Proof. The finite abelian chromatic Smith theorem gives loss at most one for every nontrivial cyclic \(p\)-group, including target height zero [3]; see also [15]. Along a length-\(s\) cyclic subnormal chain, apply it to the finite genuine \(G_i/G_{i-1}\)-spectrum obtained by restricting \(X\) to \(G_i\) and taking genuine geometric \(G_{i-1}\)-fixed points. Transitivity identifies the underlying and full fixed points with \(\Phi^{G_{i-1}}X\) and \(\Phi^{G_i}X\). Iterating at heights \(n+s-1,\ldots,n\) proves the assertion. ◻

Proof. Inflation of a finite type-\((m+1)\) spectrum generates the kernel as a localizing tensor ideal. Equivalently, use its products with the finite orbit generators. This is a finite localization, and its compact category is the idempotent completion of the corresponding compact quotient. The point classification and localization theorem of Balmer–Sanders identify its spectrum with the displayed subspace [2]. Indeed the inflated generator belongs precisely to the primes at test heights at most \(m\).

Geometric fixed points commute with this inflated localization. Each \(\Phi^H Z\) is therefore \(L_m^f\)-local; telescope detection at the retained heights, followed by genuine geometric detection, proves the zero criterion. For a finite representative, (4) gives agreement of the two support tests. For the assertion about all compact objects, the kernel of \(K(i)\wedge\Phi^H(-)\) on compacts is a tensor-triangular prime by the Morava \(K\)-theory field property. Its inverse image in the original compact category is \(\mathcal P_i^G(H)\), since \(K(i)\wedge L_m^fS\simeq K(i)\) for \(i\le m\). The injectivity of the spectrum map for finite localization therefore identifies this kernel with the point labelled \((H,i)\). ◻

Proof. For any genuine \(L\)-spectrum \(X\), categorical fixed points commute with ordinary rationalization: their homotopy groups are represented by suspensions of the compact genuine sphere. In the rational genuine category, \(e_L^L X_{\mathbb Q}\) restricts to zero at every proper subgroup, since \(e_L^L\) does. The isotropy separation sequence for the family of proper subgroups consequently identifies \[(e_L^L X_{\mathbb Q})^L\simeq \Phi^L(e_L^L X_{\mathbb Q})\simeq(\Phi^L X)_{\mathbb Q}.\] Apply this to the already formed cofree spectrum \(X=\mathop{\mathrm{Bor}}_L(R)\), whose categorical fixed points are \(R^{BL}\). The comparison respects multiplication and identifies the full idempotent unit with the unit of \((U_L(R))_{\mathbb Q}\), proving (9). A unital ring spectrum is zero exactly when its unit is zero. Taking \(R=\mathbf 1\) and using the preceding unit criterion for \(D_L\) proves the last assertion. ◻

Proof. We first record how a subgroup summand detects occurrence. If \(I\le F\), the mark formula gives \[ \operatorname{ind}_I^F(e_I^I) =|N_F(I):I|\,e_I^F. \tag{10}\] Indeed a subgroup has a nonzero mark on the left only when it is conjugate to \(I\), and the \(I\)-mark is \(|N_F(I):I|\). Covering transfer carries the full \(I\)-unit to the right side of (10). Thus \(e_I^F A_F\ne0\) implies \(D_I\ne0\).

Let \(S\) be a Sylow \(p\)-subgroup of \(F\). The class \([F/S]\in\pi_0 A_F\) is a unit. To see this, regard it as an endomorphism of the constant unit in \(\mathop{\mathrm{Fun}}(BF,\mathrm{Sp}_{T(l)})\). Its value at the point is multiplication by \([F:S]\), a \(p\)-local unit, and equivalences of diagrams are detected at the point. If \(F\) is not a \(p\)-group, the full \(F\)-mark of \([F/S]\) is zero. Hence \(e_F^F[F/S]=0\), and invertibility forces \(D_F=0\).

For a quotient \(q:L\twoheadrightarrow Q\), pullback of \(e_Q^Q\) has full \(L\)-mark \(1\). Its image therefore acts as the identity on \(D_L\). If the full \(Q\)-unit were zero, its pullback would be zero, contradicting occurrence of \(L\). This proves quotient closure.

For subgroup closure, let \(I\le L\). Before any rationalization, there is a unit-preserving map of spectra \[ U_I(\mathbf 1)\longrightarrow U_L(\mathbf 1). \tag{11}\] Here is its construction. The cofree adjunction gives a map \[N_I^L\mathop{\mathrm{Bor}}_I(\mathbf 1)\longrightarrow\mathop{\mathrm{Bor}}_L(\mathbf 1)\] whose adjoint on underlying spectra with \(L\)-action is indexed multiplication \(\bigwedge_{L/I}\mathbf 1\to\mathbf 1\). This uses the ordinary commutative ring spectrum underlying the local unit; the displayed smash is the ordinary indexed smash. The multiplication is \(L\)-equivariant and carries the indexed unit to the unit. Apply geometric fixed points and the natural norm formula \(\Phi^L N_I^L\simeq\Phi^I\) [9]. This gives (11), with the stated unit preservation. If the source unit were zero rationally, its image would be zero rationally. 4 now proves subgroup closure.

It remains to bound generators. If an occurring elementary abelian group \(E=(C_p)^a\) had \(a>l\), the elementary abelian sharpness theorem of Arone, Dwyer, Lesh, and Mahowald, in the form [3], would give a finite genuine \(E\)-spectrum \(X\) with underlying type \(a\) and full geometric fixed points of type \(0\). Its underlying spectrum is \(T(l)\)-acyclic. Finite duality and the cofree projection formula give \[X\wedge\mathop{\mathrm{Bor}}_E(\mathbf 1) \simeq\mathop{\mathrm{Bor}}_E\bigl((\operatorname{res}X)\wedge\mathbf 1\bigr)=0.\] Here \(\operatorname{res}X\) retains its underlying \(E\)-action. Its smash with \(\mathbf 1\) is zero because this can be tested on the underlying spectrum, and smashing a finite spectrum with the local unit computes its localization. Taking geometric fixed points and rationalizing yields \[(\Phi^E X)_{\mathbb Q}\wedge(U_E(\mathbf 1))_{\mathbb Q}=0.\] The first factor is a nonzero finite rational spectrum, so it detects nonzero rational spectra under smash. This contradicts occurrence of \(E\). Finally, every section \(A\) of an occurring group occurs, and so does its Frattini quotient \(A/\Phi(A)\cong(C_p)^{d(A)}\). The elementary abelian bound and the Burnside basis theorem give \(d(A)\le l\). ◻

Proof. Enumerate the isomorphism classes of finite groups as \(F_0,F_1,\ldots\). Given an occurring group \(L_i\), choose an occurring extension \(L_{i+1}\twoheadrightarrow L_i\) maximizing \[\bigl|\mathop{\mathrm{Hom}}(L_{i+1},F_i)/F_i\bigr|\] among all occurring extensions of \(L_i\). The set of candidates is nonempty, since \(L_i\) is one. By 5, every candidate has at most \(l\) generators, so the displayed integer is at most \(|F_i|^l\) and a maximum is attained. Precomposition with a surjection injects conjugacy classes of homomorphisms. Once a target \(F_i\) has been maximized, every later occurring extension is still an extension of \(L_i\) and cannot increase the maximum. Consequently, for every sufficiently advanced stage \(L\) and every occurring extension \(M\twoheadrightarrow L\), \[ \mathop{\mathrm{Hom}}(L,F_i)/F_i\xrightarrow{\ \sim\ }\mathop{\mathrm{Hom}}(M,F_i)/F_i. \tag{14}\]

Every continuous homomorphism \(\Gamma\to F\) factors through some \(L_i\), since the stage kernels form a neighborhood basis of the identity. After \(F\) has been maximized, the preceding bijections imply that every such homomorphism factors through each later stage. This is actual factorization: a map conjugate to one factoring through a quotient kills the same quotient kernel. Surjectivity of \(\Gamma\to L_i\) also identifies all conjugating elements. This proves (12) first for \(Y=BF\), and then for finite disjoint unions of such groupoids.

If \(H\) is closed in \(\Gamma\) and \(H/K\) is finite, choose an open normal subgroup \(N\trianglelefteq_o\Gamma\) with \(H\cap N\subseteq K\). Then \(H/K\) is a quotient of the subgroup \(H/(H\cap N)\) of \(\Gamma/N\). The latter is a quotient of a tower stage, so 5 gives \(d(H/K)\le l\). For each open normal \(K\) of \(H\), the tuples in \(H^l\) whose images generate \(H/K\) form a nonempty clopen subset. These subsets have the finite intersection property, by applying the same bound to the quotient by an intersection of finitely many such \(K\). Compactness gives a tuple generating every finite quotient, hence topologically generating \(H\). This proves (i).

We prove the splitting for \(Y=BF\), at a stage \(L\) satisfying (14) for \(F\). We first show that every surviving subgroup summand comes from a graph, and then identify each graph summand by restriction. Put \(G=L\times F\) and let \(\pi:G\to L\) be the first projection. By (8), \(D_L\otimes A_F\) is the summand \(\pi^*(e_L^L)A_G\). The marks of \(\pi^*(e_L^L)\) are supported exactly on subgroups \(M\le G\) projecting onto \(L\). If \(e_M^G A_G\ne0\), then \(M\) occurs by (10). It is therefore an occurring extension of \(L\). Apply (14) to its second projection \(M\to F\): this map is conjugate to \(\alpha\circ\pi\) for some \(\alpha:L\to F\). Absorbing the conjugation into \(\alpha\) shows that \(M\) is contained in its graph, and surjectivity onto \(L\) gives \[M=M_\alpha:=\{(x,\alpha(x)):x\in L\}.\] Graphs are conjugate in \(L\times F\) exactly when the corresponding homomorphisms are conjugate in \(F\): conjugation in the first factor is already absorbed by conjugation by its image under \(\alpha\).

We next identify each graph summand, keeping track of the map. Let \(B_\alpha=e_{M_\alpha}^G A_G\) and write \(j_\alpha:L\xrightarrow{\sim}M_\alpha\hookrightarrow G\). Restriction along \(j_\alpha\), followed by the full projection, gives \[r_\alpha:B_\alpha\longrightarrow D_L.\] It has a right inverse \(s_\alpha\), obtained by pulling back along \(\pi\) and projecting to \(B_\alpha\), because \(\pi j_\alpha=\mathrm{id}\) and \(j_\alpha^*(e_{M_\alpha}^G)=e_L^L\). For the other inverse, let \(t_\alpha:D_L\to B_\alpha\) be covering transfer along \(j_\alpha\), followed by projection. The projection formula gives \[t_\alpha r_\alpha = |(G/M_\alpha)^{M_\alpha}|\,\mathrm{id}_{B_\alpha} = |C_F(\alpha(L))|\,\mathrm{id}_{B_\alpha}.\] Indeed the full graph mark of \([G/M_\alpha]\) is \(|N_G(M_\alpha):M_\alpha|=|C_F(\alpha(L))|\). This positive integer is invertible in \(\mathcal C_{\mathbb Q}\), so \(|C_F(\alpha(L))|^{-1}t_\alpha\) is a left inverse. Thus \(r_\alpha\) is an isomorphism, with inverse \(s_\alpha\). In particular every graph summand survives, since \(D_L\ne0\).

Combining the graph summands proves (13) for \(BF\). Its coordinate at \(\alpha:L\to F\) is explicitly the diagonal restriction \[ D_L\otimes A_F\longrightarrow D_L, \qquad d\otimes a\longmapsto d\,\bigl(e_L^L\alpha^*a\bigr). \tag{15}\] It is a map of \(D_L\)-algebras. Composition of homomorphisms composes these restrictions, giving naturality. Applying the disjoint-union identity in (8) proves the assertion for every finite groupoid. Only finitely many component automorphism groups occur in any finite diagram, so they can all be maximized before choosing the stage. This proves (ii). ◻

Proof. Base change identifies \(\alpha^*f_!(1)\) with the Burnside class \([S]\) in \(\pi_0 A_L\). Multiplication by \(e_L^L\) selects its full mark \(|S^L|\). This is exactly the evaluation map (15). ◻

Proof. The diagonal \(\Delta:Y\to Y\times Y\) is a finite covering: its homotopy fiber at \((y,y')\) is the finite set \(\operatorname{Iso}_Y(y,y')\). By 7, its transferred unit evaluates at a pair \((\alpha,\beta)\) as the number of isomorphisms between the two continuous functors. Thus \[\Delta_!1=\sum_{[\alpha]}a_\alpha(e_\alpha\otimes e_\alpha)\] in the coordinates for \(Y\times Y\). Composition of integrations, applied to \(\operatorname{pr}_1\Delta=\mathrm{id}_Y\), gives \[(\mathrm{id}\otimes\epsilon_Y)(\Delta_!1)=1.\] At the \(\alpha\) coordinate this reads \(a_\alpha\epsilon_Y(e_\alpha)=1\). Positive integers are invertible in \(\mathcal C_{\mathbb Q}\), proving (17).

Naturality of the evaluation splitting identifies \(f^*\) with pullback along \(\pi_0\mathcal X_\Gamma(Y)\to\pi_0\mathcal X_\Gamma(Z)\). The module identity for \(f_!\) therefore forces \(f_!(e_\alpha)\) to be supported at \([f\alpha]\). If its value there is \(c_\alpha\), composition with integration to the point gives \[\frac{c_\alpha}{a_{f\alpha}}=\frac1{a_\alpha}.\] This proves (18).

For completeness, the homotopy fiber in (19) has objects \((\alpha,\eta)\) with \(\eta:f\alpha\xrightarrow{\sim}\beta\). The contribution of a fixed isomorphism class \([\alpha]\) is the cardinality of the action groupoid \[\operatorname{Iso}(f\alpha,\beta)\mathbin{//}\mathop{\mathrm{Aut}}(\alpha),\] which is \(a_\beta/a_\alpha\) if \(f\alpha\simeq\beta\) and zero otherwise. This computation includes automorphisms in the kernel of \(\mathop{\mathrm{Aut}}(\alpha)\to\mathop{\mathrm{Aut}}(f\alpha)\), so it applies to every map of finite groupoids. ◻

Proof. We first show that \(1+px\) is a unit in \(\pi_0\mathbf 1\) for every \(x\in\pi_0\mathbf 1\). Put \(M=S/p\). In the \(p\)-local stable category, \(M\)-acyclic spectra have \(p\) invertible and hence are rational. They are therefore \(T(l)\)-acyclic for \(l>0\). It follows that every \(T(l)\)-local spectrum is \(M\)-local.

The cofiber triangle \[S\xrightarrow{p}S\xrightarrow{i}M\xrightarrow{\partial}\Sigma S\] gives \(p^2\mathrm{id}_M=0\), including when \(p=2\). Indeed, \((p\mathrm{id}_M)i=ip=0\), so \(p\mathrm{id}_M\) factors as \(a\partial\) for a map \(a:\Sigma S\to M\). Hence \[p^2\mathrm{id}_M=p a\partial=a(\Sigma p)\partial=0.\] After smashing with \(M\), the endomorphism \(1+px\) of \(\mathbf 1\) thus has inverse \(1-px\). Its cofiber is both \(M\)-local and \(M\)-acyclic, so it is zero. This proves the asserted invertibility.

An element of \(\pi_0A_F\) is an endomorphism of the constant unit in \(\mathop{\mathrm{Fun}}(BF,\mathrm{Sp}_{T(l)})\). Evaluation at the unique object of \(BF\) detects equivalences of diagrams. Consequently a cochain whose point restriction is a unit in \(\pi_0\mathbf 1\) is itself a unit in \(\pi_0A_F\).

Suppose \(r\notin\mathbb Z_{(p)}\). Then \(r^{-1}\in p\mathbb Z_{(p)}\), and the integral cochain \[u=1-r^{-1}z\in\pi_0A_F\] has point restriction of the form \(1+px\). It is a unit, whereas \(\chi_\alpha(u)=0\). An algebra map cannot send a unit to zero in the nonzero algebra \(D\), a contradiction. This proves \(r\in\mathbb Z_{(p)}\). If \(z\) has zero point restriction and \(r\notin p\mathbb Z_{(p)}\), then \(r\) is a unit of \(\mathbb Z_{(p)}\); the same cochain \(u\) now has point restriction exactly \(1\), giving the same contradiction. ◻

Proof. The integral \(\epsilon_{BF}(1)\) is an actual element of \(\pi_0\mathbf 1\). By 8, its image in a sufficiently advanced nonzero \(D\) is the scalar \[\sum_{[\alpha]\in\mathop{\mathrm{Hom}}_{\mathrm{cts}}(\Gamma,F)/F} \frac1{|C_F(\mathop{\mathrm{im}}\alpha)|} =\frac{|\mathop{\mathrm{Hom}}_{\mathrm{cts}}(\Gamma,F)|}{|F|},\] where the equality is orbit–stabilizer for conjugation by \(F\). Apply 9 with the trivial group as its \(F\). ◻

Proof. We realize weighted counts of finite \(U\)-sets equivariantly labeled in \(X\) as character values of integral pushforwards. The logarithm of their generating function isolates the transitive sets, whose stabilizers give the open subgroups in (21).

For \(j\geq0\), let \(\mathcal L_j(X)\) be the groupoid of continuous \(U\)-sets \(T\) of size \(j\) equipped with a \(U\)-equivariant map \(T\to X\). Its arrows are the equivariant bijections preserving the labeling. Since \(U\) is finitely generated, it has only finitely many homomorphisms to \(\Sigma_j\), so this groupoid is finite up to equivalence. Set \[ b_j(X)=|\mathcal L_j(X)|_{\mathrm h} =\sum_{[T\to X]}\frac1{|\mathop{\mathrm{Aut}}_U(T\to X)|}. \tag{22}\] Write \(X_{\mathrm{triv}}\) for the same underlying set with trivial action. We first prove \[ b_j(X)\in\mathbb Z_{(p)},\qquad b_j(X)-b_j(X_{\mathrm{triv}})\in p\mathbb Z_{(p)}. \tag{23}\]

Choose \(N\trianglelefteq_o\Gamma\) contained in \(U\) and in the kernel of the action on \(X\), and put \(Q=\Gamma/N\) and \(\overline U=U/N\). The finite groupoid \[Y_j(X)=X^j\mathbin{//}\Sigma_j\] classifies \(j\)-element sets labeled in \(X\); \(\overline U\) acts by changing the labels. Form its coinduction \[Z_j(X)=\operatorname{Coind}_{\overline U}^{Q}Y_j(X).\] This is a finite \(Q\)-groupoid whose underlying groupoid is a product of \([\Gamma:U]\) copies of \(Y_j(X)\). More explicitly, choosing representatives \(r_t\) for \(t\in Q/\overline U\), with \(r_{\overline U}=1\), gives this product model; if \(a r_t=r_{at}u(a,t)\), then \(a\) sends the \(t\)-coordinate to the \(at\)-coordinate using the action of \(u(a,t)\in\overline U\). Let \[\pi_j:E_j(X)=Z_j(X)\mathbin{//}Q\longrightarrow BQ, \qquad z_j(X)=(\pi_j)_!1\in\pi_0A_Q.\] These are a map of finite groupoids and an integral cochain.

Make the same construction for \(X_{\mathrm{triv}}\), using the same \(N\) and \(Q\). For this fixed \(j\), choose a common stage \(L\) sufficiently advanced for \(Q\) and every component automorphism group of both total groupoids \(E_j(X)\) and \(E_j(X_{\mathrm{triv}})\). Then \(D=D_L\ne0\), and inflation gives equivalences \[\mathop{\mathrm{Fun}}(BL,E_j(X))\simeq\mathcal X_\Gamma(E_j(X)), \qquad \mathop{\mathrm{Fun}}(BL,BQ)\simeq\mathcal X_\Gamma(BQ),\] and likewise for \(X_{\mathrm{triv}}\). In particular, the quotient homomorphism \(q:\Gamma\to Q\) factors through \(L\). Factorization gives essential surjectivity of these equivalences, and surjectivity of \(\Gamma\to L\) identifies the intertwining conditions and gives full faithfulness. The equivalences therefore identify homotopy fibers over \(q\). The quotient \(Q\) specifies the original action, while \(L\) also captures actions on the labeled objects. Each coefficient uses finitely many total groupoids, so such a common stage exists by 6.

We can now evaluate both cochains at the character indexed by \(q\) over this same \(D\). By 8, the value of \(z_j(X)\) is the homotopy cardinality of \[ \begin{aligned} \mathcal X_\Gamma(E_j(X)) \mathop{\times}^{h}_{\mathcal X_\Gamma(BQ)}\{q\} &\simeq Z_j(X)^{h\Gamma}\\ &\simeq Y_j(X)^{hU} \simeq\mathcal L_j(X). \end{aligned} \tag{24}\] Here \((-)^{h\Gamma}\) and \((-)^{hU}\) mean continuous equivariant objects, including their coherence isomorphisms. The first equivalence identifies a lift of \(q\), together with its specified identification over \(BQ\), with an equivariant object of the pulled-back bundle. For the second, evaluate an equivariant object at the identity coset. To construct an inverse, choose lifts \(g_t\in\Gamma\) of the representatives \(r_t\), with \(g_U=1\), and place a copy of the \(U\)-equivariant object in each coordinate. The coherence map for \(\gamma\in\Gamma\) from coordinate \(t\) to coordinate \(\gamma t\) is supplied by the \(U\)-equivariance map for \[g_{\gamma t}^{-1}\gamma g_t\in U.\] Its image in \(\overline U\) is \(u(q\gamma,t)\), as required by the product action above. The cocycle identity proves coherence; the same construction on morphisms shows that the two functors are inverse equivalences. Continuity follows from continuity of the \(U\)-equivariance data and the finiteness of \(\Gamma/U\). Finally, equivariance of a labeled \(j\)-element set is precisely a continuous \(U\)-action on that set for which its map to \(X\) is equivariant. Thus \(\chi_q(z_j(X))=b_j(X)\cdot1_D\). The same argument gives \(\chi_q(z_j(X_{\mathrm{triv}}))=b_j(X_{\mathrm{triv}})\cdot1_D\).

After forgetting their \(Q\)-actions, the groupoids \(Z_j(X)\) and \(Z_j(X_{\mathrm{triv}})\) are equivalent: their product factors have the same underlying labeled-set groupoid. Base change along \(*\to BQ\) therefore gives the integral equality \[z_j(X)|_* =\epsilon_{Z_j(X)}(1) =\epsilon_{Z_j(X_{\mathrm{triv}})}(1) =z_j(X_{\mathrm{triv}})|_* \quad\text{in }\pi_0\mathbf 1.\] Apply 9 to \(z_j(X)\) and to the difference of the two classes, evaluated over the same \(D\). This proves (23).

It remains to pass from all labeled sets to transitive ones. Define the formal power series \[B_X(t)=\sum_{j\geq0}b_j(X)t^j\in1+t\mathbb Z_{(p)}[[t]].\] Every finite labeled \(U\)-set is a disjoint union of transitive labeled \(U\)-sets. If a transitive isomorphism type \(a\) has size \(m_a\) and automorphism group of order \(c_a\), then \(k\) copies have automorphism group of order \(c_a^k k!\). This orbit decomposition is the mechanism behind Tamanoi’s automorphism-weighted exponential formulas [18]. Consequently, degree by degree in \(\mathbb Q[[t]]\), \[ B_X(t)=\prod_a\sum_{k\geq0}\frac{t^{k m_a}}{c_a^k k!}, \qquad \log B_X(t)=\sum_a\frac{t^{m_a}}{c_a}. \tag{25}\] There are finitely many types in each degree, which justifies these formal products and sums.

A transitive continuous \(U\)-set has size \(p^h\) for some \(h\), since \(U\) is pro-\(p\). Pointing a transitive labeled set of size \(p^h\) multiplies its homotopy cardinality by \(p^h\) and makes every automorphism trivial. A pointed object is determined, up to unique isomorphism, by the stabilizer \(V\le_oU\) of its distinguished point and its label \(x\in X^V\). It follows that \[ [t^{p^h}]\log B_X(t) =\frac1{p^h}\sum_{\substack{V\le_oU\\{}[U:V]=p^h}}|X^V|. \tag{26}\] In particular this normalization counts actual subgroups, without passing to conjugacy classes.

By (23), \[\frac{B_X(t)}{B_{X_{\mathrm{triv}}}(t)}=1+pA(t) \quad\text{for some }A(t)\in t\mathbb Z_{(p)}[[t]].\] Its formal logarithm lies in \(p\mathbb Z_{(p)}[[t]]\), because \[\log(1+pA(t)) =\sum_{q\geq1}\frac{(-1)^{q+1}p^q}{q}A(t)^q, \qquad q-v_p(q)\geq1.\] Only finitely many terms contribute to any coefficient, and the valuation inequality holds also at \(p=2\). Taking the coefficient of \(t^{p^h}\), using (26) for \(X\) and \(X_{\mathrm{triv}}\), and multiplying by \(p^h\) proves (21). ◻

Proof. For a closed subgroup \(H\leq E\) and \(N\trianglelefteq_o H\), choose \(M\trianglelefteq_o E\) with \(H\cap M\subseteq N\). Then \(H/N\) is a quotient of the subgroup \(H/(H\cap M)\) of \(E/M\). Under the asserted generator bound, for each \(N\trianglelefteq_o H\) the set of tuples in \(H^l\) generating \(H/N\) is nonempty and closed. These sets have the finite intersection property, by replacing finitely many \(N\) by their intersection. Compactness supplies a tuple generating every finite quotient of \(H\), hence topologically generating \(H\).

A topological generator gives a continuous epimorphism \(\mathbb Z_p\twoheadrightarrow Q\) onto any procyclic pro-\(p\) group \(Q\). Every closed subgroup of \(\mathbb Z_p\) is a \(\mathbb Z_p\)-submodule and is either \(0\) or \(p^a\mathbb Z_p\), proving the classification. If \(K\twoheadrightarrow\mathbb Z_p\), choose \(x\in K\) mapping to \(1\). The closed subgroup generated by \(x\) maps isomorphically to \(\mathbb Z_p\): a nonzero closed kernel in a procyclic group would make the quotient finite. This supplies a continuous section.

Finally, the abelianization of a finitely generated pro-\(p\) group is a finitely generated \(\mathbb Z_p\)-module. The elementary structure theorem over the discrete valuation ring \(\mathbb Z_p\) writes it as a finite group times \(\mathbb Z_p^b\). If it is infinite, then \(b>0\). ◻

Proof. By 13, write \(K=N\rtimes_\alpha\mathbb Z_p\), with \(N\) finitely generated. For \(s\geq0\), put \[E_s=(N\rtimes_{\alpha_s}\mathbb Z_p)\times\mathbb Z_p^d, \qquad \alpha_s(z)=\alpha(p^s z), \qquad E_\infty=N\times\mathbb Z_p^{d+1}.\] The group \(E_s\) is isomorphic to the open subgroup \((N\rtimes p^s\mathbb Z_p)\times\mathbb Z_p^d\) of \(K\times\mathbb Z_p^d\), and hence has both stated properties.

We construct a common finite quotient which captures every permutation action of a prescribed bounded degree. Fix \(M\geq1\), and let \[O_M=\bigcap_{\rho\in\mathop{\mathrm{Hom}}_{\mathrm{cts}}(N,\Sigma_M)}\ker\rho.\] There are finitely many such homomorphisms, because \(N\) is finitely generated. Thus \(O_M\) is open and characteristic in \(N\). Choose \(a\geq0\) with \(p^a\geq M\). Every \(p\)-element of \(\Sigma_M\) has order at most \(p^a\), as its order is the largest length of one of its \(p\)-power cycles. Therefore every homomorphism from \(\mathbb Z_p\) to \(\Sigma_M\) kills \(p^a\mathbb Z_p\).

The action \(\alpha\) on the finite group \(N/O_M\) is continuous, so \(\alpha(p^s\mathbb Z_p)\) acts trivially on it for all sufficiently large \(s\). For these \(s\), both \(E_s\) and \(E_\infty\) have the quotient \[ Q_M=(N/O_M)\times(\mathbb Z/p^a)^{d+1}. \tag{29}\] Every homomorphism from either group to \(\Sigma_M\) factors through this quotient: its restriction to \(N\) kills \(O_M\), and its restrictions to the \(d+1\) cyclic coordinates kill \(p^a\mathbb Z_p\). The same assertion holds for every smaller permutation degree, by adding fixed points.

Now fix \(U\leq_o E_\infty\), a finite continuous \(U\)-set \(X\), and \(h\geq0\). If \(X\) is empty, the required congruence is immediate. Otherwise set \(m=[E_\infty:U]\) and take \[M\geq\max\{m,\,m|X|,\,mp^h\}.\] The coset action on \(E_\infty/U\) factors through \(Q_M\), so \(U\) is the inverse image of a subgroup \(\overline U\leq Q_M\). The action on the induced finite set \(E_\infty\times_U X\) also factors through \(Q_M\). In particular, the quotient kernel acts trivially on its fiber \(X\) over \(U\), making \(X\) a \(\overline U\)-set. Let \(U_s\leq E_s\) be the inverse image of \(\overline U\), and give the same set \(X\) the resulting \(U_s\)-action.

Every open subgroup \(V\leq U\) of index \(p^h\) has index \(mp^h\) in \(E_\infty\). Its coset action therefore factors through \(Q_M\), so \(V\) contains the quotient kernel. Exactly the same argument applies to every open subgroup of index \(p^h\) in \(U_s\). Thus the two collections of subgroups are both in bijection with \[\{\overline V\leq\overline U: [\overline U:\overline V]=p^h\}.\] The corresponding fixed-point counts on \(X\) agree term by term. The congruence for \(U_s\) consequently proves the congruence for \(U\). This proves the required assertion for every open subgroup of \(E_\infty\) and every test on it.

For the generator bound, let \(F\) be a finite quotient of \(E_\infty\). Its regular permutation action shows, using \(M\geq|F|\), that the quotient map factors through (29). Thus \(F\) is also a quotient of \(E_s\) for sufficiently large \(s\). Every finite section of \(E_\infty\) is a section of such an \(F\) by 13; it therefore has generator number at most \(l\). ◻

Proof. For \(d=1\) there is exactly one subgroup of each index. For \(d>1\), project a sublattice of \(\mathbb Z_p\times\mathbb Z_p^{d-1}\) onto the first coordinate. If its image is \(p^a\mathbb Z_p\) and its intersection with the remaining coordinates has index \(p^b\), it is determined by that intersection and a lift of \(p^a\) modulo it. There are \(p^b\) choices of lift, and the total index is \(p^{a+b}\). Consequently \[a_j(d)=\sum_{a+b=j}p^b a_b(d-1),\] which gives the displayed generating function by induction.

Write \[\prod_{u=1}^{d-1}(1-p^u t)^{-1} =\sum_{m\geq0}b_m t^m.\] Each \(b_m\) is divisible by \(p^m\), and the generating function gives \(a_j(d)=\sum_{m=0}^j b_m\). These partial sums converge \(p\)-adically. The convergent geometric series in the finite product identify their limit with \(c_d\). Every factor \(1-p^u\), \(u\geq1\), is a \(p\)-adic unit, so \(c_d\) is nonzero. The empty product for \(d=1\) is \(1\). ◻

Proof. Let \(\Phi(K)\) be the intersection of the kernels of homomorphisms \(K\to C_p\). There are finitely many such homomorphisms, so \(F=K/\Phi(K)\) is a finite elementary abelian group. It is nontrivial: a nontrivial pro-\(p\) group has a nontrivial finite \(p\)-group quotient, and every nontrivial finite \(p\)-group has a quotient \(C_p\). We will use the following precise property of \(F\): \[ J\leq K\text{ closed and }J\twoheadrightarrow F \quad\Longrightarrow\quad J=K. \tag{31}\] Indeed, if \(J\) were proper, its image would be proper in some finite quotient of \(K\). A maximal subgroup containing that image is normal of index \(p\), as in every finite \(p\)-group. Its inverse image is the kernel of a map \(K\twoheadrightarrow C_p\) containing \(J\), contradicting surjectivity onto \(F\).

The rational full-mark idempotent \(e_F^F\) is a rational virtual \(F\)-set with mark \(1\) at \(F\) and mark \(0\) at every proper subgroup. Since \(F\ne1\), its underlying virtual cardinality is zero. Choose a positive integer \(D\) clearing its denominators and inflate \(De_F^F\) to \(E=K\times\mathbb Z_p^d\) through \(E\to K\to F\). For a closed subgroup of \(E\), its projection to \(K\) is closed. By (31), the inflated virtual set has mark \(D\) exactly when that projection is all of \(K\), and mark zero otherwise.

Let \(c_h\) be the number of open subgroups of index \(p^h\) in \(E\) projecting onto \(K\). The congruences extend by additivity to integral virtual sets. Applying them to \(De_F^F\) at \(U=E\) gives \[ Dc_h\in p^{h+1}\mathbb Z_{(p)}, \qquad v_p(c_h)\geq h+1-v_p(D). \tag{32}\] In particular, \(c_h\to0\) in \(\mathbb Q_p\).

We compute this count in a second way. Set \(A=\mathbb Z_p^d\). For an open subgroup \(H\leq K\times A\) projecting onto \(K\), put \(V=H\cap K\) and \(W=\operatorname{pr}_A(H)\). Then \(V\trianglelefteq_o K\): a lift in \(H\) of any element of \(K\) normalizes \(V\). Also \(W\leq_o A\), and \(H/V\) is the graph of a unique continuous surjection \[f:W\twoheadrightarrow K/V.\] Conversely, every such triple \((V,W,f)\) gives one subgroup \(H\). Its index is \([K:V][A:W]\). Every open subgroup \(W\) of \(A\) is isomorphic to \(A\), and hence \[ c_h= \sum_{\substack{V\trianglelefteq_o K\\{}[K:V]=p^i,\ i\leq h}} a_{h-i}(d)\,|\mathop{\mathrm{Sur}}_{\mathrm{cts}}(\mathbb Z_p^d,K/V)|. \tag{33}\] There is no automorphism denominator: the actual graph map is uniquely determined by the actual subgroup.

Suppose now that \(K^{\mathrm{ab}}\) were finite. Any \(K/V\) appearing with a nonzero summand in (33) is abelian, so \(V\) contains the closed commutator subgroup. There are therefore only finitely many contributing \(V\). By 15, \[\lim_{h\to\infty}c_h =c_d\sum_{V\trianglelefteq_o K} |\mathop{\mathrm{Sur}}_{\mathrm{cts}}(\mathbb Z_p^d,K/V)|\ne0,\] where the sum has only finitely many nonzero terms and is a positive integer: \(V=K\) contributes \(1\). This contradicts (32), proving the lemma. ◻

Proof of 12. If \(\Gamma=1\), the empty series suffices. Otherwise \(\Gamma\) has infinite abelianization. For if \(\Gamma^{\mathrm{ab}}\) were finite of order \(p^b\), then for every sufficiently large \(r\), \[|\mathop{\mathrm{Hom}}_{\mathrm{cts}}(\Gamma,C_{p^r})|=p^b.\] This follows by decomposing the finite abelianization into cyclic factors, each of which has as many maps to \(C_{p^r}\) as it has elements once \(p^r\) is divisible by its exponent. Equation (27) would then assert \(p^{b-r}\in\mathbb Z_{(p)}\) for arbitrarily large \(r\), a contradiction. Thus 13 supplies an epimorphism \(K_0=\Gamma\twoheadrightarrow\mathbb Z_p\) with kernel \(K_1\).

By 14, \(K_1\times\mathbb Z_p\) satisfies the subgroup congruences and has the same finite-section bound. More generally, after \(d\) steps we obtain \(K_d\times\mathbb Z_p^d\) with these properties. If \(K_d\ne1\), it is finitely generated by 13, and for \(d\geq1\), 16 supplies an epimorphism \(K_d\twoheadrightarrow\mathbb Z_p\). Take its kernel \(K_{d+1}\) and apply 14 again.

If \(K_d\ne1\), it has a quotient \(C_p\), so \(K_d\times\mathbb Z_p^d\) has a quotient \((C_p)^{d+1}\). The section bound therefore forces \(d+1\leq l\). The construction must terminate with \(K_t=1\) for some \(t\leq l\), and \(K_{i-1}/K_i\cong\mathbb Z_p\) at every step. Taking the images of these subgroups in a finite quotient \(P\) gives a subnormal series with cyclic factors. Deleting repetitions proves \(\ell(P,1)\leq l\). ◻

Proof. For \(l\geq1\), 6 constructs a pro-\(p\) group \(\Gamma\twoheadrightarrow P\) with the required finite-section bound. [lem:hom-integrality,thm:subgroup-congruence] give (27) and (28). The conclusion follows from 12.

For \(l=0\), let \(P\ne1\). Rational group cohomology, computed by averaging, gives \((H\mathbb Q)^{BP}\simeq H\mathbb Q\). The transfer class of \(P/1\) acts on these cochains as multiplication by \(|P|\), which is invertible, whereas its full \(P\)-mark is zero. The full-mark summand is consequently zero. The rational categorical-to-geometric fixed-point comparison of 4 gives \(\bigl(U_P(H\mathbb Q)\bigr)_{\mathbb Q}=0\). Cofree completion and geometric fixed points preserve rational objects, since multiplication by every nonzero integer remains an equivalence. Hence \(U_P(H\mathbb Q)=0\). Thus nonvanishing at \(l=0\) forces \(P=1\), as required. ◻

Proof. Write \(G_r=P\times(C_{p^r})^k\). At each level let \(\mathcal S_r\) be the finite nonempty set of all cyclic-quotient subnormal chains from \(1\) to \(G_r\) of length exactly \(l\), allowing repetitions. Projection induces maps \(\mathcal S_{r+1}\to\mathcal S_r\). There is a compatible choice of chains: in the compact product of the finite sets \(\mathcal S_r\), any finite collection of compatibility conditions is satisfied by projecting a chain from the largest level mentioned. The finite intersection property applies even though the maps between the sets of chains need not be surjective.

Denote the selected chain at level \(r\) by \((H_{i,r})_{0\leq i\leq l}\). Compatibility means that every map \(H_{i,r+1}\to H_{i,r}\) is surjective. The inverse limits give closed subgroups \[1=H_0\trianglelefteq H_1\trianglelefteq\cdots \trianglelefteq H_l=P\times\mathbb Z_p^k.\] Moreover, \[H_i/H_{i-1}\cong \varprojlim_r H_{i,r}/H_{i-1,r}.\] To see surjectivity of this comparison, take a compatible tuple of cosets. Its nonempty finite sets of representatives at the finite levels have compatible choices by compactness, giving a representative in \(H_i\). The kernel is \(H_{i-1}\). Each displayed inverse limit is procyclic: the surjective transition maps of its cyclic finite quotients send generators to generators, so compatible generators exist by the same compactness argument, and their resulting closed cyclic subgroup surjects to every level.

Let \(A_i\) be the image of \(H_i\) in \(\mathbb Z_p^k\), and set \(J_i=H_i\cap P\). These subgroups are closed, and there is an exact sequence \[ 1\longrightarrow J_i/J_{i-1} \longrightarrow H_i/H_{i-1} \longrightarrow A_i/A_{i-1}\longrightarrow1. \tag{34}\] For the kernel assertion, an element of \(H_i\) projecting into \(A_{i-1}\) can be multiplied by an element of \(H_{i-1}\) with the same projection; the resulting element belongs to \(J_i\), and \(J_i\cap H_{i-1}=J_{i-1}\).

Each \(A_i\) is a \(\mathbb Z_p\)-submodule of \(\mathbb Z_p^k\). The procyclic quotient \(A_i/A_{i-1}\) has rank at most one, and the rank increments add to \(k\). Exactly \(k\) steps therefore increase rank. At any such step, \(H_i/H_{i-1}\) is an infinite procyclic pro-\(p\) group, hence \(\mathbb Z_p\) by 13. Its finite subgroup \(J_i/J_{i-1}\) is trivial. Thus \(J_i=J_{i-1}\) at each of these \(k\) steps.

The \(J_i\) form a subnormal chain from \(1\) to \(P\). Their factors embed in the procyclic factors of the \(H_i\) and, being finite, are cyclic. Deleting the \(k\) repetitions just found leaves a cyclic-quotient chain of length at most \(l-k\), proving the claim. ◻

Proof. The assertion at \(k=0\) is immediate. For \(k>0\), choose a positive-degree periodic self-map \(v:\Sigma^dV\to V\) of a finite type-\(k\) spectrum whose telescope represents \(T(k)\). Suppose that \(K(k)\wedge R=0\). In the associative ring spectrum \[A=\mathop{\mathrm{End}}(V)\wedge R\] consider the element \(x=v\otimes1\in\pi_dA\). Its image in \(K(j)_*A\) is nilpotent at every finite height \(j\ne k\), since the corresponding image of \(v\) is already nilpotent. At height \(k\) it is zero by hypothesis. The mod-\(p\) Hurewicz image of \(v\) is nilpotent as well: it has positive degree in the bounded graded ring \(H_*(\mathop{\mathrm{End}}(V);\mathbb F_p)\). This remains true after mapping to \(H_*(A;\mathbb F_p)\).

The ring form of the nilpotence theorem therefore gives \(x^N=0\) for some \(N\) [12]. Here the tests include rational homology and mod-\(p\) homology, and the theorem imposes no boundedness condition on \(A\). Acting on \(V\wedge R\) shows that \((v\wedge1_R)^N\) is null. Its telescope is consequently zero, so \(T(k)\wedge R=0\), as required. ◻

Proof. The category \(\mathrm{Sp}_{K(k)}\) is stable, \(p\)-local, and \(1\)-semiadditively symmetric monoidal [5]. Apply the torsion-nilpotence theorem [5] with its coalgebra equal to the unit and its commutative algebra equal to \(B\). It says that every torsion element of \(\pi_0B\) is nilpotent. If the unit were torsion, it would be zero. The unitality identity would then make \(\mathrm{id}_B\) zero, contradicting \(B\ne0\). ◻

Proof. By 4, it suffices to show that the full-mark projector on \(B^{BC}\) is nonzero after rationalizing morphisms. We will detect it by computing its trace over \(E\) and then extending scalars to \(B\).

With degree-zero coordinates for the formal group, the complex-oriented cyclic calculation gives \[E^0(BC)=E_0[[x_1,\ldots,x_k]]/ ([p^r](x_1),\ldots,[p^r](x_k)).\] It is finite free of rank \(p^{rk^2}\) over \(E_0\), and \(E^*(BC)\) is concentrated in even degrees [10]. Choosing representatives for a degree-zero basis gives an equivalence \[E^{BC}\simeq E^{\oplus p^{rk^2}}\] of \(E\)-modules. Thus the categorical trace of an endomorphism of \(E^{BC}\) is its ordinary matrix trace over \(E_0\).

Lubin–Tate theory satisfies the complete-local and finite-height hypotheses of the generalized character theorem [10]. After a faithfully flat extension of \(E_0[1/p]\), the character isomorphism identifies the base change of \(E^0(BC)\) with functions on \(\mathop{\mathrm{Hom}}_{\mathrm{cts}}(\mathbb Z_p^k,C)\). A finite \(C\)-set \(S\) acts at a character \(\alpha\) by \(|S^{\mathop{\mathrm{im}}\alpha}|\), by the transfer formula [10]. Consequently the full mark \(e_C^C\) is the projector retaining precisely the surjective characters. Its trace is \[ \mathop{\mathrm{tr}}_E(e_C^C\mid E^{BC}) =N_{k,r}\cdot1, \qquad N_{k,r}=p^{k^2(r-1)}\prod_{i=0}^{k-1}(p^k-p^i)>0 \tag{35}\] in \(E_0\otimes\mathbb Q\). Indeed every homomorphism to \(C\) factors through \((\mathbb Z/p^r)^k\); it is onto precisely when its matrix reduces to an invertible matrix over \(\mathbb F_p\). Each such matrix has \(p^{k^2(r-1)}\) lifts. Faithful flatness descends this trace equality from the character extension to \(E_0[1/p]\).

Now apply extension of scalars \[B\widehat\otimes_E(-): \mathop{\mathrm{Mod}}_E(\mathrm{Sp}_{K(k)})\longrightarrow\mathop{\mathrm{Mod}}_B(\mathrm{Sp}_{K(k)}).\] Both module categories are \(1\)-semiadditive: their finite-groupoid norms are computed after forgetting the module structure, where they are equivalences. Extension of scalars preserves colimits and hence finite-groupoid limits [5]. We obtain the canonical equivalence \[ B\widehat\otimes_E E^{BC}\simeq B^{BC}. \tag{36}\] It respects the cochain algebra structures, restrictions, and finite-cover transfers; compatibility of the canonical norms with such functors gives the transfer assertion [5]. In particular, it carries the Burnside projector over \(E\) to that over \(B\).

This symmetric monoidal functor preserves categorical traces of dualizable objects. After tensoring morphism groups with \(\mathbb Q\), (35) therefore becomes \[\mathop{\mathrm{tr}}_B(e_C^C\mid B^{BC})=N_{k,r}\cdot1_B \quad\text{in }\pi_0B\otimes\mathbb Q.\] This is nonzero by 20. The projector is thus a nonzero endomorphism in the rationalized module category. It acts by multiplication by \(a=e_C^C\cdot1\in\pi_0(B^{BC})\otimes\mathbb Q\), so \(a\ne0\). The rational full-mark comparison of 4 identifies this element with the unit of \((U_C(B))_{\mathbb Q}\), which proves the proposition.

All tensor products and groupoid limits in (36) were formed before rationalizing morphisms. The last step concerns the ordinary rational homotopy of the already formed cofree spectrum. ◻

Proof. Write \(G=P\times C\). For a genuine \(G\)-spectrum \(X\), let \(\operatorname{res}_{\mathrm{mix}}X\in\mathop{\mathrm{Fun}}(BC,\mathrm{Sp}^P)\) be its restriction to a complete \(P\)-universe with trivial \(C\)-action. Let \(\operatorname{res}_C\) take a genuine \(C\)-spectrum to its underlying coherent \(C\)-diagram. We first justify the natural comparison \[ \operatorname{res}_C(\Phi^P X) \simeq\Phi^P(\operatorname{res}_{\mathrm{mix}}X). \tag{37}\] On the left, geometric fixed points retain genuine quotient symmetry; on the right, they are applied pointwise in \(\mathop{\mathrm{Fun}}(BC,\mathrm{Sp}^P)\).

Let \(\rho_P,\rho_C\) be the real regular representations, and let \(\overline\rho_P\) be the reduced regular representation of \(P\). Put \(V=\overline\rho_P\otimes1_C\) and \(W=\overline\rho_P\otimes\rho_C\). For every \(H\le G\), \[\dim W^H=[C:\operatorname{pr}_C H]\bigl([P:H\cap P]-1\bigr).\] Thus \(S^{\infty W}\) is the localization sphere for the family of subgroups not containing \(P\). The invariant line \(1_C\to\rho_C\) induces a unital localization comparison \(S^{\infty V}\to S^{\infty W}\) which is a \(P\)-equivalence. After restriction to \(\mathop{\mathrm{Fun}}(BC,\mathrm{Sp}^P)\), this map is an equivalence and compares the two isotropy-separation formulas. For the right side of (37), smash with \(S^{\infty V}\) and take derived categorical \(P\)-fixed points pointwise. For the left side, use \(S^{\infty W}\) and evaluate the normal geometric-fixed-point formula on trivial \(C\)-representations. The comparison of spheres therefore gives (37), retaining the coherent \(C\)-action and the commutative algebra structures.

Now set \(Q=\Phi^P\mathop{\mathrm{Bor}}_G(D)\), retaining its genuine \(C\)-structure. Model the cofree algebra as \[\mathop{\mathrm{Bor}}_G(D)=F((EP\times EC)_+,\operatorname{inf}_G D).\] Based-space cotensors are computed levelwise, so restriction to the chosen \(P\)-universe identifies its mixed restriction with \(F_P((EP\times EC)_+,\operatorname{inf}_P D)\), including the coherent \(C\)-action. Projection \(EP\times EC\to EP\) induces a map of commutative algebras \[\operatorname{const}_C\mathop{\mathrm{Bor}}_P(D) \longrightarrow\operatorname{res}_{\mathrm{mix}}\mathop{\mathrm{Bor}}_G(D).\] Forgetting \(C\), the projection is a \(P\)-equivalence: the spaces are contractible at the trivial subgroup and have empty fixed points at every nontrivial subgroup of \(P\). The displayed map is therefore an equivalence in \(\mathop{\mathrm{Fun}}(BC,\mathrm{Sp}^P)\). Applying pointwise geometric \(P\)-fixed points and (37) gives an equivalence of commutative algebras \[\operatorname{res}_C Q\simeq\operatorname{const}_C U_P(D) \quad\text{in }\mathop{\mathrm{Fun}}(BC,\mathrm{Sp}).\] Now apply \(\Phi^C\) to the unit \(Q\to\mathop{\mathrm{Bor}}_C(\operatorname{res}_C Q)\) of the underlying/cofree adjunction. Transitivity of geometric fixed points identifies its source with \(U_{P\times C}(D)\) and its target with \(U_C(U_P(D))\). The construction takes place in commutative algebras, so the map preserves units. ◻

Proof. The case \(k=0\) is 17. For \(l=0\), the cofree rational calculation gives \(U_P(H\mathbb Q)=0\) for \(P\ne1\), while \(U_1(H\mathbb Q)=H\mathbb Q\); the latter is killed by every positive-height telescope. Hence we may assume \(k,l>0\).

Put \(R=U_P(L_{T(l)}S)\). This is an ordinary commutative ring spectrum, and 19 gives \(K(k)\wedge R\ne0\). Set \[B=L_{K(k)}(E_k\wedge R).\] The algebra \(B\) is nonzero. To check this without any commutativity assumption on Morava \(K\)-theory, use the graded-field module decomposition [12] \[K(k)\wedge E_k\simeq\bigvee_{a\in I}\Sigma^{d_a}K(k).\] Here \(I\ne\varnothing\), since \(E_k\) is nonzero and \(K(k)\)-local. Smashing with \(R\) leaves a nonempty wedge of nonzero spectra, so \(K(k)\wedge E_k\wedge R\ne0\). Localization preserves this homology. Moreover, \(B\) is a commutative \(E_k\)-algebra in the local symmetric monoidal category, and the natural map \(R\to B\) is unital.

For each \(r\ge1\), take \(C=(C_{p^r})^k\). 22 and the map \(R\to B\) give unital maps \[U_{P\times C}(L_{T(l)}S) \longrightarrow U_C(R)\longrightarrow U_C(B).\] The target has a nonzero rational unit by 21. The source unit therefore survives rationalization as well. Applying 17 to \(P\times C\), we obtain \[\ell(P\times(C_{p^r})^k,1)\le l\qquad(r\ge1).\] By 18, these inequalities imply \(k+\ell(P,1)\le l\). ◻

Proof. Fix \(k\ge0\) and write \(d=\ell(P,1)\). We induct on \(m\), retaining all values of \(k\), including \(k>m\). At \(m=0\), the calculation \(U_P(H\mathbb Q)=0\) for \(P\ne1\) proves the assertion when \(d>0\). If \(P=1\), the required cases have \(k>0\), and \(T(k)\wedge H\mathbb Q=0\).

Let \(m>0\) and assume \(m<k+d\). Apply the exact functor \(U_P\) and then smash with \(T(k)\) in the sphere fracture square (5). The corner \(U_P(L_{m-1}^fS)\) vanishes after this test by induction. The corner \(U_P(L_{T(m)}S)\) vanishes by 23.

For the overlap, smashing finite localization makes \(L_{m-1}^fL_{T(m)}S\) a module over \(L_{m-1}^fS\). Lax monoidality therefore makes its image under \(U_P\) a unital module over \(U_P(L_{m-1}^fS)\). If \(T\wedge A=0\) and \(M\) is any unital \(A\)-module, the unit and action exhibit \[M\longrightarrow A\wedge M\longrightarrow M\] as a retraction, proving \(T\wedge M=0\). Thus the overlap also vanishes after smashing with \(T(k)\). Three corners of the resulting pullback square are zero, so its remaining corner is zero. This completes the induction. ◻

Proof. First work before chromatic localization. With \(N=N_J\), \(W=W_J\), factor the functor as \[\mathrm{Sp}^G\xrightarrow{\operatorname{Res}_N^G}\mathrm{Sp}^N \xrightarrow{\Phi_N^J}\mathrm{Sp}^W \longrightarrow\mathop{\mathrm{Fun}}(BW,\mathrm{Sp}).\] Here the middle functor retains genuine quotient symmetry. Let \(\mathscr F_J=\{L\le N:J\not\le L\}\), and let \(\widetilde E\mathscr F_J\) be the cofiber of \((E\mathscr F_J)_+\to S\). Normal geometric localization identifies \(\mathrm{Sp}^W\) with the full subcategory of genuine \(N\)-spectra whose geometric fixed points vanish at subgroups not containing \(J\) [2]. Its fully faithful right adjoint is \[\rho_J(Y)=\widetilde E\mathscr F_J\otimes\pi^*Y, \qquad \pi:N\longrightarrow W,\] where \(\pi^*\) is genuine inflation. Consequently the required right adjoint before chromatic localization is the explicit composite \[ F_JM=\operatorname{Coind}_{N}^{G}\bigl(\rho_J\mathop{\mathrm{Bor}}_W(M)\bigr). \tag{42}\]

For completeness, compute its geometric \(K\)-fixed points. Coinduction equals induction for finite groups. The fixed-orbit formula therefore gives one term for each coset \(gN\in(G/N)^K\). Such a coset satisfies \(g^{-1}Kg\le N\). The factor \(\widetilde E\mathscr F_J\) kills this term unless \(J\le g^{-1}Kg\). Thus the surviving cosets correspond bijectively to conjugates \(J'=gJg^{-1}\) normal in \(K\). Above such a subgroup, normal geometric inflation has the quotient fixed-point formula; restriction of a cofree spectrum is again cofree. Its coefficient diagram is the restriction \(M_g\) specified in the statement, so the resulting term is \(\Phi^{K/J'}\mathop{\mathrm{Bor}}_{K/J'}(M_g)\). For the constant diagram \(E\), this is exactly \(U_{K/J'}(E)\), proving (40). If the index set is empty, the fixed points vanish for every \(M\).

All these constructions retain the residual actions. More explicitly, \(N_G(K)\) acts on the surviving cosets by left translation, or equivalently on the normal conjugates \(J'\) by conjugation. It transports each term by the conjugation and restriction maps in (42); the action factors coherently through \(W_K\). Formula (40) records the underlying spectrum of this diagram. At \(K=J\), a surviving conjugate \(J'\le J\) has the same order as \(J\), so \(J'=J\). Only the identity coset of \(N\) remains. The counit \(\phi_JF_JM\to M\) then identifies with the cofree counit on underlying \(W_J\)-diagrams and is an equivalence. This proves full faithfulness, including when \(J\) is not normal in \(G\).

The unlocalized right adjoint already takes \(E\)-local diagrams to \(\mathcal C\). Indeed, for such an \(M\) and every genuine \(G\)-spectrum \(X\), adjunction gives \[\begin{align*} \mathop{\mathrm{Map}}(e\otimes X,F_JM) &\simeq\mathop{\mathrm{Map}}(E\otimes\phi_JX,M)\\ &\simeq\mathop{\mathrm{Map}}(\phi_JX,M) \simeq\mathop{\mathrm{Map}}(X,F_JM). \end{align*}\] These equivalences identify precomposition with the localization map \(X\to e\otimes X\); this is exactly the criterion for \(F_JM\) to be local. Hence the adjunction and its formulas restrict to the categories under consideration.

In the restricted adjunction, duality gives, for \(X\in\mathcal C\), \[\begin{align*} \mathop{\mathrm{Map}}(X,z\otimes F_JM) &\simeq\mathop{\mathrm{Map}}(z^\vee\otimes X,F_JM)\\ &\simeq\mathop{\mathrm{Map}}(\phi_J(z^\vee)\otimes\phi_JX,M)\\ &\simeq\mathop{\mathrm{Map}}(\phi_JX,\phi_Jz\otimes M). \end{align*}\] This proves (41). It uses dualizability in \(\mathcal C\), and places no finiteness condition on \(M\). The projection equivalence also identifies the unit on \(z\otimes w\) with \(z\) tensored with the unit on \(w\): their adjoints agree by the triangle identity \(\varepsilon_{\phi_Jw}\circ\phi_J(\eta_w)=\mathrm{id}_{\phi_Jw}\). Finally, the right adjoint of a strong symmetric monoidal functor is lax symmetric monoidal; its unit value is therefore a commutative algebra and each of its values is a unital module over that algebra. ◻

Proof. Let \(X\) be the fiber of \(z\to Q\). By the projection formula, \[X=z\otimes w_0,\qquad w_0=\operatorname{fib}(\mathbf 1_{\mathcal C}\to \mathcal A_B).\] Full faithfulness of \(F_B\) gives \(\phi_BX=0\).

List the other subgroup conjugacy classes as \(J_1,\ldots,J_t\), in nondecreasing order of subgroup size. Set \(X_0=X\) and inductively form fiber sequences \[ X_s\longrightarrow X_{s-1}\longrightarrow Y_s:=F_{J_s}\phi_{J_s}X_{s-1} \qquad(1\le s\le t). \tag{45}\] We verify that the process preserves three properties: \(X_s=z\otimes w_s\) for some \(w_s\), its \(B\)-component is zero, and its \(J_1,\ldots,J_s\)-components are zero.

If \(X_{s-1}=z\otimes w_{s-1}\), the projection formula identifies \[Y_s\simeq z\otimes F_{J_s}\phi_{J_s}w_{s-1}.\] It also identifies the map to \(Y_s\) with \(z\) tensored with the adjunction unit on \(w_{s-1}\). Thus its fiber has the same form. The \(J_s\)-component of the fiber vanishes by full faithfulness. A previously removed subgroup has smaller order, or the same order in a different conjugacy class. It cannot contain a conjugate of \(J_s\), so 25(i) shows that its component of \(Y_s\) is zero. Its vanishing therefore persists in the fiber.

To check the \(B\)-component, put \(M_s=F_{J_s}\phi_{J_s}w_{s-1}\). This is an \(\mathcal A_{J_s}\)-module. For any unital \(A\)-module \(M\), the unit and action exhibit \(M\) as a retract of \(A\otimes M\). Consequently \(\phi_Bz\otimes\phi_BM_s\) is a retract of \[(\phi_Bz\otimes\phi_B\mathcal A_{J_s})\otimes\phi_BM_s=0\] by (44). Hence \(\phi_BY_s=0\), and \(\phi_BX_s=0\) follows. This argument applies to the possibly noncompact objects \(w_s\) and \(Y_s\).

At the end every subgroup component of \(X_t\) is zero; geometric detection gives \(X_t=0\). On the other hand, (43) and adjunction give, for each \(s\), \[\mathop{\mathrm{Map}}(Q,Y_s) \simeq\mathop{\mathrm{Map}}(\phi_{J_s}Q,\phi_{J_s}X_{s-1})=0.\] Here \(\mathop{\mathrm{Map}}\) denotes the mapping spectrum, so this vanishing holds in every degree. Applying \(\mathop{\mathrm{Map}}(Q,-)\) to (45) and descending from \(X_t=0\) gives \(\mathop{\mathrm{Map}}(Q,X)=0\). In particular the boundary \(Q\to\Sigma X\) in the original fiber sequence is zero, proving the splitting.

The stated component vanishings follow from construction and (43). They imply \(\Phi^K(X\otimes Q)=0\) for every \(K\), hence \(X\otimes Q=0\). When \(z\) is compact, its retracts \(X\) and \(Q\) are compact as well. ◻

Proof. There is no point \((B,k)\in\mathop{\mathrm{supp}}(z)\) with \(k<m\): otherwise the second hypothesis would contradict the first. Thus only the test at height \(m\) can be nonzero on \(\Phi^Bz\).

Fix \(K\not\sim B\). In a summand of \(\Phi^K\mathcal A_B\), let \(B'\sim B\) be normal in \(K\), and put \(d=\ell(K,B')=\ell(K/B',1)\). The equality follows by the subgroup correspondence for \(K\to K/B'\). Since \(K\not\sim B\), this quotient is nontrivial and \(d>0\). Consider a height \(0\le k\le m\) with \((K,k)\in\mathop{\mathrm{supp}}(z)\). If \(k=m\), then \(m<k+d\). If \(k<m\), concatenate cyclic subnormal chains and use conjugacy invariance to obtain \[m-n<\ell(G,B') \le\ell(K,B')+\ell(G,K) \le d+k-n.\] Again \(m<k+d\), so 24 gives \(T(k)\wedge U_{K/B'}(E)=0\).

At a height outside \(\mathop{\mathrm{supp}}(z)\), the finite representative of \(z\) instead gives \(T(k)\wedge\Phi^Kz=0\), by 3. Thus every retained telescope kills \(\Phi^K(z\otimes \mathcal A_B)\). This spectrum is \(L_m^f\)-local, so it is zero. Forgetting a coherent action detects zero, which proves (43).

For \(J\not\sim B\), every summand of \(\Phi^B\mathcal A_J\) comes from a proper normal subgroup \(J'\sim J\) of \(B\). Since \(\ell(B/J',1)>0\), 24 at \(k=m\) kills that summand after smashing with \(T(m)\). All smaller-height tests are killed by the \(z\)-factor. The same detection argument proves (44). ◻

Proof. Fix \(G,n\) and put \(y=(G,n)\). Any specialization \((H,m)\) of \(y\) has \(m\ge n\). Indeed, if \(m<n\), inflate a finite spectrum of exact type \(m+1\). It belongs to \(\mathcal P_m(H)\) and not to \(\mathcal P_n(G)\), contradicting (7).

Suppose that the theorem fails. Choose the least violating height \(m\) and a violating subgroup \(B\) at that height. Thus \[(B,m)\in\overline{\{y\}},\qquad m-n<\ell(G,B),\] and \(B\ne G\). By 3, the spectrum \(X_m=\mathop{\mathrm{Spc}}((\mathcal C_m^G)^c)\) is a finite subspace containing \(y\) and \((B,m)\). Let \(Z=\overline{\{y\}}\cap X_m\), the closure of \(y\) in that subspace.

We construct one compact object with support exactly \(Z\), keeping a finite genuine representative. For every \(q\in X_m\setminus Z\), reverse prime inclusion gives \(q\not\subseteq y\). Choose a finite genuine spectrum \(a_q\in q\setminus y\). Its support contains \(y\), hence all of \(Z\), and excludes \(q\). The finite tensor product \[a=\bigotimes_{q\in X_m\setminus Z}a_q,\qquad z=e\otimes a\] (with the unit as empty product) satisfies \(\mathop{\mathrm{supp}}(z)=Z\). It is compact and dualizable, and its displayed finite representative allows us to apply 27.

Minimality of \(m\) shows that every \((K,k)\in Z\) with \(k<m\) satisfies \(\ell(G,K)\le k-n\). Therefore [lem:layer-orthogonality,lem:layer-splitting] give \[z\simeq X\oplus Q,\qquad Q=F_B\phi_Bz,\qquad X\otimes Q=0,\] with both summands compact, \(\phi_BX=0\), and \(\phi_KQ=0\) for \(K\not\sim B\). Their compact supports are disjoint closed subsets whose union is \(Z\). Both are nonempty: \((B,m)\in\mathop{\mathrm{supp}}(Q)\), since \(\phi_BQ\simeq\phi_Bz\), and \(y\in\mathop{\mathrm{supp}}(X)\), since \(B\ne G\) and \(\phi_GQ=0\). This contradicts irreducibility of the closure of a point. ◻

Proof of 1. For an integer \(r\ge0\), the universal implication defining \(r_n(G,H)\) is precisely \[\mathcal P_{n+r}^G(H)\subseteq\mathcal P_n^G(G).\] By (7) and 28, this inclusion forces \(r\ge\ell(G,H)\). Equivalently, for every smaller \(r\), its failure supplies a finite genuine spectrum in the first prime and outside the second, witnessing failure of the proposed implication. 2 supplies the opposite inequality. Thus \(r_n(G,H)=\ell(G,H)\) for all the stated groups, subgroups, and heights. The rational height \(n=0\) has been retained throughout, and when \(H=G\) both sides are zero. ◻

Proof. Balmer–Sanders [2] reduce the displayed inclusion to the existence of such a conjugate and the inclusion \(\mathcal P_m^{H/N}(K^g/N)\subseteq \mathcal P_n^{H/N}(H/N)\). Their prime indices are \(m+1,n+1\). The common \(p\)-residual condition is equivalent to \(p\)-subnormality of \(K^g\) in \(H\), and \(H/N\) is a \(p\)-group. The relative specialization bound and the cyclic upper bound give precisely the stated inequality. When \(m<n\), an inflated finite spectrum of type \(m+1\) rules out the reduced inclusion, as in the proof of 28. For the upper direction at \(m>n+\ell(H/N,K^g/N)\), use also the nesting of the Morava-acyclic ideals of finite spectra. ◻