Proof. Products and filtered colimits of stable categories preserve the stable structure. To see the mapping formula and the assertion about finite diagrams directly, use simplicial presentations of the categories. Inner horn fillers and any prescribed finite diagram involve only finitely many simplices. Such data therefore pass through the filtered colimit. Mapping spaces are obtained from finite limit tests on these presentations, and filtered colimits of spaces commute with finite limits. This gives the mapping-space version of Equation (2); applying it with every fixed integer suspension gives the assertion for mapping spectra. The same mapping-space tests show that primewise finite limits and colimits have the required universal properties. The tensor product and its coherent finite-arity structure pass through the construction as well. These facts also follow from the general categorical ultraproduct results in the author version of (Barthel et al. 2020, Proposition 3.12, Corollary 3.16, and Lemma 3.17).

For every fixed nonzero integer \(m\), multiplication by \(m\) is an equivalence at every prime not dividing \(m\). It is consequently an equivalence on each mapping spectrum in Equation (2). These spectra are therefore rational. Homotopy groups commute with products of spectra and with filtered colimits, so their groups are the corresponding group ultraproducts. Each fixed positive stable stem of the sphere is finite; this is the stable consequence of Serre’s finiteness theorem (Serre 1951, V, §3, Proposition 3). Its \(p\)-localization therefore vanishes for all but finitely many primes. Negative stable stems vanish, and the zeroth stem gives \(D\). Hence the endomorphism algebra of the unit is the discrete commutative ring spectrum \(HD\).

The inverse of any fixed nonzero integer exists in \(D\), giving the inclusion \(\mathbb Q\subset D\). If \(\varpi a=0\), choose a sequence representing \(a\). On a large set one has \(p a_p=0\) in the domain \(\mathbb Z_{(p)}\), so \(a_p=0\) there. Thus \(a=0\). Finally, compactness of the sequence objects in the Ind-completion is part of the defining Yoneda embedding into that completion. ◻

Proof. The symmetric monoidal equivalence between \(HR\)-module spectra and the derived category of \(R\)-modules is (Lurie 2017, Theorem 7.1.2.13). The commutative differential graded algebra model and its comparison with \(E_\infty\)-algebras require \(\mathbb Q\subset R\), precisely our hypothesis (Lurie 2017, Propositions 7.1.4.10–7.1.4.11). For clarity, the comparison uses a resolution of the commutative operad by an \(R\)-linear \(E_\infty\)-operad. Since the order of each finite symmetric group is invertible in \(R\), its coinvariant functor is exact. Comparison of free algebras is therefore a quasi-isomorphism; the filtration by free cells extends this comparison to cellular resolutions. With the additional \(\Gamma\)-grading the same argument is applied weight by weight, summing the weights of the inputs. Under a prescribed scalar algebra one first chooses a cofibrant model of that algebra and then uses derived base change. All these comparisons are equivalences of categories with their mapping spaces, so they also apply to diagrams and their specified homotopies. ◻

Proof. Let \(F\) be the free commutative \(R\)-algebra on a cycle \(u\) of degree \(r\) and weight \(\eta\), and map \(u\) to \(x\). The required pushout is \(B\otimes_F R\), where \(F\to R\) sends \(u\) to zero. Resolve this augmentation by the free disk algebra \[F[z],\qquad \partial z=u.\] The complex spanned by \(u,z\) is contractible. Every positive symmetric power of this disk is acyclic, since tensor powers are contractible and symmetric-group coinvariants are exact over \(R\). Thus the disk algebra maps by a quasi-isomorphism to \(R\). As an \(F\)-module it has an increasing filtration by the number of copies of \(z\), whose successive quotients are free shifted \(F\)-modules. This filtration computes the derived tensor with \(B\). After base change the result is exactly \(B[z]\) with \(\partial z=x\). The argument respects the specified weight throughout.

If \(r=0\), the variable \(z\) is odd, hence exterior. The underlying module is the two-term extension of \(B\) by the weight-\(\eta\), degree-one copy of \(B\), with differential given by multiplication by \(x\). This is Equation (3). To identify the chosen null without an ambiguity, one can first perform this calculation over the polynomial cycle algebra \(F=R[u]\). Its augmentation is the cofiber of multiplication by \(u\), and its discrete target has no degree-one ambiguity in the null of the generator. Tensoring to \(B\) gives the displayed comparison through the universal null. A map out of the disk algebra sends \(z\) to the chosen target null; on the underlying two-term module its value on a base multiple of \(z\) is therefore that same null multiplied by the image of the base element. ◻

Proof. The Picard space of \(\mathcal D\) is a grouplike \(E_\infty\)-space and hence is the infinite loop space of a connective spectrum \(U\); the recognition statement is (Lurie 2017, Remark 5.2.6.26). A monoidal grading by \(\Gamma\) is equivalently a spectrum map \(H\Gamma\to U\). For \(k\geq2\), \[\pi_k U\cong\pi_{k-1}\mathop{\mathrm{map}}_{\mathcal D}(\mathbf 1,\mathbf 1),\] so \(\tau_{\geq2}U\) is rational.

Put \(Q_s=\mathop{\mathrm{cofib}}(S\to H\mathbb Z)\). The rationalization of \(S\to H\mathbb Z\) is an equivalence. Moreover \(Q_s\) is 2-connective, and \(\pi_2 Q_s=\pi_1^s=\mathbb Z/2\) (Hatcher 2002, Corollaries 4J.3–4J.4). Since maps from a rationally zero spectrum to a rational spectrum vanish, there is an equivalence \[\mathop{\mathrm{map}}(Q_s,U)\simeq\mathop{\mathrm{map}}(Q_s,\tau_{\leq1}U).\] Connectivity and truncation imply that its homotopy groups in degrees at least zero vanish. In degree minus one its only possible contribution is \[\pi_{-1}\mathop{\mathrm{map}}(Q_s,\tau_{\leq1}U) \cong\mathop{\mathrm{Hom}}(\mathbb Z/2,\pi_1 U),\] which is killed by two. The fiber sequence \[\mathop{\mathrm{map}}(Q_s,U)\longrightarrow\mathop{\mathrm{map}}(H\mathbb Z,U) \longrightarrow\mathop{\mathrm{map}}(S,U)=U\] therefore shows that evaluation is an equivalence of spaces onto a union of components of \(\Omega^\infty U\). The obstruction to a component lying in that union takes values in a group killed by two. Every double in \(\pi_0 U\) lies in the image, so a specified tensor square extends to a grading on \(\mathbb Z\).

The equivalence onto a union of components also identifies all path spaces between eligible objects. Hence specified equivalences of the generator objects lift to equivalences of gradings, with their higher compatibilities. Finally, \(H\Gamma\) is the finite direct sum of copies of \(H\mathbb Z\) indexed by the basis. The space of its maps to \(U\) is the corresponding product. Applying the preceding argument factor by factor proves both the assertion and its relative form. ◻

Proof. Mapping out of a fixed sphere is exact. Given two input maps, tensor them and use the chosen equivalence \(S(\alpha)\otimes S(\beta)\simeq S(\alpha+\beta)\); then compose with the target pairing. This defines the pairing at the pair of weights \((\alpha,\beta)\). Monoidality of the sphere grading and separate exactness of tensor make these maps coherent at every finite tuple of weights. Their sums give convolution pairings in the category of graded mapping spectra. This construction requires no infinite coproduct in \(\mathcal C\) itself. The endomorphism algebra of the unit acts centrally through tensor, so the pairings are \(D\)-balanced. If the targets carry another diagram index, the same construction commutes with its structure maps and its additive pairings. ◻

Proof. Invertibility gives \[h_\Omega(S(\eta))_\theta \simeq\mathop{\mathrm{map}}_{\mathcal C}(S(\theta-\eta),\mathbf 1) =A_{\theta-\eta}.\] Thus a sphere generator maps to \(A\langle\eta\rangle\). For any \(X\in\mathcal C\), mapping from this free graded module evaluates \(h_\Omega(X)\) at \(\eta\), and therefore agrees with \(\mathop{\mathrm{map}}_{\mathcal C}(S(\eta),X)\). Exactness extends this comparison from the generators to every finite-cell source. It follows that \(h_\Omega\) is fully faithful on \(\mathcal C_{\mathrm{cell}}\). A finite-cell graded module is realized inductively: lift its attaching maps by full faithfulness and take the corresponding cofibers in \(\mathcal C\). This gives precisely the stated essential image.

The pairings of Lemma 7 are balanced over \(A\). Their coherent balanced comparison is obtained from the tensor bar construction: use the pairings with an arbitrary finite number of inserted \(A\)-factors and then take its realization in graded mapping spectra. If \(X=S(\eta)\), Equation (6) is the weight-shift equivalence supplied by invertibility. Both sides are exact in \(X\), so the comparison remains an equivalence for every finite-cell \(X\), with \(Y\) arbitrary. Symmetry gives the other case. The same bar pairings for several inputs and iterated relative tensors retain compatibility with composition and symmetry. Hence the equivalence on the full tensor-closed finite-cell subcategories is symmetric monoidal. Commutative algebra objects in those full subcategories transport across it, with all their coherence. ◻

Proof. Each variable \(e_h\) is exterior. The underlying \(A\)-module has the finite basis of ordered exterior monomials \(e_S\) for \(S\subseteq\{0,\ldots,j-1\}\). Order these subsets by the largest entry at which they differ. In the differential of \(e_S\), replacing an index \(h\) by indices strictly less than \(h\) decreases this order; terms with a repeated index vanish. The differential therefore provides a finite-cell filtration on this basis. Proposition 9 transports the algebra and its unit to \(Y_j\).

A degree-zero cycle at weight \(w_j\) represents a class in \(\pi_0\mathop{\mathrm{map}}_{\mathcal C}(S(w_j),Y_j)\). Its product with the identity of \(Y_j\) corresponds to multiplication by \(b_j\) on the algebraic module. Lemma 4 identifies the extension by \(e_j\), through its specified null, with the cofiber of this multiplication map. This proves Equation (8), including its unit and cofiber-map identifications. Finally, \(Y_j\) is finite-cell, so Equation (6) with the other input \(M\) gives Equation (9), compatibly with all the algebra maps and specified nulls. ◻

Proof. At an index \(i\), algebraic inversion of \(t\) takes the telescope of \(B(X)_{i+s\tau}\). Moving the factor \(T^{-2s}\) from the source to the target identifies it with the mapping telescope into \(T^{2s}\) times the target. Lemma 11 and compactness identify this with the mapping spectrum in \(\mathcal E'\). These identifications respect multiplication by the coherent tensor powers of \(T^2\). For the second assertion, at \((0,\delta,0)\) the source sphere is \(\mathop{\mathrm{Tr}}S(\delta)\), and the map is simply the one induced by the symmetric monoidal trivial-action functor. ◻

Proof. Apply the exact functor of Lemma 13 to mapping spectra out of \(S(\lambda)\), and then use \[N\bigl(X_p(\lambda)\bigr)\longrightarrow \mathop{\mathrm{Tr}}X_p(p\lambda).\] The latter map is the \(p\)-ary multiplication. It is equivariant, coherently permutation invariant, and compatible with multiplying tuples, by the commutative algebra structure at each prime. It can equivalently be read from the algebra codiagonal, since tensor product is the coproduct of commutative algebras. The exact strong monoidal mapping-spectrum construction and these coherent multiplications give the asserted algebra map. Naturality holds for actual primewise algebra maps. The description on represented classes follows before localization from the same construction. Rational linearity follows from stable enrichment and rationality of the mapping spectra; the operation can change the action of the larger scalar ring \(D\). ◻

Proof. The two-periodic resolution of \(C_p\) gives \[H^*(BC_p;\mathbb F_p)=\Lambda(\eta)\otimes\mathbb F_p[\xi], \qquad |\eta|=1,\quad |\xi|=2.\] The class \(\xi\) is the Bockstein of \(\eta\), or equivalently the reduction of a first Chern class. Under an automorphism of \(C_p\), both classes transform with multiplier weight one, up to replacing the character by its inverse throughout. As a \(C_p\)-representation, \(V\) is a sum of \((p-1)/2\) nontrivial complex lines. Its Euler class is a nonzero scalar times \(\xi^{(p-1)/2}\). The sphere-bundle sequence therefore gives \[H^*(S(V)_{hC_p};\mathbb F_p) =\Lambda(\eta)\otimes \mathbb F_p[\xi]/\bigl(\xi^{(p-1)/2}\bigr).\] Every positive-degree monomial has multiplier weight strictly between zero and \(p-1\). Passage to \(G\) takes invariants under \(\mathbb F_p^\times\), whose order is invertible in \(\mathbb F_p\), so only the degree-zero class remains.

The \(p\)-local homology is finitely generated in every degree, since the group is finite and the sphere is finite dimensional. The universal coefficient sequence and Nakayama’s lemma now imply that its reduced \(\mathbb Z_{(p)}\)-homology vanishes. The augmentation cofiber is bounded below; the stable Hurewicz theorem applied to its first possible nonzero homotopy group makes it contractible. ◻

Proof. For even \(r\), the determinant character of \(rV\) is trivial. After fixing the fiber degree, its classifying map to the classifying space of sphere automorphisms lifts to the simply connected cover. All higher homotopy groups of this cover are \(p\)-local abelian groups. By Lemma 16, reduced cohomology of \(Z_p^{\mathrm{orb}}\) with every such constant coefficient group vanishes. Induction along the Eilenberg–Mac Lane fibers of the Postnikov tower shows that the corresponding based mapping spaces are contractible. Mapping into the limit gives a based null of the classifying map. Dualization handles virtual negative multiples as well.

Maps between free twisted rank-one \(J^\vee\)-modules are sections of the corresponding sphere mapping local system over \(S(V)_{hG}\). Indeed, this follows from free-module adjunction and duality for \(S(V)_+\); composition of these maps is pointwise composition of sections. The trivializations just constructed therefore give the required module equivalences primewise on generators. In \(W_i\), the twisting exponent of \(V\) is \(d(\lambda)-2m\), which is even, and its total underlying dimension is Equation (15). Apply Lemma 6 in the rational category of \(J^\vee\)-modules in \(\mathrm{Ind}(\mathcal C_G)\) to extend the generator equivalences coherently over \(I\). All the required generator objects are squares. The relative assertion in that lemma keeps the \(\delta\) identifications fixed. Finally apply the symmetric monoidal localization. ◻

Proof. For a finite group, the norm is the natural map from homotopy orbits to homotopy fixed points; see (Nikolaus and Scholze 2018, Definition I.1.10 and Example I.1.11). One concrete description uses the group ring spectrum \(\mathcal H=S_{(p)}[G]\). To specify its derived construction, regard \(\mathcal H\) as a \(G\times G\)-spectrum with action \((g,h)\cdot k=gkh^{-1}\). It is induced from the trivial sphere along the diagonal subgroup \(\Delta G\). Finite induction agrees with coinduction, so the unit of restriction–coinduction gives a map from the trivial \(G\times G\)-sphere to \(\mathcal H\). On underlying spectra this is the diagonal into the finite product of spheres. Equivalently, it is the \(\mathcal H\)-bimodule norm-element map that sums all group elements.

Take its derived tensor over the right \(\mathcal H\)-action with \(U_p\). The remaining left action makes the resulting map from the trivially acted-on spectrum \(S_{(p)}\otimes_{\mathcal H}U_p\) to \(U_p\) equivariant. Its adjoint is \[S_{(p)}\otimes_{\mathcal H}U_p \longrightarrow\mathop{\mathrm{map}}_{\mathcal H}(S_{(p)},U_p).\] In particular, composing the inclusion of an underlying fiber with this map and then evaluating at a point sums the group actions. On an induced object \(\mathcal H\otimes Z\), the orbit and fixed-point adjunctions identify both sides with \(Z\); the diagonal entry at the identity element identifies the norm with \(\mathop{\mathrm{id}}_Z\). Both functors are exact, so the same equivalence holds on finite stable extensions of induced objects; compare (Nikolaus and Scholze 2018, Lemma I.3.8).

Apply this to \(E_{b+}\otimes U_p\), whose equivariant cellular filtration consists of free cells. The join triangle, followed by homotopy fixed points, thus identifies its first term with homotopy orbits. Take the ultraproduct and then the colimit in \(b\). Equation (11) identifies the last term, giving Equation (17). ◻

Proof. First omit \(Z_p\). The partial Thom spectrum \((E_{b+}\otimes T_p^{r_p})_{hG}\) has cells in degrees \(0,\ldots,b-1\) relative to \(r_p(p-1)\). This assertion also applies to a virtual representation, after shifting by its virtual dimension. Since \(E_b\) is \((b-2)\)-connected, its Thom homology in relative degrees less than \(b-1\) agrees with group homology with coefficient \(\mathbb F_p(\det^{r_p})\).

For fixed \(b\) and sufficiently large \(p\), these group homology groups consist only of degree zero when \(r_p\) is even, and vanish in this range when \(r_p\) is odd. To see this, use the cyclic-group generators \(\eta,\xi\) from Lemma 16 and take multiplier invariants; the corresponding homology calculation is its degreewise dual. The sign of the multiplier permutation by \(\alpha\) is \(\alpha^{(p-1)/2}\) modulo \(p\), as follows by applying the permutation to the Vandermonde product. Thus the odd determinant twist shifts the required multiplier weight by \((p-1)/2\). For a fixed degree range this weight, and every positive untwisted invariant weight, eventually lies beyond that range.

In the even case, choose the identity vertex in the first join factor at every stage. The fiber-point inclusions then commute with the maps from stage \(b\) to stage \(b+4\), and hence induce maps of their cofibers. Take these cofibers as remainders; in the odd case, keep the entire partial Thom spectrum. For \(b\geq2\) each remainder is a finite \(p\)-local cell spectrum with relative cells in degrees at most \(b-1\), and it has no \(p\)-local homology below degree \(b-1\). Here mod-\(p\) vanishing gives \(p\)-local vanishing because the homology groups are finitely generated. Stable Hurewicz after desuspension therefore gives the same lower connectivity bound. The remainder at \(b+4\) is connective in relative degree \(b+3\), so the transition from the remainder at \(b\) to the remainder at \(b+4\) is null by comparison with the source cell dimensions. This null holds at almost all primes for each fixed \(b\).

The common virtual dimension shift does not affect the argument, even if \(r_p\) varies without bound. Moreover the transition remains null after smashing with any \(Z_p\). Since \(Z_p\) has trivial action, this smash commutes with the homotopy orbit construction. The remainder telescope is consequently zero after the ultraproduct. The compatible fiber-point maps in the even case give the asserted equivalence. ◻

Proof. Dualizing the first Euler triangle gives \[T^{-1}\longrightarrow\mathbf 1\longrightarrow J^\vee.\] Lemma 19 and exactness identify \(\operatorname{Part}(J^\vee\mathop{\mathrm{Tr}}Z)\) with \(\mathcal H(Z)\), using the fiber point and the unit of \(J^\vee\). On the other hand, \[(J_p^\vee\otimes\mathop{\mathrm{Tr}}Z_p)^{hG_p} \simeq\mathop{\mathrm{map}}\bigl(S(V)_{hG_p+},Z_p\bigr) \simeq Z_p\] by Lemma 16. Evaluation at a point is the inverse of the constant-class equivalence. By Lemma 18, norm followed by this evaluation on the constant-unit copy sums all group actions. It is therefore multiplication by \(|G_p|=p(p-1)\).

To check Equation (18), pair the norm triangle for \(J^\vee\) with \(\mathcal H(Z)\) by tensoring trivial maps. Do this first on the partial homotopy fixed point terms, before using their norm identifications with homotopy orbits. These pairings are balanced over \(D\), acting by trivial sphere scalars, and give a map of the entire join triangles. Passage to the telescope gives the localization pairing on their cofibers.

The comparisons on the first two terms are equivalences. For the partial term, the point-and-unit class at \(Z=\mathbf 1\), followed by the norm, multiplies a class from \(\mathcal H(Z)\) to the corresponding point-and-unit-norm class for \(Z\), by naturality at each prime. Lemma 19, also applied to suspended sphere classes, identifies this generator with the free rank-one \(D\)-module and identifies its external product with \(\mathcal H(Z)\). For the full term the same assertion follows from the constant generator. Exactness gives the equivalence on cofibers.

At \(Z=\mathbf 1\), both initial terms are \(D\) in degree zero, and their map is multiplication by \(\varpi(p-1)_p\). The factor \((p-1)_p\) is a unit and \(\varpi\) is a nonzerodivisor. The cofiber algebra thus has homotopy only in degree zero, equal to \(D/\varpi\), and its unit induces the quotient map. Choose a null of \(\varpi\) in this algebra. The resulting algebra map from \(D//\varpi\) is an equivalence on underlying modules, hence an algebra equivalence. ◻

Proof. The dual Euler triangles give \(J^\vee\simeq\Sigma T^{-1}J\). It suffices to apply Equation (17) to \(T^{r-1}J\mathop{\mathrm{Tr}}Z\) and show that its first map is an equivalence.

At each prime, \(S(V)\) has a finite equivariant cell decomposition with prime-to-\(p\) stabilizers. One may use the boundary of the permutation simplex: a subgroup with \(p\)-torsion has no fixed point in \(S(V)\). Induction from each such stabilizer is \(p\)-locally a retract of free induction by Lemma 12. The ordinary full orbit-to-fixed-point norm on \(T^{r-1}J\mathop{\mathrm{Tr}}Z\) is therefore an equivalence, by its finite equivariant cellular filtration.

Its full homotopy orbit spectrum is identified with the underlying fiber by inclusion of a fiber point: \(r-1\) is even, so use the trivialization of Lemma 17, followed by the acyclic augmentation of Lemma 16. Its partial orbit spectrum is identified by the same point. Indeed projection \(J\to\mathbf 1\) and the Euler triangle reduce that assertion to the even calculation for \(T^{r-1}\mathop{\mathrm{Tr}}Z\) and the odd vanishing for \(T^r\mathop{\mathrm{Tr}}Z\) in Lemma 19. The projection from partial to full orbits respects these fiber-point maps, so it is an equivalence. Composing with the full norm proves the assertion. ◻

Proof. For the right-hand equivalence, replace \(W_i\) coherently by \(\mathop{\mathrm{Tr}}S(Fi)\) using Lemma 17 and free-module adjunction. Apply Equation (18) with the target shifted by \(S(-Fi)\). Multiplication of the coefficient algebra at index zero with ordinary maps from these spheres gives the comparison in every degree and weight. The same construction respects all products, since the sphere trivializations are coherent and the external pairing is multiplication in the mapping algebra. The chosen agreement on the \(\delta\) summand makes its scalar action ordinary evaluation. Naturality in targets is built into all of these maps. In particular no algebra structure on the target is needed for the underlying coefficient comparison.

For the left-hand equivalence, first replace \(J^\vee\) by \(J_2^\vee\) in the definition of \(\Phi(X)\). In that algebra, null \(t\) by the tautological Euler null on \(S(2V)\), multiplied by the target unit. Concretely the tautological nonzero vector gives a radial path from the zero section of the disk bundle to its boundary, which becomes the point at infinity in the representation sphere. This is the null in the dual Euler triangle \[T^{-2}\longrightarrow\mathbf 1\longrightarrow J_2^\vee.\] The universal property of the algebra quotient gives an algebra map from \(B(X)//t\) to this \(J_2^\vee\) mapping algebra.

We verify its underlying map with the null specified, since the existence of some module equivalence would not suffice. By Lemma 4, its source is the cofiber of multiplication by \(t\), and the map on the added module cell is the chosen null multiplied by the input from \(B(X)_{i-\tau}\). The coherent identifications of \(W_\tau\), \(W_{i-\tau}\), and \(W_i\) identify multiplication by \(t\) with the map obtained from \(T^{-2}\to\mathbf 1\) and the target unit. The multiplied null is, by the same unitality, the geometric null in the dual Euler triangle tensored with the target. The original input maps are constant along \(S(2V)\). Exactness consequently identifies this particular map with the cofiber equivalence \[ \mathop{\mathrm{cofib}}\bigl(B(X)_{i-\tau}\xrightarrow{t}B(X)_i\bigr) \simeq\mathop{\mathrm{map}}_{\mathcal E} \left(W_i,J_2^\vee\otimes \mathop{\mathrm{Tr}}\bigl(X_p(p\lambda+\delta)\bigr)_p\right). \tag{20}\]

Now restrict along the inclusion of the first copy \(S(V)\to S(2V)\). Its quotient is stably \(T\otimes J\): the open complement with nonzero second component is an open \(V\)-ball times \(S(V)\), and its one-point compactification gives this description. After dualizing, the fiber of restriction therefore has an extra factor \(T^{-1}J^\vee\). Every \(W_i\) has even twisting exponent in \(V\), so moving it to the target leaves an odd twist. Lemma 21 makes this fiber zero, including any ordinary sphere shift. Restriction is thus an equivalence. Its composite with Equation (20) proves the first map of Equation (19). All choices of nulls have been multiplied with units and transported by the stated maps, so the construction has the claimed unit and target compatibilities. ◻

Proof. For constant input \(X\), \(Q(X)=F^*h_\Omega(X)\). The first Koszul attachment kills \(b_0=\varpi u_0\). Its quotient is base change by \(D//\varpi\): send the null of \(b_0\) to \(u_0\) times the null of \(\varpi\). The chosen-null cofiber description and Proposition 9 give this identification also after tensoring with \(M\). Scalar base change supported in weight zero is componentwise tensor on the other input, so it commutes with the pullback of weights by \(F\). Apply Proposition 22 and its unit compatibility. ◻

Proof. Derived reduction modulo \(t^r\) is tensoring over the polynomial algebra with its two-term perfect quotient. It commutes with limits. After this reduction, the cones of the maps from the initial module \(C\) to its quotients modulo \(t^s\) form a pro-zero tower: their transitions are multiplications by \(t\) on the corresponding shifts, and any \(r\) consecutive transitions are null because \(t^r\) has its quotient nullhomotopy. Their inverse limit is zero. Taking the limit of the reduced quotient maps therefore gives the asserted equivalence. Applying this for all \(r\) proves completeness of \(\widehat C\). A perfect \(\widehat C\)-module is dualizable; tensoring it commutes with this limit and with the finite quotient cofibers. The same calculation proves the final assertion. ◻

Proof. Binary order compares two subsets at their largest differing index. In a term of \(\partial e_S\), one replaces an index \(h\in S\) by a subset of \([h)\). If the resulting exterior monomial is nonzero, its index set is strictly smaller than \(S\) in binary order. Therefore the coefficient equation for \(e_U\) in \(\partial b_j^u\) expresses \(\partial z_{j,U}\) in terms of coefficients \(z_{j,S}\) with \(S>U\) and coefficients \(z_{h,T}\) with \(h<j\). The signs are fixed by \[\partial(z_{j,S}e_S) = (\partial z_{j,S})e_S +(-1)^{|S|}z_{j,S}\partial e_S.\] Each nonzero summand in the expression for \(\partial z_{j,U}\) contains one coefficient of each of these two kinds. It is thus decomposable, with the required triangular ordering.

Inductively suppose that \(\partial^2 e_h=0\) for \(h<j\). The square of an odd derivation is a derivation, since its mixed terms cancel. Applying \(\partial\) to \(\partial b_j^u=0\) consequently gives \[0=\partial^2 b_j^u =\sum_{S\subseteq[j)}(\partial^2 z_{j,S})e_S.\] Independence of the exterior basis gives \(\partial^2z_{j,S}=0\) for every \(S\), and the defining cycle equation gives \(\partial^2e_j=0\). This proves the assertion by induction. The actual coefficients of \(b_j\) satisfy precisely these equations, so they define Equation (21); in particular \(z_{0,\emptyset}\) maps to \(\varpi u_0\). All equations for an earlier stage are retained when a new value of \(j\) is added. ◻

Proof. The differential of a coefficient generator increases length, by Lemma 26. The summand \(z_{j,S}e_S\) of \(\partial e_j\) has length \(1+|S|\), so only \(S=\emptyset\) contributes at the original length one. For the module cell \(e_S\), differentiating one factor \(e_h\) and using the term indexed by \(T\subseteq[h)\) gives total length \(|S|+|T|\). This is at least \(|S|\), and the resulting exterior basis vector is earlier in binary order. Thus the differential is a filtered attachment to earlier module cells. There are \(2^{a+1}\) basis vectors, including the initial one indexed by the empty set. ◻

Proof. The associated graded pairing is obtained by taking the cofiber of the pushout-product of the filtration arrows. On steps this gives the claimed tensor equivalence. Steps generate the filtered category under colimits, and both constructions preserve colimits, so the equivalence holds in general, with its symmetric monoidal compatibilities. The assertions for free algebras and their pushouts follow.

For the last assertion, fix a finite tuple of weights and filtration orders. The tensor cube for the filtration arrows maps to the arrow at the sum of those orders: every vertex with at least one order increased maps into filtration at least one higher than that sum. Taking the cube’s pushout-product and cofiber constructs the associated graded pairing. The exact lax pairings defining \(h_\Lambda\) commute with these finite cofibers and produce the same construction. This verifies the assertion for all finite tuples, together with the compatibility under composition of pairings. It does not require \(h_\Lambda\) to commute with infinite filtration colimits. ◻

Proof. We first construct the algebras in the triangular generator order of Lemma 26. Start at the unit, in weight and filtration order zero. Suppose that the preceding generators have been constructed, and let \(z=z_{j,S}\), \(r=|S|\), and \(\gamma=\gamma_{j,S}\). Its boundary \(\partial z\) is a cycle of chain degree \(-r-1\), weight \(\gamma\), and filtration at least two. The comparison already established at this fixed weight and filtration lifts the boundary to maps \[ S^{d(\gamma)-r-1} \longrightarrow(\mathcal F_2R_p^{\mathrm{old}})(\gamma), \tag{23}\] together with a path matching the normalized class with \(\partial z\). Compose with the map to \(\mathcal F_1\) and attach a free commutative algebra cell that nulls this map. Explicitly, push out the free algebra on the indicated sphere, placed as a step at order one, along its augmentation to the unit. The support and filtration bounds are preserved.

After taking associated graded, the attaching map in order one is null by its specified factorization through \(\mathcal F_2\). Lemma 28 identifies the result with free adjunction of the sphere \(S^{d(\gamma)-r}\) in weight \(\gamma\) and order one. Repeating this construction gives a primewise free algebra on all the generator spheres as the associated graded of \(R_p^a\).

We verify carefully that the comparison with the filtered dg algebra extends at each step. Over the \(\mathbb Q\)-algebra \(D\), free algebras on free step generators are computed by ordinary graded symmetric powers, with their step orders added. For a cycle generator \(x\), the free disk algebra with a generator \(z\) satisfying \(\partial z=x\), both placed at order one, resolves the filtered unit over the free cycle algebra. Indeed every component of positive word length is contractible. If the null generator is even, the needed contraction divides by its positive multiplicity, which is invertible in \(D\). Moreover, filtering the disk algebra by the number of null generators gives successive free step modules over the cycle algebra. The disk therefore computes the derived relative tensor, and its base change is exactly the extension by \(z\) with the specified differential and length filtration.

These computations use derived tensor products. To see directly that the strict monomial models compute them, order monomials at each weight by their highest differing generator multiplicity in the triangular list. There are finitely many such monomials by positive mass, and their differentials use earlier ones. For \(K^a\), place the \(e_j\) after all coefficient generators, in increasing \(j\). The resulting filtered complexes are built from free \(D\)-module steps. Their strict tensor pairings thus represent the derived pairings as well.

The old comparison and the tautological null in the spectral attachment now induce a map from the dg pushout to the normalized spectral pushout. We use the chosen matching path from Equation (23) inside \(\mathcal F_2\). In order one modulo order two, the new generator is the difference of two nulls: the attached null and the null obtained by projecting the boundary from \(\mathcal F_2\). The matching path respects the latter null. Consequently the comparison sends the new associated graded generator to the spectral generator supplied by the attachment. Taking cofibers and projecting the filtration commute with normalized Hom, so this is also an identification of the chosen generator classes, not merely of the objects.

At any fixed weight, the associated graded spectral algebra is a finite sum of extended powers of the generator spheres, with arities bounded by the mass of that weight independently of \(p\). For sufficiently large \(p\), the symmetric groups in these terms have invertible order. An even sphere repeated \(k\) times contributes a sphere of the summed degree: the fiber inclusion into its homotopy orbits is an equivalence after \(p\)-localization. An odd sphere repeated at least twice contributes zero. To justify both assertions, its extended power is the Thom spectrum of the permutation sphere over \(B\Sigma_k\), and its ordinary homology is the group homology with the permutation orientation character. When \(p\nmid k!\), higher group homology vanishes. The degree-zero coinvariants are \(\mathbb Z_{(p)}\) in the even case and zero in the odd case, since a transposition acts by \(-1\) and \(2\) is a unit. The spectra are bounded below after translating the sphere degree, so the homology statements imply the claimed equivalences by stable Hurewicz.

Products of distinct generator factors are treated separately. Multiplication of their generator classes uses the fiber inclusions. The coherent sphere grading and \(\mathop{\mathrm{map}}_{\mathcal C}(\mathbf 1,\mathbf 1)=HD\) identify their normalized Hom with the expected free graded-commutative basis over \(D\). The extra chain shifts are fixed at each fixed weight, so the fixed-stem calculation applies. Our comparison is therefore an equivalence on associated gradeds, and hence on the bounded filtration at every weight. This completes the induction constructing the filtered algebra comparison.

It remains to realize the evaluation, including its specified nulls. If \((X_p)\) is any sequence of graded commutative algebras, there is a comparison \[ \prod_{\mathcal U}\mathop{\mathrm{Map}}_{\mathrm{CAlg}^\Lambda}(R_p^a,X_p) \longrightarrow \mathop{\mathrm{Map}}_{\mathrm{CAlg}_D^\Lambda}(R^a,h_\Lambda(X_p)), \tag{24}\] where the source uses maps of whole graded systems, after forgetting filtration. This is an equivalence by induction over the finite list of free algebra cells. For one cell, extending a map means choosing a null of the image of its attaching map. Its mapping space is the corresponding homotopy pullback of mapping spaces from the generator sphere at its specified weight. Those sphere mapping spaces agree by the definition of \(h_\Lambda\). The initial unit case agrees as well, and finite homotopy pullbacks commute with the ultraproduct. The comparison constructed above sends attached-null data to the attached-null data in these pullbacks, since it used both the tautological null and the specified matching path.

Forgetting filtration in this argument does preserve the free-cell pushouts: it is colimit towards decreasing filtration order, a colimit-preserving symmetric monoidal functor, as one sees on steps. Here it takes the constant value at orders at most zero. Apply Equation (24) to the constant graded sphere target and the evaluation in Equation (21). It supplies the required primewise evaluations and the matching homotopy. The same comparison commutes with restriction to earlier cells. It is therefore an equivalence on the homotopy fibers over already chosen maps, including their comparison paths. The new evaluation can thus extend the preceding evaluation with its chosen compatibility, which proves the assertion about diagrams across stages.

Finally, construct \(K_p^a\) using the finite filtered module cells from Lemma 27. A free module cell over \(R_p^a\) with generator of weight \(\gamma\), order \(s\), and ordinary degree \(d(\gamma)+u\), for fixed \(u\in\mathbb Z\), normalizes to the corresponding free module shift over \(h_\Lambda(R_p^a)\) with chain degree \(u\). This uses the coherent sphere grading. A boundary is, by adjunction, a map from one sphere into the specified weight and filtration of the preceding module. Lift that map and choose a path matching the boundary. Taking its cofiber extends the module comparison. Exactness proves the comparison at each stage and gives the compatible associated graded comparison. ◻

Proof. After applying \(h_\Lambda\), the right side becomes the tensor of two-term cofibers for the classes \(z_{j,\emptyset}\) in \(\mathop{\mathrm{gr}}R^a\). The tensor comparison is an equivalence for these finite free module cells: it is the free-shift identification for one cell and extends by finite cofibers. This is the Koszul module with differential \(\partial e_j=z_{j,\emptyset}\), hence agrees with the normalized left side by Lemma 27. A change in a sphere identification can multiply a chosen basis element by a unit and does not change this equivalence.

To lift that equivalence, consider the ultraproduct of categories of whole \((\text{weight},\text{order})\)-graded modules over \(\mathop{\mathrm{gr}}R_p^a\). There is a natural comparison from its mapping spectra to mapping spectra of graded modules over \(h_\Lambda(\mathop{\mathrm{gr}}R_p^a)\). For a free source at one fixed weight, order, and extra integer suspension, both mapping spectra are exactly the same normalized evaluation of the target at that weight and order. Induction through cofiber sequences proves full faithfulness for sources built from finitely many such free shifts, against any whole-system target.

Both modules in Equation (25) satisfy this finiteness condition. For the left side, take associated graded of the finite filtered cell construction. For the right side, expand the tensor of the finitely many two-term cofibers. Their formal weight and order shifts and their extra integer suspensions are fixed independently of \(p\). Full faithfulness lifts the normalized equivalence, its inverse, and the two homotopies identifying their composites with the respective identities. These are finitely many maps and homotopies of whole systems. After restricting to one \(\mathcal U\)-large prime set, they are inverse equivalences at each prime as whole graded module maps. Thus the same prime set works for all weights; no intersection over the lattice of weights is taken. ◻

Proof. The exponent of \(p\) in \(k!\) is \(l\), since \(k<p^2\). Consequently \(H\) is a Sylow subgroup. Its nontrivial orbits are exactly the \(p\)-element blocks, so a normalizing permutation permutes these blocks and the \(b\) remaining letters separately. On each block the normalizer of translations consists of affine maps \(x\mapsto ux+v\). Dividing out the translations gives the displayed normalizer quotient. Its order is \((p-1)^l l!b!\), which is prime to \(p\) because \(l,b<p\).

We recall the transfer identities needed here, including their effect on coefficients. Cohomological transfer is obtained by summing a cochain over representatives of the relevant finite coset set, using the group action to identify the coefficient fibers. It follows that transfer after restriction from a group to a subgroup is multiplication by the subgroup index. For the opposite composite, decomposing that coset set into subgroup orbits gives the double-coset formula: each double coset contributes restriction to the corresponding intersection, conjugation with its coefficient action, and transfer from that intersection. These descriptions also give the projection formula for transfer and cup products.

Since \([\Sigma_k:H]\) is prime to \(p\), transfer after restriction proves that restriction to \(H\) is injective. Its image is invariant under the normalizer: conjugation together with the coefficient action acts trivially on cohomology of the full group, and restriction respects this action.

To prove that every invariant is in the image, consider the other transfer composite. The restriction of either \(M\) to \(H\) is trivial: each \(p\)-cycle is even. If \(L\) is a proper subgroup of \(H\), it is a proper linear subspace of the elementary abelian group \(H\). A linear retraction \(H\to L\) shows that \(H^*(H;\mathbb F_p)\to H^*(L;\mathbb F_p)\) is surjective. Choose a basis of the one-dimensional, trivial \(H\)-module \(M\). Every class with coefficients in \(M\) over \(L\) is then the restriction of a class over \(H\). The projection formula gives \[\operatorname{tr}_L^H\operatorname{res}_L^H(x) =x\,[H:L]=0.\] Hence transfer from every proper intersection subgroup in the double-coset formula vanishes, in all degrees. An intersection \(H\cap gHg^{-1}\) equals \(H\) exactly when \(g\) normalizes \(H\). The remaining terms are therefore \[\operatorname{res}_H^{\Sigma_k} \operatorname{tr}_H^{\Sigma_k}(x) =\sum_{\bar g\in N/H}\bar g\cdot x.\] For an invariant \(x\) this is \(|N/H|x\). The scalar is invertible in \(\mathbb F_p\), proving surjectivity onto the invariants. ◻

Proof. The Thom spectrum has a cellular filtration with its cells in degrees \(kd+j\), where \(j\geq0\) is a base-cell degree. This proves the first bound, including for virtual bundles. Since \(p>C\) and \(k\leq Cp\), we have \(k<p^2\) and \(l\leq C<p\), so Lemma 31 applies.

The two-periodic cyclic-group resolution, together with its periodicity class and Bockstein, gives \[H^*(C_p;\mathbb F_p)=\Lambda(\eta)\otimes\mathbb F_p[\xi], \qquad |\eta|=1,\quad |\xi|=2.\] Both \(\eta\) and \(\xi\) transform with multiplier weight one under \(\mathbb F_p^\times\), or both with its inverse if the conjugation convention is reversed. The periodic resolution gives the polynomial degree-two class, the Bockstein identifies its multiplier action with that of the degree-one class, and \(\eta^2=0\) follows from graded commutativity since \(p\) is odd.

The sign of multiplication by \(u\in\mathbb F_p^\times\) on the \(p\) letters of a block is \(u^{(p-1)/2}\), viewed in \(\mathbb F_p\). Indeed, applying the permutation to the Vandermonde product on the \(p\) field elements multiplies that product both by its sign and by \(u^{p(p-1)/2}=u^{(p-1)/2}\). Translations have sign one. A block transposition has sign \((-1)^p=-1\), and permutations of the remaining \(b\) letters have their ordinary sign.

Suppose first that \(d\) is odd. If \(b\geq2\), a transposition of the remaining letters acts trivially on \(H\) and by \(-1\) on the coefficient line. The normalizer invariants are consequently zero in every degree. If \(b=0\) or \(1\), invariance under the independent multipliers on each block requires multiplier weight \((p-1)/2\) modulo \(p-1\) in each factor of \(H^*(H;\mathbb F_p)\). The least degree with this weight is \(p-2\), represented by \(\eta\xi^{(p-3)/2}\); the least even degree is \(p-1\). Reversing the multiplier convention gives the same condition. Therefore the normalizer invariants have no classes below degree \(l(p-2)\). The further block-permutation condition can only remove classes. Since \(b\leq1\) in this case, \[l(p-2)=k-(2l+b)\geq k-(2C+1).\] This gives the cohomological vanishing required for Equation (27).

For even \(d\), put \[U=H^*(C_p;\mathbb F_p)^{\mathbb F_p^\times},\qquad V=\ker(U\longrightarrow\mathbb F_p).\] The least positive degree in \(U\) is \(2p-3\), represented by \(\eta\xi^{p-2}\). By Lemma 31, trivial coefficients give \[H^*(\Sigma_k;\mathbb F_p)\cong(U^{\otimes l})^{\Sigma_l}.\] The permutations of the tensor factors here include the Koszul signs of their cohomological degrees. The leftover \(\Sigma_b\) acts trivially and imposes no further condition.

We now compute restriction to \(\Sigma_{k-1}\). If \(b>0\), choose the fixed letter outside the \(p\)-blocks. Both symmetric groups then have the same Sylow subgroup \(H\), and their normalizer invariants are the same: the only change is from \(\Sigma_b\) to \(\Sigma_{b-1}\) on the unused letters. The restriction is an isomorphism, since its further restriction to \(H\) is the identity.

If \(b=0\) and \(k>0\), choose the fixed letter in the last block. A Sylow subgroup of \(\Sigma_{k-1}\) consists of the first \(l-1\) blocks. Naturality of restriction and its injectivity on these Sylow subgroups identify the restriction map with \[ (U^{\otimes l})^{\Sigma_l} \longrightarrow (U^{\otimes(l-1)})^{\Sigma_{l-1}}, \qquad \mathop{\mathrm{id}}^{\otimes(l-1)}\otimes\epsilon, \tag{29}\] where \(\epsilon:U\to\mathbb F_p\) is augmentation. To determine this map without a choice of normalization, decompose \(U=\mathbb F_p\oplus V\). A tensor with exactly \(r\) entries from \(V\) has its nonunit entries in an \(r\)-element subset of \(\{1,\ldots,l\}\). An invariant on all such subsets is uniquely determined by its component on one subset, which belongs to \((V^{\otimes r})^{\Sigma_r}\). Conversely, place such a component in each \(r\)-element subset and sum the resulting tensors, without dividing the sum by its number of terms. This gives all the invariants and respects the graded signs. Applying augmentation to the last factor kills exactly the summands whose subset contains that factor. For \(r<l\), the remaining sum is precisely the corresponding sum over the \(r\)-element subsets of \(\{1,\ldots,l-1\}\). For \(r=l\), every summand is killed. Consequently Equation (29) is surjective and its kernel is \((V^{\otimes l})^{\Sigma_l}\). Every class in this kernel has degree at least \(l(2p-3)\).

The Thom isomorphism is compatible with the block inclusion defining \(T_{p,k}(d)\): on \(\Sigma_{k-1}\) the bundle \(d\rho_k\) restricts to \(d\rho_{k-1}\) plus a trivial bundle of rank \(d\). Both Thom spectra therefore have rank \(kd\). The preceding restriction maps are surjective in every degree. The cohomology long exact sequence of their cofiber identifies its cohomology with the kernel of restriction in the same degree, without an additional degree shift. Thus it vanishes entirely if \(b>0\); if \(b=0\) its least possible degree above \(kd\) is \[l(2p-3)=2k-3l\geq2k-3C.\] The case \(k=0\) of Equation (28) follows directly from \(T_{p,0}(d)=S^0\).

For completeness, these mod-\(p\) calculations imply the stated homotopy bounds, not just cohomology bounds. Classifying spaces of the finite groups used above have finitely many cells in each degree, and the virtual Thom spectra are bounded below after suspension. Their \(p\)-local integral homology groups, and those of the indicated cofibers, are finitely generated in each degree. Cohomology vanishing over the field \(\mathbb F_p\) gives homology vanishing in the same range. The universal coefficient sequence then shows that each integral homology group in that range has zero quotient modulo \(p\). Finite generation over \(\mathbb Z_{(p)}\) forces that group to vanish. Finally, for a bounded-below spectrum, the first nonzero homotopy group, if there is one below the claimed bound, maps isomorphically to the first nonzero integral homology group by the stable Hurewicz theorem. This would contradict the integral homology vanishing. The same argument in every degree proves contractibility in the cases where all mod-\(p\) cohomology vanishes. ◻

Proof. Write \(\gamma=p\lambda+\delta\). Restrict to the set of primes where Equation (25) holds for whole graded modules. If \(\gamma\) is outside the support monoid of the universal model, its component is zero and there is nothing to prove. Otherwise decompose the free algebra \(\mathop{\mathrm{gr}}R_p^a\) as the tensor product of the free algebras on the individual generator spheres. There are \(\sum_{j=0}^a2^j=N_a\) such generators.

For a nonempty subset \(S\subseteq[j)\), set \(r=|S|\) as before. The factor with multiplicity \(k=k_{j,S}\) is \[\mathop{\mathrm{Sym}}^k\big(S^{d(\gamma_{j,S})-r}\big).\] For \(S=\varnothing\), the corresponding quotient factor in Equation (25) has multiplicity-\(k\) component \(T_{p,k}(d_j)\). Here multiplicity includes the generator in the source of the multiplication map, or equivalently the possible Koszul cell, as well as the ordinary polynomial copies. Its filtration order is \(k\) on both sides of that map. Distinct factors tensor together, and the weight-\(\gamma\) component is the finite sum of the terms satisfying \[ \sum_{j,S}k_{j,S}\gamma_{j,S}=\gamma. \tag{32}\]

Every generator weight has positive integral mass. Taking mass in Equation (32) therefore gives \[0\leq k_{j,S}\leq\sigma(\gamma) =p\sigma(\lambda)+\sigma(\delta)\leq Cp.\] This also proves finiteness of the sum at each prime. For \(p>C\), Proposition 32 applies to every factor. Since \(d(\gamma_{j,S})\) is even, the odd-degree improvement applies exactly when \(r\) is odd. For \(r=0\), the multiplication-cofiber improvement applies instead. Tensoring lower degree bounds adds them: after desuspending each factor to a connective spectrum, their tensor product is connective. The bound for each tensor term, relative to \(d(\gamma)\), is consequently at least \[ -\sum_{j,S}r k_{j,S} +\sum_{r\text{ odd}} k_{j,S} +2\sum_{r=0}k_{j,S}-E. \tag{33}\] Indeed the odd factors lose at most \(2C+1\) each, the multiplication cofibers lose at most \(3C\) each, and there are at most \(N_a\) factors; the displayed value of \(E\) bounds their total loss, including factors of multiplicity zero.

The elementary inequality \[-r+\mathbf 1_{\{r\text{ odd}\}} +2\mathbf 1_{\{r=0\}}\geq2(1-r) \qquad(r\geq0)\] and the identity \(\nu(\gamma_{j,S})=1-r\) turn Equation (33) into the lower bound \(2\nu(\gamma)-E\). Thus every associated graded piece at weight \(\gamma\) lies in the range asserted in Equation (31).

At this weight the filtration is zero above \(\sigma(\gamma)\) and constant in filtration degrees at most zero. Starting at its zero top term, the cofiber sequences relating successive filtration terms and associated graded pieces prove the same bound for the unfiltered component. This uses only closure of \(\mathrm{Sp}_{\geq L}\) under extensions. The filtration length is finite at each prime; no uniform bound on that length is required. ◻

Proof. Let \(s\in\mathbb Z\) be any fixed homotopy degree. The \(s\)th homotopy group of \(Q(K_p^a)_i\) is computed by maps from the sphere of ordinary degree \(d(Fi)+s\) to \(K_p^a(p\lambda+\delta)\). Subtracting this requested degree from the lower bound of Proposition 33 gives \[\begin{align*} &d(p\lambda+\delta)+2\nu(p\lambda+\delta)-E -\big(d(Fi)+s\big)\\ &\hspace{1cm}= 2p\big(\nu(\lambda)+m\big) +2\nu(\delta)-2m-E-s. \end{align*}\] Because \(\nu(\lambda)+m\) is a positive integer, this expression tends to \(+\infty\) with \(p\). The requested primewise homotopy group therefore vanishes on a set in \(\mathcal U\). It follows that \(\pi_s Q(K_p^a)_i=0\). This argument applies separately to every fixed \(s\), so the mapping spectrum is zero. The arbitrary-object base-change comparison of Equation (19) identifies \[\Phi(K_p^a)_i\simeq Q(K_p^a)_i\otimes_D(D//\varpi),\] which proves the second assertion. ◻

Proof. Fix \(\lambda\) and \(m\). Trivial-action maps in the \(\delta\)-direction, followed by the primewise module action, give the balanced pairing \[ K^a\otimes_{R^a}\Phi(R_p^a)_{\lambda,*,m} \longrightarrow \Phi(K_p^a)_{\lambda,*,m}. \tag{35}\] In constructing it we use the chosen comparisons \(h_\Lambda(R_p^a)\simeq R^a\) and \(h_\Lambda(K_p^a)\simeq K^a\) from Section 5. The compatibility of the module action with these comparisons makes the pairing balanced over the indicated \(\delta\)-scalar action.

We verify that Equation (35) is an equivalence using the finite unfiltered module cells of \(K_p^a\). For a free cell generated in fixed weight \(\alpha\) and ordinary sphere degree \(d(\alpha)+u\), with \(u\in\mathbb Z\) fixed, the pairing identifies both sides with the shift of \(\Phi(R_p^a)_{\lambda,*,m}\) by weight \(\alpha\) and chain degree \(u\). Concretely, the primewise module input at \(\delta\) on the right is \(S^{d(\alpha)+u}\otimes R_p^a(p\lambda+\delta-\alpha)\). After applying \(J^\vee\otimes\mathop{\mathrm{Tr}}\), the Picard grading moves the trivially acting \(S(\alpha)\) into the source, changing its index to \((\lambda,\delta-\alpha,m)\) and leaving the fixed suspension \(u\). This is exactly the free-cell shift on the left. Both sides of Equation (35) are exact in the module variable: tensor products preserve finite cofibers, and the mapping constructions are exact componentwise. Induction over the finitely many cells now proves the equivalence for \(K_p^a\).

If \(m>-\nu(\lambda)\), Lemma 34 shows that the right side is zero at every fixed \(\delta\). Finally, Equation (19) identifies \(B(R_p^a)//t\) with \(\Phi(R_p^a)\) compatibly with the scalar action. Since \(t\) has chain degree zero, its algebra quotient has the multiplication cofiber \(B(R_p^a)/t\) as underlying module, with the identification specified by its universal null. Substituting this comparison into Equation (35) proves Equation (34). ◻

Proof. The cosimplicial structure inserts the unit of \(K\) in the coface maps and multiplies adjacent \(K\) factors in the codegeneracy maps. Tensoring with \(K\) commutes with totalization because a perfect module is dualizable. After that tensor, the augmented cosimplicial object is split: multiplication into the extra, distinguished \(K\) factor gives the extra codegeneracy. It follows that \[K\otimes_R U\ \xrightarrow{\simeq}\ K\otimes_R\mathcal T(U).\]

Every nonaugmented term is a \(K\)-module. If \(W\) is a \(K\)-module and \(V\) is \(K\)-acyclic, extension and restriction of scalars give \[\mathop{\mathrm{map}}_R(V,W)\simeq\mathop{\mathrm{map}}_K(K\otimes_R V,W)=0.\] Thus those terms are \(K\)-local. Local objects are closed under limits, so their totalization is local as well. The fiber of \(c_U\) is \(K\)-acyclic by the preceding split calculation. If \(U\) is local, that fiber is also local, and hence is zero: its endomorphism spectrum vanishes by applying the definition of locality to the fiber itself. This proves the assertion about local inputs. If \(U\) is acyclic, each term in its totalization is zero.

Tensoring over \(R\) is exact in the stable module category, and limits preserve finite limits. Therefore \(\mathcal T\) is exact. Finally, for algebra input the cosimplicial terms are commutative algebras and all structure maps are algebra maps. Their limit has the asserted algebra structure and coaugmentation. This is the usual dualizable-algebra completion argument; compare (Mathew et al. 2017, Example 2.18 and Proposition 2.21). ◻

Proof. Under Equation (36), restriction of the right-hand algebra to \((0,\delta,0)\) is \[h_\Lambda(Y_{a+1}) =\bigl(A[e_0,\ldots,e_a]\bigr)|_\Lambda.\] There is an algebra map from \(K^a\) to this restriction: send the universal coefficients to the chosen coefficients in \(A|_\Lambda\), and send each \(e_j\) to the actual null cell \(e_j\). Its differential identities are exactly the universal-cycle identities after evaluation. The scalar compatibility in Equation (19) and the chosen path matching primewise and actual coefficient evaluations identify its restriction to \(R^a\) with the \(R^a\)-action on \(L_a//t\). Transporting along Equation (36) therefore makes every row of that layer a \(K^a\)-module, and hence a local module.

The cofiber of a composite of \(s\) multiplications by \(t\) is a finite extension of shifts of the cofiber of one multiplication by \(t\). Applied row by row, this shows that \(L_a/t^s\) is local for every \(s\geq1\). Here \(/\) denotes the module cofiber; its underlying module agrees with that of the corresponding algebra quotient. Local modules are closed under finite extensions and inverse limits. Since \(L_a\) is derived \(t\)-complete, it is the inverse limit of these local finite quotients and is itself local. The conclusion follows from Lemma 36. This proof uses the already attached cells, and makes no assumption about any Hurewicz image. ◻

Proof. By Equation (34), the cofiber of multiplication by \(t\) into a row \((\lambda,*,m)\) is \(K\)-acyclic whenever \(m> -\nu(\lambda)\). Exactness of \(\mathcal T_a\) sends this cofiber to zero. Multiplication by the coaugmented \(t\) on \(\mathcal T_a B\) is the image under \(\mathcal T_a\) of multiplication by \(t\) on \(B\): this holds termwise in its cosimplicial construction and therefore after totalization. Consequently the transitions \[(\mathcal T_a B)_{(\lambda,\delta,m-1)} \xrightarrow{t} (\mathcal T_a B)_{(\lambda,\delta,m)}\] are equivalences whenever \(m> -\nu(\lambda)\).

The telescope for localization is thus constant up to equivalence starting at the boundary row \(m=-\nu(\lambda)\). In particular, that row maps equivalently to the same row after localization. Multiplication by \(t^{\nu(\lambda)}\) identifies the latter row with the localized row at \(m=0\). This proves the assertion in every weight. When \(\nu(\lambda)<0\), the last multiplication is simply a negative Laurent power; the telescope still reaches the same invertible tail. Multiplicativity follows from Equation (37). ◻

Proof. The first assertion says that the following square commutes: \[\begin{array}{ccc} R^a&\xrightarrow{\ P^a\ }&L_a^\#\\[2pt] {\scriptstyle P_{R_p^a}}\downarrow&& \downarrow{\scriptstyle s_{L_a}}\\[2pt] (B[t^{-1}])^\flat&\xrightarrow{\ (v[t^{-1}])^\flat\ }& (L_a[t^{-1}])^\flat . \end{array}\] To verify it, naturality of \(s\) for \(c_{L_a}\) and \(\mathcal T_a v\), followed by naturality of the coaugmentation for \(v\), gives \[\begin{aligned} &s_{L_a}(c_{L_a}^\#)^{-1}(\mathcal T_a v)^\# s_{\mathcal T_a B}^{-1}(c_B[t^{-1}])^\flat\\ &\qquad=\bigl((c_{L_a}[t^{-1}])^\flat\bigr)^{-1} \bigl((\mathcal T_a v)[t^{-1}]\bigr)^\flat (c_B[t^{-1}])^\flat \simeq (v[t^{-1}])^\flat. \end{aligned}\] The inverse in this display is the localization of an already invertible coaugmentation on \(L_a\).

For the second assertion, the compatible universal models and evaluations give a map from the preceding diagram \(K^{a-1}\leftarrow R^{a-1}\to B(R_p^{a-1})\to L_{a-1}\) to the current diagram \(K^a\leftarrow R^a\to B(R_p^a)\to L_a\). In cosimplicial degree \(r\), it induces an algebra map \[(K^{a-1})^{\otimes_{R^{a-1}}(r+1)} \otimes_{R^{a-1}}B(R_p^{a-1}) \longrightarrow (K^a)^{\otimes_{R^a}(r+1)}\otimes_{R^a}B(R_p^a).\] The same construction applies with \(L_{a-1}\) and \(L_a\) in the last positions. These maps commute with the units and multiplications defining the cosimplicial structure, so they induce maps of coaugmented totalizations. They also preserve \(t\).

The maps \(s\) are natural for these maps of totalizations and for their localizations. Both normalization maps being inverted in Equation (38) are equivalences, so their inverses respect these naturality squares. The same is true of the two target coaugmentations. Finally, \(P_{R_p^a}\) is natural for the map of actual primewise algebras, and the chosen small-weight comparisons and evaluation paths extend those of the preceding stage. Comparing all the arrows in Equation (38) therefore proves the claimed homotopy of algebra maps. This comparison uses actual maps between the two Amitsur diagrams. It does not interchange totalization with \(t\)-inversion or identify a totalization with its extension of scalars. ◻

Proof. Algebra pushouts commute with one another, so reducing the first attachment modulo \(t\) gives \[L_{a+1}//t\simeq (L_a//t)//\overline g_a.\] The old layer equivalence, the chosen path matching attaching classes, and the actual null \(e_{a+1}\) define an algebra map from this pushout to \(F^*h_\Omega(Y_{a+2})\). We check this particular map on its underlying module.

At an index \(i\in I\), the source module is the cofiber of multiplication by \(\overline g_a\) from \((L_a//t)_{i-i_a}\) to \((L_a//t)_i\). Under the old equivalence these terms become \[h_\Omega(Y_{a+1})_{F(i)-w_{a+1}} \xrightarrow{\ b_{a+1}\ } h_\Omega(Y_{a+1})_{F(i)}.\] Their cofiber is \(h_\Omega(Y_{a+2})_{F(i)}\). The map on the new cell is the chosen null \(e_{a+1}\) multiplied by the base input. Lemma 4 therefore identifies the constructed map with this cofiber equivalence. This argument uses only the additivity of \(F\) and a two-term cofiber at each index. It does not require \(F\) to be injective or its pullback to preserve arbitrary relative algebra tensors.

For the \(M\)-tower, base change gives \[L_{a+1}^M//t\simeq (L_a^M//t)\otimes_{L_a//t}(L_{a+1}//t).\] Map its first input into \(F^*h_\Omega(M\otimes Y_{a+2})\) by the old \(M\)-comparison followed by the actual \(Y\)-attachment. Map its second input there by the new sphere comparison followed by the \(M\)-unit. Their agreement on \(L_a//t\) is the previous unit compatibility together with the same attaching-class path. The resulting pushout map is again the underlying cofiber comparison: the new null is the image of \(e_{a+1}\) under the \(M\)-unit, and the finite-module tensor comparison identifies the target with the cofiber of multiplication by this image of \(b_{a+1}\). This proves the second equivalence and its compatibility with the first. The construction uses the canonical maps from the old algebras, so it also preserves the initial-layer and tower maps.

The new attachment is a finite perfect-module extension of its base. Completeness follows from Lemma 24, for both the sphere tower and its \(M\)-base change. ◻

Proof. The initial layer was established at the beginning of the section. Given stage \(a<n\), Propositions 29 and 35 supply the universal models and Equation (34). Lemmas 37 and 38 define \(P^a\); Lemma 39 and the older attached nulls extend it to \(f^a\). Equation (40) defines \(g_a\), and Lemma 40 constructs the next cycle and preserves the invariant. There are exactly \(n\) such steps. ◻

Proof. The actual evaluation \(R^a\to A|_\Lambda\) sends the universal coefficients to those of the chosen cycles. Consequently \[\bigl(A[e_0,\ldots,e_{a-1}]\bigr)|_\Lambda \simeq A|_\Lambda\otimes_{R^a}K_-^a,\] with each universal null sent to its actual counterpart. Tensoring with \(h_\Lambda(M)\) over \(A|_\Lambda\) gives the equivalence on the left of Equation (41). Its identification with \(h_\Lambda(M\otimes Y_a)\) follows from the finite-module tensor comparison of Proposition 9. Only the finite Koszul module is used here, so the same comparison applies after restriction from \(\Omega\) to \(\Lambda\).

Define a map on the first pushout input \(h_\Lambda(M)\) using \(P_M\) and the maps \(B(M)[t^{-1}]\to L_0^M[t^{-1}]\to L_a^M[t^{-1}]\). On the other input use \[K_-^a\xrightarrow{f^a}L_a^\#\longrightarrow(L_a^M)^\# \xrightarrow{s_{L_a^M}} (L_a^M[t^{-1}])^\flat.\] These maps agree on \(R^a\). Indeed, restriction of the second map to \(R^a\) recovers the evaluated, unnormalized power by Equation (39). Naturality of the primewise power for \(R_p^a\to S_{(p)}\to\mathrm{MU}_{(p)}\) identifies this with \(P_M\) applied to the evaluated coefficients. The chosen path between spectral evaluation and the actual coefficient evaluation is part of this comparison. It gives agreement as algebra maps, including the scalar action and the boundary data.

The universal property of the pushout now gives the arrow in Equation (41). Although its source was described using \(D\)-algebras, the same span is a pushout of rational algebras: the common algebra \(R^a\) supplies the scalar identifications. No additional \(D\)-linearity condition on the map into the powered target is required.

Finally, the Hurewicz image of \(b_a\) corresponds in that pushout to the universal cycle \(b_a^u\in K_-^a\). Its image under \(f^a\) is \(g_a\) by definition. Since \(\nu(w_a)=1\), the map \(s_{L_a^M}\) multiplies this class by \(t\), changing its index from \((w_a,0,-1)\) to \((w_a,0,0)\). This proves the asserted value \(t g_a^M\). ◻

Proof. In the standard \(p\)-typical coordinate, the logarithm of the Brown–Peterson formal group is \[\log(Z)=\sum_{h\geq 0}\ell_hZ^{p^h},\qquad \ell_0=1, \qquad p\ell_h=\sum_{0\leq s<h}\ell_s v_{h-s}^{p^s}.\] This is the Hazewinkel recursion (Hazewinkel 1977, sec. 3.1, Equations (3.1.1)–(3.1.3)). Over the rationals, the \(p\)-series is \(\log^{-1}(p\log(Z))\). Its indecomposable term at \(Z^{p^h}\) is \[(p-p^{p^h})\ell_h =(1-p^{p^h-1})v_h \pmod{\text{lower generators}},\qquad h>0.\] By homogeneity no generator \(v_j\) with \(j>h\) occurs in this coefficient; every term other than the displayed multiple of \(v_h\) uses lower generators. The coefficient itself is integral, so these remaining terms lie in \(\mathbb Z_{(p)}[v_1,\ldots,v_{h-1}]\). The scalar \(1-p^{p^h-1}\) is a \(p\)-local unit. Also, modulo \((p,v_1,\ldots,v_{h-1})\), degree considerations exclude every coefficient of the \(p\)-series below \(Z^{p^h}\).

These leading-coefficient assertions are unchanged, up to a scalar unit, by a change of Euler coordinate. Indeed, the projective-space formula identifies a second coordinate with an integral power series in the first, with invertible linear coefficient. Conjugating a series whose terms below order \(p^h\) vanish preserves that order and multiplies its leading coefficient by a power of this invertible linear coefficient. For normalized complex orientations the change is strict, and the leading coefficient is unchanged. Induction on \(h\) therefore gives the asserted images of the ideals \(I_h\) and of their next generators, also for the Euler coordinate induced from \(M_p\).

We next prove regularity in \((M_p)_*\) itself. For \(h\geq1\), homogeneity in the polynomial coefficient ring gives \[x_h=c_h a_{p^h-1} +D_h(a_1,\ldots,a_{p^h-2}),\qquad c_h\in\mathbb Z_{(p)},\] where \(D_h\) is decomposable. Under the projection to \(\mathrm{BP}_*\), the image of a generator of smaller degree cannot involve \(v_h\). Consequently the coefficient of \(v_h\) in the image of \(x_h\) is the product of \(c_h\) and the coefficient of \(v_h\) in the image of \(a_{p^h-1}\). The preceding calculation says that this product is a unit modulo \(p\). Thus \(c_h\) is a \(p\)-local unit. Replacing \(a_{p^h-1}\) successively by \(x_h\) is a triangular change of polynomial generators. Since \(p\) is a nonzerodivisor, this proves regularity of \(p,x_1,x_2,\ldots\).

Finally, work modulo \(I_h\), with \(h>0\). This coefficient ring has characteristic \(p\). Suppose that the \(p\)-series has a first nonzero term \(cZ^s\). Since it is a formal group endomorphism, comparison of total degree \(s\) in its compatibility with formal addition gives \[c(X+Y)^s=cX^s+cY^s.\] Hence \(\binom{s}{i}c=0\) for \(0<i<s\). If \(s=p^rm\) with \(p\nmid m\) and \(m>1\), then \(\binom{s}{p^r}\equiv m\pmod p\), which is a unit in any \(\mathbb F_p\)-algebra. This is impossible for nonzero \(c\). Thus \(s\) must be a \(p\)-power. All possible such coefficients below \(p^h\) are among \(x_0,\ldots,x_{h-1}\) and have already been killed. This proves \(k_u\in I_h\) in the stated range. The argument uses no assumption that the quotient ring is a domain. ◻

Proof. The classes \(1,Z,\ldots,Z^r\) form a free basis for \(M_p^*(\mathbb CP^r)\). To see this at the module level, the cellular spectral sequence collapses because \(M_p\) has even coefficients. The class \(Z\) is a generator in filtration two, and its powers generate the successive even filtrations by the ordinary cup-product calculation. The resulting map \[\bigoplus_{j=0}^r\Sigma^{-2j}M_p \longrightarrow\mathop{\mathrm{map}}(\mathbb CP^r_+,M_p)\] of \(M_p\)-modules is an equivalence. The class \(Z^{r+1}\) vanishes because its filtration exceeds the dimension of \(\mathbb CP^r\). Applying cochains from \(BK\) to this finite splitting proves the product formula at finite \(r\). Passage to the inverse limit gives the power-series formula; the truncation maps are surjective, so there is no additional derived-limit term.

The unit circle bundle of the \(p\)-th tensor power of the universal line over \(\mathbb CP^\infty\) models \(BC_p\). Its sphere-bundle triangle, together with the Thom isomorphism, identifies the relevant cochain map with multiplication by \([p](x)\). This power series is a nonzerodivisor in \((M_p)^*(\mathrm{pt})[[x]]\), since the coefficient ring is a polynomial domain. The resulting exact sequence gives the first formula in Equation (43).

For the second formula, compute homotopy fixed points first by \(C_p\) and then by its complement \(\mathbb F_p^\times\). If a finite group \(K\) has order invertible on a spectrum, homotopy fixed points calculate the invariants on homotopy groups without a boundedness hypothesis. Indeed, the map to the coinduction of the underlying spectrum and the sum map back have composite \(|K|\). Finite induction and coinduction agree, so dividing this composite by \(|K|\) gives a retraction. After taking homotopy fixed points, the coinduced term is the underlying spectrum, and the opposite composite is the averaging operator. Apply this to \(K=\mathbb F_p^\times\) and to the \(C_p\)-homotopy fixed points of the constant spectrum \(M_p\).

Finally, a degree-zero Borel class acts as an \(M_p\)-linear endomorphism of the constant rank-one \(M_p\)-module over \(BG\). If its value at a point is a unit, this endomorphism is a fiberwise equivalence, hence an equivalence of local systems. Its inverse is again such an endomorphism, proving the assertion about units. ◻

Proof. For every \(s>0\), the quotient \((x^s)/(x^{s+1})\) in \(H_{C_p}^*\) is annihilated by \(p\): multiply the relation \([p](x)=0\) by \(x^{s-1}\). A multiplier \(\alpha\in\mathbb F_p^\times\) acts on this quotient by \(\alpha^s\), or its inverse if the opposite action convention is used. For \[p\leq s<2(p-1),\] this character is nontrivial. An invariant element in \((x^s)\) therefore maps to zero in \((x^s)/(x^{s+1})\), since some \(\alpha^s-1\) is a unit modulo \(p\). Iterating over this finite range shows that \(y|_{C_p}\) lies in \((x^{2(p-1)})\).

Since \(\epsilon|_{C_p}\) is \(x^{p-1}\) times a unit, there is a homogeneous \(z\in H_{C_p}^*\) with \(y|_{C_p}=\epsilon^2z\). Average \(z\) over \(\mathbb F_p^\times\). Both \(y|_{C_p}\) and \(\epsilon\) are invariant, so the averaged divisor still has product \(y|_{C_p}\) with \(\epsilon^2\). By Lemma 44, this invariant divisor comes from \(H_G^*\). Notice that this argument neither asserts that \(\epsilon\) is a nonzerodivisor nor cancels a factor of \(\epsilon\). ◻

Proof. Naturality follows from the diagonal construction. Before the Thom correction, the product rule follows from the symmetric monoidal permuted smash and the commutative multiplication of \(M_p\). Compare the product of the two chosen Thom trivializations with the chosen trivialization for the sum of the degrees. Their ratio is an automorphism of a constant suspended rank-one \(M_p\)-module over \(BG\), hence a unit in \(H_G^0\). This comparison is made before introducing \(X\), so the factor has no dependence on its cohomology classes.

To examine a finite sum of classes in one degree, factor the sum map through a finite biproduct and distribute the \(p\)-fold smash over that biproduct. The summands are indexed by label tuples on the \(p\) letters. A constant label tuple gives exactly the power of the corresponding summand. The stabilizer of a nonconstant tuple has order prime to \(p\): otherwise it contains the unique subgroup \(C_p\) of order \(p\), whose transitive action on the letters would force the labels to be constant. The other orbits therefore factor through inductions from prime-to-\(p\) subgroups.

A composite between trivially acting endpoints that factors through an induction from a subgroup is the corresponding additive transfer. This follows by the induction–restriction adjunction and the coinduction–restriction adjunction, using equality of finite induction and coinduction; their unit and counit give the transfer of the composite at the subgroup. Tensoring with a Thom trivialization and multiplying do not change the property of factoring through such an induction. This proves additivity modulo \(\mathfrak t_G(X)\) and also explains directly its stability under multiplication by classes from the larger group.

For the last assertion, use the finite \(M_p\)-module splitting of the cochains of \(\mathbb CP^r\) from Lemma 44. Coefficient extraction is the projection onto one of its summands. Taking cochains from a space with trivial group action commutes with finite induction, since that induction is a finite sum. Consequently these projections commute with transfers and take a transfer error to a transfer error on the coefficient spectrum. ◻

Proof. Represent \(Z\) by the zero section of \(P\) followed by its Thom map. Fiberwise, the normalized Thom map is a map from the local sphere \(S^{P-\mathbb C}\) to \(M_p\) that becomes an equivalence after extending to \(M_p\)-modules. Thom spectra can be formed as colimits of these sphere local systems. Thus the external permuted smash of the Thom maps, which commutes with their colimits, is the corresponding fiberwise construction on the external sum of the bundles. On pulling back to the diagonal and making the Thom correction in Equation (45), the zero section becomes that of \(P\otimes(\rho\otimes_{\mathbb R}\mathbb C)\).

The subsequent map of its sphere local system, normalized by \(\mathbb C^p\), is an \(M_p\)-linear equivalence after extension: on each fiber it is the product of the chosen units. Its ratio to the ordinary Thom equivalence of this bundle is therefore a unit over \(BG\times\mathbb CP^\infty\). This is the class \(U(Z)\) in Equation (46).

Restricted to \(C_p\), the complex permutation representation splits into the characters indexed by \(\alpha\in\mathbb F_p\). Multiplicativity of Euler classes and the formal group law give the product in Equation (47). The trivial summand gives the factor \(Z\), so the constant term vanishes; evaluating every other factor at \(Z=0\) gives the linear coefficient \(U(0)\epsilon\). With compatible canonical Thom choices this is the usual exact Euler-product formula (Johnson and Noel 2010, Equations (4.11) and (4.14)); the unit-factor version suffices here. ◻

Proof. Set \(r_*=p^{a+1}\), and work on \(\mathbb CP^{r_*}\). Tensoring the universal line \(p\) times gives a classifying map to \(\mathbb CP^\infty\) that pulls \(Z\) back to \([p](Z)\). Naturality of \(\Pi_p\) therefore gives \(h([p](Z))\) for the power of this Euler class. All substitutions can be made at finite truncation: by cellular approximation, the classifying map from \(\mathbb CP^{r_*}\) factors up to homotopy through its \(2r_*\)-dimensional skeleton, and \(Z^{r_*+1}=0\) on this space.

Alternatively, expand \([p](Z)\) as in Equation (42). Each summand \(k_uZ^u\) has cohomological degree two, so the additivity assertion of Lemma 46 applies. Repeated use of its product rule gives \[ h([p](Z))\equiv \sum_{1\leq u\leq r_*} \alpha_u\Pi_p(k_u)h(Z)^u \pmod{Z^{r_*+1},\ \mathfrak t_G(\mathbb CP^{r_*})}, \tag{49}\] where every \(\alpha_u\) is a unit in \(H_G^0\) pulled back from \(BG\). These factors may depend on \(u\) and on the Thom choices; no compatibility between them is required below.

Extract the coefficient of \(Z^{r_*}\). Since both sides of Equation (49) have degree \(2p\), this coefficient has degree \[2p-2r_*=-p d_a.\] The transfer error remains an element of \(\mathfrak t_G\) by Lemma 46.

First consider the term \(u=p^a\) on the right. Put \(A_u=[Z^{r_*}]h(Z)^u\). Its degree at \(u=p^a\) is zero. At a point of \(BG\), the permutation bundle is trivial and Equation (46) becomes \(h(Z)=U(Z)Z^p\). Consequently \[A_{p^a}|_{\mathrm{pt}}=U(0)|_{\mathrm{pt}}^{\,p^a},\] a scalar unit. Lemma 44 shows that \(A_{p^a}\) is a unit of \(H_G^0\). Multiplication by \(\alpha_{p^a}\) gives the coefficient \(C\) of \(\Pi_p(x_a)\).

If \(u>p^a\), restrict to \(C_p\). Each of the \(pu\) formal-addition factors in the product expression for \(h(Z)^u\) lies in the ideal \((Z,x)\). The additional factor \(U(Z)^u\) is an ordinary power series in these variables, and hence cannot decrease this ideal order. Therefore \[A_u|_{C_p}\in(x^{pu-r_*}),\qquad pu-r_*\geq p.\] This assertion can be checked using any power-series lifts before quotienting by \([p](x)\); all exponents of \(x\) and \(Z\) are nonnegative. Since \(A_u\) comes from \(BG\), Lemma 45 gives \(A_u\in\epsilon^2H_G^*\). After multiplication by \(\alpha_u\Pi_p(k_u)\) and summation, all such terms contribute to \(\epsilon^2D_*\).

For \(1\leq u<p^a\), Lemma 43 gives a finite homogeneous expression \[k_u=\sum_{0\leq j<a}c_{u,j}x_j \quad\text{in }(M_p)_*.\] All summands have the same degree as \(k_u\). Applying the sum and product rules of Lemma 46 expresses \(\Pi_p(k_u)\), modulo transfers, as a sum of multiples of \(\Pi_p(x_j)\) for \(j<a\). Multiplying by \(\alpha_uA_u\) and collecting terms yields the last ideal sum in Equation (48). When \(a=0\) this range is empty.

It remains to describe the coefficient on the left of Equation (49). Write \(h(Z)=\sum_{s\geq 1}h_sZ^s\). Its linear term contributes \[h_1[Z^{r_*}][p](Z) =U(0)\epsilon x_{a+1}.\] This is the displayed main term, with a unit coefficient. Every contribution from \(h_s[p](Z)^s\) for \(s\geq 2\) is a product of at least two coefficients \(k_u\) with \(1\leq u<r_*\). Each is in \(I_{a+1}\) by Lemma 43. Thus these contributions belong to \(I_{a+1}^2H_G^*\) and can be written as the displayed sum of \(x_jx_k\) for \(j,k\leq a\).

Move the other right-hand terms to the left or absorb their signs into their coefficients. The unit multiplying the main term remains a unit, giving Equation (48). Every sum used above is finite at this fixed prime. After collection there are only the displayed finitely many ideal factors, although their coefficients and the expression for \(T_*\) may vary with \(p\) without a uniform bound on the number of intermediate terms. ◻

Proof. The map on \(E\)-homology induced by any spectrum map is \(R\)-linear. Its source here is the shifted cyclic module \(\Sigma^dR/J\), and the given homotopy says that it takes its generator to \(a x\). It is therefore multiplication by \(a x\) on all homogeneous elements. This map is injective. The cofiber long exact sequence consequently gives, for every integer \(k\), \[0\longrightarrow (R/J)_{k-d} \xrightarrow{\,a x\,}(R/J)_k \longrightarrow E_k\mathop{\mathrm{cofib}}(\mu)\longrightarrow 0.\] The quotient map in this sequence is induced by the cofiber map, so the resulting identification is induced by the stated map from the sphere. Since \(a\) is a unit, its cokernel is \(R/(J,x)\). ◻

Proof. The coefficient ring of \(M_p\) and its quotient by \(I_{a+1}\) are concentrated in even degrees. Every degree \(d(F(i))\) is even. Equation (52) therefore gives \(\pi_1(L_a^M/t)_i=0\) for every fixed \(i\). In the exact sequence of the cofiber \[(L_a^M)_{i-\tau}\xrightarrow{t}(L_a^M)_i \longrightarrow (L_a^M/t)_i,\] this vanishing proves the asserted injectivity. Localization is the componentwise telescope along multiplication by \(t\). Homotopy groups commute with that filtered colimit, and every transition on \(\pi_0\) is injective, which proves the last assertion. Notice that this argument uses only the evenness of the layer; it makes no evenness assertion about the completed algebra \(L_a^M\). ◻

Proof. Apply Proposition 42 and Equation (41). The classes \(x_j\) for \(j<a\) vanish in its source \(h_\Lambda(M\otimes Y_a)\), and their images in the target are the localizations of \(\mathcal P(x_j)\). The same comparison takes the Hurewicz image of \(b_a\) to \(t g_a\). Since it is a map of rational graded algebras and that Hurewicz image is \(u x_a\), its value is the localization of \(\mathcal P(u)\mathcal P(x_a)\). Lemma 50 now proves both equalities before inversion. No linearity over \(D\) is required of the powered comparison.

Raw powering is multiplicative on the represented scalar maps: the product of the permuted powers of two maps is the permuted power of their product, followed by the commutative multiplication of \(M_p\). Apply this to primewise representatives of \(u\) and \(u^{-1}\). It gives \(\mathcal P(u)\mathcal P(u^{-1})=1\) already in \(B(M)\), and hence in \(L_a^M\). ◻

Proof. For the first assertion, recall that \(\epsilon\) is the Euler class of the complex representation \(V\otimes_{\mathbb R}\mathbb C\), whose underlying real representation is \(2V\). Its zero-section map, after the Thom identification, is the map \(\mathbf 1\to T^2\) defining \(t\). Changing the Thom equivalence changes it by a degree-zero unit. The second assertion follows by undoing the Thom conversion in the definition of \(\Pi_p\): the source \(W_{s_j}\) is exactly the permuted sphere \(N(S(w_j))\). Both constructions then use the same permuted smash of the coefficient map and multiplication in \(M_p\).

At \(\delta_j\) the source is the untwisted sphere \(\mathop{\mathrm{Tr}}S(w_j)\). The scalar compatibility of Equation (19), followed by its extension to Equation (36), identifies the reduction of \(\theta_{\delta_j}(x_j)\) with the usual coefficient action on \(M\otimes Y_{a+1}\), up to the allowed unit. This action is zero because \(x_j\in I_{a+1}\). Exactness of the cofiber sequence for \(t\) proves the third assertion.

For the fourth assertion, first note that \[F(i_a)=w_{a+1},\qquad d(F(i_a))=p d_a+q=d_{a+1}.\] It remains to check the coefficient, since equality of these degrees alone does not identify the two maps. Recall that Equation (19) is constructed through \(\Phi(M)\) using the chosen \(J^\vee\)-linear sphere comparison \[J^\vee\otimes W_i\simeq J^\vee\otimes\mathop{\mathrm{Tr}}S(F(i)).\] Extend this comparison to \(J^\vee\otimes\mathop{\mathrm{Tr}}M\). Also extend the Borel Thom comparison defining \(\theta_i\) along the unit of \(J^\vee\). The two are trivializations of the same free twisted rank-one module. Their ratio is an invertible endomorphism, hence a unit in \(\pi_0\Phi(M)_0\). With the first trivialization, the constant coefficient map \(S^{d(F(i))}\to M_p\), followed by the \(J^\vee\)-unit, is precisely the coefficient supplied by the right-hand comparison of Equation (19). The second trivialization therefore gives that coefficient times the stated unit. Extending from this base layer to Equation (52) uses the actual unit and tower maps, so it preserves this comparison.

Finally, an additive transfer from \(H\leq G\) factors through an induced object \(\operatorname{Ind}_H^G Z\). If \(|H|\) is prime to \(p\), then \(Z\) is a retract of \(\operatorname{Ind}_{\{1\}}^H\mathop{\mathrm{Res}}_{\{1\}}^H Z\): the diagonal and sum maps have composite \(|H|\), which is invertible \(p\)-locally. Inducing to \(G\) makes the original induction a retract of a free \(G\)-induction. At each prime, factor a finite sum of transfers through the direct sum of these induced objects. That entire sum is a retract of one free induction on the direct sum of the underlying spectra. Although its size may grow with \(p\), these sums form a single sequence object, and their free inductions are among the objects killed by the defining localization to \(\mathcal E\). Tensoring with a representation sphere preserves induction by the projection formula, and composing with the Thom maps preserves the factorization. Thus the whole transfer error has zero image already in \(B(M)\), before Euler inversion or completion. ◻

Proof. At \(j=0\), the quotient assertions hold because \(Y_0=\mathbf 1\), and the Hurewicz assertion holds because \(b_0=\varpi u_0\) represents the sequence \(p=x_0\) at weight \(w_0\). Lemmas 43 and 49, applied to Equation (51), give both quotient assertions for \(Y_1\).

Suppose now that \(0\leq a<n\), that the Hurewicz assertions hold through \(a\), and that the quotient assertions hold through \(a+1\). Lemmas 50 and 51 apply. We will use Proposition 48 to identify the reduction of \(g_a\), which by construction is the Hurewicz image of \(b_{a+1}\).

Map Equation (48) to \(L_a^M\) at total index \(s_a=i_a+\tau\). Its total cohomological degree is \(-p d_a\). The index assignments for its factors and error coefficients are \[\begin{array}{c|c@{\qquad}c|c} \text{class}&\text{index}&\text{class}&\text{index}\\ \hline C,C'&0&\Pi_p(x_j)&s_j\\ \epsilon&\tau&x_j\ (j\leq a)&\delta_j\\ x_{a+1}&i_a&D_*&s_a-2\tau\\ D_{jk}&s_a-\delta_j-\delta_k&H_j&s_a-s_j. \end{array}\] These are fixed indices in \(I\). They have exactly the required ordinary degrees: for example, \(d(F(i_a))=d_{a+1}\), \(d(F(s_a-2\tau))=p d_a+2q\), and \(d(F(s_a-\delta_j-\delta_k))=p d_a-d_j-d_k\). Thus each coefficient in the primewise relation defines a sequence in the source of the appropriate \(\theta_i\). There is no requirement that these coefficients themselves have formulas independent of \(p\).

Lemma 52 and Equation (53) now account for every term. The earlier powered coefficients \(\mathcal P(x_j)\), \(j<a\), vanish. The \(\epsilon^2D_*\) term is divisible by \(t^2\). Each \(x_j\), \(j\leq a\), assigned to \(\delta_j\), is divisible by \(t\), so every \(D_{jk}x_jx_k\) term is also divisible by \(t^2\). The complete transfer error vanishes, including when the number of its primewise summands grows. Products under the Thom comparisons introduce only index-zero units. Since \(C,C'\) and \(\mathcal P(u^{(a)})\) are units, the two remaining leading terms give \[ t\bigl(U_1g_a-U_2X_{a+1}\bigr)=t^2z \quad\text{in }\pi_0(L_a^M)_{s_a}, \tag{56}\] where \(U_1,U_2\in\pi_0(L_a^M)_0^\times\) and \(z\in\pi_0(L_a^M)_{s_a-2\tau}\). After the error coefficients in Equation (48) have been collected, the list of terms just used has size bounded in terms of \(a\).

Multiplication by \(t\) from index \(i_a\) to index \(s_a\) is injective by Lemma 50. Canceling it in Equation (56) gives \[U_1g_a-U_2X_{a+1}=tz.\] Reduce modulo \(t\) and use Equation (52). The reduction of \(g_a\) is the \(M\)-Hurewicz image of \(b_{a+1}\), because the construction of the new cycle and its chosen attaching-map path commutes with the unit from the sphere to \(M\). Lemma 52 identifies the reduction of \(X_{a+1}\) with a unit times the ordinary coefficient \(x_{a+1}\). We have therefore proved that the Hurewicz image of \(b_{a+1}\) is a unit at index zero times that coefficient, at weight \(w_{a+1}\).

This unit can be replaced by a scalar in \(D^\times\). Indeed, at index zero the actual layer has \[ \pi_0(L_a^M/t)_0 \cong\prod_{\mathcal U}(M_p)_0Y_{a+1,p} \cong\prod_{\mathcal U}\mathbb F_p \cong D/\varpi. \tag{57}\] The identifications are ring identifications via the unit: the quotient assertion identifies the underlying degree-zero module, and the unit map is already a ring map in the ultraproduct. Here \(I_{a+1}\) contains \(p\), and its other generators have positive ordinary degree. A unit in \(D/\varpi\) has a representative whose residue at \(p\) is nonzero on a \(\mathcal U\)-large set. Lift those residues to elements of \(\mathbb Z_{(p)}^\times\), and choose arbitrary units at the remaining primes. This gives the required lift in \(D^\times\). Its action agrees with that of the original layer unit. Denote the resulting scalar by \(u^{(a+1)}\).

The resulting Hurewicz equality is an equality in the one specified mapping group \[\pi_0\mathop{\mathrm{map}}_{\mathcal C}(S(w_{a+1}),M\otimes Y_{a+1}) \cong \prod_{\mathcal U}(M_p)_{d_{a+1}}Y_{a+1,p}.\] It consequently holds on a \(\mathcal U\)-large set at the actual primes, with scalar representatives in \(\mathbb Z_{(p)}^\times\). Naturality under the multiplicative projection \(M_p\to\mathrm{BP}\) gives the analogous Hurewicz equality in \(\mathrm{BP}\)-homology. Lemma 43 makes both coefficient classes nonzerodivisors in the current quotients. Applying Lemma 49 gives the two quotient assertions for \(Y_{a+2,p}\), in all degrees at once.

This completes the induction. At each step the Hurewicz equality requires only one new large-set restriction, together with the finitely many comparisons used at that step. The calculation of the entire homology module then holds at every prime of that set by coefficient linearity and the cofiber exact sequence. Since \(n\) is fixed, intersecting the finitely many resulting sets proves the simultaneous assertion. In particular, no intersection over all homological degrees, weights, or infinitely many stages is taken. ◻

Proof. Suspensions preserve this class. For finite CW spectra \(F_0,F_1\), compactness of \(F_0\), and the expression of localization as a filtered colimit of multiplication maps, give \[[ (F_0)_{(p)},(F_1)_{(p)} ] \cong [F_0,F_1]\otimes\mathbb Z_{(p)}.\] Thus a map between these localizations has the form \(f/s\), where \(f:F_0\to F_1\) is a stable map and \(p\nmid s\). Precomposing or postcomposing with multiplication by \(s\), an equivalence after localization, identifies its cofiber with \(\mathop{\mathrm{cofib}}(f)_{(p)}\). The cofiber of a stable map of finite CW spectra is again a finite CW spectrum: represent the map after a finite suspension by a cellular map between finite CW models and take its finite mapping cone, then desuspend. This proves the claim and applies inductively starting from the sphere. ◻

Proof of Theorem 1. For \(n=0\), choose any prime \(p\) and take the cofiber of \(p:S_{(p)}\to S_{(p)}\). Multiplication by \(p\) is injective on \(\mathrm{BP}_*\), so the cofiber exact sequence gives \(\mathrm{BP}_*X\cong\mathrm{BP}_*/(p)\), via the map from the sphere. This is a finite \(p\)-local spectrum.

For \(n\geq1\), fix the construction above and choose a prime in the common \(\mathcal U\)-large set of Proposition 53. Set \(X=Y_{n+1,p}\). Equations (51) and Lemma 54 show that \(X\) is the \(p\)-localization of a finite CW spectrum, and hence is finite \(p\)-local in the sense of the theorem. By Equation (55) and Lemma 43, its actual unit map from the \(p\)-local sphere induces a surjection \[\mathrm{BP}_*\longrightarrow\mathrm{BP}_*X \quad\text{with kernel }(p,v_1,\ldots,v_n).\]

For both cases, the polynomial cooperation algebra \(\mathrm{BP}_*\mathrm{BP}=\mathrm{BP}_*[t_1,t_2,\ldots]\) is flat over the coefficient ring (Hazewinkel 1977, sec. 4.4). Thus \(\mathrm{BP}\)-homology of a spectrum carries its natural \(\mathrm{BP}_*\mathrm{BP}\)-coaction, and the map from the sphere is a comodule map. Since it is surjective on homology, naturality determines the coaction on every element of the quotient: it is exactly the coaction induced from the coefficient comodule \(\mathrm{BP}_*\). Thus the displayed isomorphism has the canonical quotient comodule structure, and its generator is the image of \(1\in\mathrm{BP}_0\), in degree zero. This proves the assertion for every \(n\), with the prime allowed to depend on \(n\). ◻