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 \[b_\ell=\sum_{j=1}^a c_{\ell j}\beta_j, \qquad c_{\ell j}\in k\quad (|\ell|\leq m).\] There is a subsequence of positive integers \(N\) on which \(q^N\)-power Frobenius fixes all the \(c_{\ell j}\): finitely many elements of \(\overline{\mathbb F_p}\) lie in a common finite field. On this subsequence, \[\sum_{|\ell|\leq m}b_\ell^{1/q^N}s^\ell =\sum_{j=1}^a\beta_j^{1/q^N}v_j, \qquad v_j=\sum_{|\ell|\leq m}c_{\ell j}s^\ell.\] 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 \(w_j\in\mathcal O_C\). At least one \(\rho_j\) is nonzero, and \[\eta:=\max_j|w_j-\rho_j|<1.\] 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 \[\left|\sum_j\beta_j^{1/q^N}\rho_j\right| =\left|\sum_j\beta_j\rho_j\right|^{1/q^N} \longrightarrow1.\] On the other hand, \[\limsup_N \left|\sum_j\beta_j^{1/q^N}(w_j-\rho_j)\right| \leq\eta<1,\] since \(|\beta_j|^{1/q^N}\to1\) for every \(j\). The ultrametric inequality therefore implies \[\left|\sum_j\beta_j^{1/q^N}v_j\right|\longrightarrow V\] along this subsequence. Equation (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 \[\left|\sum_{|\ell|\leq m}\lambda(b_\ell)s^\ell\right| =\left|\sum_j\lambda(\beta_j)v_j\right| \leq V\leq\|s\|^{m+1}.\] 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 (1) for \(m=0\) forces \(b_0=0\), giving the same conclusion. ◻

Proof. Every \(f\in R\) has a unique finite decomposition \[f=\sum_{0\leq a_i<p}T_1^{a_1}\cdots T_r^{a_r}f_a(T)^p.\] An ordinary \(k\)-derivation kills all the \(p\)th 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 \[ \Omega^1_{B/k} \cong B^r\big/ \left\langle \bigl(\partial_1f,\ldots,\partial_rf\bigr):f\in I \right\rangle_B. \tag{2}\] In particular, this ordinary differential module is finite over \(B\).

Consider the completed product ring \[B\widehat\otimes_k B =k[[X_1,\ldots,X_r,Y_1,\ldots,Y_r]]/(I(X),I(Y))\] and its diagonal ideal \(J=(Y_i-X_i)_i\). Writing \(Y=X+Z\) modulo \((Z)^2\) gives \[f(X+Z)=f(X)+\sum_i\partial_i f(X)Z_i\pmod{(Z)^2}.\] Thus \(J/J^2\) has exactly the presentation in Equation (2). This computation makes no reducedness assumption on the completed product. The group-difference automorphism \[(x,y)\longmapsto(x,y-_{\mathcal H}x)\] 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 \[ \Omega^1_{B/k}\cong B\otimes_k(\mathfrak m_B/\mathfrak m_B^2)\cong B^e. \tag{3}\]

Let \(\mathfrak p\) be a minimal prime of \(B\) and set \(D=B/\mathfrak p\), \(K=\mathop{\mathrm{Frac}}(D)\), and \(d=\dim D\). Since \(B\) is reduced, \(B_{\mathfrak p}\) is the field \(K\). Localization of ordinary differentials in Equation (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 \(L=\mathop{\mathrm{Frac}}(k[[x_1,\ldots,x_d]])\). 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 \[[K:K^p] =\frac{[K:L]\,[L:L^p]}{[K^p:L^p]}=p^d.\] 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 5, 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 \(q\)th power and fix the variables. For \(P\in I\), stability under multiplication by \(p\) gives \[P(T_1^q,\ldots,T_r^q)=(\sigma^{-1}P(T))^q\in I.\] Radicality implies \(\sigma^{-1}I\subseteq I\). Applying \(\sigma\) repeatedly produces the ascending chain \[I\subseteq\sigma I\subseteq\sigma^2 I\subseteq\cdots.\] 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 \[J_j=(\overline T_1^{q^j},\ldots,\overline T_r^{q^j}),\] then \[ \mathfrak m_B^{r(q^j-1)+1}\subseteq J_j \subseteq\mathfrak m_B^{q^j}. \tag{4}\] 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 \[\varphi_{\mathbb Q_p}:\Gamma_h^r[p^\infty] \longrightarrow\Gamma_h[p^\infty]\] 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 (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 \[Q(s,T')=\sum_\ell b_\ell(T')^\ell.\] 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 \(q^N\)th power is multiplication by \(p^N\); hence \(Q(s,t^{q^N})=0\). Truncation at degree \(m\) gives \[\left|\sum_{|\ell|\leq m}b_\ell t^{q^N\ell}\right| \leq\|t\|^{q^N(m+1)}.\] Taking \(q^N\)th roots and applying Lemma 3 shows that, for every \(k\)-linear \(\lambda:C\to k\), the coefficientwise transform \[Q_{s,\lambda}(T'):=\sum_\ell\lambda(b_\ell)(T')^\ell\] 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 \[D_m=k[[T']]/\bigl(I(T')+(T')^{m+1}\bigr).\] Choose a \(k\)-basis \(e_1,\ldots,e_b\) of \(D_m\). Modulo its defining ideal, write \[Q(T,T')=\sum_{a=1}^b c_a(T)e_a, \qquad c_a(T)\in k[[T]].\] 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 \[Q\in I(T)+I(T')+(T')^{m+1} \qquad\text{for every }m.\] Krull intersection (The Stacks Project Authors 2026, 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 \(\mathop{\mathrm{Spf}}(R/I)\) is a proper reduced closed formal subgroup of \(\Gamma_h^r\).

Lemma 6 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 \[\varphi(T)=a_1(T_1)+_{\Gamma_h}\cdots+_{\Gamma_h}a_r(T_r).\] 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 \(\mathcal O_{X_C}^r\to\mathcal E_n\), and let \(b=\mathop{\mathrm{rank}}\mathcal F\). 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 \[0\leq\deg\mathcal F\leq \frac{b}{n}<1.\] Its integral degree is therefore zero. The divisor of the wedge section is effective of degree zero, hence empty by Theorem 8. 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 \(\mathbb Q_p^b\), so their assumed \(\mathbb Q_p\)-independence gives \(r\leq b\). Consequently \(b=r\), as required. ◻

The identification and full faithfulness. For part (i), let \(E^{(s)}\subset\mathbb C_p\) be the unramified extension of \(\mathbb Q_p\) of degree \(s\), and let \(G^{(s)}\) be a Lubin–Tate formal group for \(E^{(s)}\), with uniformizer \(p\) (Lubin and Tate 1965, 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 \(\mathbb C_p\) and over \(C\) (Le Bras 2018, Theorem 2.7). Thus each \(S=\operatorname{Spa}(R,R^+)\) has its specified untilt \(S^\sharp=\operatorname{Spa}(R^\sharp,R^{\sharp+})\) over \(\mathbb C_p\). Fargues–Scholze identify the Lubin–Tate universal cover with relative sections by a natural isomorphism (Fargues and Scholze 2021, Proposition II.2.2) \[\varprojlim_{[p]} G^{(s)}\bigl((R^\sharp)^{\circ\circ}\bigr) \ \cong\ H^0\bigl(X_{S,E^{(s)}},\mathcal O(1)\bigr).\] 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 \(p^\flat\in\mathcal O_C\) with \((p^\flat)^\sharp=p\). The tilting comparison identifies \[(R^\sharp)^\circ/p \ \cong\ R^\circ/(p^\flat),\] 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 \[[p]^j(a_{i+j}),\qquad j\geq0,\] 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 \[\|[p](w)\| \leq\max\{|p|\epsilon,\epsilon^2\} \qquad (\|w\|\leq\epsilon),\] 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 (Fargues and Scholze 2021, sec. II.2.1) and (Le Bras 2018, Proposition 2.14, Lemma 2.15, and Remark 2.21).

Applying this comparison on both sides of the displayed tilting isomorphism gives \[\varprojlim_{[p]} G^{(s)}\bigl((R^\sharp)^{\circ\circ}\bigr) \ \cong\ \varprojlim_{[p]}\Gamma_s(R^{\circ\circ}) \ \cong\ \Gamma_s(R^{\circ\circ}).\] 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 \(\mathbb Q_p\). Along the unramified coefficient extension \(\pi_s:X_{S,E^{(s)}}\to X_S\), one has \[(\pi_s)_*\mathcal O(1)\cong(\mathcal E_s)_S.\] This is the unramified pushforward description of \(\mathcal O(1/s)\) (Le Bras 2018, sec. 5.1); see also (Fargues and Scholze 2021, 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 \[\operatorname{Coh}_{X_C}^{-} =\left\{F\in D^b(\operatorname{Coh}_{X_C}): H^i(F)=0\ (i\ne-1,0),\quad H^{-1}(F)<0,\quad H^0(F)\geq0\right\}\] by the functor \(R^0\tau_*\) (Le Bras 2018, 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 (Le Bras 2018, 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 (Le Bras 2018, 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 \(\mathop{\mathrm{Perf}}_C\). ◻

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 7 would give \(a_1,\ldots,a_r\in A_h\), not all zero, such that \[ \sum_{i=1}^r a_i\nu d(z_i)=0 \qquad\text{for every }d\in A_n. \tag{5}\] The sum in this equation is the target group sum.

By Theorem 10, 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 \[\Delta_n\otimes_{\mathbb Q_p}K_X \longrightarrow \mathop{\mathrm{End}}_{K_X}\bigl((\mathcal E_n)_{K_X}\bigr).\] 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 (5) 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 9, 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 \[Mz_{i_0}=w,\qquad Mz_i=0\quad(i\ne i_0).\] 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 \[\tau(qf)=(1+L)q,\qquad L(q)=\tau(qe).\] 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 \[ [p^j]_H(T)=U_j(T)W_j(T), \tag{7}\] 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 \[ \|P_y^{\circ j}(a)-P_y^{\circ j}(b)\| \le \|a-b\|^{q^j} \qquad(a,b\in R^{\circ\circ}). \tag{10}\] For \(m\ge0\) put \[c_{j,m}=P_y^{\circ j}\bigl(z^{1/Q^{j+m}}\bigr).\] The discrepancy before the last \(j\) compositions is bounded by \(\delta\), so Equation (10) gives \[\|c_{j+1,m}-c_{j,m}\|\le\delta^{q^j}.\] Thus \(c_{j,m}\) converges, uniformly in \(z\), and its limit satisfies \[ \|\Phi_y(z)_m-z^{1/Q^m}\|\le\delta, \qquad \|\Phi_y(z)_m\| \le\max\{\|z\|^{1/Q^m},\delta\}<1. \tag{11}\] 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 \[\Psi_y((t_m))=\lim_{m\to\infty}t_m^{Q^m}.\] Indeed, \[\|t_{m+1}^{Q^{m+1}}-t_m^{Q^m}\| =\|(t_{m+1}^Q-P_y(t_{m+1}))^{Q^m}\| \le\delta^{Q^m}.\] The limit \(z\) therefore exists and obeys \[ \|z-t_m^{Q^m}\|\le\delta^{Q^m}, \qquad \|z\|\le\max\{\|t_0\|,\delta\}<1. \tag{12}\] No common bound less than one on the norms of all the \(t_m\) is needed. Equation (11), raised to the \(Q^m\)-th power, proves \(\Psi_y\Phi_y(z)=z\). Conversely, Equation (12) at \(m+j\) gives \[\|z^{1/Q^{m+j}}-t_{m+j}\|\le\delta.\] Applying Equation (10) 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 \[\Phi_y(\Gamma_n(z,w))_m =H_y(\Phi_y(z)_m,\Phi_y(w)_m).\] 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 \(\mathbb Z_p\)-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 (8) 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 (9), 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 12 every such point is small, and the étale height is \(r\). Over the algebraically closed field \(C\) the resulting Tate module is \(\mathbb Z_p^r\).

For surjectivity, apply Weierstrass preparation to \([p]_H(T)-V\) over \(R_0[[V]]\). After setting \(y=x\) and \(V=w\in\mathfrak m_C\), 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 13 proves surjectivity of \(\Theta_x\). Finally, multiplying a putative rational relation among a \(\mathbb Z_p\)-basis by a sufficiently large power of \(p\) proves the last assertion. ◻

Proof. We use the square-zero deformation theorem of (Lau 2010, Theorem 5.1), in the cotangent-theoretic tradition of Grothendieck and Illusie (Illusie 1985, 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 \[\mathop{\mathrm{Hom}}_A(\omega_{G^\vee},J\otimes_A\mathop{\mathrm{Lie}}G).\] The associated Kodaira–Spencer map is characterized by the arbitrary-ring-lift equivariance of (Lau 2010, 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 \[R_0[\epsilon_1,\ldots,\epsilon_{n-1}]/ (\epsilon_i\epsilon_j:1\le i,j\le n-1)\] 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 \[a=\sum_{0\le e_i<p}u^e a_e^p,\] so ordinary differentials over \(\mathbb F_p\) are free on \(du_1,\ldots,du_{n-1}\). The target is free of the same rank because \(\mathop{\mathrm{Lie}}\mathcal G\) 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 (Lubin and Tate 1966, 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 \[([p^j]_F(T))\subseteq(T^{p^j}),\qquad T^{aQ^j}\in([p^j]_F(T)).\] 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 \(a\)th 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 (Tate 1967, sec. 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 \[a\longmapsto a(x)+\epsilon\sum_i v_i\partial_i a(x) \quad\text{from }R_0\text{ to }C[\epsilon]/\epsilon^2.\] 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 \(\Omega^1_{R_0/\mathbb F_p}\otimes_{R_0}C\), not the intrinsic module \(\Omega^1_{C/\mathbb F_p}\), 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 (Lubin and Tate 1966, 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 (Lubin and Tate 1966, Theorem 3.1) respects the marking. For \(h=1\) there are no parameters and the same conclusion holds. ◻

Proof. Analyticity follows from Proposition 13, 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 15, 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 \[t_{i,j}(y)=\Theta_y(p^{-j}z_i),\qquad t'_{i,j}=t_{i,j}(y')\in A,\] 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 (7) 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 \[U_{j,y}(t_{i,j}(y))\,W_{j,y}(t_{i,j}(y)) =\Theta_y(z_i).\] 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 \[[p]t'_{i,j+1}=t'_{i,j},\qquad [p^j]t'_{i,j}=0.\] 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 \[\|a+\epsilon b\|_s=\max\{|a|,s|b|\},\] 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 \[\mathcal G_{y'}\cong \mathcal G_{y'}^0\times(\mathbb Q_p/\mathbb Z_p)^r.\]

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 (7) identifies that completion with \[B_j^0\cong A[[T]]/([p^j]_{H_{y'}}(T)).\] 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 16, 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 15 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 (14), 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 (14) 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 18 therefore shows that every coefficient of \(\alpha_x\) lies in \(\mathcal O_C\).

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 13, this proves additivity and naturality of \(\nu_x\) on all such affinoids, hence as a sheaf morphism. Continuity extends integer linearity to \(\mathbb Z_p\)-linearity. Multiplication by \(p\) is invertible on both Honda sheaves, so the morphism is \(\mathbb Q_p\)-linear. If scalars are expressed as the sheaf of continuous \(\mathbb Q_p\)-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 10 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 \[|a_jt^j|\le|t|^2<|a_1t|,\] so \(\alpha_x(t)\ne0\). Lemma 14 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 \[\delta(X,Y)=\widetilde d(F(X,Y)) -_F F(\widetilde d(X),\widetilde d(Y))\] 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 (10), 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 (16). It gives \(gT_x\subseteq T_{y_g}\); the inverse isomorphism gives equality. ◻

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 (The Stacks Project Authors 2026, 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 \(\sqrt J=\mathfrak m_R\), 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 \(\sqrt{aD}=\mathfrak m_D\). 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 \(\mathcal O_C\): 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 \(\mathop{\mathrm{Frac}}(D)/k_n((a))\) 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 \(z\in\mathfrak m_D\), some power \(z^N\) belongs to \(aD\), whence \[|z|^N\le |c|<1.\] Thus the composite \(R\to D\to\mathcal O_C\) 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 \(\mathop{\mathrm{Frac}}(D)/\mathop{\mathrm{Frac}}(W(k_n))\) into \(\mathbb C_p\) over the fixed base embedding. Integrality again gives boundedness, and \(\sqrt{pD}=\mathfrak m_D\) gives strict smallness of \(\mathfrak m_D\).

Finally, continuity follows in both cases from finite generation of \(\mathfrak m_R\): its generators have images of norm at most some \(\rho<1\), so \(\mathfrak m_R^N\) 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 \[(\tau_x f)_J =\sum_{I\ge J}\binom IJ a_I x^{I-J}.\] 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 \[\sum_{I\ge K} a_I x^{I-K}\binom IK \sum_{K\le J\le I} (-1)^{|J-K|}\binom{I-K}{J-K} =a_K.\] 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 \[\|f\|\,\|g\|\,\rho^{|P|+|Q|-|H|},\] so they may be regrouped. The multi-index Vandermonde identity \[\sum_{\substack{J+K=H\\J\le P,\ K\le Q}} \binom PJ\binom QK=\binom{P+Q}{H}\] 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 \(W(k)\to\mathcal O_{\mathbb C_p}\). The marked universal deformation and its \(A_n^\times\)-action commute with this base change, and the resulting \(\mathbb C_p\) generic fiber is \(\mathfrak X_{\mathbb C_p}\). The Gross–Hopkins period theorem (Hopkins and Gross 1994b, sec. 1.1, Theorem 1), with its proof in (Hopkins and Gross 1994a, Proposition 23.5), gives an equivariant étale map \[\Phi:\mathfrak X_{\mathbb C_p}\longrightarrow\mathbb P^{n-1}_{\mathbb C_p}.\] The action on the target is the projectivization of the split division-algebra representation \[\Delta_n\otimes_{\mathbb Q_p}\mathbb C_p \simeq\mathop{\mathrm{End}}_{\mathbb C_p}(\mathbb C_p^n).\] 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 \(T_\xi\mathbb P^{n-1}_{\mathbb C_p}\). Since \(A_n\) spans \(\Delta_n\) over \(\mathbb Q_p\), choose \(d_1,\ldots,d_{n-1}\in A_n\) whose infinitesimal projective directions form a \(\mathbb C_p\)-basis of this tangent space. For small \(t=(t_1,\ldots,t_{n-1})\in\mathbb C_p^{n-1}\), form \[\eta(t)=\left(1+\sum_i t_i d_i\right)\xi,\] 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 \(t\in(p^a\mathbb Z_p)^{n-1}\) 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 \[(\Phi|_V)^{-1}(\eta(t))=g_t x, \qquad (Q\circ\eta)(t)=0.\] An analytic function on a \(\mathbb C_p\)-polydisc that vanishes on \((p^a\mathbb Z_p)^{n-1}\) 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 24 gives a point \(x:R\to\mathcal O_C\) 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 22 applies.

If \(\mathfrak p/I_h\ne0\), choose a nonzero series \[f\in\mathfrak p/I_h \subset k_n[[u_h,\ldots,u_{n-1}]].\] Its coefficients are bounded, and Lemma 25 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 22. Consequently \(\mathfrak p=I_h\).

Now suppose \(p\notin\mathfrak p\). The mixed-characteristic part of Lemma 24 supplies a point \(x:E_0/\mathfrak p\to\mathcal O_{\mathbb C_p}\) with all parameters small. If \(\mathfrak p\ne0\), take \(0\ne f\in\mathfrak p\). The fixed embedding \(W(k_n)\hookrightarrow\mathcal O_{\mathbb C_p}\) regards \(f\) as a nonzero bounded-coefficient series in the ambient generic polydisc. By Lemma 25, its germ at \(x\) is nonzero. It vanishes on the \(U\)-orbit, contradicting Proposition 26. 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 \[g(a)\in P,\qquad g^{-1}(a)\in P \qquad(a\in P)\] 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 27 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 (Barthel et al. 2022, 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 (Barthel et al. 2022, Proposition 4.6); neither statement has a prime bound. These inputs use Morava-\(E\) descent and the perfect-module comparison of Mathew (Mathew 2016, 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 (Dell’Ambrogio and Stanley 2016, Theorem 1.1 and Lemma 3.10), applied to Morava \(E\)-theory in (Barthel et al. 2022, Lemmas 4.3–4.4), identifies \[\mathop{\mathrm{Spc}}(\mathop{\mathrm{Perf}}(E))\simeq\mathop{\mathrm{Spec}}(E_0)\] 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 (Barthel et al. 2022, Lemma 4.5).

Tensor-nilpotence detection gives a surjection \[f:\mathop{\mathrm{Spc}}(\mathop{\mathrm{Perf}}(E))\longrightarrow\mathop{\mathrm{Spc}}(\mathcal D)\] by Balmer’s surjectivity theorem (Balmer 2018, Theorem 1.3), in the form (Barthel et al. 2022, Theorem 2.6 and Corollary 4.8). Writing \(\operatorname{supp}\) for tensor-triangular support, functoriality gives \[\operatorname{supp}(\mathcal F(X)) =f^{-1}\bigl(\operatorname{supp}(X)\bigr).\] 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 (Balmer 2005, Theorem 4.10), in the support-containment form of (Barthel et al. 2022, 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 (Barthel et al. 2022, Corollary 2.7).

For the finite type-\(k\) objects the coefficient support is \(V(I_k)\), as in (Barthel et al. 2022, 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 2. By Corollary 28 and Proposition 29, the support of every object of \(\mathcal D\) belongs to the finite chain \[\mathop{\mathrm{Spec}}(E_0)=V(I_0)\supsetneq V(I_1)\supsetneq\cdots \supsetneq V(I_n)\supsetneq\varnothing.\] Indeed, \(S(X)=V(\mathop{\mathrm{Ann}}_{E_0}M_X)\), 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 \[\mathcal J=\langle Y\rangle_\otimes.\] 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 29 in both directions gives \[\mathcal J =\langle L_{K(n)}F(k)\rangle_\otimes =\mathcal D_k.\] 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 (Hopkins and Smith 1998). 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 \(V(I_0)=\mathop{\mathrm{Spec}}(E_0)\) 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 29, 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 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. ◻