Counterexamples to weak chromatic splitting: sphere kernels and descent exponents

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

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 2.1 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

Both commute by naturality of localization units. Equivalently,

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. Apply Lemma 3.2 to the ordinary spectral sequence just described. The class of \eta_*(\beta_1)^{h^2} in F^{2h^2}\pi_{dh^2}Y/F^{2h^2+1}\pi_{dh^2}Y is

by (2) and (3). Hence the homotopy class itself is nonzero. Since \eta is a ring map, this power is exactly \eta_*(\beta_1^{h^2}), the image of the specified sphere product.□

Proof of Theorem 3.1. The spectrum Y is K(h)-local, since E_h is local and local objects are closed under homotopy limits. Its unit therefore factors through L_{K(h)}S, where the image of z_p must be nonzero by Proposition 3.3. Lemma 2.1 makes L_{K(h)}(S\to X) an equivalence; naturality carries that nonzero image to L_{K(h)}X. For every r\geq h, the same lemma factors the detecting unit through S\to X\to L_rX\to Y. Its nonzero value on z_p proves the remaining assertion.□

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

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 \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,

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

Lemma 4.1 kills its positive Q-columns, giving

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

Both outer terms vanish for s>D+1. At the remaining boundary degree, the same sequence gives

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 \Q_p of dimension a^2, so it is compact p-adic analytic of that dimension [24, Sections 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 \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 \cd_p(\St_a)=a^2 for discrete p-primary coefficients [29, Section 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/\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 [7, 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 8.1 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=\Gal(k_a/\F_p) acts semilinearly by the Witt lift. At a=p, Lemma 4.2 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 \St_p.

At a=p+1 with p\geq7, the Galois quotient has order p+1, prime to p. An order-p element of \G_a would therefore lie in \St_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_{\cts}(\G_a,M_j)=0 for s>a^2. Apply only the inverse-limit part of Lemma 4.2 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. Suppose a map f_s:T_s\to T'_s over V has been constructed, and put g_s=q'_sf_s. Choose a retract

and a splitting \sigma_s:E\otimes Q_s\to E\otimes T_s with \sigma_s(1_E\otimes q_s)=1. The composite

extends g_s along q_s. Indeed, unit naturality gives \eta_{Q_s}q_s=(1_E\otimes q_s)\eta_{T_s}. The splitting removes 1_E\otimes q_s, and a second use of unit naturality followed by the module unit law gives

Complete this square to a map of fiber triangles to obtain f_{s+1}:T_{s+1}\to T'_{s+1}. Starting with f_0=\mathrm{id}_V gives maps over V at every finite stage, and hence one inclusion between the stage-s images in [W,V]. Repeating the construction in the other direction proves equality. The comparison begins at stage zero on V, so the indices agree. Only these finite-stage maps are used; no equivalence of towers is asserted.□

Proof. Devinatz–Hopkins give (6) with strong convergence and identify it with the local E_a-Adams spectral sequence [7, Theorem 1(iii)–(iv)]; the full extended group is open in itself. The coefficient topology is the full maximal-ideal-adic topology of [7, Remark 1.3]. Their ideal (p,v_1,\ldots,v_{a-1}) agrees degreewise with \m_a=(p,u_1,\ldots,u_{a-1}), since v_i=u_i u^{1-p^i} and u is invertible.

The tensor powers T_s=I_a^{\otimes_a s} form resolution triangles

Their right terms are E_a-injective, and multiplication on E_a retracts E_a\otimes_a q_s. Thus they form a relative Adams resolution in the convention just used. By Lemma 5.1, their stage-s images agree with those in the Devinatz–Hopkins resolution at the same index. This proves (7). In particular T_0=B_a and T_1=I_a give the full group and the kernel of \pi_*B_a\to\pi_*E_a, respectively. Strong convergence supplies separation and the stated associated-graded quotients for these actual image groups.

A map from a suspended sphere to a local object is equivalently a map from the suspended local unit. Tensoring two lifts through I_a^{\otimes_a r} and I_a^{\otimes_a s} therefore gives a lift of their product through I_a^{\otimes_a(r+s)}. This proves multiplicativity. Monoidal localization carries the sphere unit and its products to the local unit, so these are also the products of the actual sphere images. Smashing localization is not needed.□

Proof. Proposition 4.3, with the finite test V_0=S, gives

At height p this uses the pure stabilizer dimension and the semilinear Witt trace-one contraction for the order-p Galois quotient. At height p+1 the full extended group has no p-torsion, so its dimension gives the finite-quotient bound. Both cases retain the possible adic inverse-limit contribution in degree a^2+1.

By Lemma 5.2, the successive quotients of the actual filtration from a^2+2 onward are zero. That tail is consequently constant, and separatedness makes its common value zero. This proves (9). It is a statement about homotopy images; the exponent argument will require a separate proof of map nullity.□

Proof. Let a be either upper height in its stated range, and let x_a\in\pi_dB_a be the unit image of x. Positive stable sphere stems are torsion: suspend into an odd-dimensional sphere in the stable range and apply Serre’s finiteness theorem [28, Chapter V, Section 3, Proposition 3]. Since (E_a)_d is torsion-free, x_a maps to zero in \pi_dE_a and belongs to F^1\pi_dB_a.

Since d is even and E_a has only even homotopy,

Thus x_a\in F^2, and multiplicativity in Lemma 5.2 gives x_a^k\in F^{2k}\pi_{dk}B_a. The numerical margins above the last possible filtration are

Therefore 2k\geq a^2+2, and (9) gives x_a^k=0. This is the unit image of the original sphere product x^k, by the product identification in Lemma 5.2.□

Proof of Theorems 1.2 and 1.1, and Corollary 1.3. Use the single named product z_p=\beta_1^{(p-1)^2}. Proposition 3.3 detects it under the actual unit to the K(p-1)-local spectrum Y=E_{p-1}^{hG}. Proposition 5.4, applied to x=\beta_1, kills that same product in B_p for p\geq5 and in B_{p+1} for p\geq7.

Apply Proposition 2.2 with t=p-1 and n=p or n=p+1, respectively. The result is a nonzero image of z_p in \pi_{dk}L_{n-1}X killed by the actual canonical map i_{n,X}. Its two naturality squares identify that map after completion. A retraction would induce a left inverse on this homotopy group, which is impossible.

For p=5 the product is \beta_1^{16} in degree 608, and its upper image lies in F^{32}\pi_{608}B_5, while the zero tail starts at F^{27}. Taking n=5 gives the named prime-five statement as well.□

Proof. Let \mathcal P_{\ne\varnothing}([s]) be the poset of nonempty subsets of \{0,\ldots,s\}. The functor

uses the order-preserving injection induced by each inclusion of subsets. At [l], with l\leq s, the comma category \theta_s/[l] consists of nonempty subsets equipped with a weakly increasing map to [l]. It is the poset of nonempty faces of the simplicial complex P(s,l) whose simplices are lists

This complex is contractible. The base case P(0,0) is a point. Every simplex either omits the first-coordinate value s or contains exactly one vertex (s,j). Begin inductively with the cone of vertex (s,l) on P(s-1,l). This cone is contractible even when l=s, when its base is not covered by induction. For each j<l, attach the cone of vertex (s,j) along its base P(s-1,j). This is its exact intersection with the previous complex and is contractible by induction, since j\leq s-1. The intersections are simplicial subcomplexes, so each attachment preserves contractibility. The functor is therefore homotopy initial.

The partial totalization is consequently the limit of the punctured (s+1)-cube obtained by tensoring copies of \one\to A and then tensoring with V. Its omitted vertex is V. Taking successive fibers and using exactness of tensor gives J^{\otimes(s+1)}\otimes V, with its canonical arrow to V. Removing the last direction of the cube gives the stated transition. This is a genuinely finite cubical limit, so any exact \Phi preserves it. Finiteness of the list of objects in \Delta_{\leq s} alone would not justify that assertion.□

Proof. The unit map of a module splits by its action, making its fiber map null. Naturality gives retract invariance, and tensoring a null map gives tensor invariance. In the cofiber sequence, the composite J^{\otimes c}\otimes V\to V\to W is null, so choose a lift \ell:J^{\otimes c}\otimes V\to U. After precomposition by b more \epsilon factors, naturality identifies the resulting map into U with e_b(U)(1\otimes\ell), which is null. Its composite to V is e_{b+c}(V). Finally the consecutive maps J^{\otimes(i+1)}\otimes V\to J^{\otimes i}\otimes V have the stated cofibers. Their composites to V filter \cofib(e_m(V)) by these m layers. Nullity of e_m(V) makes V a retract of that cofiber.□

Proof. In coordinates (s,d) with d=q-s, the derived couple has

and maps

Here the second E-index denotes total degree; no convergence claim is used. Let c be an actual exponent bound. Every length-c transition in the fiber tower U_l is null, being \Phi applied to a tensor multiple of e_c(V). For s\geq0 the boundary square from (11) is

The image from P_{s+c} has zero boundary and lifts to \pi_dV'; conversely an augmentation class comes from P_{s+c}. Hence

Both sides are zero for negative s. These image groups have surjective transition i. The augmentation into P_s is injective on homotopy for s\geq c-1, since its kernel is the image of a map factoring through e_c(V). Thus i is an isomorphism for s\geq c. Exactness of the derived couple gives j=0 and, for s\geq0, identifies

In particular this is zero for s\geq c. Taking c=b proves (i). For (ii) retain any c. The E_2 line makes these same quotients zero for s\geq b. Since F^c=0, descending from c, or monotonicity when c<b, gives F^b=0. Equation (12) is exactly the asserted zero image. This argument uses only finite cubes and never assumes that \Phi preserves an infinite limit.□

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

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 6.3(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 [26, Section 15, Proposition 15.1 and Corollary 15.4] and [14, Lemma 4.4]; Morava E-theory over BP is included in [16, 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 [16, 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

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

is surjective. To see this, lift the first component to M; its error against the second component lies in \m^jN, and this error lifts from \m^jM. Each quotient is a finite set. Given a continuous map T\to N, first lift its reduction modulo \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_{\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_{\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_{\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 \m-completion, and flat homotopy has no higher derived completion. The natural localization map induces the ordinary completion map [27, 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/\m^lU is degreewise surjective. By flatness of U, its first homology is

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\m^lP_0)/\m^lK. Artin–Rees [30, Tag 00IN] gives a d such that K\cap\m^lP_0\subseteq\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

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/\m^j; for l\geq j the modules on the right are U/\m^jU. This proves the reduction formula.□

Proof of Proposition 8.1. The completed cooperation theorem gives

At g, the map is \mu(1\wedge g); the first factor acts by constant scalars. This is the evaluation map of [14, Section 1 and Theorems 4.11 and 6.8]. The equivalent convention in [7, 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 [7, Remark 1.3].

Put P_*=\pi_*(E\wedge E), regarded as an O_*-module through the first factor. Lemma 8.2 makes P_* flat. The actual localization map, the reduction in Lemma 8.4, and (18) applied to O_*/\m^jO_* give

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

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_*/\m^jO_* is finite. Continuous functions to it are unions of function modules on finite quotient sets of \G_a. Tensors of the function modules on finite sets are the function modules on their products, and every continuous function on \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_{\cts}(\G_a^r,O_*/\m^jO_*). Apply Lemma 8.4 only now, to this ordinary (r+1)-fold smash. The result is

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 8.4 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 8.3, 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

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,

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 \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=\fib(L_aS\to L_{a-1}S), and let I_{\mathrm{BC}} be the ordinary Brown–Comenetz dualizing spectrum for the injective group \Q/\Z_{(p)}. Gross–Hopkins duality, announced in [12, Theorem 6] and proved in the form used here in [15, Theorem 10.2(b),(e)], says that

is K(a)-local and invertible for \otimes_a. The identification with this monochromatic object uses [15, 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 [15, Theorem 7.1]. Invertibility of D_a, followed by ordinary closed adjunction, gives equivalences of ordinary spectra

The displayed smash is ordinary. Put C=D_a\wedge M_aS. Brown–Comenetz representability now gives, naturally in U,

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

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 9.1. 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 [7, Appendix A, Proposition A.3]. This is the local nilpotence consequence of Hopkins–Ravenel; its descendability form is [20, 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 6.2. 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 8.1 identifies its second page with

Proposition 4.3 makes this page zero for s\geq h^2+2=m. The tested-image assertion of Proposition 6.3(ii) applies using the already known finite exponent c. It concerns the full finite partial totalizations and their finite fiber cubes, and gives

Thus the actual arrow e_m(B_h), regarded in ordinary spectra, vanishes after every finite ordinary precomposition. Lemma 9.2 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

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

Finally F_m=\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:

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

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

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

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

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

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

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 [19, 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\Tr(r)=\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

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\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 10.1. Lemmas 10.3 and 10.4 give \bar z\in R with \Tr(\bar z)=1. Lift it to z_0\in\pi_0D, and put v=\sum_g g(z_0). Then

The mixed module \pi_*D is flat over (E_t)_* by Lemma 8.2. Lemma 8.4 therefore identifies the homotopy of its actual localization D\to D' with the ordinary \m_t-adic completion of \pi_*D.

At the finite adic stage \pi_0D/\m_t^j\pi_0D, the element

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 10.2 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\Sp 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 9.1 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 10.1 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 9.1, and add the m layer bounds using Lemma 6.2. 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 2.1 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

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 11.1 and preservation of exponent bounds under retracts would give \expn_H(E_t)\leq m. The finite-page lower result Proposition 7.2 gives the strict inequality \expn_H(E_t)>m, a contradiction.□

Proof of Theorem 1.4. Proposition 7.2 gives the strict lower bound and Proposition 11.1 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 3.2. 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

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 [2, 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

They are nonnegative and each has sum one. Equivalently, their ith 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

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

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

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

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

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

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

Multiplication on R and precomposition give

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 jth output vector, divide the lengths of its intersections with [c_{i-1},c_i] by the length of the jth interval. The preceding naturality and equivariance proof applies verbatim. If k_j is an index whose coordinate interval contains the jth 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

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

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 7. 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.□