\to\Gamma_h. In fact, the ambient and target kernel ideals are respectively (T_1^{q^j},\ldots,T_r^{q^j}) and (T^{q^j}), and inverse limits of their coordinate rings are the corresponding power-series rings. The finite-level homomorphism identities pass to these separated inverse limits. The resulting formal morphism is nonzero since \varphi is nonzero. Its restriction to \mathcal H vanishes by Equation (2.4) and \bigcap_j J_j=0. This proves the assertion.□

Proof. Let R=k[[T_1,\ldots,T_r]], let F be the law of \Gamma_h^r, and let I\subset R be the ideal of all series vanishing on S. Evaluation into the field C shows that I is radical. It is nonzero by hypothesis and proper because 0\in S. We show that it defines a formal subgroup.

For P\in I, put Q(T,T')=P(F(T,T')). Fix s\in S and write

Each b_\ell is the evaluation at s of a series over k, so |b_\ell|\leq1. For q=p^h and every t\in S, coordinatewise q^Nth power is multiplication by p^N; hence Q(s,t^{q^N})=0. Truncation at degree m gives

Taking q^Nth roots and applying Lemma 2.1 shows that, for every k-linear \lambda:C\to k, the coefficientwise transform

vanishes on S, and therefore belongs to I(T').

To descend this statement in the first variable, fix m and consider the finite-dimensional k-algebra

Choose a k-basis e_1,\ldots,e_b of D_m. Modulo its defining ideal, write

Every c_a(T) is a finite k-linear combination of the coefficients of (T')^\ell in Q, with |\ell|\leq m. Consequently \lambda(c_a(s)) is exactly the e_a-coordinate of the image of Q_{s,\lambda} in D_m. It is zero. Since algebraic k-linear functionals separate the points of C, we obtain c_a(s)=0 for every s\in S, and thus c_a\in I(T). This argument moves \lambda across a finite linear combination of already evaluated coefficients; it does not move it through an infinite evaluation.

It follows that

Krull intersection [TSPA26, Tag 00IP] in the Noetherian local quotient by I(T)+I(T') gives Q\in I(T)+I(T'). The identity and inversion conditions follow from 0\in S and -_{\Gamma_h^r}S=S. Therefore \Spf(R/I) is a proper reduced closed formal subgroup of \Gamma_h^r.

Lemma 2.4 supplies a nonzero formal homomorphism \varphi:\Gamma_h^r\to\Gamma_h killing this subgroup. Its restrictions to the factors are endomorphisms a_i\in A_h, and the commutative group law gives

They are not all zero. The coordinate series of \varphi belongs to I, so evaluating on S proves the required identity.□

Proof. Let \mathcal F\subseteq\mathcal E_n be the saturation of the image of the section map \OO_{X_C}^r\to\mathcal E_n, and let b=\rank\mathcal F. On a regular Noetherian curve saturation is a subbundle, and 1\leq b\leq r<n. Choose b of the given sections that are independent at the generic point. Their wedge is a nonzero global section of \det\mathcal F, so \deg\mathcal F\geq0. Semistability of \mathcal E_n gives

Its integral degree is therefore zero. The divisor of the wedge section is effective of degree zero, hence empty by Theorem 3.1. The chosen b sections thus trivialize \mathcal F by the determinant criterion.

All the original sections belong to H^0(X_C,\mathcal F). Under this trivialization that space is \Qp^b, so their assumed \Qp-independence gives r\leq b. Consequently b=r, as required.□

The identification and full faithfulness. For part (i), let E^{(s)}\subset\Cp be the unramified extension of \Qp of degree s, and let G^{(s)} be a Lubin–Tate formal group for E^{(s)}, with uniformizer p [LT65, Theorem 1]. Its reduction over k has height s as a p-divisible formal group. Choose a formal group isomorphism from this reduction to \Gamma_s.

Tilting gives an equivalence between the big pro-étale sites over \Cp and over C [LB18, Theorem 2.7]. Thus each S=\operatorname{Spa}(R,R^+) has its specified untilt S^\sharp=\operatorname{Spa}(R^\sharp,R^{\sharp+}) over \Cp. Fargues–Scholze identify the Lubin–Tate universal cover with relative sections by a natural isomorphism [FS21, Proposition II.2.2]

Here X_{S,E^{(s)}} denotes the relative curve with coefficient field E^{(s)}. Their integral formulation gives the same universal-cover points, since the topologically nilpotent elements belong to every allowed integral subring.

We spell out the comparison of the left side with \mathcal H_s(S). Choose p^\flat\in\OO_C with (p^\flat)^\sharp=p. The tilting comparison identifies

with compatible residue embeddings of k. On these quotient rings the chosen special-fiber isomorphism identifies the two formal group laws. It remains to pass between universal-cover points before and after reduction.

The relevant reduction comparison can be seen directly. Suppose an integral formal group law is evaluated in a complete uniform algebra, and reduce its power-bounded ring modulo a fixed small base parameter of norm \delta<1. A nilpotent element of the quotient lifts to a topologically nilpotent element: if a lift a satisfies a^m in the parameter ideal, its spectral norm satisfies \|a\|^m\leq\delta. Given compatible formal points (\bar a_j)_{j\geq0} of the quotient under [p], choose such lifts a_j. For each fixed i, the sequence

converges. Indeed the formal difference between two successive terms is obtained by pushing down j times a difference of norm at most \delta. For a difference of norm \epsilon<1, integrality and the linear coefficient p give the bound

and its iterates tend to zero. In characteristic p the linear term is zero, giving the same conclusion. These estimates use an equivalent complete spectral norm, which is submultiplicative and power-multiplicative; no field hypothesis on the test algebra is needed. Integral formal translation also turns these bounds on formal differences into bounds on ordinary coordinate differences.

For fixed i the limit has norm at most \max\{\|a_i\|,\delta\}<1. The scalar ideal is closed for the spectral norm, so the limits retain the prescribed reductions. The limits therefore form a compatible lift of (\bar a_j) by topologically nilpotent points. Two choices of lifts give the same limits by the identical contraction estimate, and the same estimate proves uniqueness of a compatible lift. The comparison therefore commutes with algebra maps and with the group operations. It also commutes with formal scalar multiplication. This proves the required natural reduction invariance of the universal cover; compare [FS21, § II.2.1] and [LB18, Proposition 2.14, Lemma 2.15, and Remark 2.21].

Applying this comparison on both sides of the displayed tilting isomorphism gives

The last isomorphism sends z to the compatible sequence (z^{1/q_s^j})_{j\geq0} and uses perfectness of R.

We now pass from the coefficient field E^{(s)} to \Qp. Along the unramified coefficient extension \pi_s:X_{S,E^{(s)}}\to X_S, one has

This is the unramified pushforward description of \OO(1/s) [LB18, § 5.1]; see also [FS21, proofs of Theorem II.2.14 and Corollary II.3.3]. The description is relative and natural in S: on the Frobenius cover, pushforward cycles s lines, with the s-fold isocrystal Frobenius having multiplier p^{-1}. It therefore gives the rank-s isocrystal of slope -1/s, and hence the bundle of slope +1/s. Taking sections commutes with this pushforward, proving part (i) on the whole site.

For part (ii), Le Bras identifies Banach–Colmez sheaves with the tilted heart

by the functor R^0\tau_* [LB18, Theorem 1.2, Theorem 7.1]. The inequalities here refer to all Harder–Narasimhan slopes. A nonnegative-slope vector bundle, placed in degree zero, belongs to this heart, and its R^0\tau_* is the relative section sheaf just defined [LB18, Proposition 6.8 and its proof]. The heart is full in the derived category, so Hom between two such bundles is their usual bundle Hom. Moreover, the proof of [LB18, Theorem 7.1] establishes the degree-zero Ext comparison in the category of abelian sheaves on the big site. Consequently it applies to the Hom group in part (ii), without an additional analytic-morphism or scalar-continuity hypothesis. The fixed tilting equivalence transports this statement to \Perf_C.□

Proof. The set S_\nu is a subgroup because the ring addition in A_n and the map \nu respect the formal group laws. If a nonzero k-formal series vanished on it, Proposition 2.5 would give a_1,\ldots,a_r\in A_h, not all zero, such that

The sum in this equation is the target group sum.

By Theorem 3.3, regard the z_i as global sections of \mathcal E_n and \nu as a nonzero bundle map \mathcal E_n\to\mathcal E_h. The operators in A_n and A_h also act by bundle maps. Passing to the generic point gives a unital K_X-algebra homomorphism

The source is central simple over K_X. The homomorphism is therefore injective, and both algebras have dimension n^2 over K_X, so it is an isomorphism. In particular, splitting of \Delta_n over K_X follows from the displayed action; it has not been assumed in constructing that action.

Equation (3.1) is linear in d. First extend it from A_n to A_n[1/p]=\Delta_n. At the generic point it then extends K_X-linearly to all of the displayed matrix algebra. This is the extension of a vanishing linear identity; no assertion that an arbitrary matrix preserves the order A_n is involved.

By Lemma 3.2, the generic vectors z_i are independent. Choose an index i_0 with a_{i_0}\ne0. This operator is invertible in the division algebra \Delta_h, hence acts invertibly on \mathcal E_h. The generic map \nu_{K_X} is nonzero: a nonzero morphism between vector bundles on an integral curve cannot have zero generic fiber. Choose w\in(\mathcal E_n)_{K_X} with \nu_{K_X}(w)\ne0, and choose a matrix M satisfying

Such a matrix exists by independence. Substituting it into the extended identity gives a_{i_0}\nu_{K_X}(w)=0, a contradiction.□

Proof. The preparation used here follows from a short division argument. Let (A,\mathfrak a) be a complete Noetherian local ring, let d\geq1, and let f(T)=T^d+e(T) with e\in\mathfrak a A[[T]]. Let \tau discard terms of degree below d and shift the remaining degrees down by d. Then

The operator L raises \mathfrak a-adic order, so (1+L)^{-1}=\sum_{i\geq0}(-L)^i converges on A[[T]]. For any g, take q=(1+L)^{-1}\tau(g); the unique remainder g-qf has degree less than d. Dividing T^d gives T^d=qf+r with q\equiv1\pmod{\mathfrak a} and r\in\mathfrak a A[T] of degree less than d. Thus q is a unit and f=q^{-1}(T^d-r) is prepared. The same division proves that A[[T]]/(f) is free over A on 1,T,\ldots,T^{d-1}.

Let \mathfrak m=(u_1,\ldots,u_{n-1})\subset R_0. Reduction modulo \mathfrak m sends [p^j]_H(T) to T^{Q^j}. Weierstrass preparation therefore writes

with U_j\in R_0[[T]]^\times and W_j a distinguished monic polynomial of degree Q^j. The level algebra is consequently finite free of rank Q^j. It is \mathfrak m-adically complete, and its coordinate T is topologically nilpotent. Substitution of the formal group law in the finite level algebras thus defines genuine group-scheme operations.

Apply preparation once more to [p]_H(T)-V over R_0[[V]]. This shows that substitution V=[p]_H(T) is finite free of rank Q. Reducing by [p^j]_H(V) proves that the multiplication maps between consecutive levels are finite faithfully flat of that rank. The kernel and inclusion identities follow from the identities of the p-series. These are exactly the p-divisible-group conditions.

After evaluation at x, all nonleading coefficients of W_j have norm less than one. At a point of norm at least one its leading term strictly dominates the others, so every root is small. The evaluated U_j is a unit on the open unit disc: its constant coefficient has norm one and its remaining coefficients have norm at most one. The roots of W_j are therefore precisely the small zeros of [p^j]_{H_x}, with the same group operations.

The initial degree of [p^j]_{H_x} is q^j, with nonzero coefficient. Completing the finite algebra at the identity thus gives C[[T]]/(T^{q^j}), with the group law induced by H_x. These kernels recover the formal group H_x: their defining ideals are cofinal with the powers of (T). Its height is h, so the étale height of the full height-n group is n-h.□

Proof. First fix y and choose \delta<1 bounding the norms of its parameters. The coefficients of H_y-\Gamma_n and P_y(T)-T^Q then have norm at most \delta. Integral power series are 1-Lipschitz on the open unit ball. Since P_y(T)=f_y(T^q), power-multiplicativity gives

For m\ge0 put

The discrepancy before the last j compositions is bounded by \delta, so Equation (4.5) gives

Thus c_{j,m} converges, uniformly in z, and its limit satisfies

Shifting the index in the limit proves P_y(\Phi_y(z)_{m+1})=\Phi_y(z)_m.

Here is an explicit inverse. If (t_m)_{m\ge0} is a compatible sequence in the inverse limit, define

Indeed,

The limit z therefore exists and obeys

No common bound less than one on the norms of all the t_m is needed. Equation (4.6), raised to the Q^m-th power, proves \Psi_y\Phi_y(z)=z. Conversely, Equation (4.7) at m+j gives

Applying Equation (4.5) and using P_y^{\circ j}(t_{m+j})=t_m proves \Phi_y\Psi_y((t_m))=(t_m).

To verify the group law, the coefficients of \Gamma_n lie in k_n, so Q-th powers and their inverses commute with its addition. For a=z^{1/Q^{j+m}} and b=w^{1/Q^{j+m}}, the two inputs \Gamma_n(a,b) and H_y(a,b) differ by at most \delta. After j compositions their difference is at most \delta^{q^j}. Since P_y^{\circ j} is an H_y-homomorphism, passage to the limit gives

This proves additivity, and hence additivity of the inverse as well. The uniform tail estimates and continuity of roots and convergent series prove continuity of the maps. All constructions commute with continuous morphisms of perfectoid C-algebras. They therefore give the asserted natural maps, and commute with \Zp-multiplication by continuity from integer multiplication.

Finally, fix z\in\mathcal H_n(C) and a closed parameter polydisc strictly inside the open polydisc. Choose \delta<1 bounding all its parameter norms. Each finite stage of Equation (4.3) belongs to its Tate algebra: the substituted constant has norm less than one and all coefficients of the law are bounded. The same bound \delta^{q^j} holds in the parameter Gauss norm. The stages consequently converge in that complete Tate algebra. This proves analyticity. Partial differentiation is continuous on a fixed closed polydisc, with operator norm bounded by the inverse coordinate radius. The convergence therefore also permits first differentiation.□

Proof. Under Equation (4.4), the kernel consists of sequences with t_0=0 and [p]t_{j+1}=t_j. These are precisely the compatible p^j-torsion points of \mathcal G_x. By Lemma 4.1 every such point is small, and the étale height is r. Over the algebraically closed field C the resulting Tate module is \Zp^r.

For surjectivity, apply Weierstrass preparation to [p]_H(T)-V over R_0[[V]]. After setting y=x and V=w\in\mm_C, its distinguished polynomial is monic with all lower coefficients small. Algebraic closedness gives a root, and every root is small. Thus every small point admits division by p within the open unit disc. Repeated choices give a compatible sequence with any prescribed zeroth point, and Proposition 4.2 proves surjectivity of \Theta_x. Finally, multiplying a putative rational relation among a \Zp-basis by a sufficiently large power of p proves the last assertion.□

Proof. We use the square-zero deformation theorem of [Lau10, Theorem 5.1], in the cotangent-theoretic tradition of Grothendieck and Illusie [Ill85, Theorem 4.4 and Corollary 4.7], fixing its torsor structure throughout. If B\to A has square-zero kernel J and p is nilpotent, marked lifts of a p-divisible group G/A form a torsor under

The associated Kodaira–Spencer map is characterized by the arbitrary-ring-lift equivariance of [Lau10, Equation (5.1)]. We use this same construction at the closed point and at every later specialization. The theorem applies in characteristic p at every prime, including p=2.

Apply this comparison to the two maps into

given by u_i\mapsto u_i and u_i\mapsto u_i+\epsilon_i. They define the R_0-linear map \mathsf K. The source is the full module of parameter derivations: every series in R_0 has a finite expression

so ordinary differentials over \Fp are free on du_1,\ldots,du_{n-1}. The target is free of the same rank because \Lie\mathcal G has rank one and \omega_{\mathcal G^\vee} has rank n-1.

At the closed point, it is enough to show injectivity of the tangent map from formal-group deformations to p-divisible-group deformations: both tangent spaces have dimension n-1, by [LT66, Proposition 2.6 and Theorem 3.1] and the chosen square-zero torsor theorem. Suppose a parameter tangent vector has zero image. Then its p-divisible deformation is marked-isomorphic to the constant one, so its finite kernels have compatible marked isomorphisms.

Those kernels recover the formal-group isomorphism. More explicitly, for a deformation F over a local Artin characteristic-p algebra B with maximal ideal J and J^a=0, one has

The first inclusion follows by iterated Frobenius factorization. For the second, preparation gives a monic polynomial of degree Q^j whose lower coefficients lie in J; in its quotient T^{Q^j} lies in J, so its ath power vanishes. The kernel ideals are therefore cofinal with the coordinate-adic topology. Taking inverse limits of the compatible finite Hopf-algebra isomorphisms yields a marked formal-group isomorphism. This is the formal reconstruction underlying Tate’s equivalence [Tat67, Section 2.2, Proposition 1 and Lemma 0]. Lubin–Tate tangent injectivity forces the original vector to be zero. Thus \mathsf K is invertible modulo the maximal ideal of R_0. Its determinant is a unit, so it is an isomorphism over R_0.

For a vector v=(v_i)\in C^{n-1}, the analytic first-order substitution is the actual ring homomorphism

All evaluations converge because the coordinates of x are small. The arbitrary-map functoriality in the cited deformation theorem identifies its deformation class with the specialization of \mathsf K. This proves the final assertion. In cotangent notation the specialized map involves \Omega^1_{R_0/\Fp}\otimes_{R_0}C, not the intrinsic module \Omega^1_{C/\Fp}, which is zero because C is perfect.□

Proof. Factorization through T^{p^h} is preserved by a coordinate change: the p^h-th power of any series has only exponents divisible by p^h. Choose an isomorphism of the special fiber with \Gamma_h and apply the marked Lubin–Tate classification over A. It gives parameters a_1,\ldots,a_{h-1} in the maximal ideal of A. Successively, modulo preceding parameters the coefficient of T^{p^i} in the p-series is a_i times a unit. This is the height-parameter property recalled above; it also follows directly from the leading law term a_i C_{p^i} in [LT66, Proposition 1.1]. Here C_{p^i} is the reduction of the integral polynomial ((X+Y)^{p^i}-X^{p^i}-Y^{p^i})/p; its p-fold sum has coefficient p^{p^i-1}-1, a unit modulo p. The assumed factorization therefore forces every a_i to vanish. The zero parameter tuple is the constant deformation, and the isomorphism supplied by [LT66, Theorem 3.1] respects the marking. For h=1 there are no parameters and the same conclusion holds.□

Proof. Analyticity follows from Proposition 4.2, and \theta(x)=0. Let v be a direction in the kernel of its differential. Write A=C[\epsilon]/\epsilon^2 and y' for the corresponding first-order parameter point. Its preceding parameters remain zero. By Proposition 4.4, it suffices to prove that \mathcal G_{y'} is the constant deformation of \mathcal G_x with its residue marking. We will use the vanishing differential to lift the Tate basis as compatible torsion sections, splitting the étale quotient. The fixed-height condition will then trivialize the connected factor.

Lifting the Tate basis. For j\ge1 set

where evaluation at y' means first-order Taylor evaluation.

We verify membership in the finite group schemes before passing to a p-divisible group. Evaluate Equation (4.2) at the analytic function t_{i,j}(y). Its value at x is small, so on a neighborhood of x this gives an identity of convergent analytic functions

The first factor has nonzero reduced value at x. Taylor evaluation in A makes the right side zero, by the choice of v, and the first factor remains a unit. Hence W_{j,y'}(t'_{i,j})=0: the Taylor coordinate is an actual point of \mathcal G_{y'}[p^j]. The analytic multiplication identities also give

These identities agree with the algebraic finite-level operations. For any finite collection of Taylor points a+\epsilon b, equip A with the complete submultiplicative norm

choosing s>0 so that the points and parameter values are small. The universal congruences defining the polynomial finite-level operations then converge; their terms divisible by the appropriate source-level polynomials vanish, so their evaluations agree with the Taylor-evaluated formal operations.

Each finite identity is evaluated on a neighborhood where its finitely many series converge. Those neighborhoods may depend on j; all their Taylor values lie in the same algebraic ring A. In particular no bound on the derivatives uniform in j is required.

Sending a basis vector of (\mathbb Z/p^j)^r to t'_{i,j} gives a homomorphism of finite group schemes. These maps respect multiplication by p and also the level inclusions: the source inclusion sends the basis vector at level j to p times that at level j+1, exactly as in the displayed identity. Over the strictly henselian local Artin ring A, the connected–étale sequence exists and its étale quotient is the unique constant lift of the residue quotient. The constructed maps induce an isomorphism onto that quotient at each level, since they do so over C. They therefore split it and give

Trivializing the connected factor. At level j, first base-change the finite algebra, obtaining B_j=A[T]/(W_{j,y'}(T)). The idempotents of its residue algebra lift uniquely across \epsilon, and select the identity factor B_j^0. This factor is a finite local Artin algebra, so it equals its completion at (\epsilon,T). Equation (4.2) identifies that completion with

Indeed the preparation identity holds coefficientwise over A, and all residue factors supported away from T=0 become units in this completion. The group law is H_{y'}, by Taylor evaluation of the same preparation and group-law identities. Since the preceding parameters are zero and b(x)\ne0, the series [p^j]_{H_{y'}} is T^{q^j} times a unit. Thus the displayed algebra is also A[[T]]/(T^{q^j}). This describes the connected factor by taking the identity factor after algebraic base change.

Its formal group H_{y'} satisfies the hypothesis of Lemma 4.5, so it is the constant deformation of H_x with the residue marking fixed. That formal isomorphism restricts to all its finite kernels: the coordinates there are nilpotent, so every formal series evaluates by a finite sum. The connected p-divisible factor is consequently constant with its marking.

Recovering the marked deformation. The splitting of the étale factor also has the required marking. Over algebraically closed C, each connected finite group scheme has only its identity as a C-point. A map from a constant finite group scheme is determined by the images of its C-points. The actual torsion points given by the chosen Tate basis therefore select the unique section of the étale quotient on the special fiber. The constructed splitting reduces to this section. Combining it with the marked connected trivialization gives the constant deformation of the entire \mathcal G_x, with the original marking.

Proposition 4.4 now implies v=0. Both the parameter tangent space and the target of D\theta_x have dimension r=n-h, so this differential is invertible.□

Proof. Both sides of the identity contain only powers of T divisible by q. Their coefficients at T^{qj} give exactly Equation (4.9), since the contribution of a_j on the left is b^ja_j. There are no other coefficient equations.

For the integrality assertion, argue inductively on j. The right side of Equation (4.9) is integral if the preceding coefficients are integral. A root with |a_j|>1 would make |a_j|^q strictly larger than both |b^ja_j| and the norm of the right side, which is impossible. This includes j=1, whose right side is zero. The polynomial is monic, so roots exist in the algebraically closed field C, and the same argument bounds every possible choice.□

Proof. The formal isomorphism exists by the height classification over the algebraically closed field C. It intertwines [p]_{H_x}(T)=f_x(T^q) with [p]_{\Gamma_h}(T)=T^q. Lemma 4.7 therefore shows that every coefficient of \alpha_x lies in \OO_C.

An integral series with zero constant term converges on R^{\circ\circ} for every affinoid perfectoid C-algebra R, with value again in R^{\circ\circ}. Its formal homomorphism identity converges on those arguments as well. Combined with Proposition 4.2, this proves additivity and naturality of \nu_x on all such affinoids, hence as a sheaf morphism. Continuity extends integer linearity to \Zp-linearity. Multiplication by p is invertible on both Honda sheaves, so the morphism is \Qp-linear. If scalars are expressed as the sheaf of continuous \Qp-valued functions, the same equality follows at geometric valued-field points by naturality; these detect equality of coordinates in a uniform affinoid algebra. Equivalently, Theorem 3.3 permits working throughout with additive sheaf morphisms.

Write a_1=\alpha_x'(0)\ne0 and choose t\in C with 0<|t|<|a_1|. For j\ge2 the integral coefficients satisfy

so \alpha_x(t)\ne0. Lemma 4.3 supplies z\in\mathcal H_n(C) with \Theta_x(z)=t. Thus \nu_x(z)\ne0, proving nonzeroness of the sheaf morphism. Only the forward series has been evaluated on the whole open unit disc; no convergence assertion about its inverse is needed.□

Proof. Let B=E_0/\mathfrak m_E^M, let J be its maximal ideal, and write F for the universal formal group over B. For d\in A_n, lift its coefficients in k_n to B, obtaining a series \widetilde d(T) with zero constant term. Its homomorphism defect

has all coefficients in J, because d is a homomorphism on the special fiber. For every a\geq1, substitution in [p]_F(Z)=pZ+\sum_{j\geq2}c_jZ^j sends a series with coefficients in J^a and zero constant term to one with coefficients in J^{a+1}. This uses p\in J and 2a\geq a+1. Thus [p]_F^N(\delta)=0 for N\geq M.

Since [p]_F^N is itself a homomorphism, this identity says exactly that f_N=[p]_F^N\circ\widetilde d is an endomorphism of F. Its special fiber is p^Nd, and \operatorname{id}_F-_F f_N has linear coefficient 1-p^N\widetilde d'(0), a unit of B. It is therefore an automorphism lifting 1-p^Nd. The marked Lubin–Tate classification now identifies the deformation with its translate by 1-p^Nd, so that this element fixes the parameter map E_0\to B. The assertion is uniform in d; replacing d by -d gives the stated congruence subgroup. These subgroups form a neighborhood basis of the identity in A_n^\times, proving continuity.□

Proof. The coefficients of g lie in k_n, so g(z^{1/Q^j})=(gz)^{1/Q^j}. Choose \delta<1 bounding the evaluated parameter ideals. The coefficients of \iota_g-g are then bounded by \delta. On small arguments the two series differ by at most that amount. By Equation (4.5), their images after j compositions with P_{y_g} differ by at most \delta^{q^j}. Using P_{y_g}^{\circ j}\iota_g=\iota_g P_x^{\circ j} and passing to the limit proves Equation (4.11). It gives gT_x\subseteq T_{y_g}; the inverse isomorphism gives equality.□

Proof. The formal definition of U is valid because \theta_i\in(X_1,\ldots,X_r), so only finitely many summands contribute to each total degree. Formal inversion applies because L is invertible, and gives the asserted unique expansion of P.

For fixed m\geq1, the finite sum

is an actual analytic germ. Its Taylor polynomial through total degree m is J_mU_i, the corresponding truncation of U_i. Analyticity and \theta(0)=0 give \|\theta(\delta)\|\leq c\|\delta\| near 0. For sufficiently large N, coefficient matching and integrality give

Here \|\theta(\delta_N)\|<1, which also proves convergence of the numerical series defining U_N. The Taylor estimate for the finitely many analytic functions F_{m,i} therefore yields

Explicitly, an analytic series H=\sum_{|\beta|\geq m+1}h_\beta X^\beta converging on a closed polydisc of radius \rho satisfies |H(\delta)|\leq M\rho^{-(m+1)}\|\delta\|^{m+1} for \|\delta\|\leq\rho, where M=\sup_\beta |h_\beta|\rho^{|\beta|}<\infty.

For m=1, write U_N=L\delta_N+R_{1,N}. If \delta_N\neq0 and N is sufficiently large, then

The ultrametric inequality gives \|U_N\|=\|L\delta_N\|, and hence \|\delta_N\|\leq\|L^{-1}\|\|U_N\|. For \delta_N=0 the same bound holds directly. For sufficiently large N, U_N=0 therefore implies \delta_N=0.

Put Q_m(Z)=\sum_{|\ell|\leq m}B_\ell Z^\ell and V_m=J_mU. The analytic germ P-Q_m(V_m) has zero Taylor coefficients through degree m: this follows from the formal identity for P and V_m\equiv U\pmod{(X)^{m+1}}. Consequently,

Both U_N and V_m(\delta_N) tend to zero. On the unit polydisc the finite polynomial Q_m is Lipschitz: telescoping each monomial gives the constant \max(1,\max_{1\leq|\ell|\leq m}|B_\ell|). Equation (5.2), the preceding bound, and P(\delta_N)=0 now imply

The inverse norm estimate proves Equation (5.1) for m\geq1. For m=0, continuity gives B_0=P(0)=0, so the estimate also holds. Every use of the formal identity in this argument has taken place after a finite truncation.□

Proof of Theorem 5.1. Put r=n-h and q=p^h. Choose a \Zp-basis z_1,\ldots,z_r of T_x as in Lemma 4.3. Proposition 4.6 says that

is analytic with invertible differential at x. Fix the formal group isomorphism

of Proposition 4.8. Its forward coefficients belong to \OO_C, its linear coefficient is nonzero, and \nu_x=\alpha_x\Theta_x is a nonzero natural map \mathcal H_n\to\mathcal H_h.

Formal coordinates. Write [p]_{H_y}(T)=f_y(T^q) and b(y)=f'_y(0), so that b(x)\neq0. By Lemma 4.7, a series \alpha(T)=\sum_{j\geq1}A_jT^j intertwines this multiplication by p with qth powers exactly when

Inductively choose an analytic germ a_j(y) satisfying this equation and a_j(x)=a_j^x. The derivative in the new variable is -b(x)^j\neq0, so the analytic implicit function theorem applies. One can see the analytic branch directly: with the earlier coefficients fixed and A_j=a_j^x+t, the equation becomes

On a sufficiently small closed parameter polydisc, choose \eta>0 with \|b^{-j}\|\eta^{q-1}<1 and \|b^{-j}e\|\leq\eta. The right side is a strict contraction on the ball of analytic functions of norm at most \eta; its fixed point has value zero at x. This proves the required germ at every finite stage.

Set \delta=y-x, where this subtraction is ordinary parameter subtraction. Define

Each coefficient of \delta uses only finitely many a_j. The linear matrix is a_1^x d\theta_x, which is invertible. Thus, if P were nonzero, there would be a fixed family (B_\ell)_{\ell\in\mathbb N^r} in C, not all zero, such that

The germs a_j, the formal coordinates U, and the coefficients B_\ell are now fixed. No common analytic domain for all a_j has been chosen.

Matching actual coefficients on an orbit. Fix d\in A_n. For all sufficiently large N, the element g_N=1-p^Nd is a unit in G, and its orbit point y_N=y_{g_N} tends to x by the continuity discussed before Lemma 4.10. The subtraction in g_N is in the endomorphism ring. Transport sends g_Nz_i into T_{y_N}; applying the group homomorphism \Theta_{y_N} therefore gives

This identity is obtained using formal-group addition.

We next choose a numerical series at each y_N. For every M\geq1 choose an integer N_M\geq M, strictly increasing with M, so that for every N\geq N_M the following conditions hold: the first M germs a_j are defined at y_N, their instances of Equation (5.3) hold there, and b(y_N) and a_1(y_N) are nonzero. Such thresholds exist because y_N\to x. For N\geq N_1 put

Then K(N)\to\infty. Prescribe a_{N,j}=a_j(y_N) for 1\leq j\leq K(N), and extend the sequence inductively by choosing roots in C of Equation (5.3) at y_N. Lemma 4.7 shows that every chosen coefficient, including the prescribed ones, belongs to \OO_C. The resulting actual series

converges on \mm_C and satisfies the full identity

This intertwining identity is the only property of \alpha_N needed below. In particular, for every fixed j its coefficient satisfies a_{N,j}=a_j(y_N) eventually and hence a_{N,j}\to a_j^x.

Define convergent numerical quantities

Iterating the intertwining identity in Equation (5.4) gives the exact coordinate identity

All evaluations are on topologically nilpotent elements; the integral series involved preserve this domain.

Continuity in the parameter gives \Theta_{y_N}(dz_i)\to\Theta_x(dz_i). Choose c<1 bounding the norms of these finitely many arguments for all sufficiently large N and of their limits. Integrality bounds the tails of \alpha_N past degree J by c^{J+1}, uniformly in N; the same bound holds for \alpha_x. The first J coefficients converge term by term. Letting first N\to\infty and then J\to\infty proves

This argument also covers a zero coordinate, or a zero limiting tuple.

Frobenius roots and one formal equation. Apply Lemma 5.2 with \delta_N=y_N-x, the germs a_j(x+\delta), and the actual coefficients a_{N,j}. The vanishing hypothesis gives P(y_N)=0 eventually. Thus, for every fixed m\geq0,

eventually. The constant and threshold may depend on the fixed d as well as on m. Since C is perfect of characteristic p, Equation (5.5) permits taking the unique q^Nth root of this finite sum, giving

For this fixed m, the finitely many numbers B_\ell^{1/q^N} are uniformly bounded. By Equation (5.6), replacing s_N by s(d) in the finite sum changes its value by a quantity tending to zero. As D_m^{1/q^N}\to1, the ultrametric inequality gives the full-sequence estimate

When s(d)=0, the right side of the preceding numerical bound and the replacement error both tend to zero, so the same conclusion holds.

For each d\in A_n, Equation (5.7) holds for every m\geq0. Lemma 2.1 therefore gives

for every k-linear functional \lambda:C\to k and every d\in A_n. Choose one such functional nonzero on a fixed nonzero coefficient B_{\ell_0}. Since the family B_\ell was fixed before d, this produces a single nonzero formal series over k vanishing on the whole set

The z_i are \Qp-linearly independent, r=n-h<n, and \nu_x is a nonzero natural map. Proposition 3.4 excludes precisely this formal equation. The contradiction proves the theorem.□

Proof. We first construct a one-dimensional domain quotient on which a specified nonzero nonunit a survives. We use the Noetherian local dimension and parameter theorems in [TSPA26, Tags 00KQ and 00KD]. Put d=\dim R, and extend a to a system of parameters a,b_2,\ldots,b_d. Let J=(b_2,\ldots,b_d). Some minimal prime \mathfrak q over J avoids a. Indeed, otherwise a\in\sqrt J, and the parameter property would give \sqrt J=\mm_R, contradicting the height bound for an ideal generated by d-1 elements.

The ring D=R/\mathfrak q is a complete local domain in which a is a nonzero nonunit and \sqrt{aD}=\mm_D. The principal ideal theorem gives \dim D\le1, and the existence of that nonzero nonunit gives equality. The a-adic and maximal-ideal topologies on D coincide. This construction includes d=1, when J=0 and we may take \mathfrak q=0.

In the first case, substitution defines an injective map k_n[[t]]\to D, t\mapsto a. To check injectivity, write a nonzero series as t^m v(t), where v(0)\ne0: its image is a nonzero power of a times a unit. Moreover, D/aD is finite-dimensional over k_n, so completeness makes D a finite k_n[[a]]-module. Explicitly, lift a k_n-basis of D/aD; successively subtract linear combinations of these lifts modulo a,a^2,\ldots. The coefficient series converge in k_n[[a]] and express every element of D in their finite span.

For the prescribed c, evaluation embeds k_n[[t]] in \OO_C: the first nonzero term of a series is uniquely dominant when 0<|c|<1. Its fraction field is a complete discretely valued subfield of C, with uniformizer c. The finite extension \Frac(D)/k_n((a)) embeds into the algebraically closed field C over this embedding of the base. Every element of D is integral over k_n[[a]], so its image has norm at most one. For z\in\mm_D, some power z^N belongs to aD, whence

Thus the composite R\to D\to\OO_C has all the required properties.

For the second case apply the same construction with a=p. The resulting D is finite over W(k_n), by the identical successive approximation argument. The structural map from this DVR is injective because p\ne0 in the domain D. Embed the finite extension \Frac(D)/\Frac(W(k_n)) into \Cp over the fixed base embedding. Integrality again gives boundedness, and \sqrt{pD}=\mm_D gives strict smallness of \mm_D.

Finally, continuity follows in both cases from finite generation of \mm_R: its generators have images of norm at most some \rho<1, so \mm_R^N maps to elements of norm at most \rho^N.□

Proof. The cases r=0 or x=0 are immediate. Otherwise put \rho=\|x\|, so 0<\rho<1, and M=\sup_I|a_I|. The coefficient of T^J in the translated series is

Here inequalities between multi-indices are coordinatewise. Each term has norm at most M\rho^{|I|-|J|}, so the sum converges and the new coefficients are bounded by M.

For completeness, all reorderings needed for substitution can be checked coefficientwise. The coefficient of T^K in \tau_{-x}\tau_x f is a sum over I\ge J\ge K. Its individual terms have norm at most M\rho^{|I|-|K|}, and only finitely many pairs occur below any fixed total degree. The family is therefore summable independently of order. Regrouping gives

The inner sum is zero unless I=K. Replacing x by -x proves the other inverse identity.

Multiplicativity has the same convergence justification. If g=\sum_Q d_QT^Q, the coefficient of T^H in \tau_x f\,\tau_x g is a sum over J+K=H, P\ge J, and Q\ge K. The terms are bounded by

so they may be regrouped. The multi-index Vandermonde identity

then identifies the coefficient with that of \tau_x(fg). Both inverse maps are contractions for the coefficient supremum norm, so they are isometries.

A bounded-coefficient series converges on every closed polydisc of radius less than one. Its expansion at x is the series \tau_xf, by the same convergent binomial expansion. A zero analytic germ would have all Taylor coefficients zero and would therefore give \tau_xf=0, hence f=0.□

Proof. At n=1 the polydisc is a point, so the assertion is immediate. Suppose n>1. We apply the period theorem after completed base change from W(k_n) to W(k) and the compatible embedding W(k)\to\OO_{\Cp}. The marked universal deformation and its A_n^\times-action commute with this base change, and the resulting \Cp generic fiber is \mathfrak X_{\Cp}. The Gross–Hopkins period theorem [HG94b, Section 1.1, Theorem 1], with its proof in [HG94a, Proposition 23.5], gives an equivariant étale map

The action on the target is the projectivization of the split division-algebra representation

Write \xi=\Phi(x). The analytic inverse function theorem supplies neighborhoods V of x and W of \xi on which \Phi is an analytic isomorphism. For this use and the one below, its local proof is the same contraction argument used in Section 5: after translating and applying the inverse linear part, write the map as z+R(z), with R(0)=0 and dR_0=0, and shrink a closed polydisc until R has Lipschitz constant less than one. For w in that polydisc, iteration of z=w-R(z) contracts in its complete Tate algebra, producing the analytic inverse. Transport the given germ P to the germ Q=P\circ(\Phi|_V)^{-1} at \xi.

The infinitesimal action of the full matrix algebra surjects onto T_\xi\mathbb P^{n-1}_{\Cp}. Since A_n spans \Delta_n over \Qp, choose d_1,\ldots,d_{n-1}\in A_n whose infinitesimal projective directions form a \Cp-basis of this tangent space. For small t=(t_1,\ldots,t_{n-1})\in\Cp^{n-1}, form

using the split representation to interpret the matrix on the right. This is an analytic map with invertible differential at zero. Shrink its domain so that its image lies in W and Q\circ\eta is defined.

For sufficiently large a, every t\in(p^a\Zp)^{n-1} defines an element g_t=1+\sum_i t_i d_i\in U. Continuity of the stabilizer action allows us also to require g_t x\in V, within the neighborhood on which P vanishes along the orbit. Equivariance and the fixed inverse branch give

An analytic function on a \Cp-polydisc that vanishes on (p^a\Zp)^{n-1} is zero: fix all but one variable in that set, apply one-variable uniqueness using a sequence tending to zero, and repeat for each variable. Indeed a nonzero one-variable germ has the form t^r u(t) with u(0)\ne0, so its zeros cannot accumulate at zero. Thus Q\circ\eta is the zero germ. The invertible differential of \eta implies that Q, and therefore P, is the zero germ.□

Proof. Let \mathfrak p be such a prime, stable under the open subgroup U.

Suppose first that p\in\mathfrak p, and choose the largest h such that I_h\subseteq\mathfrak p. If h=n, the ideal I_n is maximal, so \mathfrak p=I_n. Otherwise the image of u_h in R=E_0/\mathfrak p is a nonzero nonunit. Lemma 6.1 gives a point x:R\to\OO_C at which all parameters are small and u_h(x)\ne0. It is therefore a point of the height-at-least-h polydisc to which Theorem 5.1 applies.

If \mathfrak p/I_h\ne0, choose a nonzero series

Its coefficients are bounded, and Lemma 6.2 shows that it defines a nonzero ambient analytic germ at x. Stability of \mathfrak p makes f vanish at every point of the U-orbit of x. This contradicts Theorem 5.1. Consequently \mathfrak p=I_h.

Now suppose p\notin\mathfrak p. The mixed-characteristic part of Lemma 6.1 supplies a point x:E_0/\mathfrak p\to\OO_{\Cp} with all parameters small. If \mathfrak p\ne0, take 0\ne f\in\mathfrak p. The fixed embedding W(k_n)\hookrightarrow\OO_{\Cp} regards f as a nonzero bounded-coefficient series in the ambient generic polydisc. By Lemma 6.2, its germ at x is nonzero. It vanishes on the U-orbit, contradicting Proposition 6.3. Thus \mathfrak p=0.□

Proof. Let J be a proper U-invariant radical ideal. Noetherianity gives finitely many minimal primes, and the group permutes them. The stabilizer of each such prime P is closed. Indeed, ideals in the complete Noetherian local ring E_0 are adically closed, and each of the conditions

is closed by continuity of the action. Together these conditions say gP=P. This closed stabilizer has finite index in U and is therefore open in U: its complement is a finite union of closed cosets. Since U is open in A_n^\times, the stabilizer is open in A_n^\times as well. Theorem 6.4 applies to every minimal prime of J. Their intersection is J; since the standard primes form a chain, this intersection is its smallest member. Hence J is standard.

An ideal invariant under G_n is invariant under A_n^\times. Conversely, every ideal in the displayed list is invariant under the full canonical action. For I_h, this follows from its intrinsic description as the locus where the universal formal group has height at least h, or equivalently from the successive height parameters of its p-series. The zero and unit ideals are invariant as well.□

Proof. Here we assemble the established results and spell out the direction of support descent. Base change is symmetric monoidal into the category of K(n)-local E-modules. Its dualizable objects are exactly the ordinary perfect E-modules by [BHN22, Proposition 4.7]. Thus \mathcal F has the asserted target even when X is dualizable but not compact as a K(n)-local spectrum. Tensor-nilpotence detection is [BHN22, Proposition 4.6]; neither statement has a prime bound. These inputs use Morava-E descent and the perfect-module comparison of Mathew [Mat16, Propositions 3.27 and 10.10–10.11].

The coefficient ring E_* is even-periodic, regular, and Noetherian. The comparison theorem of Dell’Ambrogio–Stanley [DS16, Theorem 1.1 and Lemma 3.10], applied to Morava E-theory in [BHN22, Lemmas 4.3–4.4], identifies

and identifies the tensor-triangular support of \mathcal F(X) with S(X). The tensor-product identity for that support gives S(X\otimes Y)=S(X)\cap S(Y). The module M_X is finite over E_0, so this support is closed. Its canonical semilinear stabilizer action makes it invariant; equivalently one may use [BHN22, Lemma 4.5].

Tensor-nilpotence detection gives a surjection

by Balmer’s surjectivity theorem [Bal18, Theorem 1.3], in the form [BHN22, Theorem 2.6 and Corollary 4.8]. Writing \operatorname{supp} for tensor-triangular support, functoriality gives

If S(X)\subseteq S(Y), surjectivity of f therefore implies \operatorname{supp}(X)\subseteq\operatorname{supp}(Y). Every object of \mathcal D is dualizable, so its thick tensor ideals are radical. Balmer’s classification [Bal05, Theorem 4.10], in the support-containment form of [BHN22, Corollary 2.4], gives X\in\langle Y\rangle_\otimes. The converse follows from exactness, tensor compatibility, and the same support criterion. This is also the transfer formulated in [BHN22, Corollary 2.7].

For the finite type-k objects the coefficient support is V(I_k), as in [BHN22, Proposition 3.3 and the proof of Theorem 4.9]. This is the usual finite chromatic support calculation; it is unrestricted in p. The two remaining support calculations follow from \mathcal F(0)=0 and \mathcal F(L_{K(n)}S)=E.□

Proof of Theorem 1.2. By Corollary 6.5 and Proposition 6.6, the support of every object of \mathcal D belongs to the finite chain

Indeed, S(X)=V(\Ann_{E_0}M_X), and its defining radical ideal is G_n-invariant. Empty support occurs only for the zero object, by applying the support comparison with Y=0.

Let \mathcal J\ne0 be any thick tensor ideal. The supports occurring among its objects form a nonempty subset of this finite chain, and so have a greatest member that is attained by some Y\in\mathcal J. For every X\in\mathcal J, support comparison gives X\in\langle Y\rangle_\otimes. The reverse containment follows from Y\in\mathcal J, hence

There is no assumption that \mathcal J was singly generated: attainment follows solely from finiteness of the possible supports.

Write S(Y)=V(I_k). The given object L_{K(n)}F(k) has the same support, so applying Proposition 6.6 in both directions gives

The zero ideal is \mathcal D_{n+1}.

The same comparison proves independence of the generator: any other finite type-k spectrum has support V(I_k) and therefore generates the same ideal. This also agrees with the finite thick-subcategory theorem of Hopkins–Smith [HS98]. The strict inclusions of the displayed supports imply \mathcal D_k\supsetneq\mathcal D_{k+1}, since L_{K(n)}F(k) belongs to the first ideal and its support is not contained in that of a generator of the second. For k=n, its support V(I_n) is nonempty, so \mathcal D_n\ne0. Finally V(I_0)=\Spec(E_0) is the support of the tensor unit. Thus \mathcal D_0 contains the unit and equals \mathcal D. The list has exactly n+2 distinct entries, as asserted.□

Proof. The possible coefficient supports form a total chain, including the empty set. Hence S(X)\cap S(Y) is one of S(X) and S(Y). By the tensor-product and containment statements in Proposition 6.6, if X\otimes Y belongs to a proper \mathcal D_j, then one of X and Y does as well. This also includes \mathcal D_{n+1}=0: two nonempty supports both contain V(I_n), so their intersection is nonempty. Every proper member of the chain is therefore prime, and Theorem 1.2 leaves no other prime ideals.

In the Balmer topology, a point \mathcal Q lies in \overline{\{\mathcal P\}} precisely when \mathcal Q\subseteq\mathcal P. Indeed every support containing \mathcal P must contain \mathcal Q, which says that X\notin\mathcal P implies X\notin\mathcal Q for every object X. The strict ideal chain gives the stated closure formula. Finally L_{K(n)}F(k)\in\mathcal D_j exactly when k\geq j, by the coefficient-support comparison, proving the support formula.□