Proof. Take the inverse limit of the cofiber sequences \(S\xrightarrow{p^j}S\to S/p^j\), with multiplication by \(p\) on the left spheres and identity on the middle spheres. The fiber of \(c\) is \[C\simeq\operatorname{holim}(\cdots\xrightarrow pS\xrightarrow pS) \simeq F(S[1/p],S).\] Multiplication by \(p\) is an equivalence on this function spectrum, because it is an equivalence on its first argument. It remains an equivalence after ordinary smash with \(K(a)\). But \(\pi_*(K(a)\wedge C)\) is a module over \(K(a)_*\) and is annihilated by \(p\). Thus it is zero, proving the first claim.

A map to a \(K(t)\)-local target consequently extends across \(c\). The target is also \(L_r\)-local when \(t\leq r\), since an \(L_r\)-acyclic spectrum is \(K(t)\)-acyclic. The extension therefore factors through \(L_rX\). ◻

Proof. Lemma 5 factors the detecting map through \(L_{n-1}X\), so \(\bar z\) is nonzero. Write \(\eta\) for \(K(n)\)-localization and \(\lambda\) for \(L_{n-1}\)-localization. The relevant two squares are \[\begin{CD} S @>{c}>> X\\ @V{\eta_S}VV @VV{\eta_X}V\\ L_{K(n)}S @>{L_{K(n)}c}>> L_{K(n)}X \end{CD} \qquad \begin{CD} X @>{\eta_X}>> L_{K(n)}X\\ @V{\lambda_X}VV @VV{\lambda_{L_{K(n)}X}}V\\ L_{n-1}X @>{i_{n,X}}>> L_{n-1}L_{K(n)}X. \end{CD}\] Both commute by naturality of localization units. Equivalently, \[\eta_Xc\simeq L_{K(n)}(c)\eta_S, \qquad i_{n,X}\lambda_X\simeq\lambda_{L_{K(n)}X}\eta_X.\] The first sends \(z\) to zero, and the second then gives the asserted kernel of the exact map (1). A retraction would induce a left inverse on this homotopy group and is therefore impossible. ◻

Proof. Use homogeneous group cochains: a degree-\(q\) cochain is an equivariant function \(f:Q^{q+1}\to P\), with differential the alternating deletion of coordinates. For \(q\geq1\) define \[(hf)(g_0,\ldots,g_{q-1}) =\sum_{\gamma\in Q}\gamma(w)f(\gamma,g_0,\ldots,g_{q-1}).\] For \(g\in Q\), replacing \(\gamma\) by \(g\gamma\) proves that \(hf\) is equivariant: semilinearity converts \((g\gamma)(w)\,g f\) into \(g(\gamma(w)f)\). In \(hd+dh\), the terms deleting one of the displayed \(g_i\) cancel in pairs. The term deleting the inserted coordinate is \(\sum_\gamma\gamma(w)f=f\). Thus \(hd+dh=\mathrm{id}\) in positive degrees, proving acyclicity for arbitrary \(P\).

For the Witt-ring case, the finite-field trace is onto. Lift a residue trace-one element to \(w_0\in W\). Its Witt trace lies in \(\mathbb Z_p\) and is congruent to one modulo \(p\), hence is a unit. Dividing \(w_0\) by this invariant unit supplies \(w\). We have not divided by \(|Q|\). ◻

Proof. The pure-subgroup cochain complex is \(W\)-linear and has semilinear \(Q\)-action, including the conjugation action on its arguments: for an inhomogeneous \(b\)-cochain, \[(\gamma f)(s_1,\ldots,s_b)= \gamma\bigl(f(\gamma^{-1}s_1\gamma,\ldots, \gamma^{-1}s_b\gamma)\bigr).\] The pure differential is \(W\)-linear and commutes with this action. Its cohomology is therefore semilinear, whether or not that cohomology has separately been shown finite. The continuous Hochschild–Serre sequence at the discrete coefficient level is \[H^i\bigl(Q,H^b_{\mathop{\mathrm{cts}}}(S_0,M_j)\bigr) \Longrightarrow H^{i+b}_{\mathop{\mathrm{cts}}}(G,M_j).\] Lemma 10 kills its positive \(Q\)-columns, giving \[H^s_{\mathop{\mathrm{cts}}}(G,M_j)=H^s_{\mathop{\mathrm{cts}}}(S_0,M_j)^Q.\] This proves the first bound. Continuous cochains into \(M\) are compatible cochains into its finite quotients. Their transition maps are degreewise surjective: a continuous cochain has a finite value set, and any choice of lifts of those values gives a continuous cochain into the next finite quotient. No equivariant section is needed for inhomogeneous cochains. The countable inverse-limit sequence is therefore \[ 0\longrightarrow\varprojlim_j{}^1 H^{s-1}_{\mathop{\mathrm{cts}}}(G,M_j) \longrightarrow H^s_{\mathop{\mathrm{cts}}}(G,M) \longrightarrow\varprojlim_j H^s_{\mathop{\mathrm{cts}}}(G,M_j)\longrightarrow0. \tag{4}\] Both outer terms vanish for \(s>D+1\). At the remaining boundary degree, the same sequence gives \[H^{D+1}_{\mathop{\mathrm{cts}}}(G,M) \cong\varprojlim_j{}^1 H^D_{\mathop{\mathrm{cts}}}(G,M_j).\] Thus the bound retains the possible extra inverse-limit degree. ◻

Proof. The pure stabilizer is the maximal-order unit group in the central division algebra over \(\mathbb Q_p\) of dimension \(a^2\), so it is compact \(p\)-adic analytic of that dimension (Morava 1985, secs. 2.1–2.2). It has no element \(g\) of order \(p\) in either stated range. Indeed, \((g-1)\Phi_p(g)=0\) in a division algebra and \(g\ne1\) imply \(\Phi_p(g)=0\). The polynomial \(\Phi_p(X+1)\) is Eisenstein, so \(g\) generates the degree-\((p-1)\) subfield \(\mathbb Q_p(\zeta_p)\). The division algebra would then be a vector space over this subfield, forcing \(p-1\mid a^2\). For \(a=p\) this forces \(p-1\mid1\); for \(a=p+1\) it forces \(p-1\mid4\). These are impossible in the stated ranges. The Lazard–Serre theorem therefore gives \(\mathop{\mathrm{cd}}_p(\mathbb S_a)=a^2\) for discrete \(p\)-primary coefficients (Serre 1965, sec. 1, Corollary (1)).

For each \(q\), the coefficient module \(M\) is finitely generated over the complete Noetherian ring \(O_a\), by the finite construction of \(DV_0\). It is complete and separated; its quotients \(M_j=M/\mathfrak m_a^jM\) are finite discrete \(p\)-primary modules with surjective transitions. This uses no flatness of \(M\) and allows torsion and both parities. The maximal ideal is invariant. For \(V_0=S\), continuity of the action on these quotients is part of the Morava action used by Devinatz–Hopkins (Devinatz and Hopkins 2004, Theorem 1(ii) and Remark 1.3). For every finite \(V_0\), the actual action and its continuity are also obtained by the ordinary cooperation construction of Proposition 23 below.

The pure stabilizer fixes \(W(k_a)\). It fixes the residue field and thus the unique Hensel lifts of the roots of \(X^{p^a}-X\), which are the Teichmüller representatives; continuity then fixes their convergent Witt expansions. Hence its action on \(M_j\) is Witt-linear, while \(Q=\mathop{\mathrm{Gal}}(k_a/\mathbb F_p)\) acts semilinearly by the Witt lift. At \(a=p\), Lemma 11 with \(D=p^2\) gives the bound. The full group has the split Galois subgroup of order \(p\); this conclusion is specific to the stated coefficients. The cohomological-dimension theorem was applied only to \(\mathbb S_p\).

At \(a=p+1\) with \(p\geq7\), the Galois quotient has order \(p+1\), prime to \(p\). An order-\(p\) element of \(\mathbb G_a\) would therefore lie in \(\mathbb S_a\), where we have just excluded it. The full group is a finite extension of the compact \(p\)-adic analytic pure group and has the same dimension \(a^2\). The same dimension theorem now applies to the full group itself and gives \(H^s_{\mathop{\mathrm{cts}}}(\mathbb G_a,M_j)=0\) for \(s>a^2\). Apply only the inverse-limit part of Lemma 11 to this finite-quotient bound. In both cases the finite quotients vanish for \(s>a^2\), and (4) gives the asserted bound \(s>a^2+1\) for \(M\). ◻

Proof. We have \(m<2t^2+1\), since the difference is \(p^2-4p+1>0\). Choose a positive \(N\) divisible by \(p\) with \(2N>m\). We show that \(\beta^N\ne0\) on \(E_{m+1}\) in bidegree \((2N,2ptN)\). For \(2\leq r\leq m\) all possible source and target filtrations are positive, so (14) applies. Even \(r\) gives odd internal degree and a zero group. For odd \(r\), internal-degree divisibility forces \(r=2ta+1\) with \(1\leq a<t\).

An outgoing target must be \(\alpha\beta^{N+ta}\Delta^l\). Equality of internal degrees gives \(a-1=pt(a+l)\), which forces \(a=1\). An incoming source must be \(\alpha\beta^{N-ta-1}\Delta^l\), whose internal-degree equation is \(a-t=pt(a-l)\), impossible for \(1\leq a<t\). Thus only the outgoing differential \(d_{2t+1}\) remains. The same degree argument shows that \(\beta\) exists up to that page; its only possible incoming differential has length two and odd internal source degree. The derivation rule on that page gives \[d_{2t+1}(\beta^N)=N\beta^{N-1}d_{2t+1}(\beta)=0,\] because \(\beta\) has even total degree, \(p\) divides \(N\), and the target is \(p\)-torsion. With all incoming differentials and all other outgoing possibilities excluded through page \(m\), this class is nonzero on \(E_{m+1}\). Its filtration is \(2N>m\), contradicting Proposition 20(i) if the exponent were at most \(m\). ◻

Proof. Apply the ordinary Landweber cooperation calculation to the \(p\)-typical flat Hopf algebroid \((BP_*,BP_*BP)\). This gives the displayed \(BP_*\) formula. It is the \(p\)-typical form of the \(MU_*\) formulas displayed in (Rezk 1998, sec. 15, Proposition 15.1 and Corollary 15.4) and (Hovey 2004, Lemma 4.4); Morava \(E\)-theory over \(BP\) is included in (Hovey and Strickland 2005, Example 0.1(d)). The two tensor products use the two different unit maps into \(BP_*BP\). For a Landweber-exact \(BP_*\)-algebra \(C\), the module \(BP_*BP\otimes_{BP_*}C\) is flat over the other \(BP_*\)-action. Indeed, tensoring with it factors as the exact cofree-comodule functor followed by the exact Landweber functor on comodules. This is also the opposite-unit criterion of (Hovey and Strickland 2005, Lemma 2.2), with conjugation exchanging the units when necessary. Base change along \(BP_*\to(E_a)_*\) proves the asserted flatness. The displayed coefficient and cooperation rings are even graded, which proves evenness. In particular this argument does not require \((E_b)_*\) to be flat over \(BP_*\). ◻

Proof. First suppose \(W\) finite. The equality \[D_0\wedge W\simeq D_0\wedge_E(E\wedge W)\] is an equality of ordinary derived constructions. The comparison is the balanced homotopy cross product for this relative smash, so it is natural in both variables before choosing any resolution. Realize a finite graded free surjection onto \(\pi_*(E\wedge W)\) by an \(E\)-module map from a finite wedge of suspensions of \(E\). Surjectivity makes the homotopy of its fiber the first syzygy. Repeat along a finite free resolution. At the last stage, lifts of a free basis give an equivalence from a finite free \(E\)-module, since they give an isomorphism on all homotopy groups. Relative smash with \(D_0\) is exact. Induction back through these fiber sequences, using the assumed tensor exactness on each finitely generated syzygy, proves (17) for finite \(W\).

Every ordinary \(p\)-local spectrum is a filtered homotopy colimit of finite ordinary \(p\)-local spectra. Both sides of (17) preserve those colimits: ordinary smash and homotopy groups do so, as does tensoring a module with \(\pi_*D_0\). Naturality extends the comparison to arbitrary \(W\).

For the assertion about functions, let \(M\to N\) be a surjection of finitely generated \(O\)-modules. For every \(j\geq1\) the map \[M/\mathfrak m^{j+1}M\longrightarrow (M/\mathfrak m^jM)\mathbin{\times}_{N/\mathfrak m^jN}(N/\mathfrak m^{j+1}N)\] is surjective. To see this, lift the first component to \(M\); its error against the second component lies in \(\mathfrak m^jN\), and this error lifts from \(\mathfrak m^jM\). Each quotient is a finite set. Given a continuous map \(T\to N\), first lift its reduction modulo \(\mathfrak m\), and then use these surjections to lift successively while preserving the previous choice. A lift on each finite value set is continuous. The compatible system defines a continuous lift to \(M\), since finitely generated modules are complete. Kernels carry their subspace topologies: the relevant modules are compact Hausdorff, and the induced continuous bijection from a kernel to its image is a homeomorphism. It follows that \(C_{\mathop{\mathrm{cts}}}(T,-)\) is exact on these modules. Apply this exact functor to a finite presentation of \(M\), and compare with tensoring the same presentation by \(C_{\mathop{\mathrm{cts}}}(T,O)\). The comparison is an isomorphism on the finite free terms and hence on \(M\). Applying this argument in each parity proves (18). In particular \(C_{\mathop{\mathrm{cts}}}(T,O_*)\otimes_{O_*}(-)\) has precisely the tensor exactness used in the first assertion. ◻

Proof. The first assertion is the localization theorem for ordinary Morava \(E\)-modules: \(K(a)\)-localization is derived \(\mathfrak m\)-completion, and flat homotopy has no higher derived completion. The natural localization map induces the ordinary completion map (Rezk 2009, Propositions 3.4, 3.6 and Corollary 3.8).

Here is the algebraic reduction argument, including its needed inverse-limit step. Choose a resolution \(P_\bullet\to N_0\) of a finitely generated \(O\)-module \(N_0\) by finite free modules. The tower \(P_\bullet\otimes_O U/\mathfrak m^lU\) is degreewise surjective. By flatness of \(U\), its first homology is \[\operatorname{Tor}_1^O(N_0,O/\mathfrak m^l)\otimes_O U.\] This homology tower is pro-zero. Indeed, for a finite first presentation \(0\to K\to P_0\to N_0\to0\), the corresponding Tor group is \((K\cap\mathfrak m^lP_0)/\mathfrak m^lK\). Artin–Rees (The Stacks Project Authors 2026, Tag 00IN) gives a \(d\) such that \(K\cap\mathfrak m^lP_0\subseteq\mathfrak m^{l-d}K\) for \(l\geq d\). For any fixed \(j\), a transition from \(l\geq j+d\) to \(j\) is therefore zero. Tensoring these zero transitions with \(U\) preserves this conclusion.

The homology exact sequence for the countable inverse limit of this degreewise-surjective tower now gives \[N_0\otimes_O\widehat U \cong\varprojlim_l(N_0\otimes_O U/\mathfrak m^lU);\] the possible \(\varprojlim^1\) of the first homology is zero because that tower is pro-zero. Finite freeness of the \(P_i\) identifies their tensor with \(\widehat U\) with their inverse-limit tensor. Take \(N_0=O/\mathfrak m^j\); for \(l\geq j\) the modules on the right are \(U/\mathfrak m^jU\). This proves the reduction formula. ◻

Proof of Proposition 23. The completed cooperation theorem gives \[ \pi_*L_{K(a)}(E\wedge E) \cong C_{\mathop{\mathrm{cts}}}(\mathbb G_a,O_*). \tag{19}\] At \(g\), the map is \(\mu(1\wedge g)\); the first factor acts by constant scalars. This is the evaluation map of (Hovey 2004, sec. 1 and Theorems 4.11 and 6.8). The equivalent convention in (Devinatz and Hopkins 2004, Proposition 2.2, Equation (2.3), and Equations (2.5)–(2.7)) uses \(\mu(g^{-1}\wedge1)\); exchanging the factors and inverting the group coordinate gives the convention above. The coefficient topology is the adic one (Devinatz and Hopkins 2004, Remark 1.3).

Put \(P_*=\pi_*(E\wedge E)\), regarded as an \(O_*\)-module through the first factor. Lemma 24 makes \(P_*\) flat. The actual localization map, the reduction in Lemma 26, and (18) applied to \(O_*/\mathfrak m^jO_*\) give \[ P_*/\mathfrak m^jP_* \cong C_{\mathop{\mathrm{cts}}}(\mathbb G_a,O_*/\mathfrak m^jO_*). \tag{20}\] This isomorphism is induced by the actual ordinary evaluation \(E\wedge E\xrightarrow{1\wedge g}E\wedge E\xrightarrow{\mu}E\) followed by reduction. It is not just an abstract isomorphism of the two reduction modules.

To form higher powers, use a common first factor in the ordinary relative smash products \[(E_0\wedge E_1)\wedge_{E_0}\cdots \wedge_{E_0}(E_0\wedge E_r) \simeq E_0\wedge E_1\wedge\cdots\wedge E_r.\] Each \(E_i\) denotes the same spectrum \(E\); the subscripts mark positions. Iterating (17), including its extension to arbitrary ordinary \(W\), identifies the homotopy of this ordinary smash with \(P_*^{\otimes_{O_*}r}\). The comparison inserts each pair into the first and one other position and multiplies at the common position. These modules are flat, since \(P_*\) is flat.

In each degree the coefficient ring \(O_*/\mathfrak m^jO_*\) is finite. Continuous functions to it are unions of function modules on finite quotient sets of \(\mathbb G_a\). Tensors of the function modules on finite sets are the function modules on their products, and every continuous function on \(\mathbb G_a^r\) to a finite set factors through finite quotients in the coordinates. It follows from (20) that the reductions of \(P_*^{\otimes_{O_*}r}\) are \(C_{\mathop{\mathrm{cts}}}(\mathbb G_a^r,O_*/\mathfrak m^jO_*)\). Apply Lemma 26 only now, to this ordinary \((r+1)\)-fold smash. The result is \[ \pi_*Z_r\cong C_{\mathop{\mathrm{cts}}}(\mathbb G_a^r,O_*). \tag{21}\] For \(r=0\) this is the locality of \(E\).

The maps in (21) remain the evaluations on the non-first factors followed by multiplication. In fact each ordinary evaluation extends through localization because its target \(E\) is local. On homotopy its first-factor \(O_*\)-linearity makes it adically continuous. Lemma 26 makes the ordinary image dense, and the reductions above show that the extension agrees with the displayed continuous function. Thus the construction preserves the evaluation maps as well as the coefficient modules.

For finite \(W\), the ordinary \(E\)-module \(Z_r\) satisfies the exactness assumption in Lemma 25, by (21) and (18). Those comparisons give (15). This step allows any finitely generated \(\pi_*(E\wedge W)\), including torsion and both parities; no flatness of that module has been used.

The terms of the local unit Amitsur object are exactly \(Z_r\). For finite \(V_0\), ordinary duality identifies \(F(V_0,Z_r)\simeq Z_r\wedge D V_0\), so the previous formula computes each cosimplicial homotopy group. Relabel the independent evaluations by their cumulative coordinates \[1,\quad g_1,\quad g_1g_2,\quad\ldots,\quad g_1\cdots g_r.\] The cochain identities below can be checked on ordinary evaluations before localization. For finite \(W\), their target \(E\wedge W\) is \(K(a)\)-local, since finite dualizable smash preserves local objects. They therefore extend through localization and remain identities on the tested coefficients. Insertion of the first unit acts on the coefficient; an interior unit combines two consecutive coordinates, and insertion of the last unit drops the last coordinate. Explicitly, for a degree-\(r\) cochain \(c\), \[\begin{aligned} (\delta^0c)(g_1,\ldots,g_{r+1}) &=g_1c(g_2,\ldots,g_{r+1}),\\ (\delta^ic)(g_1,\ldots,g_{r+1}) &=c(g_1,\ldots,g_i g_{i+1},\ldots,g_{r+1}) &&(1\leq i\leq r),\\ (\delta^{r+1}c)(g_1,\ldots,g_{r+1}) &=c(g_1,\ldots,g_r). \end{aligned}\] Multiplication of adjacent factors gives each codegeneracy, which inserts the identity in the corresponding coordinate. These are all the inhomogeneous cochain operators, including the degeneracies of the full cosimplicial object. The first unit insertion also displays the orbit function of every coefficient. Those functions are continuous, and reduction gives continuous actions on the finite invariant quotients by powers of \(\mathfrak m_a\). The \(E_2\)-page of the tower of finite partial totalizations is cosimplicial cohomology of these homotopy groups, which is (16). ◻

Proof. Let \(M_aS=\mathop{\mathrm{fib}}(L_aS\to L_{a-1}S)\), and let \(I_{\mathrm{BC}}\) be the ordinary Brown–Comenetz dualizing spectrum for the injective group \(\mathbb Q/\mathbb Z_{(p)}\). Gross–Hopkins duality, announced in (Hopkins and Gross 1994, Theorem 6) and proved in the form used here in (Hovey and Strickland 1999, Theorem 10.2(b),(e)), says that \[D_a=F(M_aS,I_{\mathrm{BC}})\] is \(K(a)\)-local and invertible for \(\otimes_a\). The identification with this monochromatic object uses (Hovey and Strickland 1999, Theorem 6.19).

For a local target \(Z\), the ordinary spectrum \(F(R,Z)\) is local. A map into it from a \(K(a)\)-acyclic spectrum \(V\) corresponds to a map \(V\wedge R\to Z\), and \(V\wedge R\) is still acyclic. Hence ordinary function spectra compute the local internal Hom, as in (Hovey and Strickland 1999, Theorem 7.1). Invertibility of \(D_a\), followed by ordinary closed adjunction, gives equivalences of ordinary spectra \[B_a\simeq F(D_a,D_a) \simeq F(D_a\wedge M_aS,I_{\mathrm{BC}}).\] The displayed smash is ordinary. Put \(C=D_a\wedge M_aS\). Brown–Comenetz representability now gives, naturally in \(U\), \[[U,B_a]\cong \operatorname{Hom}_{\mathbb Z}\bigl(\pi_0(U\wedge C),\mathbb Q/\mathbb Z_{(p)}\bigr).\]

Write \(U\) as a filtered homotopy colimit of finite ordinary \(p\)-local spectra \(U_i\). Ordinary smash with \(C\) and ordinary homotopy groups preserve this colimit, so \[\pi_0(U\wedge C)=\varinjlim_i\pi_0(U_i\wedge C).\] The functional corresponding to \(f\) vanishes on every term by the assumed finite precompositions. It therefore vanishes on the colimit, and \(f=0\). This argument tests by ordinary finite spectra and uses no interchange of \(K(a)\)-localization with this filtered colimit. ◻

Proof of Proposition 27. First obtain some finite exponent for the actual local unit, independently of any finite test. Devinatz–Hopkins prove that every \(K(h)\)-local spectrum belongs to the thick tensor ideal generated by \(E_h\) in \(\mathcal C_h\) (Devinatz and Hopkins 2004, Appendix A, Proposition A.3). This is the local nilpotence consequence of Hopkins–Ravenel; its descendability form is (Mathew 2016, Proposition 10.10). The objects of finite \(E_h\)-exponent form a thick tensor ideal containing \(E_h\): modules have exponent at most one, tensoring preserves a bound, and cofiber sequences and retracts preserve finite bounds by Lemma 19. It follows that \(B_h\) has some finite exponent \(c\). This is one bound on an actual map before any test is chosen; the nilpotence theorem supplies no numerical value needed here.

Apply the full local-unit Amitsur construction in \(\mathcal C_h\) to \(B_h\), and then the exact ordinary functor \(F(V_0,-)\) for a finite ordinary \(V_0\). Proposition 23 identifies its second page with \[H^s_{\mathop{\mathrm{cts}}}\bigl(\mathbb G_h,\pi_q(E_h\wedge D V_0)\bigr).\] Proposition 12 makes this page zero for \(s\geq h^2+2=m\). The tested-image assertion of Proposition 20(ii) applies using the already known finite exponent \(c\). It concerns the full finite partial totalizations and their finite fiber cubes, and gives \[[V_0,I_h^{\otimes_h m}]\longrightarrow[V_0,B_h] \quad\text{zero for every finite ordinary }V_0.\] Thus the actual arrow \(e_m(B_h)\), regarded in ordinary spectra, vanishes after every finite ordinary precomposition. Lemma 28 makes it null as an ordinary map and hence null in the full subcategory \(\mathcal C_h\).

For clarity, all layers needed next occur at finite stages. Set \(I_h^{\otimes_h0}=B_h\), and for \(0\leq j\leq m\) put \[F_j=\mathop{\mathrm{cofib}}\bigl( I_h^{\otimes_h m}\longrightarrow I_h^{\otimes_h(m-j)} \bigr).\] The consecutive maps form a finite filtration \(0=F_0\to F_1\to\cdots\to F_m\). The cofiber of \(I_h^{\otimes_h(i+1)}\to I_h^{\otimes_h i}\) is \(E_h\otimes_h I_h^{\otimes_h i}\). The cofiber comparison for consecutive composites therefore identifies \[\mathop{\mathrm{cofib}}(F_j\to F_{j+1}) \simeq E_h\otimes_h I_h^{\otimes_h(m-j-1)} \quad(0\leq j<m).\] Finally \(F_m=\mathop{\mathrm{cofib}}(e_m(B_h))\). A nullhomotopy of \(e_m(B_h)\) makes \(B_h\) a retract of this cofiber. This proves both assertions. ◻

Proof. Since \(H\) is finite, the underlying object \(A_H\wedge U\) is \(F(H_+,U)\), with diagonal action. Untwisting the value at \(g\) by the action of \(g^{-1}\) on \(U\) identifies it with the coinduction of the underlying spectrum: \[\theta:A_H\wedge U\xrightarrow{\ \simeq\ } \operatorname{Coind}_{1}^{H}(\operatorname{Res}_{1}^{H}U).\] Indeed, on functions the diagonal action is \((h f)(g)=h f(h^{-1}g)\), and \((\theta f)(g)=g^{-1}f(g)\) turns it into translation \((h\theta f)(g)=(\theta f)(h^{-1}g)\). For finite \(H\), the finite coproduct-product equivalence in spectra identifies induction with coinduction. Induction adjunction takes the underlying multiplication map \(\mu_z:U\to U\) to a map with coherent \(H\)-action \[r_z:A_H\wedge U\longrightarrow U.\]

Let \(j:U\to A_H\wedge U\) be the unit map. Under \(\theta\), its component at \(g\) is \(g^{-1}:U\to U\). The induction map has component \(g\mu_z\) at \(g\), so the underlying composite is \[r_zj=\sum_{g\in H}g\mu_zg^{-1} =\mu_{\sum_g g(z)}=\mu_v.\] The middle equality uses the compatibility of the \(D'\)-module structure with the \(H\)-action. Since \(v\) is a unit, this is an equivalence of underlying spectra. Equivalences of \(H\)-diagrams are detected on underlying objects, so \(r_zj\) is an equivalence in \(\mathcal D_H\). Hence \((r_zj)^{-1}r_z\) retracts \(j\) there. Its fiber map is null, which is exponent at most one. The class \(z\) was only an underlying class; no homotopy-fixed lift of it is needed. ◻

Proof. We first recall explicitly the naturality of the formal group used here. For a complex-oriented even periodic commutative ring spectrum \(B\), an orientation and a periodic unit give a coordinate in degree zero. The finite projective-space calculation and its surjective restriction tower give \[B^0(\mathbb{CP}^{\infty})=(\pi_0B)[[z_B]].\] Products of projective spaces give the corresponding group law. A unital ring-spectrum map \(f:B\to C\) carries an orientation to an orientation and a periodic unit to a unit. Its coordinate therefore has an invertible linear coefficient in the chosen coordinate of \(C\). The inverse of this coordinate substitution gives an isomorphism from the base change of the formal group of \(B\) to the formal group of \(C\). The product calculation gives compatibility with group laws, and this construction respects composites of ring-spectrum maps. All later base changes are base changes of these algebraic formal groups along maps of coefficient rings.

Let \(\rho:\pi_0D\to\Omega\) be the composite through \(x\). Write \(\iota_t:E_t\to D\) and \(\iota_h:E_h\to D\) for the two factor maps, and let \(\rho_t,\rho_h\) be their coefficient composites with \(\rho\). The map \(\rho_t\) factors through \[O_t/\mathfrak m_t=k_t\longrightarrow\Omega.\] The last map is injective, since it is a unital map from a field. After base change along \(\rho_t,\rho_h,\rho\), the factor maps give formal-group isomorphisms \[\phi_t:\mathcal F_{t,\Omega}\xrightarrow{\ \cong\ } \mathcal F_{D,\Omega},\qquad \phi_h:\mathcal F_{h,\Omega}\xrightarrow{\ \cong\ } \mathcal F_{D,\Omega}.\]

Let \(g_D\) denote the action of \(g\) on \(D\). The equality \(x\circ g=x\) implies \(\rho\circ\pi_0(g_D)=\rho\), so naturality makes \(g_D\) an automorphism \(u_g\) of the single formal group \(\mathcal F_{D,\Omega}\). Since \(g_D\iota_h=\iota_h\), naturality with the second factor gives \(u_g\phi_h=\phi_h\). Thus \(u_g\) is the identity. But \(g_D\iota_t=\iota_t g\); naturality with the first factor gives \[u_g\phi_t=\phi_t\,g_\Omega,\] where \(g_\Omega\) is the specialization of the prescribed stabilizer automorphism. Consequently \(g_\Omega\) is the identity.

On the special fiber the pure stabilizer element \(g\) is its specified automorphism of the Honda formal group over \(k_t\). This is the special-fiber content of Lubin–Tate deformation (Lubin and Tate 1966, Proposition 3.3 and paragraph 3.4), with its coherent realization as above. If \(g\ne1\), some coefficient of the power series \(g(X)-X\) is nonzero in \(k_t\). It remains nonzero under the injective map \(k_t\to\Omega\), contrary to \(g_\Omega=1\). Therefore \(g=1\).

This proof used only the coefficient map to \(\Omega\) for algebraic base change; it did not require a ring-spectrum map to a spectrum with coefficient field \(\Omega\). The assertion concerns equality of actual field-valued maps; invariance of their kernels alone would not give the conclusion needed next. ◻

Proof. The trace image is an ideal of \(R^H\), since trace is additive and \(a\mathop{\mathrm{Tr}}(r)=\mathop{\mathrm{Tr}}(ar)\) for \(a\in R^H\). Suppose this ideal is proper, and choose a maximal ideal \(\mathfrak q\) containing it. For every \(r\in R\), the monic orbit polynomial \(\prod_{g\in H}(X-g(r))\) has coefficients in \(R^H\). Thus \(R\) is integral over \(R^H\). Lying over gives a prime \(\mathfrak p\) of \(R\) above \(\mathfrak q\). The quotient and inclusion into its fraction field give a unital map \(x:R\to\Omega=\operatorname{Frac}(R/\mathfrak p)\) killing \(\mathfrak q\).

The multiplicative unital maps \(\chi_g:R\to\Omega\), \(\chi_g(r)=x(g(r))\), are pairwise distinct by the hypothesis. Such distinct maps are linearly independent as functions. To prove this, suppose \(\sum_{i=1}^v a_i\chi_i=0\) is a nonzero relation with the least possible number of nonzero coefficients. One term is impossible by evaluation at \(1\). If \(v>1\), choose an element \(b\in R\) distinguishing \(\chi_1\) from \(\chi_v\). Subtract \(\chi_v(b)\) times the relation at \(r\) from the relation at \(br\). The result is \[\sum_{i<v}a_i\bigl(\chi_i(b)-\chi_v(b)\bigr)\chi_i(r)=0.\] Its coefficient at \(i=1\) is nonzero in the field, giving a shorter nonzero relation, a contradiction.

In particular the function \(\sum_{g\in H}\chi_g\) is not zero: each of its coefficients is \(1\) in the field, even if the characteristic divides \(|H|\). On the other hand it equals \(x\circ\mathop{\mathrm{Tr}}\), which is zero since \(x\) kills the trace ideal. This contradiction proves the claim. If \(R\) is the zero ring, the conclusion holds directly with \(1=0\). ◻

Proof of Proposition 29. Lemmas 31 and 32 give \(\bar z\in R\) with \(\mathop{\mathrm{Tr}}(\bar z)=1\). Lift it to \(z_0\in\pi_0D\), and put \(v=\sum_g g(z_0)\). Then \[v-1\in\mathfrak m_t\pi_0D.\] The mixed module \(\pi_*D\) is flat over \((E_t)_*\) by Lemma 24. Lemma 26 therefore identifies the homotopy of its actual localization \(D\to D'\) with the ordinary \(\mathfrak m_t\)-adic completion of \(\pi_*D\).

At the finite adic stage \(\pi_0D/\mathfrak m_t^j\pi_0D\), the element \[w_j=\sum_{k=0}^{j-1}(1-v)^k\] satisfies \(v w_j=1-(1-v)^j=1\). These inverses are compatible under reduction. Here \(j\) is a finite truncation level; the quotient rings themselves are not required to be finite sets. The comparison with completion is induced by the actual ring map \(D\to D'\). Multiplication by the image of \(v\) is \(O_t\)-linear and adically continuous and agrees with multiplication by \(v\) on the dense ordinary image. It hence agrees with the levelwise multiplication on the completion. The compatible \(w_j\) give an inverse to the image of \(v\) in \(\pi_0D'\).

Localization is equivariant, so that image of \(v\) is the \(H\)-trace of the image of \(z_0\). Lemma 30 now applies. If the completed ring is zero, its unit is zero and every unital module is contractible, giving the same conclusion. ◻

Proof. The inclusion \(\mathcal C_h\to\mathrm{Sp}_{(p)}\) is exact and lax symmetric monoidal. Since \(E_h\) is local, the localization unit identifies the included local algebra \(E_h\) with its given ordinary algebra. Each local layer \(E_h\otimes_h I_h^{\otimes_h i}\) in Proposition 27 consequently has an ordinary \(E_h\)-module structure. Ordinary smash with \(E_t\) makes it a \(D=E_t\wedge E_h\)-module with compatible \(H\)-action.

The \(K(t)\)-acyclic spectra form a smash ideal. Thus the localization into \(\mathcal C_t\) is symmetric monoidal for its localized tensor, and the inclusion back into ordinary spectra is lax symmetric monoidal. Applying localization to the algebra and this module gives, after that inclusion, a \(D'\)-module structure on the transformed layer in \(\mathcal D_H\). Proposition 29 gives every such layer \(H\)-exponent at most one.

The functor \(T\) preserves cofiber sequences and retracts. Apply it to the finite filtration and retraction of Proposition 27, and add the \(m\) layer bounds using Lemma 19. This yields the stated bound. The construction used ordinary module structures on the individual layers; it required no preservation by \(T\) of the \(\mathcal C_h\) tensor product. ◻

Proof. The functor \(T\) inverts \(K(t)\)-equivalences: the cofiber of such a map is \(K(t)\)-acyclic, and remains so after ordinary smash with \(E_t\). Every \(L_t\)-localization unit is a \(K(t)\)-equivalence. Lemma 5 says that the actual completion \(c:S\to X\) is a \(K(a)\)-equivalence for each positive \(a\), in particular for \(a=t,h\). Write \(\lambda_Z:Z\to L_tZ\) for the lower localization unit. Consider the square of actual localization and completion maps \[\begin{CD} S @>{\eta_S^{(h)}}>> B_h\\ @V{\lambda_X\,c}VV @VV{\lambda_{L_{K(h)}X}\,L_{K(h)}(c)}V\\ L_tX @>{i_{h,X}}>> L_tL_{K(h)}X . \end{CD}\] It commutes by naturality of the two localization units. Both vertical arrows become equivalences under \(T\): for the left one use the \(K(t)\)-equivalences just stated, and for the right one use that \(L_{K(h)}(c)\) is an equivalence followed by a \(K(t)\)-equivalence. Moreover \(T(S)\simeq E_t\), with coherent \(H\)-action, since \(E_t\) is \(K(t)\)-local.

A homotopy retraction of \(i_{h,X}\) would therefore, after applying \(T\) and these identifications, exhibit \(E_t\) as a retract of \(T(B_h)\) in \(\mathcal D_H\). Proposition 33 and preservation of exponent bounds under retracts would give \(\mathop{\mathrm{exp}}_H(E_t)\leq m\). The finite-page lower result Proposition 22 gives the strict inequality \(\mathop{\mathrm{exp}}_H(E_t)>m\), a contradiction. ◻

Proof of Theorem 4. Proposition 22 gives the strict lower bound and Proposition 33 gives the upper bound, both for \(m=p^2+2\) in the single category \(\mathcal D_H\). The preceding corollary proves that a retraction of the canonical map would turn the lower object into an \(H\)-equivariant retract of the upper object and contradict those bounds. This proves every assertion of the theorem. The lower bound uses finite-page survival only; the upper bound uses the actual null map and its finite layers. ◻

Proof of Lemma 8. Use the homogeneous free bar construction \(EQ_\bullet\), with \(EQ_q=Q^{q+1}\), and let \(EQ^{(s)}\) be its \(s\)-skeleton. The full totalization tower of the group-action resolution is \[V_s=F_Q((EQ^{(s)})_+,R),\qquad \operatorname{holim}_sV_s=F_Q(EQ_+,R)=R^{hQ}.\] Here \(F_Q\) denotes the derived spectrum of equivariant maps. The skeletal filtration gives the usual homogeneous cochains and hence the displayed \(E_2\) page.

For the prismatic geometry below, compare Basterra–Mandell (Basterra and Mandell 2013, proof of Theorem 7.1, pp. 23–26). Their bar diagonal for a (partial) non-unital \(E_1\) algebra over a base ring spectrum \(H\) uses the same half-cut regions and inverse barycentric coordinates. Here we construct the equivariant relative maps for \(EQ\) and identify the resulting \(E_\infty\) product with the associated-graded product on actual homotopy groups under the lemma’s stated filtration hypothesis. For barycentric coordinates \(t=(t_0,\ldots,t_q)\in\Delta^q\), set \(c_{-1}=0\) and \(c_i=t_0+\cdots+t_i\). Define two barycentric vectors by \[ \begin{split} \ell_i(t)&=2\bigl(\min(c_i,\tfrac12)-\min(c_{i-1},\tfrac12)\bigr),\\ \rho_i(t)&=2\bigl(\max(c_i,\tfrac12)-\max(c_{i-1},\tfrac12)\bigr). \end{split} \tag{22}\] They are nonnegative and each has sum one. Equivalently, their \(i\)th coordinates are twice the lengths of the intersections of \([c_{i-1},c_i]\) with \([0,\tfrac12]\) and \([\tfrac12,1]\). The formulas are continuous, including when a cumulative sum equals \(\tfrac12\).

For an order-preserving map \(\theta:[q]\to[q']\), let \((\theta_*t)_j=\sum_{\theta(i)=j}t_i\). Each interval belonging to \(\theta_*t\) is the union of the consecutive intervals belonging to the corresponding fiber of \(\theta\); an empty fiber gives an interval of length zero. Additivity of intersection lengths gives \[\ell(\theta_*t)=\theta_*\ell(t),\qquad \rho(\theta_*t)=\theta_*\rho(t).\] The realization relation is \([\theta^*\sigma,t]=[\sigma,\theta_*t]\) for \(\sigma\in EQ_{q'}\). Thus for every bar simplex \(\sigma\in EQ_q\) the formula \[\delta[\sigma,t]=([\sigma,\ell(t)],[\sigma,\rho(t)])\] is compatible with all faces and degeneracies and defines a continuous map \(\delta:EQ\to EQ\times EQ\). It is \(Q\)-equivariant because the action changes \(\sigma\) and leaves the barycentric coordinates fixed. Moreover \[[\sigma,t]\longmapsto \bigl([\sigma,(1-\tau)t+\tau\ell(t)], [\sigma,(1-\tau)t+\tau\rho(t)]\bigr), \qquad 0\leq\tau\leq1,\] is compatible with the same identifications and is equivariant. It is a homotopy from the ordinary diagonal to \(\delta\).

Choose \(k\) with \(c_{k-1}\leq\tfrac12\leq c_k\). The support of \(\ell(t)\) is contained in the front face \([0,\ldots,k]\), and the support of \(\rho(t)\) is contained in the back face \([k,\ldots,q]\). These faces have dimensions \(k\) and \(q-k\). Degenerate faces can only lower those dimensions. Consequently \[\delta(EQ^{(s)})\subset \bigcup_{i+j\leq s}EQ^{(i)}\times EQ^{(j)}.\] This also identifies the cellular chain map. On the region \(c_{k-1}\leq\tfrac12\leq c_k\), the map \(t\mapsto(\ell(t),\rho(t))\) is a homeomorphism onto \(\Delta^k\times\Delta^{q-k}\), with inverse \[t_i=\ell_i/2\ (i<k),\qquad t_k=(\ell_k+\rho_k)/2,\qquad t_i=\rho_i/2\ (i>k).\] In the oriented coordinates \(t_1,\ldots,t_q\) on the source and \(\ell_1,\ldots,\ell_k,\rho_{k+1},\ldots,\rho_q\) on the product, its Jacobian determinant is \(2^q>0\). The regions meet on their boundaries. The image of the cellular fundamental chain is therefore \[[\sigma]\longmapsto \sum_{k=0}^q [\sigma|_{[0,\ldots,k]}]\otimes [\sigma|_{[k,\ldots,q]}],\] with degenerate terms omitted in normalized chains. This is the Alexander–Whitney diagonal, now obtained from the explicit map.

Write \(EQ/EQ^{(s)}\) for the based cofiber of \((EQ^{(s)})_+\to EQ_+\), and set \(EQ^{(-1)}=\varnothing\). The skeletal inclusions are \(Q\)-CW cofibrations, so these quotients represent their homotopy cofibers. For \(a,b\geq0\), the skeletal containment above makes the composite of \(\delta\) with the quotient to \((EQ/EQ^{(a-1)})\wedge(EQ/EQ^{(b-1)})\) constant on \(EQ^{(a+b-1)}\): if \(i+j<a+b\), then \(i<a\) or \(j<b\). We get a \(Q\)-equivariant relative map \[EQ/EQ^{(a+b-1)}\longrightarrow (EQ/EQ^{(a-1)})\wedge(EQ/EQ^{(b-1)}).\] Multiplication on \(R\) and precomposition give \[\begin{split} F_Q(EQ/EQ^{(a-1)},R)\wedge F_Q(EQ/EQ^{(b-1)},R) \longrightarrow F_Q(EQ/EQ^{(a+b-1)},R). \end{split}\] These maps are compatible as \(a\) and \(b\) vary, since they are induced by the same \(\delta\) and the nested skeletal quotients.

For completeness, the associativity needed for powers also holds within this relative filtration. Partition \([0,1]\) into any \(m\) successive intervals of positive lengths. For the \(j\)th output vector, divide the lengths of its intersections with \([c_{i-1},c_i]\) by the length of the \(j\)th interval. The preceding naturality and equivariance proof applies verbatim. If \(k_j\) is an index whose coordinate interval contains the \(j\)th cut, the supports lie in the successive faces \([0,\ldots,k_1]\), \([k_1,\ldots,k_2]\), …, \([k_{m-1},\ldots,q]\). Their dimensions sum to \(q\); boundary cases can only reduce the sum. Thus the same construction gives \(m\)-fold relative maps. Composing cut maps is exactly refinement of the partition. The positive interval lengths form a convex simplex, so the partitions for two parenthesizations are joined by varying those lengths. This gives a homotopy through maps with the same relative skeletal property. Thus the relative products are associative up to filtered homotopy, as required for their iterated products.

Put \(I_s=F_Q(EQ/EQ^{(s-1)},R)\) for \(s\geq0\). The quotient cofiber sequence for two successive skeleta gives \[\operatorname{cofib}(I_{s+1}\to I_s) \simeq F_Q(EQ^{(s)}/EQ^{(s-1)},R) \simeq \operatorname{fib}(V_s\to V_{s-1}).\] The fiber sequences \(I_s\to R^{hQ}\to V_{s-1}\) form a diagram with the identity on \(R^{hQ}\) and the tower transition on the right. The stable fiber diagram identifies the connecting maps of the displayed cofiber sequences with those of the \(V_s\) tower, with the usual rotation signs. Thus the paired relative spectral sequence is the ordinary tower spectral sequence with the same filtration index. The relative products pair these sequences and their exact couples. On cellular cochains the displayed chain formula is the ordinary cup product; in inhomogeneous coordinates its value is \[(v\smile w)(g_1,\ldots,g_{a+b}) =v(g_1,\ldots,g_a)\,(g_1\cdots g_a) w(g_{a+1},\ldots,g_{a+b}),\] with the usual graded signs. The cellular boundary formula for a product, together with the internal Koszul sign, gives the derivation rule in the paired exact couples and hence on every spectral-sequence page. This \(E_2\) product is the ordinary cup product used by the ordinary-to-Tate comparison in Section [sec:lower-exponent]. Since each subsequent product is induced on homology, the products agree on every later page as well.

The fiber sequence for \(I_a\) identifies its image in \(\pi_*R^{hQ}\) with \(\ker(\pi_*R^{hQ}\to\pi_*V_{a-1})\). After forgetting the relative conditions, the product just constructed is homotopic to the actual product on \(F_Q(EQ_+,R)\) by the equivariant homotopy from \(\delta\) to the diagonal. Thus these relative products multiply the actual filtration groups. Passing to successive relative quotients is exactly the product in the paired spectral sequence. When the stated convergence identifies those quotients with \(E_\infty\), this is the associated-graded homotopy product as stated. ◻