Proof. We verify flatness, affineness, and then coverage. The Landweber flatness criterion applies to each localization of \(B_2\) inverting \(p\), \(v_1\), or \(v_2\), and to \(A_{\mathrm{Haz}}\): the invariant regular sequence remains regular up to the point where its ideal becomes the unit ideal. The interpretation of this criterion as flatness of law charts is the standard equivalence between Landweber exact comodules and flat descent; see [18] and [23].

One can pass explicitly from exactness on graded comodules to the flatness assertion being used. Apply exact coinduction \(\Gamma_P\otimes_P(-)\) to a sequence of graded \(P\)-modules, and then apply base change to the chart ring \(A\). This tests flatness of \(A\otimes_{P,\eta_L}\Gamma_P\) over \(P\) via \(\eta_R\); inversion of arrows interchanges the two units if necessary. Graded flatness here implies ordinary flatness. For an ungraded test module \(N\), put a copy of \(N\) in every weight and let each homogeneous element act through its ordinary action into the correspondingly shifted copy. A fixed weight of the graded tensor product recovers the ungraded tensor product. Thus the graded exactness test also tests arbitrary modules.

After pulling back to an affine scheme of laws with its grading frame trivialized, the fiber of \(f\) is the affine scheme of strict arrows whose other end is a \(B_2\)-law. The deleted locus cannot occur at that end by (4). This proves affineness of \(f\). Fiber products of affine law charts are affine as well: in addition to the strict-arrow coordinates, they retain only an invertible grading parameter. In particular the relevant chart maps and diagonals are affine.

It remains to check that the indicated flat charts cover. This may be checked on geometric points. In characteristic zero the logarithm gives strict isomorphisms of the required \(p\)-typical form. In characteristic \(p\), let \(h=1\) or \(2\) be the height, and compare two laws over an algebraically closed field with the same nonzero coefficient \(v_h\). For a fixed left law, solve for an arrow by prescribing the right coefficients in order. After the preceding equations have been imposed, the equation at step \(j\geq1\) is \[ \eta_Rv_{j+h} =v_h t_j^{p^h}-v_h^{p^j}t_j +\text{a polynomial already determined by }t_{<j}. \tag{5}\] Indeed, compare degree \(p^{j+h}\) in \([p]_{F_L}(\phi(T))=\phi([p]_{F_R}(T))\). All exponents in the left \(p\)-series are divisible by \(p^h\), so only \(\phi\) through degree \(p^j\) can enter there. The first occurrence of \(t_j\) is additive in degree \(p^j\). On the right, apart from the displayed coefficient \(\eta_Rv_{j+h}\), only previously fixed coefficients enter: the formal-sum coefficients needed at this step have total exponent at most \(p^j\) and hence, by homogeneity, use only generators of weight at most \(m_j\). This gives (5). Its leading coefficient is nonzero, so it can be solved at every step. Both charts contain laws of the prescribed height and leading coefficient; for the Hazewinkel chart use the homogeneous triangular coordinate change above. This proves coverage. Finally the displayed arrow ring is the usual Landweber base-change calculation for cooperations, and is also the ring obtained directly by pulling back the two ends of the law groupoid. ◻

Proof. We first descend the free action to finite quotients. We then prove the finite torsor assertion directly and pass to the limit. Set \(A_m=A^{Z_m}\). Every element has finite orbit, so \(A\) is integral over each \(A_m\). Two geometric points with the same restriction to \(A_m\) differ by \(Z_m\). To see this, restrict first to a subalgebra generated by finitely many finite orbits. For a finite group, distinct orbits of geometric points are separated by invariant functions: apply the Chinese remainder theorem and take an invariant product of separating functions. Formation of finite-group invariants commutes with field extension, since it is a finite kernel and a field extension is flat. The finite-level assertions then give nonempty closed sets of possible elements of \(Z_m\); compactness supplies a compatible element for the whole algebra.

Integrality lifts every geometric point of \(A_m\) to one of \(A\). The induced \(Z/Z_m\)-action on \(\mathop{\mathrm{Spec}}A_m\) is free. In fact, if a coset fixes such a point, lift it to a point of \(A\). The translate of that lift and the original lift differ by \(Z_m\) by the preceding paragraph. Geometric freeness upstairs forces the original coset to be trivial.

For completeness, a finite constant group \(K\) acting geometrically freely on an affine scheme gives a torsor over the invariant ring even when \(|K|\) is not invertible. At a lift of a geometric point of the invariant spectrum, choose elements \(b_1,\ldots,b_{|K|}\) whose evaluations on its orbit form an invertible matrix. Such elements exist by the Chinese remainder theorem after field extension, and hence a determinant of elements from the original algebra is nonzero there. If \(D_b=\det(g(b_j))_{g\in K,j}\), then \(D_b^2\) is invariant. After inverting it, solve for the coefficients of any element in the basis \(b_j\) using this matrix. Uniqueness makes the coefficients invariant. The same matrix gives the tensor torsor isomorphism. These neighborhoods cover the invariant spectrum by integrality.

Applying this argument to \(A_m\) shows that \(\mathop{\mathrm{Spec}}A_m\) is a finite faithfully flat \(Z/Z_m\)-torsor over \(\mathop{\mathrm{Spec}}A^Z\). Passing to the filtered union gives faithful flatness and the torsor identity for \(A\). The resulting affine map is quasi-compact, hence fpqc. ◻

Proof. We construct the quotient on an affine presentation, and then check the chart and the weight line. Take \(S=\mathop{\mathrm{Spec}}(A_2\otimes k)\) and quotient its affine arrow scheme \(S\times_{\mathcal M_k}S\) by vertical composition with \(Z\). The hypotheses of Lemma 3 hold. The action is geometrically free by the leading orders of finite-height \(p\)-series, and it is locally finite on the arrow coefficients by uniform continuity on each formal-variable jet. Thus the quotient is affine and the original arrow scheme is an fpqc torsor over it. Composition descends by centrality: after locally lifting two composable arrows, changing either lift changes their composite by the corresponding vertical element. Identity and inverse descend as well. Source and target remain flat by faithfully flat descent. This defines the asserted groupoid and \(\mathcal N\). Descent of a sheaf’s arrow isomorphism through the torsor gives the claimed description of \(r^*\) and its exactness.

Similarly, \(S\times_{\mathcal N}\mathcal X_k\) is the quotient of \(S\times_{\mathcal M_k}\mathcal X_k\) by vertical composition. An isomorphism in the quotient has local lifts whose choices form a \(Z\)-torsor, as is visible on the arrow chart. There is no residual automorphism from \(\mathcal X_k\) in this vertical subgroup: an automorphism from \(\mathcal X_k\) is a grading element, and its grading parameter must be identity to lie in the strict subgroup. This assertion holds fpqc locally over arbitrary bases, not only on geometric points. The quotient is therefore represented by the invariant affine of Lemma 3. Its map to \(S\) is flat, since the map upstairs is flat and the torsor is faithfully flat. This proves the assertion about \(f'\).

The torsor descriptions commute with base change. The central action is trivial on the weight line, so that line descends. Equation (4) in characteristic \(p\) makes \(v_1\) invariant. It is a non-zero-divisor on the law charts, proving the Cartier assertion. ◻

Proof. Normalize \(v_1=1\) by its weight-one grading. We will show that every normalized law is locally isomorphic to the stated law and then determine its stabilizer. An arrow from the stated law into an arbitrary normalized law is obtained by imposing \(\eta_Rv_{j+1}=0\) successively. Modulo the preceding equations, \[\eta_Rv_{j+1}=t_j^p-t_j +\text{a polynomial in }t_{<j}.\] This is the height-one case of the coefficient comparison in (5); equivalently, only \(\phi^p\) on the left and \(\phi(T^p)\) on the right can contribute the new \(t_j\). The equations are successively finite étale, so their limit is faithfully flat and supplies the cover.

On the splitting law the automorphism equations reduce recursively to \(t_j^p=t_j\). Their group of solutions is exactly \(Z\): \([1+ap^j](T)-_F T\) supplies any prescribed next digit in \(\mathbb F_p\), and successive correction and completeness identify all strict automorphisms. There is no remaining grading stabilizer, since strict arrows preserve the normalized coefficient \(v_1=1\). The first equivalence follows. Base change to \(k\) and quotient by the central group give the second. ◻

Proof. We normalize \(v_2\) and then identify arrows by lifting their special-fiber labels. Uniqueness of the lifts will also identify composition over every nilpotent base.

On \(V(t^d)\) the coordinate \(v_2\) is invertible. Normalize it to 1 by adjoining a \(q\)-th root of the grading parameter. This gives a flat cover by the stated chart: the map obtained by adjoining a grading parameter \(\lambda\) has coordinates \((v_1,v_2)=(\lambda x,\lambda^q)\) and is the composite of the original flat cover with the grading action. On \(\mathcal X_k\) the remaining grading group is \(\Lambda\), giving the asserted chart there; its ambient \(v_2\ne0\) open is \([\mathop{\mathrm{Spec}}k[x]/\Lambda]\).

We identify all arrows and their composition, not just their geometric fibers. If \(\rho\) is the grading parameter and \(t_j\) are the strict-arrow parameters with left end \(x\), the right chart point is \(y=x/\rho\), since \(\eta_Rv_1=x\). The remaining equations are \[ \eta_Rv_2=\rho^q,\qquad \eta_Rv_{j+2}=0\quad(j\geq1). \tag{12}\] Modulo \(x\) the first right unit is 1. Modulo the preceding equations the subsequent ones are \(t_j^{p^2}-t_j=0\). This follows by comparing \(\phi(T)^{p^2}\) with \(\phi(T^{p^2})\) for the Honda law; its coefficients lie in \(\mathbb F_p\), and all preceding digits are fixed by the square of Frobenius. Thus the special-fiber arrow scheme is proconstant with labels \[(\rho_0,t_1^0,t_2^0,\ldots) \in\Lambda\times k^{\mathbb N_{>0}}.\] Labels with \(\rho_0=1\) give \(G^1\), and grading-only arrows split off \(\Lambda\).

For a prescribed left end \(x\), each label lifts uniquely through the nilpotent layers. At a square-zero layer \((x^b)/(x^{b+1})\), only the special-fiber Jacobian acts. Its first diagonal entry is a unit because \(q\) is prime to \(p\); the other diagonal entries are \(-1\). After adding combinations of earlier rows, the system is lower triangular. Each equation is a finite polynomial, so the linear correction system on the infinite product of coefficient modules is solved recursively from its first row. This proves existence and uniqueness on every layer and over every chart algebra. Proconstant labels are unchanged by a nilpotent thickening, so the construction is functorial. In particular composition is determined by composition of the special-fiber labels.

Write the \(g\)-arrow as an arrow from the right end \(y\) to the left end \(x\), and define \(g(x)=y\). If an \(h\)-arrow is composed on its source side, its left end is \(y\), and the composite label is \(gh\). The resulting coefficient pullback is \(g\circ h\). This convention therefore gives left semilinear actions: the moving source coordinate \(g(x)\) supplies their coefficient action. Strict labels are tangent to identity, and the grading labels act as stated. Central \([z]\) fixes \(x\) and has trivial grading: its unique lift is \([z]_{F_x}\) itself. Dividing by it gives the second presentation in (10).

For the last assertion, a strict-label lift has \(\rho\in1+xR\). There is a unique \(\chi\in1+xR\) with \(\chi^{1-p}=\rho\), since \(1-p\) is a unit. The linear coordinate change \(T\mapsto\chi T\) sends \[v_i\longmapsto\chi^{1-p^i}v_i=\rho^{m_i}v_i.\] It carries the law at \(y=x/\rho\) to the source of the strict arrow. Composing gives the required formal isomorphism over \(R\). Finally invariance of \(t=xu\) gives (11) over the domain \(R\). Reducing this identity gives the formula on each \(R_d\); thus no division by a nilpotent coordinate is involved. ◻

Proof. The maps from \(\mathcal X_k/(t^d)\) in Proposition 6 are the subgroup maps for \(\Lambda\), so their pushforward is coinduction. To check the semilinear formula explicitly, let \(M\) be a semilinear \(G\)-module. An \(R_d\)-linear \(\Lambda\)-equivariant functional \(\epsilon:M\to R_d\) has adjoint \[m\longmapsto\bigl(s\longmapsto s(\epsilon(s^{-1}m))\bigr).\] The function is constant on cosets, is locally constant by smoothness, and is equivariant for the stated action. Evaluation at \(1\Lambda\) is the inverse adjunction. This proves the formula. Multiplication by \(t\) is injective on \(\mathcal O_{\mathcal X_k}\). Affineness and the projection formula show it remains injective after either chart pushforward, and identify the quotients with the pushforwards from \(\mathcal X_k/(t^d)\). Monicity makes these ordinary quotients the multiplication cofibers of pure complexes. ◻

Proof. The special-fiber labels, with composition truncated on each jet, make \(G^1\) a compact Hausdorff inverse limit of finite groups. The central subgroup \(Z\) is compact and hence closed, so \(H\) is compact Hausdorff. Each \(H(i)\) is closed. Nested cosets in this compact group show that \(H\to\varprojlim_i H/H(i)\) is surjective. To see that its kernel is trivial, represent an element in every \(H(i)\) by \(g\in G^1\). The nonempty closed sets \[\{z\in Z:g z^{-1}\in G^1(i)\}\] are nested. Their intersection is nonempty, and any element in it shows that \(g\in Z\), since the digit filtration of \(G^1\) is separated. This proves completeness and separatedness of the quotient filtration.

Work first in the ring of actual endomorphisms of \(F_0\), with its formal-group addition and composition. An automorphism beginning in digit \(i\) is \(1+h\), where \(h\) has order at least \(p^i\). Composition orders give the commutator inclusion. In the binomial expansion of \((1+h)^p-1\), the leading term at order \(p^{i+2}\) is \(ph\): here \([p]_{F_0}(T)=T^{p^2}\), and for \(p>3\) the remaining terms have larger order. Its coefficient is the original digit, since that digit lies in \(k=\mathbb F_{p^2}\). All digits occur by the arrow calculation above. The central subgroup has its first nonzero digit in \(\mathbb F_p\) at an even index, as follows from \([p^a]_{F_0}(T)=T^{p^{2a}}\). Quotienting gives (13) and the power assertion.

For the coordinates, proceed from \(s\) to \(s+1\). The \(p^s\)-th powers of \(g_1,g_2\) correct the two independent digits of \(H(2s+1)/H(2s+2)\). They do so also when inserted in their ordered positions, because the resulting commutators lie in the next filtration step. The \(p^s\)-th power of \(g_3\) then corrects the remaining digit of \(H(2s+2)/H(2s+3)\). These corrections are unique. Completeness and separatedness of the digit filtration give the coordinates on the inverse limit and all the asserted subgroup descriptions. The weight \(m_2=q\) is trivial on \(\Lambda\) and the other digits of \(\gamma\) vanish, proving its commutation assertion. ◻

Proof. The coordinates of Lemma 8 show that ordered monomials in \(U_1,U_2,U_3\) form a topological \(k\)-basis of \(\Omega\). At level \(H/\Gamma_s\), the ordered group elements and the ordered difference monomials with exponents less than \(p^s\) are related by an invertible binomial change of basis; larger \(p^s\)-th powers vanish in characteristic \(p\). Including powers of \(x\) gives the corresponding basis of \(\Delta\).

Initially filter by the closed spans of all words of weight at least a given integer. These spaces tend to zero. Indeed at each finite quotient the action preserves the ideal \((x)\), so \(x\) is normal there. It is also nilpotent, and modulo \(x\) the ideal generated by the remaining letters is the augmentation ideal of a finite \(p\)-group algebra. Thus the ideal generated by all the letters is nilpotent at each finite quotient. Conversely, the word-filtration spaces contain deep open neighborhoods by the ordered topological basis.

There is a strict gain in weight whenever letters are interchanged. For \(U_1,U_2\), the commutator lies in \(H(2)\), whose first two coordinates are divisible by \(p\). Characteristic-\(p\) binomial expansion therefore gives weight at least \(3\) for its difference from 1, exceeding the original weight \(2\). Commuting \(U_1\) or \(U_2\) with \(U_3\) gives an element of \(\Gamma_1\), so the difference has weight at least \(p>4\). Finally \[U_i x-xU_i=(g_i(x)-x)g_i\] has weight at least \(8\), since \(g_i(x)-x\in x^2R\). This is larger than either \(\operatorname{wt}(U_i)+4\).

We can consequently reorder every word modulo strictly higher weight, and iterate to all weights by completeness. Every element of a word-filtration stage expands in ordered monomials of at least that weight. Uniqueness in the topological basis proves the reverse inclusion and shows that there are no further associated-graded relations. This proves (14).

Normality of \(x\) follows from \(g x=x(g(x)/x)g\). Weighting each group coefficient by the continuous unit \(g(x)/x\), or by its inverse, defines the requisite continuous change of coefficients even in the completion: at precision \(x^d\) choose a group quotient detecting the action modulo \(x^{d+1}\). Thus the ideal \((x)\) is two-sided and quotienting simply removes the \(x\) coordinate and its weight. ◻

Proof. The sheaf categories in question are Grothendieck categories. Colimits and AB5 follow from flat affine descent. For a generating set, choose an infinite cardinal \(\kappa\) dominating the sizes of the base and arrow rings. Starting with a bounded-size family of elements of a comodule, repeatedly adjoin components in finite expressions for their coactions and close under the module operations. Flatness injects the tensor of the resulting submodule into that of the original module, so the coaction identities restrict. The resulting subcomodules have bounded cardinality and form a generating set. For \(\mathcal X\) one may use its finitely many grading-trivialized principal charts; their disjoint union is affine and the same argument applies. The other stacks have the affine presentations constructed above.

Twisted periodic complexes are Grothendieck as well. Disk objects formed from sheaf generators in adjacent indices, with identity differential and then repeated with the twists, give generators. We use Smith’s recognition theorem in the form of [8]; the solution-set condition follows from accessibility by [8]. Quasi-isomorphisms satisfy the two-out-of-three property and are closed under retracts, as is seen on homology. They are accessibly embedded and accessible. For an explicit size argument, enlarge \(\kappa\) to include the twist data. In an arrow of complexes start with boundedly many elements, close under coactions and differentials, and then adjoin preimages for cycles in its cone. Iteration expresses any quasi-isomorphism as a \(\kappa^+\)-directed union of quasi-isomorphic subarrows of size at most \(\kappa\). Objects and arrows of that size are \(\kappa^+\)-presentable: their module presentations, coaction identities, and finitely many periodic differential identities can all be tested at that size. These tests can be performed on the flat affine charts.

Acyclic monomorphisms are preserved by pushout and transfinite composition, by AB5. Monomorphisms are cofibrantly generated: for a generator \(G_0\), adjoin successively the images of maps from \(G_0\) by pushing out inclusions of its subobjects, taking the inverse image of the subobject already attained at each step. There is a set of these subobject inclusions. A map right orthogonal to all monomorphisms is a split epimorphism with injective kernel. An injective object in the category of complexes is contractible: its monomorphism into the cone of its identity splits, making it a retract of a contractible complex. Such an epimorphism is therefore a quasi-isomorphism. This verifies the remaining recognition condition.

Ordinary chain mapping cylinders are compatible with the twists; they identify ordinary mapping cones with homotopy cofibers. The chain shift is an equivalence, so the model structure is stable. Testing fibrancy against disk inclusions shows that fibrant complexes have injective terms. Since all objects are cofibrant, the chain cylinder also gives the claimed computation of maps by chain homotopy.

For the last assertion, a fibrant replacement computes the derived maps. A quasi-isomorphism between complexes with \(\mathop{\mathrm{Hom}}(W,-)\)-acyclic terms remains a quasi-isomorphism after applying \(\mathop{\mathrm{Hom}}(W,-)\). Indeed its cone is acyclic with such terms. Dimension shifting along the short exact sequences of cycles shifts the possible obstruction to exactness past degree \(a\), where it vanishes. The argument requires only the uniform bound on \(\mathop{\mathrm{Ext}}(W,-)\) and therefore works for the unbounded periodic unfolding. ◻

Proof. Choose an affine fpqc chart \(l:S\to\mathcal S\) with \(S\) affine. The functor \(I=l_*l^*\) is exact, its unit is monic, and its unit cokernel \(J\) is exact. Its values are sections-acyclic by flat affine adjunction. Iterating \(I\) and \(J\) gives the usual coinduced resolution. On the chart all these functors, including ordinary sections of the coinduced terms, commute with filtered colimits; AB5 therefore proves the same for their cohomology. After restricting to a closed substack, the base-changed chart gives exactly the corresponding resolution. This proves the assertion about sheaves pushed forward from closed loci without requiring derived closed pushforward to be fully faithful.

Under the dimension bound, the resolution can end in \(J^a\), which is sections-acyclic by dimension shifting. These functors commute with weight twists. The finite totalization thus computes sections on periodic complexes, by Lemma 10; no infinite totalization is needed. The colimit assertion follows termwise. ◻

Proof. The key step is to lift the three-step Koszul resolution from the associated graded ring and use it to compute continuous invariants. Lift the graded Koszul resolution of \(k\) over \(\mathop{\mathrm{gr}}\Omega=k[A,B,C]\) to a resolution of length three by finite free compact left \(\Omega\)-modules. Here is the lifting argument. At each step the actual kernel is complete. Lift the homogeneous columns generating its associated-graded kernel to elements of that kernel and map in a free module with the corresponding shifted filtration. Successive subtraction of initial symbols converges and makes this a strict surjection. Its complete kernel has associated graded equal to the next Koszul kernel. At the final step that graded kernel vanishes, so separatedness makes the actual kernel zero. The resulting differentials compose to zero because their columns were lifted into the actual kernels.

For a smooth discrete module \(M\), continuous \(\mathop{\mathrm{Hom}}_\Omega(-,M)\) applied to this resolution has terms finite sums of \(M\), and hence gives a delta functor. It is acyclic in positive degrees on \(C_{\mathrm{lc}}(H,M_{\mathrm{vec}})\), coinduced from the underlying vector space: evaluation and translation identify the resulting complex with continuous \(k\)-linear Hom from the compact resolution into \(M_{\mathrm{vec}}\). The adjunction is continuous because a map from a compact free module to a discrete vector space kills a sufficiently deep stable filtration stage. Exact sequences of compact \(k\)-vector spaces split continuously, as is seen by dualizing to discrete vector spaces; thus this Hom complex is exact in positive degrees. Every smooth module embeds in such a coinduction. Effaceability identifies the delta functor with derived invariants and proves the bound.

For \(G\), take \(H\)-invariants and then exact \(\Lambda\)-invariants. Restriction of injectives from \(G\) to \(H\) preserves injectives, because finite-index induction is exact. Finally, forgetting the \(R_d\)-module structure of a semilinear representation preserves injectives: it has the exact left adjoint given by the free \(R_d\)-module with diagonal action. Applying the quotient-chart description proves the assertion about \(\mathcal N/(t^d)\). ◻

Proof. We first pass from the nilpotent charts to \(\mathcal N\), then restore the central group, and finally pass from characteristic \(p\) to the integral stack.

A \(t^\infty\)-torsion sheaf on \(\mathcal N\) is a filtered union of sheaves annihilated by powers of \(t\), with the evident weight twists in the kernel maps. Lemmas 11 and 12 show that its sections cohomology vanishes above degree \(3\). The open of Proposition 5 has exact sections. For an arbitrary sheaf \(M\), the kernel and cokernel of its localization map \(M\to M[1/t]\) are \(t^\infty\)-torsion. The open is affine over the stack, so localization is exact open restriction followed by affine pushforward. The two short exact sequences associated to the kernel and cokernel give the extra degree and prove the bound \(4\).

The right adjoint \(r_*\) is vertical invariants. Smoothness and centrality ensure that the kernel and cokernel of \(c-1\) have trivial vertical action and descend; they give a two-term delta functor. To prove that it computes the derived functor, efface its degree-one term by embedding a sheaf into \(I=l_*l^*\) for an affine law chart \(l:S\to\mathcal M_k\). On the arrow chart, the action is translation on the arrow algebra tensored over the other endpoint with the module coefficients. The map \(c-1\) is surjective there. Indeed the arrow scheme is a \(Z\)-torsor over its quotient, and fpqc locally it is split. A locally constant function is represented at a finite central period. Increase that period by a factor \(p\); the sum of the repeated old period is zero in characteristic \(p\), so the function can be integrated as a discrete difference on the larger cycle. This proves surjectivity locally, hence on the sheaf. Effaceability proves (16).

Because \(r^*\) is exact, \(r_*\) preserves injectives. Derived sections composed with (16) therefore give the bound \(5\) on \(\mathcal M_k\). Finite flat extension from \(\mathbb F_p\) to \(k\) commutes with the coinduced chart resolution of sections and detects vanishing, giving the same bound on \(\mathcal M_0\).

For \(\mathcal M\), first consider \(p\)-power-torsion sheaves. A sheaf annihilated by \(p^a\) has a finite filtration with characteristic-\(p\) subquotients; a general \(p^\infty\)-torsion sheaf is the filtered union of its kernels of powers of \(p\). The preceding bound and Lemma 11 give vanishing above degree \(5\). On the rational open the logarithm trivializes the strict law groupoid, leaving only \(B\mathbb G_m\), whose sections are exact. Apply the same kernel–cokernel argument to \(M\to M[1/p]\) to obtain the last bound in (15). ◻

Proof. The composite \(rl:S\to\mathcal N\) is an affine chart, so \(I\) has \(r_*\)-acyclic values by affine adjunction. Equation (16) and the sequence \(0\to M\to IM\to JM\to0\) show that \(J\) also has \(r_*\)-acyclic values. Both \(r_*I\) and \(r_*J\) are exact: \(I\) and \(J\) are exact, and the acyclicity of their values removes the connecting obstruction after applying \(r_*\). Hence the finite totalization (17) preserves quasi-isomorphisms. On injective-term fibrant replacements it agrees up to quasi-isomorphism with ordinary \(r_*\), by the same two-term acyclic resolution. It therefore computes \(Rr_*\) on periodic complexes, not just on sheaves. The twist line is pulled back from downstairs, so all these functors and identifications respect the periodicity.

Flat affine adjunction along \(f_k\) and \(f'\) identifies \(Rr_*T_k\) with \(T'\). Derived adjunction for \(r^*\), with \(r^*\mathcal O=\mathcal O\), gives the equality of sections homotopy. For a sheaf pulled back from \(\mathcal N\), the vertical kernel and cokernel in (16) are both the original sheaf. Thus there is a natural four-term exact sequence \[0\longrightarrow U\longrightarrow r_*Ir^*U \longrightarrow r_*Jr^*U\longrightarrow U\longrightarrow0.\] It applies termwise also to periodic complexes. Its two connecting maps give (18), with the consistent homological signs. In particular the triangle is natural, but no splitting of its final connecting map is asserted. ◻

Proof. The localization telescope is \(P\to P[-D]\to P[-2D]\to\cdots\), with transition maps induced by \(t\). Exactness of filtered colimits represents this telescope by the ordinary termwise colimit. On the affine charts this is localization at the principal open defined by \(t\), so it is the asserted open push-pull functor. Flat affine adjunction identifies the open category with its essential image.

Taking the inverse limit of the multiplication-cofiber triangles shows that the fiber of \(P\to\widehat P\) is the limit of the tower \[\cdots\xrightarrow{t}P[(d+1)D] \xrightarrow{t}P[dD]\xrightarrow{t}\cdots.\] Shifting this tower by one position does not change its limit. Consequently \(t\) is invertible on this fiber. On the other hand, \(t^d\) acts nullhomotopically on \(P/t^d\), by the standard homotopy on its multiplication cone. If \(A\) is \(t\)-invertible, naturality of \(t\) therefore gives \([A[j],P/t^d]=0\) for all \(j\): multiplication by \(t^d\) is simultaneously invertible from the source and zero on the target. The same assertion for mapping spectra, followed by the inverse limit, gives \([A[j],\widehat P]=0\).

Inverse limits are exact in the stable category, so completion is exact. If \(Q\) is right orthogonal to the \(t\)-invertible objects, then the fiber of \(Q\to\widehat Q\) is both \(t\)-invertible and right orthogonal to the \(t\)-invertible objects. It is therefore zero. This proves idempotence and characterizes the complete objects by that orthogonality. The \(t\)-invertible fiber of \(P\to\widehat P\) has zero quotient by every \(t^d\), giving the asserted equivalence of reductions. Localizing this same fiber triangle leaves its fiber unchanged, which proves the pullback assertion in (20).

Finally, the finite cohomological bounds of Proposition 13, together with exact filtered colimits, allow sections-homotopy to commute with this telescope. Equivalently, compute derived sections with the bounded cohomological range established there and commute the filtered colimit through the resulting calculation. This gives (21). ◻

Proof. The chart identifications give \(\mathcal N[1/t]=\mathop{\mathrm{Spec}}k\) and \(\mathcal M_0[1/t]=BZ\), and the nowhere-vanishing section \(t\) trivializes the periodicity line. The assertion for vector spaces follows by splitting cycles and boundaries.

Fix the chosen topological generator \(c\) of \(Z\) and write \(s=c-1\). Smooth representations identify with locally \(s\)-nilpotent \(\mathbb F_p[s]\)-modules: the identity \((1+s)^{p^a}=1+s^{p^a}\) identifies annihilation by a power of \(s\) with invariance under a sufficiently small open subgroup. This torsion category is hereditary. Indeed, embed a torsion module in a divisible \(\mathbb F_p[s]\)-module and take its \(s\)-power-torsion submodule. That submodule is still divisible: division by \(s\) can be performed within the torsion submodule, and every polynomial prime to \(s\) acts invertibly on it. It is thus injective. Its quotient by the original module is again divisible and torsion, giving an injective resolution of length one in the torsion category.

Represent a periodic complex by a fibrant complex \(I\) with injective terms. Write \(Z_j\), \(B_j\), and \(H_j\) for its cycles, boundaries, and homology. From \(0\to Z_{j+1}\to I_{j+1}\to B_j\to 0\) and heredity we obtain \[\mathop{\mathrm{Ext}}^1(H_j,B_j)=\mathop{\mathrm{Ext}}^2(H_j,Z_{j+1})=0.\] The identity of \(H_j\) therefore lifts to a map \(H_j\to Z_j\). Choose these lifts for one representative of each period residue and extend them periodically. They give a quasi-isomorphism from the finite sum of shifted pure \(H_j\) to \(I\). Finally, periodizing a length-one injective resolution of a pure object computes its maps; when \(D>3\), neither its resolution degree \(0\) nor its resolution degree \(1\) can contribute to a self-map with shift \(2\). ◻

Proof. We first exclude the extension and inverse-limit terms so that all operations have ordinary compatible representatives. We then normalize the line twists and compute composition at every finite precision.

The fiber of \(T'\to\widehat T'\) is \(t\)-invertible, so Proposition 15 identifies maps into \(\widehat T'\) out of \(T'\) or out of \(\widehat T'\). Maps into the limit can consequently be computed from the tower of targets \(T'_d\).

Flat affine adjunction along \(f'\) computes \([T'[n],T'_d]\) on \(\mathcal X_k\), with target \(\mathcal O_{\mathcal X_k/(t^d)}\). This target is pushed forward from the open \(v_2\ne0\), so we may work on \([\mathop{\mathrm{Spec}}k[x]/\Lambda]\). Let \(M\) be the restriction of \((f')^*T'\) to this chart. It is \(x\)-monic and \[M/x^dM=C_{\mathrm{lc}}(H,R_d).\] As an \(R_d\)-module the latter is the scalar extension of \(C_{\mathrm{lc}}(H,k)\), hence is free. Since \(|\Lambda|=q\) is invertible in \(k\), averaging a lift over \(\Lambda\) proves equivariant projectivity as well.

Choose an equivariant projective resolution of \(M\) of length one over \(k[x]\). Start with a surjection from a free module over the skew group ring \(k[x]\#\Lambda\); its underlying \(k[x]\)-module is free. Since \(k[x]\) is hereditary, its kernel is projective as a \(k[x]\)-module; averaging again promotes this to equivariant projectivity. Reduction modulo \(x^d\) preserves exactness, since \(M\) is \(x\)-monic. Applying \(\mathop{\mathrm{Hom}}\) into \(R_du^s\) factors through this reduced resolution. Its positive cohomology vanishes by the projectivity of \(M/x^dM\) just proved. Periodize the length-one resolution, with the specified line twists. Its bounded resolution length and projective terms make it valid for computing maps to twisted periodic complexes: mapping from it takes acyclics to acyclics. Since \(D>2\), the augmentation degree and the relation degree cannot coincide modulo \(D\). Thus the only possible groups are the ordinary maps in degrees \(sD\) and their first extensions; the latter vanish. In particular, \[ [T'[sD],T'_d] =\mathop{\mathrm{Hom}}\bigl(T',T'_d(s)\bigr), \qquad [T'[n],T'_d]=0\quad(n\notin D\mathbb Z). \tag{24}\]

These ordinary Hom groups have surjective transition maps. Indeed, a map at level \(d\) can be viewed as a functional on \(M/x^{d+1}M\) with values in \(R_du^s\); equivariant projectivity of \(M/x^{d+1}M\) lifts it through \(R_{d+1}u^s\twoheadrightarrow R_du^s\). The Milnor exact sequence therefore has zero derived-limit term. Each ordinary map in (24) kills \(t^d\) and factors uniquely through \(T'_d\). Compatibility in \(d\) gives actual maps of towers of pure complexes, and composition is their ordinary composition. This factorization is a statement about sheaf maps; it does not assert full faithfulness of closed pushforward on derived categories.

We next identify these representatives and their product. Evaluation at the identity coset in coinduction identifies a map with a \(\Lambda\)-equivariant \(R_d\)-linear functional \[C_{\mathrm{lc}}(H,R_d)\longrightarrow R_du^s.\] Without the twist, dualizing the functions on each finite group quotient gives an \(R_d\)-valued measure, with masses added under passage to a smaller quotient. Taking both group and coefficient limits gives \(\Delta\). To include the twist, use \[ R_du^s\ \cong\ x^{-s}R/x^{d-s}R, \qquad u^s\longmapsto x^{-s}. \tag{25}\] This is compatible with the group action: the formula \(g(u)=(x/g(x))u\) of Proposition 6 becomes the ordinary action on the fractional coefficient \(x^{-s}\). For the tame action, \(\lambda(u)=\lambda u\) follows from the invariance of \(t=xu\), and agrees with \(\lambda(x^{-s})=\lambda^s x^{-s}\). Thus the normalization is equivariant for every positive or negative \(s\). Equivariance of the functional is therefore invariance of its measure under the coefficient and conjugation action of \(\Lambda\). This gives the groups in (22).

For clarity, write integrals in left-coefficient notation. They are finite sums after passage to a sufficiently deep coefficient and group quotient, and their values are compatible as those quotients are refined. If \(b\) is the measure describing a map and \(F\) its input function, equivariance reconstructs its value at \(h\in H\) as \[ h\left(\int_z b(z)(h^{-1}F)(z)\right) =\int_z h(b(z))F(hz). \tag{26}\] The same formula applies to fractional coefficient lattices using (25). In our diagonal semilinear coinduction coordinates the projection formula is the pointwise map \[C_{\mathrm{lc}}(H,R_d)\otimes L_j \longrightarrow C_{\mathrm{lc}}(H,L_j),\qquad F\otimes l\longmapsto\bigl(h\longmapsto F(h)l\bigr), \qquad L_j=R_du^j.\] It is equivariant: applying \(g\) to either side gives \(g(F(g^{-1}h))g(l)\) at \(h\). In particular the second factor here is \(l\), not \(h(l)\). Tensoring both sides of a map by the \(j\)th line twist therefore multiplies both its input function and its output at the identity by \(x^{-j}\). Its normalized measure is unchanged. This is the \(R\)-linear extension of the evaluation functional to the shifted lattice; it is not a claim that \(x\) commutes with the resulting skew operator on sections.

Apply the map \(b\) first and then the map \(a\). Evaluation of their composite, using (26), is \[\int_h\int_z a(h)h(b(z))F(hz),\] which is precisely left-coefficient skew convolution \(ab\). All these identities hold at each finite level, including for negative twists. For example, if the degrees of \(a\) and \(b\) are \(sD\) and \(jD\), changing their representatives respectively by coefficients in \(x^{d-s}R\) and \(x^{d-j}R\) changes the product only by \(x^{d-s-j}R\). This follows because every \(h\) preserves \(x\)-adic order. Group quotients may always be refined to detect the relevant units \(h(x)/x\) and their inverses to the required coefficient order. Thus composition and reduction are strictly compatible, as asserted.

On invariant sections the input function is the orbit function of its value \(f\) at the identity. The action consequently becomes \[ a(f)=\int_H a(h)h(f). \tag{27}\] Multiplication by \(t=xu\) normalizes to the coefficient \(1\), proving the assertion about \(t\). In particular it is central in the graded operation algebra, although \(x\) need not be central in \(\Delta\). Finally, inverting this degree-\(D\) inclusion allows arbitrarily large fractional coefficient lattices in every fixed degree. Their union is \(\Delta[x^{-1}]\), and taking \(\Lambda\)-invariants commutes with this union. This proves (23). ◻

Proof. For \(\widehat T'\) this is immediate from (22). The finite-level calculation (24) applies equally to the full-center chart sheaf \(T_k\), replacing the functions on \(H\) by the functions for \(G_{\mathrm{up}}\). They are again free over \(R_d\) and equivariantly projective for the tame subgroup \(\Lambda\). Hence the same extension and Milnor terms vanish.

For \(T_0\), use extension of scalars \(e^*:\mathbf D(\mathcal M_0)\to\mathbf D(\mathcal M_k)\). Since \(k/\mathbb F_p\) is finite separable, trace duality identifies this extension functor also as a right adjoint to restriction. It preserves homotopy limits, and therefore commutes with completion. Moreover the unit \(F\to e_*e^*F\) is naturally split as an \(\mathbb F_p\)-linear map, so extension of scalars detects zero maps. The asserted vanishing follows from that for \(\widehat T_k\). ◻

Proof. First let \(\mathsf P\) be a finite-cell module. Choose a homogeneous basis of \(\mathsf P\) compatible with its cell filtration. Replace a basis vector \(e_b\) of degree \(b\) by the shifted pure complex \(T'_d[b]\) at every level \(d\). A degree-\(j\) right-module matrix entry \[e_b\longmapsto e_{b'}a, \qquad a\in\mathscr E_{b+j-b'},\] is represented in chain degree \(m\) by \[ a:(T'_d)_{m-b}\longrightarrow(T'_d)_{m+j-b'}. \tag{28}\] Use the unique representatives of Proposition 17 for all entries, including the differential entries, where \(j=-1\). The source and target use the same line-twist periodicity identifications in every degree and at every level \(d\). On the normalized charts these are the powers of the descent unit for \(u\) already used in (25).

Strict composition makes the matrix relation \(d^2=0\) an equality of actual maps. All nonzero degrees of \(\mathscr E\) are even, so no coefficient sign enters this relation. For a degree-\(j\) matrix \(f\), the Hom differential is \[d_{\mathsf Q}f-(-1)^j f d_{\mathsf P}.\] Formula (28) respects this differential, and therefore respects chain maps, chain homotopies, the signed shift, and ordinary mapping cones. It also makes all these complexes and maps compatible under the transitions \(d+1\to d\). Every matrix entry uses its compatible tower map, so its representative is fixed at all levels at once.

The order of multiplication agrees with the right-module convention. An endomorphism of the free right module determined by \(1\mapsto a\) sends \(z\mapsto az\), by left multiplication. Composing the endomorphism for \(b\) and then the one for \(a\) gives \(ab\), the same order as the operators in Proposition 17. Tensoring a right module with a left module consequently uses the left action of these same operators.

Take the homotopy limit of the constructed tower. Limits are additive and exact in the stable category, so the realization of a levelwise cone is the cone of the realized map. Compatible chain homotopies induce equal maps. A change of homogeneous basis gives isomorphic towers through its change-of-basis matrices. We obtain an exact functor on the homotopy category of finite cells. Mapping a shifted free module to an acyclic module gives an acyclic Hom complex, and induction along a finite free-cell filtration preserves this property. Finite cells are therefore homotopically projective, so their homotopy maps compute their derived maps.

On the free generator and every shift, the functor induces precisely the isomorphisms of Proposition 17. Fix a free source and use the long exact sequences of maps along a cell filtration of the target to obtain the isomorphism for every finite-cell target. Then apply the long exact sequences along a cell filtration of the source. This proves full faithfulness for all finite cells. The target derived category is idempotent complete, so the functor extends fully faithfully to retracts. Its image is closed under shifts, finite sums, cones, and retracts and contains \(\widehat T'\). Conversely, every object in its image is constructed by those operations from \(\widehat T'\). Its image is therefore exactly \(\mathop{\mathrm{Thick}}(\widehat T')\). ◻

Proof. Suppose first that \(\mathsf P\) is a finite-cell module and write \(Y_d\) for its realization tower. Each term of \(Y_d\) is a finite sum of twists of \(T'_d\). Those twists are sections-acyclic: flat affine adjunction reduces sections to the nilpotent chart of \(\mathcal X_k\), where taking \(\Lambda\)-invariants is exact. The bounded derived-sections calculation from Proposition 13 therefore computes \(\pi_*Y_d\) by the termwise sections complex. By (27), that complex is \(\mathsf P\otimes_{\mathscr E}\mathsf L_d\). This identifies differentials and maps, as well as the underlying groups.

Write \(C_d=\mathsf P\otimes_{\mathscr E}\mathsf L_d\). In each fixed degree its terms, cycles, boundaries, and homology are finite groups: \(k\) is finite, the fractional lattices modulo \(x^d\) are finite dimensional, and there are only finitely many cells. Each of these towers therefore satisfies the Mittag–Leffler condition, so its first derived inverse limit vanishes. Taking inverse limits in the exact sequences defining cycles, boundaries, and homology shows that \(H_*(\varprojlim_d C_d)=\varprojlim_d H_*(C_d)\). The Milnor sequence for \(Y=\mathop{\mathrm{holim}}_dY_d\) and the finite graded-free underlying module of \(\mathsf P\) now give \[\pi_*Y =H_*\left(\varprojlim_d \mathsf P\otimes_{\mathscr E}\mathsf L_d\right) =H_*\bigl(\mathsf P\otimes_{\mathscr E}\mathsf L\bigr).\] The tensor-limit interchange is degreewise and involves only finitely many summands. Finite-cell right modules are homotopically flat: after tensoring their cell filtration with an acyclic left module, every successive quotient is a shift of that acyclic module, so induction makes the full tensor acyclic. Thus the ordinary cellular tensor computes the derived tensor in the first formula.

Now take the \(t\)-colimit. Proposition 15 identifies it with the localized sections. Filtered colimits are exact, and \(\mathsf L[t^{-1}]=\Phi[t^{\pm1}]\), which gives the second formula with the same homotopical flatness justification. The identifications are natural for finite-cell maps. Applying them to idempotent summands proves both formulas for all perfect modules. ◻

Proof. Multiplication by \(t^d\) is strictly zero on \(Y_d\), because it is strictly zero on every shifted \(T'_d\) in its terms. For \(e\geq d\), map the ordinary cone of \(t^d:Y_e[dD]\to Y_e\) to \(Y_d\) by the tower transition on the unshifted summand and zero on the shifted summand. Strict annihilation makes this a chain map. These maps are compatible in \(e\) and natural for the matrix maps defining our finite-cell constructions. Exactness of homotopy limits gives a map \(Y/t^d\to Y_d\).

On the free generator this is the equivalence \(\widehat T'/t^d\simeq T'/t^d=T'_d\) of Proposition 15, represented by the ordinary quotient. The comparison respects mapping cones and shifts, with the usual sign isomorphism for a shifted cone. Induction on the finite cell filtration proves the equivalence for \(Y\) and identifies the tower projection with its intrinsic reduction map.

The groups \(\pi_iY_d\) are finite by the section calculation, and their first derived inverse limits vanish. Hence \(\pi_iY=\varprojlim_d\pi_iY_d\) is profinite, and the natural equivalences just proved identify this topology with the intrinsic one. For a retract \(Y'\) of \(Y\), naturality sends its defining derived idempotent to an idempotent \(e_d\) on every \(Y_d\). Reduction preserves the retract, so \(Y'/t^d\) is the summand split by \(e_d\); this statement takes place in the derived category and requires no strict idempotent representative. On the finite group \(\pi_iY_d\), the map \(\pi_i e_d\) is an ordinary idempotent. Taking its image commutes with the inverse limit, because the associated image and kernel decompositions are compatible with the transition maps. This gives the displayed formula for \(\pi_iY'\) and its intrinsic reduction topology. ◻

Proof. For \(T'\), flat affine adjunction identifies sections-homotopy with the cohomology of twists on \(\mathcal X_k=\mathcal P_k(1,q)\). The two-variable principal-open Čech complex computes it, also in the twisted periodic category. Weight by weight, its \(H^0\) is spanned by nonnegative monomials of that weight and its \(H^1\) by Laurent monomials with both powers negative. Both sets are finite in a fixed weight because the two weights \(1\) and \(q\) are positive. Thus every \(\pi_iT'\) is finite. Degreewise finiteness is preserved by shifts, finite sums, triangles, and retracts, proving the first assertion.

Now let \(P\) satisfy the stated degreewise finiteness hypothesis. The cofiber sequence for \(t^d\) makes every \(\pi_i(P/t^d)\) finite. The Milnor sequence therefore gives \[ \pi_i\widehat P=\varprojlim_d\pi_i(P/t^d), \tag{31}\] with zero first derived-limit term. Give this group its compact Hausdorff inverse-limit topology. The equivalence \(P/t^d\simeq\widehat P/t^d\) is compatible with the projections: both maps \(\widehat P\to P/t^d\) agree after precomposition by \(P\to\widehat P\), and this determines a map uniquely because its fiber is \(t\)-invertible whereas the target is complete. This proves that (31) describes the intrinsic reduction topology.

The fracture square gives a long exact sequence with the map \[ \pi_i\widehat P\oplus\pi_iLP \longrightarrow W_i(\widehat P). \tag{32}\] Its kernel is a quotient of \(\pi_iP\) and its cokernel embeds in \(\pi_{i-1}P\). In particular both are finite. The same holds in every shifted degree. Write \(C_s=\pi_{i+sD}\widehat P\) and let \(K_s\) be its image in \(W_i(\widehat P)\) under the localization identifications. The kernel \(C_s\to K_s\) is finite, by applying (32) in degree \(i+sD\). Give \(K_s\) its compact Hausdorff quotient topology.

The maps \(C_s\to C_{s+1}\) are continuous by naturality of the intrinsic reduction maps. Their cokernels are finite, by the multiplication-cofiber sequence and \(\widehat P/t\simeq P/t\). Thus the resulting inclusions \(K_s\subset K_{s+1}\) are continuous and of finite index. A continuous injection from a compact space to a Hausdorff space is a closed embedding; here finite index also makes it open. By (21), the union of these compact open subgroups is exactly \(W_i(\widehat P)\). Give the union its corresponding topology. This is a locally compact Hausdorff topological \(k\)-vector space.

Let \(B\) be the image of \(\pi_iLP\). Its intersection with \(K_0\) is finite, because a common image determines a pair in the finite kernel of (32). In the Hausdorff group \(K_0\), choose a neighborhood of zero avoiding the nonzero points of this finite intersection. It meets \(B\) only in zero, so \(B\) is discrete. A discrete subgroup of a Hausdorff topological group is closed. The quotient by \(K_0+B\) is finite, by the finite cokernel of (32); hence \(W_i(\widehat P)/B\) is covered by finitely many translates of the compact image of \(K_0\). Thus \(B\) is cocompact.

Finally, let \(Q\) and \(f:\widehat P\to\widehat Q\) be as in the statement. The map \(f\) induces maps of the intrinsic finite reduction groups, so it is continuous on the profinite homotopy groups. After localization its restriction to each compact stage is continuous, so it is continuous on the unions just constructed. No global map \(P\to Q\) inducing \(f\) is needed. The finite-level section calculation of Proposition 20 and the natural reduction equivalences of Lemma 21 identify the profinite topologies for the perfect realization of \(\widehat P\) in the statement. These identifications commute with \(t\), so they identify each compact image and the union topology after localization. This includes the case where \(\mathsf P\) is a retract of a finite-cell module. ◻

Proof. We first bound the displacement of an element from its first nonidentity digit. The comparison is made on torsion points of the deforming formal group, where the digit gives an exact valuation and the displacement gives a lower bound for the same expression.

Choose a strict representative of \(g\) whose first nonidentity digit is at index \(i\), with coefficient \(b\ne0\). For even \(i\), choose \(b\notin\mathbb F_p\), using the quotient by the central subgroup. The chart construction gives a formal isomorphism \[\widetilde g:F_y\longrightarrow F_x,\qquad y=g(x), \qquad [p]_{F_x}(T)=xT^p+_{F_x}T^{p^2}.\] Its coefficients lie in \(R\): the grading change is effected by a \((p-1)\)-st root congruent to \(1\), followed by the compatible strict arrow. The reduction of \(\widetilde g(T)-_{F_x}T\) begins with \(bT^{p^i}\).

Extend the valuation \(v(x)=1\) to a completed algebraic closure of \(K_0\). Formal group series over \(R\) can be evaluated on its positive-valuation ball. The zeros of \([p^d]_{F_x}\) in this ball form a cyclic group of order \(p^d\), and every zero has multiplicity \(p^d\). Indeed, Weierstrass preparation, using reduction \(T^{p^{2d}}\), gives a polynomial of degree \(p^{2d}\) all of whose zeros reduce to zero; its unit factor evaluates to a unit. The order at zero over \(K_0\) is \(p^d\). Translation by a torsion point is an invertible formal coordinate change and preserves multiplicity, so the group of zeros has order \(p^d\). Its subgroup killed by \(p\) has order \(p\), which proves cyclicity.

A primitive level-\(d\) point \(\zeta\) has valuation \[\lambda_d=\frac1{(p-1)p^{2d-1}}.\] For \(d=1\), the two summands of the \(p\)-series must have equal valuation. Inductively, to have \(v([p]\zeta)=\lambda_{d-1}\), one must have \(v(\zeta)<\lambda_1\); in that range the \(p^2\)-power summand dominates. This proves the formula.

Fix \(i\) and take \(d\) sufficiently large. The residue of \[\frac{\widetilde g(\zeta)-_{F_x}\zeta}{\zeta^{p^i}}\] is \(b\): coefficients of lower powers of \(T\) lie in \(xR\), and their valuations exceed \(p^i\lambda_d\) for large \(d\). The differences \(\tau-_{F_x}\zeta\), as \(\tau\) ranges over the level-\(d\) torsion points, are the multiples \([a]_{F_x}\zeta\) for \(a\bmod p^d\). If \(s=\nu_p(a)<d\), their valuation is \(p^{2s}\lambda_d\); the zero multiple has infinite valuation. When \(2s=i\), the residue after division by \(\zeta^{p^i}\) is \(a/p^s\bmod p\in\mathbb F_p^\times\), since \([p^s]_{F_0}(T)=T^{p^{2s}}\). It cannot cancel \(b\). Ordinary and formal differences have the same valuation, because \(T-_{F_x}S\) is \(T-S\) times a unit. Consequently \[v(\widetilde g(\zeta)-\tau)=p^{\min(i,2s)}\lambda_d,\] with \(s=\infty\) for \(\tau=\zeta\).

There are \((1-1/p)p^{d-s}\) multiples of exact order parameter \(s<d\). Counting these factors with multiplicity \(p^d\) in the Weierstrass factorization gives \[\begin{align*} v\bigl([p^d]_{F_x}(\widetilde g(\zeta))\bigr) &=p^{2d}\lambda_d \left((1-1/p)\sum_{0\le s<\lceil i/2\rceil}p^s +p^{i-\lceil i/2\rceil}\right)\\ &=\frac{p^{\lceil i/2\rceil}+p^{\lfloor i/2\rfloor+1}-1}{p-1} =r_i+\frac p{p-1}. \tag{37}\end{align*}\] On the other hand, substitution gives \(F_y\equiv F_x\bmod x^{\ell(g)+1}\). Since \([p^d]_{F_x}(\zeta)=0\), the value \([p^d]_{F_y}(\zeta)\) has valuation strictly greater than \(\ell(g)+1\) when \(\ell(g)\) is finite. Its image under \(\widetilde g\) has the same valuation and equals \([p^d]_{F_x}(\widetilde g(\zeta))\). Thus \(\ell(g)<r_i+1/(p-1)\), which proves (34). If \(y=x\), the latter value would instead be zero, contradicting (37). Since the digit filtration is separated, the action is faithful.

For \(r_i<l\le r_{i+1}\), the displacement bound implies \[N_l\ge[H:H(i+1)]=p^{2i-\lfloor i/2\rfloor}.\] The length of this interval is \(p^{\lfloor(i+1)/2\rfloor}\), so its contribution to \(\kappa\) is at most \(p^{-i}\). Summing proves (35).

The inequality \(r_{2s-2}<p^s\) proves the second inclusion in (36), and the digit indices give its index bound. For the first inclusion, every tangent group operator \(g-1\) raises valuation on \(R\) by at least one. In characteristic \(p\), \(g^{p^s}-1=(g-1)^{p^s}\), so \(g^{p^s}(x)\equiv x\bmod x^{p^s+1}\). Apply this to the ordered coordinates of \(\Gamma_s\). ◻

Proof. For a fixed \(i\in\{0,\ldots,m\}\), at most \(\lfloor m/N_l\rfloor\) other indices are congruent to \(i\) modulo \(N_l\). The distance formula following (33) therefore bounds the denominator’s valuation by \(r'_m\), proving integrality. By Lagrange interpolation, the divided difference of \(Z^n\) at \(u_0,\ldots,u_m\) is the coefficient of \(Z^m\) in its remainder modulo \(\prod_{i=0}^m(Z-u_i)\). It is zero for \(n<m\) and one for \(n=m\). For \(n>m\) it is a homogeneous polynomial of degree \(n-m\) in the \(u_i\). Since each \(u_i\in xR\), the last bound follows. ◻

Proof. For \(0\le m<(M+1)X\), use a scalar multiple over \(k\) of \(x^{qb-r'_m}D_m\) and average under \(\Lambda\). This averaging is defined because \(|\Lambda|=q=p+1\) is invertible in \(k\). By (35) and (40), \[ qb-r'_m\ge \left(\frac q{2M}-\kappa(M+1)\right)X>X. \tag{42}\] The measure is therefore in \(x^{X+1}\Delta\). This ideal is \(\Lambda\)-stable, so the averaged measure lies in \((x^{X+1}\Delta)^\Lambda\). Expand \(h_g^z=(1+u_g)^z\). Lemma 24 shows that the correction vanishes in all coefficient orders below \(qb\) and changes the coefficient of \(x^{qb}\) by \(\binom zm\bmod p\). Averaging preserves this coefficient, since \(x^{qb}\) is invariant and \(h_{\lambda g\lambda^{-1}}=\lambda(h_g)\).

The characteristic-\(p\) identity \[(1+v)^z=\prod_{i\ge0}(1+v^{p^i})^{z_i}\] expresses these binomial functions as products of digit binomials. For digits below \(r\), their binomial bases span all functions. At digit \(r\) the indices \(0,\ldots,M\) span the polynomials of degree at most \(M<p\). Every product of these bases occurs with \(0\le m<(M+1)X\), proving the asserted independence. Each stage has only finitely many depths and lower labels, so its total correction lies in \((x^{X+1}\Delta)^\Lambda\), even when every depth is used. For a fixed \(e\ge1\), all stages \(X\ge e\) vanish in every finite quotient \(R_e\#(H/\Gamma_s)\) for which the action is defined. Thus the partial measures stabilize in every such quotient, uniformly in \(s\) at fixed \(e\), and converge in \(\Delta^\Lambda\). The smaller-depth vanishing above separately makes every finite matrix block constant once its largest depth has been processed. Each correction vanishes modulo \(x\). ◻

Proof. Lemma 23 gives \(N_2\ge p^2\). The residues of \(u_g/x\) lie in \(k\), and distinct classes in \(H/H_2\) give distinct residues. Hence also \(N_2\le|k|=p^2\). The first \(p^2\) balanced points \(u_j/x\) consequently have distinct residues. Moreover \(u_\gamma/x\) has zero residue: \(\gamma\) commutes with \(\Lambda\), so \(h_\gamma\) is invariant, and its positive \(x\)-exponents are divisible by \(q>1\).

The measure \(\delta_\gamma-\delta_1\) integrates every power \((u/x)^n\), \(0\le n<p^2\), to zero modulo \(x\). The Vandermonde matrix at the first \(p^2\) points \(u_j/x\) is invertible over \(R\). Solving this finite system gives corrections at those points with coefficients in \(xR\) that kill these moments exactly. Average the resulting measure under \(\Lambda\). Vanishing of each moment is preserved, as is the reduction because \(\gamma\) commutes with \(\Lambda\). Expanding \(h_g^z\) now proves the coefficient bound. The later corrections preserve the reduction, and weighting by \(h_g^\alpha\equiv1\bmod x\) does too. Finally, \(x\) has weight \(4\), whereas \(U_3\) has weight \(3\) and initial symbol \(C\); this proves (43). ◻

Proof. For fixed \(j\), put \(d_j(l)=\mathsf R(l,j)-\mathsf R(l,j-1)\). Adding one column changes the rank by \(0\) or \(1\). If column \(j\) is independent of the earlier columns on rows at least \(l+1\), it remains independent after row \(l\) is added: the newly admitted earlier column \(l\) is zero on the old rows. Thus \(d_j(l)\ge d_j(l+1)\). The difference \(d_j(l)-d_j(l+1)\) is therefore \(0\) or \(1\), with at most one nonzero value in each column. The analogous argument for row increments gives at most one in each row. Telescoping over the finite block proves that \(\mathsf R(l,j)\) counts its matched pairs.

Both corner indices are still unmatched precisely when \[\mathsf R(l,j-1)=\mathsf R(l+1,j-1)=\mathsf R(l+1,j).\] To see the exceptional corner value explicitly, permute the corner column to the first position and write the block as \[\begin{pmatrix}c&r\\ u&B\end{pmatrix}.\] The displayed rank equalities say that \(r=\rho B\) and \(u=B\eta\) for some row vector \(\rho\) and column vector \(\eta\). Row and column operations reduce the block to \(B\) together with the single entry \(c-\rho B\eta\). The value \(\rho B\eta\) is independent of the choices: changing \(\rho\) by a vector annihilating \(B\), or changing \(\eta\) by a vector annihilated by \(B\), leaves it fixed. Exactly this value of \(c\) keeps the old rank; every other value creates a new match. All entries of \(r,u,B\) have smaller depth, which also proves the independence of decisions at the current depth. ◻

Proof. We begin at one initial depth and then proceed by stages. At each subsequent stage, adjacent clusters of lower labels are paired, and only specified central clusters retain next-digit replicas. We will record the surviving labels exactly. Once the stages have defined the operator, that record will both exclude late crossings of a fixed bound and count the unmatched labels at every large scale.

Give every row and column index \(j\) the label \(z=\alpha+j\). Begin with \[d=\lfloor p/M\rfloor.\] All entries of smaller depth are zero by initialization, since \(qd<p^2\). This depth belongs to the \(X=p\) stage and satisfies \(d\ge p/(2M)\). Match every column except those with \[z_0=\delta,\qquad z_1\in\{d_0,\ldots,d_{M-1}\}.\] The pivot rule permits this: at the one specified lower label protect \(M\) digits, and at all other lower labels protect none. The corresponding surviving rows have \[z_0=\delta-d,\qquad z_1\in\{d_0,\ldots,d_{M-1}\},\] without a borrow in subtraction. These statuses persist until the end of \(qb<p^2\). Surviving lower labels could meet again only at a depth congruent to \(d\) modulo \(p\), and no second such depth occurs in this range.

For the induction, take \(X=p^r\), \(r\ge2\), and suppose the surviving labels modulo \(X\) are exactly \[ \begin{array}{c|ccc} &z_0&z_j\ (1\le j\le r-2)&z_{r-1}\\ \hline \text{columns}&\delta&d(I)&d_i\ (0\le i<M)\\ \text{rows}&\delta-d&d(I-1)&d_i\ (0\le i<M). \end{array} \tag{46}\] All choices displayed are independent, and each has all \(p\) values of the new digit \(z_r\). Group these lower labels into clusters indexed by \(i\), according to their last digit \(d_i\).

The first possible band of edges in the range \(X\le qb<pX\) consists of all pairings from column cluster \(i\) to row cluster \(i-1\), with cluster indices taken cyclically. Their depths lie in \[ X(1/M-2/p)\le b\le X(1/M+2/p), \qquad b\ge X/(2M). \tag{47}\] To verify this, the allowed depth interval is \([X/q,X-X/q)\) because \(q=p+1\). The portions of a pair of lower labels below digit \(r-1\) differ by at most \[(p/4+3)X/p^2.\] For \(r=2\) this is immediate. For \(r>2\), their digits in position \(r-2\) have center indices in \(I\cup(I-1)\), of diameter at most \(2R_0\), while all lower contributions have absolute value less than \(X/p^2\). Thus same-cluster differences and their cyclic complements fall outside the allowed depth interval. For distinct clusters separated by \(k_1\) cyclic steps, \(1\le k_1<M\), the center difference is \(k_1p/M\pm1\). Adding the lower portions puts the depth in \(X(k_1/M\pm2/p)\). The bounds on \(p\) separate the \(k_1=1\) band from all later bands and give (47). Every pair of lower labels in the adjacent clusters occurs there.

Process this first band by increasing depth. Call a lower label full if all its \(p\) next-digit replicas remain. When an edge has two full endpoints, match them at every next digit, except that for column cluster \(i\in I\) retain the \(M\) digits \[ z_r\in\{d_0,\ldots,d_{M-1}\}. \tag{48}\] In this case there is no cyclic wrap to the row cluster: the depth is the positive difference of the ordinary lower representatives. Consequently the retained next digits on the two sides agree. Mark both endpoints no longer full. Whenever either endpoint is not full, there are at most \(M\) eligible next digits, since the next-digit replicas of the two endpoints correspond bijectively at a fixed depth. Require no pivot at all of them. These are precisely the choices allowed by the pivot rule. At one depth, different column lower labels have different row lower labels, so the prescriptions are compatible.

Every full label is paired full-to-full once. An edge with a nonfull endpoint creates no pivots, so it cannot remove any replica from an endpoint that is still full. Full status can only be lost, never regained. For each pair of adjacent clusters, the possible edges form a complete bipartite graph whose sides both have \(B_0^{r-2}\) lower labels. If a full label were left over on one side, equal numbers removed from the two sides would leave one on the other side too. Their edge would already have been processed while both endpoints were still full, a contradiction. After this band, every label has at most \(M\) surviving next-digit replicas. At every later depth in the stage, require no further eligible pivots. Thus precisely the column cluster indices \(I\) and the row cluster indices \(I-1\) remain, with the full center set at the next digit. This is (46) for \(r+1\).

Figure 1 records these two features of the induction: possible pairings run from a column cluster to the preceding row cluster, and central clusters preserve the center digits. The coefficient corrections converge by Lemma 25, so the stages define an actual operator \(a\) satisfying (43).

Fix now \(J\in\mathbb Z\). At a sufficiently large stage, a possible matched pair crossing \(J\) would have column index \[J<j\le J+X(1/M+2/p),\] since new matches occur only in the first band. Its lower label is \[(\alpha+j)\bmod X=(X-1)/2+j\] without wrap, and its next digit is \(\delta\). If the lower label is eligible, its cluster index lies in \(I\). In fact its representative divided by \(X\) differs from \(1/2\) by at most \(1/M+2/p+o(1)\); if its last digit is \(d_i\), the same representative differs from \((i+1/2)/M\) by at most \(1/p\). Hence \(i\) is within three places of \(s_0\) for large \(X\), which is inside \(I\) since \(M\ge81\). But the next digit \(\delta=d_{s_0}\) was explicitly retained there. Such a pair cannot be matched. The remaining finitely many depths each have only finitely many pairs crossing \(J\). This proves the first assertion.

For the second assertion, use labels with the prescribed units digit and all later digits in \(d(I)\) for columns or \(d(I-1)\) for rows, eventually all equal to \(\delta\). These survive at every stage. Changing finitely many digits of \(\alpha\) gives an integer offset, so each such label is \(\alpha+j\) for a genuine integer row or column index \(j\). At a highest modified position \(r\ge1\), choose a center digit strictly below \(\delta\) for a column or strictly above it for a row. Both sets of center digits contain such choices. The change at position \(r\) has absolute value at least \(p^r\), while all lower changes together have absolute value less than \(p^r\), so this determines the appropriate sign of the offset. The remaining \(r-1\) intermediate digits can be chosen freely, giving at least \(B_0^{r-1}\) distinct offsets of absolute value less than \(p^{r+1}\). For \(p^{r+1}\le N<p^{r+2}\), these offsets lie in the required intervals and \[B_0^{r-1}=p^{\beta(r-1)}\ge p^{-3\beta}N^\beta.\] Reducing the constant if necessary to work on both sides proves the assertion for every sufficiently large integer \(N\), including cutoffs between successive powers of \(p\). ◻

Proof. The overlap of \(\widehat T'\) is supported in degrees divisible by \(D\), with normalized coefficient space \(\Phi\) and the action of \(a\) used above. Since \(D>2\), the cofiber calculation in each summand has separate cokernel and kernel residues. In degree \(i\), the cokernel component comes from \(m[r]\) with \(i\equiv r\bmod D\), by the cofiber inclusion from \(\widehat T'[r]\). The kernel component comes from \(m[r']\) with \(i\equiv r'+1\bmod D\), by the connecting map into \(\widehat T'[r'+1]\). Summing gives (53) and its periodic compatibility.

For complete homotopy, Proposition 20 replaces \(\Phi\) by the compact coefficient stages \[(x^{-s}R)^\Lambda=F_{\lfloor s/q\rfloor}\Phi.\] The two components may use different values of \(s\) and hence of \(J\); both increase without bound under positive \(t\)-translation. On a cokernel component, the kernel of the map from the stage cokernel to \(\mathcal Q\) is exactly \((a\Phi\cap F_J\Phi)/a(F_J\Phi)\), finite by Lemma 29; its image is the compact open image of \(F_J\Phi\). On a kernel component, the stage gives simply \(\mathcal K\cap F_J\Phi\), injecting into \(\mathcal K\). This proves the finite-kernel and compact-open claims. Any two compact open subspaces here are commensurable: their intersection is open in each compact space, so the two quotient spaces are finite.

By Lemma 21, the kernel and cokernel calculations at a fixed compact stage are inverse limits of the corresponding finite-level calculations. These inverse systems of finite groups satisfy the Mittag–Leffler condition: at each fixed level, their decreasing sequences of images stabilize. Their inverse limits are therefore exact, and the resulting compact topology is the intrinsic topology from reductions. Taking the \(t\)-colimit recovers the same locally compact union topology on \(\mathcal Q\oplus\mathcal K\). Finally the coefficient stages increase without bound under positive \(t\)-translation. Choosing a single sufficiently large stage for both components proves the last assertion. ◻

Proof. The \(q_j\) are independent modulo \(\mathop{\mathrm{im}}_{\mathcal Q}(F_0\Phi)\). Otherwise a finite nontrivial relation, after subtracting a vector in \(F_0\Phi\), would be an actual image of \(a\) whose leading position is an unmatched positive row. This contradicts Lemma 29. It also proves independence of the pairs modulo \(C_0^{\rm lat}\).

Let \(n_+(N)\) count the positive unmatched rows at most \(N\), and let \(n_-(N)\) count the nonpositive unmatched columns of absolute value at most \(N\). In the ordered pairing, the first condition holds precisely for the first \(n_+(N)\) pairs and the second for the first \(n_-(N)\) pairs. Thus exactly \(\min\{n_+(N),n_-(N)\}\) pairs satisfy both conditions. Proposition 28 bounds each count below by a constant times \(N^\beta\) for every sufficiently large \(N\), giving the simultaneous bound. The leading terms of their kernel vectors are the distinct powers \(w^l=x^{q|l|}\), with \(q|l|\le qN\). A nonzero combination has one of these as its lowest \(x\)-power, so cannot lie in \(x^{qN+1}R\). ◻

Proof. By Lemma 30, the map from \(\pi_iY\) to \(\pi_iLY\) has finite kernel and compact open image; call this image \(C_i\). It is commensurable with \(C_0^{\rm lat}\), and \(\pi_iV=\mathcal L_i\) is a discrete complement to \(C_0^{\rm lat}\). Consider the joint map \[\pi_iY\oplus\pi_iV\longrightarrow\pi_iLY.\] After dividing its kernel by the finite kernel of \(\pi_iY\to C_i\), one obtains \(C_i\cap\mathcal L_i\). This intersection injects into \(C_i/(C_i\cap C_0^{\rm lat})\), which is finite. The cokernel of the joint map is a quotient of \(C_0^{\rm lat}/(C_i\cap C_0^{\rm lat})\), also finite. Thus the joint map has finite kernel and cokernel in every degree. The pullback long exact sequence expresses \(\pi_iU\) as an extension of the kernel in degree \(i\) by the cokernel in degree \(i+1\). Both are finite, so \(\pi_iU\) is finite.

Both \(V\) and \(LY\) are \(t\)-invertible and hence have zero completion. Exactness of completion and completeness of \(Y\) give \(\widehat U\simeq Y\). Localizing the pullback instead gives \(LU\simeq V\), since the localization of \(Y\to LY\) is an equivalence. These identifications preserve the structure map. The object \(Y\) is a finite sum of shifts of a two-cell cone on \(\widehat T'\), so lies in \(\mathop{\mathrm{Thick}}(\widehat T')\). ◻

Proof. Objects of \(\mathop{\mathrm{Thick}}(T')\) have finite homotopy in each degree. The same is true of \(U\) by Proposition 32. Proposition 22 therefore makes every map between their completions continuous on overlap homotopy. Under Lemma 30, the source topology is precisely the locally compact topology on \(\mathcal Q\oplus\mathcal K\) used here.

The additional lattice assertion uses that \(\widehat g\) comes from the global map \(g\). Naturality of the fracture square gives a commutative square from \(LU\to L\widehat U\) to \(LP\to L\widehat P\). Each paired vector is represented in \(\pi_iLU=\pi_iV=\mathcal L_i\). Applying \(\pi_iLg\) and then the target structure map places its image in \(\mathop{\mathrm{im}}(\pi_iLP\to W_i(\widehat P))\), which is the discrete lattice provided by Proposition 22. This argument uses the representative in \(\pi_iLU\); an individual pair need not lift to \(\pi_iU\). ◻

Proof. First, \(S_0\) is noetherian on both sides, since every one-sided ideal is principal. Nonzero right ideals \(dS_0\) and \(eS_0\) must intersect. Otherwise the right ideals \(e^i dS_0\), \(i\geq0\), would form an infinite internal direct sum in \(S_0\): in a finite relation, its term of least \(e\)-power belongs both to \(dS_0\) and to \(eS_0\) after canceling that power, and hence vanishes; continue inductively. This contradicts noetherianity. The same argument on the left gives the two-sided Ore division ring of fractions \(Q_0\). Clearing common denominators gives exact localization of modules. Consequently, for a finitely generated right module \(M\), the kernel \[M_{\mathrm{tor}}=\ker\bigl(M\longrightarrow M\otimes_{S_0}Q_0\bigr)\] consists exactly of the elements right-annihilated by a nonzero element of \(S_0\).

The quotient \(M/M_{\mathrm{tor}}\) embeds in the finite-dimensional right \(Q_0\)-vector space \(M\otimes_{S_0}Q_0\). Choose coordinates in that space and finitely many generators of the embedded module. In each coordinate, clear their common left denominators. Left multiplication in a coordinate is right \(S_0\)-linear, so this produces an injection \[M/M_{\mathrm{tor}}\longrightarrow S_0^r\] for some finite \(r\). The choice of left denominators is the one compatible with right modules.

Every submodule of \(S_0^r\) is free of finite rank. To prove this, induct on \(r\) and project to the first coordinate. The image is a right ideal. If it is zero, the induction hypothesis applies inside \(S_0^{r-1}\). If it is nonzero, its principal generator is a nonzero element of the domain, so the image is free of rank one. The projection then splits, and its kernel lies in \(S_0^{r-1}\). This proves the assertion in both cases. It follows that \(M/M_{\mathrm{tor}}\) is finite free, and hence that the sequence with this quotient splits. The kernel of a finite free presentation of \(M\) is also finite free by the same argument. This gives a finite free resolution of length at most one.

Now view a dg module over \(S_0[t^{\pm1}]\) as a \(D_0\)-periodic complex of right \(S_0\)-modules. A finite-cell module has finite free modules in each period residue, and noetherianity of \(S_0\) makes its homology finitely generated there. The same holds for a retract, hence for every perfect module \(P\). For each \(i\in\{0,\ldots,D_0-1\}\) choose a resolution \[0\longrightarrow R_{1,i}\longrightarrow R_{0,i} \longrightarrow H_i(P)\longrightarrow0\] by finite free right modules. Place \(R_{0,i}\) in degree \(i\) and \(R_{1,i}\) in degree \(i+1\). Lift the free generators of \(R_{0,i}\) to cycles in a representative complex for \(P\). The images of the relations are boundaries. Since \(R_{1,i}\) is free, lift them through the differential onto the boundaries in degree \(i\). This gives a map of the two-term complex into \(P\).

Extend the map periodically using multiplication by \(t\). The positive even degree \(D_0\) gives the usual periodization without changing the homological signs. This periodized complex has homology \(H_i(P)\) in residue \(i\) and zero in the other residues. Sum the maps over \(0\leq i<D_0\). The sum is finite and induces an isomorphism in every homology degree, so it is a quasi-isomorphism. Finally split each \(H_i(P)\) into its torsion and finite free parts. The periodizations of the free parts are finite sums of free shifts; those of the torsion parts are perfect because of their length-one finite free resolutions. This gives the asserted splitting. ◻

Proof. Call an object a finite extension of free layers if it admits such a tower. Zero, finite sums of shifts of \(T_0\), and their shifts have this property. The octahedral axiom concatenates two towers. More explicitly, suppose \(A\to E\to B\) is a triangle, and the first layer of a tower for \(A\) is \(F_0\to A\to A_1\). The cofiber \(E_1\) of the composite \(F_0\to A\to E\) fits into triangles \[F_0\longrightarrow E\longrightarrow E_1, \qquad A_1\longrightarrow E_1\longrightarrow B.\] Induction on the number of layers in \(A\), ending with the tower for \(B\), gives a tower for \(E\). In particular, if \(A\) and \(B\) have towers and \(f:A\to B\) is a map, the triangle \[B\longrightarrow\mathop{\mathrm{cofib}}(f)\longrightarrow A[1]\] gives a tower for \(\mathop{\mathrm{cofib}}(f)\). Finite sums of towers give the same closure for finite sums.

It remains to move retracts to the end. Let \(A\) and \(B\) be retracts of finite extensions \(\overline A\) and \(\overline B\). Idempotent completeness supplies complements \[\overline A\simeq A\oplus A', \qquad \overline B\simeq B\oplus B'.\] Extend a map \(f:A\to B\) to \(\overline f:\overline A\to\overline B\) by setting all other matrix entries to zero. Its cone is isomorphic to \[\mathop{\mathrm{cofib}}(f)\oplus B'\oplus A'[1].\] Thus \(\mathop{\mathrm{cofib}}(f)\) is a retract of a finite extension. Shifts, finite sums and successive retracts of such objects are also retracts of finite extensions. The resulting subcategory is thick and contains \(T_0\), which proves the lemma. ◻

Proof of Theorem 34. Suppose that \(U_0\in\mathop{\mathrm{Thick}}_{\mathcal C}(T_0)\). By Lemma 36, choose a retract insertion \(g_0:U_0\to P_0\) and a tower as in that lemma. Let \(g_j:U_0\to P_j\) be the composite with the map to its \(j\)th tail, and put \(Y_0=\Psi(U_0)\). We prove backward from \(j=s\) that \[H(\Psi(g_j))=0, \qquad H\bigl((\Psi(g_j))^*\bigr)=0.\] Both assertions hold for \(P_s=0\).

Write one extended step as \[F\longrightarrow P_1'\longrightarrow P_2' \longrightarrow F[1], \qquad g_1':Y_0\to P_1',\quad g_2':Y_0\to P_2',\] where \(F\) is a finite sum of free shifts, and suppose that the two vanishing assertions hold for \(g_2'\). By Lemma 35, choose \[P_2'\simeq V'\oplus L',\] with \(V'\) a finite sum of free shifts and \(H(L')\) torsion. Each component of the projection \(P_2'\to V'\) is a class in the homology of \((P_2')^*\). Dual-homology vanishing for \(g_2'\) says that its composite with each of these classes is zero. Hence \[Y_0\xrightarrow{g_2'}P_2'\longrightarrow V'\] is zero as a derived map. The exact sequence of maps into the fiber triangle therefore lifts \(g_1'\) to \[Z'=\mathop{\mathrm{fib}}(P_1'\longrightarrow V').\] The fiber construction, equivalently the octahedral axiom, gives a triangle \[F\longrightarrow Z'\longrightarrow L'\longrightarrow F[1].\]

In its homology sequence, the map preceding \(H(F)\to H(Z')\) has torsion source and free target in each degree. It is zero, so \(H(F)\to H(Z')\) is injective. The lifted map \(Y_0\to Z'\) is zero on homology after composition with \(Z'\to L'\), because that composite is the corresponding summand of \(g_2'\). Its homology therefore factors through the submodule \(H(F)\subset H(Z')\). The source \(H(Y_0)\) is torsion, so its map into the free module \(H(F)\) is zero. It follows that \(H(g_1')=0\).

The map \(g_1'\) is the image of a map \(U_0\to P_j\) in \(\mathcal C\), and this tower tail belongs to \(\mathop{\mathrm{Thick}}_{\mathcal C}(T_0)\). Hypothesis (56) gives \(H((g_1')^*)=0\) as well. This completes the backward induction. In particular \(\Psi(g_0)\) is zero on homology. It has a left inverse because \(g_0\) is a retract insertion and every functor preserves its retraction equation. Thus \(H(\Psi(g_0))\) is also a split injection, contradicting the nonzero homology of \(Y_0\). ◻

Proof. We first justify the finite-module filtration properties needed for the construction. If \(N\) is a submodule of a finite shifted filtered free module, choose homogeneous generators of \(\mathop{\mathrm{gr}}N\) over the noetherian ring \(\mathop{\mathrm{gr}}\mathcal A\) and lift them to \(N\). For an element of \(N\), subtract a linear combination of the lifts matching its initial symbol, and repeat. The remainder increases in filtration at each step. The coefficients of each chosen lift form a convergent series in \(\mathcal A\), so the element is a finite linear combination of the lifts. This also proves that the resulting surjection from a finite shifted free module is strict for the induced filtration.

Each quotient \(\mathcal A/F^n\mathcal A\) is finite: the graded pieces below \(n\) are finite-dimensional over the finite field \(k\). Consequently \(\mathcal A\) and its finite products are compact. The submodule just considered is therefore compact and closed, being the image of a compact finite free module. Notice that closedness was not assumed during the initial-symbol subtraction. These observations prove that \(\mathcal A\) is noetherian on both sides and that every finitely generated module has a complete separated good filtration. Submodule and quotient filtrations are good, by using their preimages in a finite free presentation. A map of finite good filtered modules is bounded up to a filtration shift, as one sees on strict generators. In particular, any two good filtrations on one module differ by bounded shifts.

Form the Rees ring and Rees module \[\widetilde{\mathcal A} =\bigoplus_{n\in\mathbb Z}F^n\mathcal A\,s^{-n}, \qquad \widetilde M =\bigoplus_{n\in\mathbb Z}F^nM\,s^{-n},\] where \(s\) is central of Rees degree \(-1\). For \(j\geq1\), work modulo \(s^j\) and invert the homogeneous elements whose reductions modulo \(s\) lie outside \((c)\). Such elements are regular, by induction through the \(s\)-layers and regularity of their nonzero reductions in the polynomial ring. They satisfy the Ore conditions on both sides. Indeed all commutators are divisible by \(s\), so for a denominator \(d\) and an element \(e\) one has \(\operatorname{ad}_d^j(e)=0\) modulo \(s^j\). Expanding this identity shows that \(d^j e\) has a right factor \(d\), and that \(e d^j\) has a left factor \(d\). Powers of a denominator are still denominators. These relations give the Ore conditions first for homogeneous elements and then for their finite sums. We may thus use ordinary two-sided Ore localization, retaining the Rees grading. Localization is exact on modules, by clearing common denominators.

Let \(M_j^\mu\) denote this localization of \(\widetilde M/s^j\widetilde M\). Denominators lift across truncation levels. The localizations therefore reduce compatibly, and localizing a module from a lower level at a higher level gives its own localization. Define \[ F^n\mu(M)=\lim_j(M_j^\mu)_n, \qquad \mu(M)=\bigcup_{n\in\mathbb Z}F^n\mu(M). \tag{57}\] The inclusions in this union come from multiplication by powers of \(s\). More precisely, regularity of \(s\) on the untruncated Rees module gives, after exact localization, \[ 0\longrightarrow(M_j^\mu)_{n+l} \xrightarrow{\ s^l\ }(M_{j+l}^\mu)_n \longrightarrow(M_l^\mu)_n\longrightarrow0 \qquad(l,j\geq1). \tag{58}\] Transitions in each fixed Rees degree are surjective. Taking limits in (58) gives the asserted inclusions and \[F^n\mu(M)/F^{n+l}\mu(M)=(M_l^\mu)_n.\] The filtration is separated and complete. In fact compatible Cauchy data eventually lie in one fixed \(F^n\mu(M)\), where both assertions follow from the inverse-limit description. Taking \(l=1\) identifies the associated graded with the homogeneous localization of \(\mathop{\mathrm{gr}}M\).

Products at the truncated levels give \(\mathcal S=\mu(\mathcal A)\) its ring structure and give \(\mu(M)\) its module structure. The natural maps from \(\mathcal A\) and \(M\) are obtained by taking their classes at each level. A bounded-shift map of good filtered modules gives a map on these unions; changing the bound does not change the map, by the transition inclusions. Thus the underlying module \(\mu(M)\) is functorial and independent of the good filtration chosen.

This functor is exact on finite modules. Use induced and quotient filtrations in a short exact sequence to make it strict. Its Rees sequence remains exact modulo \(s^j\), because the quotient Rees module is \(s\)-torsion-free. Ore localization preserves exactness. Inverse limits preserve it here because the transition maps in each module system are surjective, and taking the filtered union preserves it as well. The natural map \[M\otimes_{\mathcal A}\mathcal S\longrightarrow\mu(M)\] is an isomorphism for a free module and hence for every finite module by a finite presentation. This proves left flatness of \(\mathcal S\); over a noetherian ring, exactness on finite modules suffices for this flatness test. The identical construction for left modules proves right flatness. ◻

Proof. The graded ring of \(S\) is a domain, so multiplicativity of initial symbols proves that \(S\) is a domain. Every nonzero homogeneous element of \(\mathop{\mathrm{gr}}S\) is a power of \(C\) times a homogeneous unit. In a nonzero right ideal choose \(d_0\) minimizing the nonnegative \(C\)-order of its initial symbol. For any member \(d\) of the ideal, its initial symbol is a multiple of \(\sigma(d_0)\). Lift the multiplier and subtract the corresponding right multiple of \(d_0\). The remainder is still in the ideal and has strictly larger filtration, so its initial \(C\)-order is still at least the chosen minimum. Repeating, and summing the convergent multipliers, expresses \(d\) as a right multiple of \(d_0\). The successive multipliers tend to zero because their filtration degrees increase by the same amount as those of the remainders. This proves the right principal-ideal property; the left proof is identical.

The intersection argument in the proof of Lemma 35 now gives the two-sided Ore division ring of fractions.

The operator \(a\) still has initial symbol \(C\) in \(S\). A one-sided inverse would make \(C\) invertible in \(\mathop{\mathrm{gr}}S\), which it is not. Thus \(S/aS\ne0\). Given a nonzero representative \(b\in S\), the nonzero intersection \(bS\cap aS\) supplies a nonzero \(d\) with \(bd\in aS\). Every class in \(S/aS\) is therefore annihilated on the right by a nonzero element, proving torsion. ◻

Proof. Choose any finite family \(f_1,\ldots,f_v\) in the indicated joint kernel which is independent modulo \(x^nR\). We bound \(v\) uniformly. Consider the orbit functions \[\varphi_i(h)=h(f_i),\qquad h\in H,\] as functions with values in \(K_0\). The proof first places them in a lattice of integral-valued functions whose reduction injects into functions on \(H\). Interpolation then makes that reduction factor through a finite quotient, where the two annihilator relations give the required dimension bound.

We first bound the coefficients of integral-valued linear combinations of the \(\varphi_i\). Put \(F_i(y)=y^{L_0}f_i(y)\in k[[y]]\). For coefficients \(c_i\in K_0\) with minimum \(x\)-valuation zero, \[\sum_i c_i \varphi_i(h) =h(x)^{-L_0}\sum_i c_i F_i(h(x)).\] The reduction modulo \(x\) of \(\sum_i c_iF_i(y)\in R[[y]]\) is a nonzero series of order \(d<n+L_0\). Indeed the residues of the \(c_i\) give a nontrivial \(k\)-linear combination, and the assumed independence of the \(f_i\) bounds its order after multiplication by \(y^{L_0}\). Weierstrass preparation writes this series as a unit times a monic polynomial of degree \(d\). Evaluation at \(h(x)=xz_h\) preserves valuation of the polynomial because the evaluated unit is a unit.

As a polynomial in \(z_h\), that monic polynomial evaluated at \(xz_h\) has leading coefficient \(x^d\). Lagrange interpolation on the first \(d+1\) balanced orbit points expresses \(x^d\) as the sum of its values divided by the interpolation denominators. Every such denominator has valuation at most \(r'_d\). Therefore at least one of the values has valuation at most \(d+r'_d\). By (61), some value of the normalized orbit combination has valuation \(O(n)\), with constants independent of the chosen family and coefficients.

It follows first that the \(\varphi_i\) are independent over \(K_0\). It follows also that the coefficients of every integral-valued combination have valuations bounded below by \(-O(n)\): factor out the smallest coefficient valuation and apply the preceding upper bound to the remaining normalized combination. Let \(\mathcal L_{\mathrm{orb}}\) be the \(R\)-module of integral-valued functions in the \(K_0\)-span of the \(\varphi_i\). It contains each \(x^{L_0}\varphi_i\) and is contained in a fixed finite free lattice by the coefficient bound. Thus it is a full finite free lattice of rank \(v\). Its reduction injects into the space of functions \(H\to k\). In fact an integral-valued function whose values all lie in \(xR\) is divisible by \(x\) within \(\mathcal L_{\mathrm{orb}}\), by its definition.

Multiply an integral combination by \(z_h^{L_0}\), which is congruent to one modulo \(x\). Its residue is consequently the residue of \[x^{-L_0}\sum_i c_iF_i(xz_h).\] The uniform coefficient bound permits truncation to a polynomial in \(z_h\) of degree \(J=O(n)\): all later terms vanish modulo \(x\). The truncated polynomial is integral-valued on the whole orbit, since it differs from the preceding integral expression by an \(xR\)-valued function.

We next control the congruence level on which the residue of such a polynomial depends. The normalized Newton basis is the integral-valued-polynomial construction associated with \(P\)-orderings; see [9] for the general framework. The orbit-specific bounds and residue estimates below are proved here. The integral-valued polynomials of degree at most \(J\) on the orbit points have the \(R\)-basis \[ x^{-r'_m}\prod_{i=0}^{m-1}(z-z_{b_i^H}), \qquad 0\leq m\leq J. \tag{62}\] Balancedness gives at least \(\lfloor m/N_l\rfloor\) factors congruent to zero modulo \(x^l\) at every orbit point. Hence each polynomial in (62) is integral-valued. At \(z=z_{b_m^H}\) these counts are exact, so its value is a unit. Successively evaluating at \(z_{b_0^H},z_{b_1^H},\ldots\) now proves the basis assertion by triangular interpolation.

If two orbit points are congruent modulo \(x^l\) and \(N_l>J\), the residues of all the polynomials in (62) agree at the two points. To see this, expand the difference of a product by replacing one factor at a time. Deleting that factor loses at most one of the required matches at each level below \(l\). No matches at levels \(l\) and higher are needed, since \(N_l>m\). The difference of the replaced factors contributes valuation at least \(l\), exceeding the possible loss of \(l-1\). After division by \(x^{r'_m}\) every difference is thus in \(xR\).

Choose \(s\geq1\) with \(N_{p^s}>J\) and \(p^s=O(n^{1/3})\). Such a choice follows from the last inequality of (61), by taking the first \(s\) for which \(p^{3(s-1)}>J\). All the residue functions coming from \(\mathcal L_{\mathrm{orb}}\) then factor through \(H/\Gamma_s\).

We have placed the rank-\(v\) lattice, after reduction, inside functions on a finite quotient. The annihilator equations now impose two algebraic relations on that function space. For any \(\varphi\in\mathcal L_{\mathrm{orb}}\) and \(\mu=a_1,b_1\), the original annihilator equations imply \[ \int_{g\in H}h\bigl(\mu(g)\bigr)\varphi(hg)=0 \qquad(h\in H). \tag{63}\] For the functions \(\varphi_i\) this is obtained by applying \(h\) to \(\mu(f_i)=0\); field-linear combinations preserve the identity because their coefficients do not depend on \(h\). All integrals can be computed at finite coefficient and group precision. On reduction, \(h\) acts trivially on the coefficient field \(k\). The resulting functions on \(H/\Gamma_s\) therefore annihilate the left translates of \(\bar a_1\) and \(\bar b_1\). Their space has dimension at most \[\dim_k\frac{\Omega} {\Omega\bar a_1+\Omega\bar b_1+ \ker\bigl(\Omega\longrightarrow k[H/\Gamma_s]\bigr)}.\] The associated graded of this quotient kills at least \[C,\quad \sigma(\bar b_1),\quad A^{p^s},\quad B^{p^s},\quad C^{p^s},\] since \(g_i^{p^s}\in\Gamma_s\). The quotient filtration is finite, already on the finite group algebra. Modulo \(C\), take a leading monomial \(A^{u_0}B^{v_0}\) of the fixed nonzero polynomial \(\sigma(\bar b_1)\bmod C\). Among the monomials \(A^iB^j\) with \(0\leq i,j<p^s\), at most \((u_0+v_0)p^s\) are not divisible by \(A^{u_0}B^{v_0}\); the case of a nonzero constant gives a zero quotient. Thus the displayed dimension is \(O(p^s)=O(n^{1/3})\). Because \(\mathcal L_{\mathrm{orb}}/x\mathcal L_{\mathrm{orb}}\) has dimension \(v\) and injects into this function space, the required bound follows. ◻

Proof. For the first assertion, suppose that every member of \(J\) reduces into \(\bar a\Omega\). Starting with \(j\in J\), subtract a right multiple of \(a\) with the same reduction and write \[j=a d_0+j_1x.\] Normality of \(x\) allows the right division, and right saturation puts \(j_1\) back in \(J\). Repetition gives \[j=a(d_0+d_1x+\cdots+d_{r-1}x^{r-1})+j_rx^r.\] The remainder tends to zero, and the coefficient series converges in \(\Delta\). Thus \(j\in a\Delta\), and \(J=a\Delta\). This contradicts \(JS=S\), because \(aS\ne S\) by Lemma 38.

Choose now \(j\in J\) whose reduction is not in \(\bar a\Omega\). If its reduction has initial symbol divisible by \(C\), subtract a right multiple of \(\bar a\) matching that symbol. Continue in the complete filtered ring \(\Omega\). The process must terminate with an initial symbol outside \((C)\): otherwise the convergent sum of multipliers would express the original reduction as a member of \(\bar a\Omega\). Lift the finitely many subtractions to right multiples of \(a\) in \(\Delta\). They preserve membership in \(J\) and produce the asserted \(b\).

For the second assertion, first multiply \(b_0\) on the left by a power of \(x\) to obtain \(b\in\Delta\). This preserves annihilation of \(V\) and nonmembership in \(Sa\), since \(x\) is a unit on \(K_0\) and in \(S\). Whenever \(\bar b\in\Omega\bar a\), subtract a left multiple of \(a\) and divide on the left by \(x\). If this procedure never terminated, its equations would give \[b=(d_0+xd_1+\cdots+x^{r-1}d_{r-1})a+x^r b_r, \qquad b_r,d_r\in\Delta.\] The limit would put \(b\) in \(\Delta a\), a contradiction. Thus a reduction outside \(\Omega\bar a\) is reached. As on the right, successive initial-symbol subtraction by left multiples of \(\bar a\) then terminates with a symbol outside \((C)\). Throughout, subtracting a left multiple of \(a\) preserves annihilation of \(V\), and division on the left by \(x\) does also. The final integral operator is the required \(b_1\). Neither of these two convergent expansions commutes \(x\) through \(a\). ◻

Proof. The pairing identifies \(K_0\) with its continuous \(k\)-linear dual. Indeed a continuous functional kills some \(x^jR\); its values on the monomials \(x^i\) are exactly the coefficients of a Laurent series bounded below, read in the reverse order. For a coefficient \(c_0\in R\) and a group element \(g\in H\), formal residue substitution in \(dx/x\) gives \[ (c_0\delta_g)^\dagger =g^{-1}(c_0)\, \frac{x\,(g^{-1}(x))'}{g^{-1}(x)}\, \delta_{g^{-1}}. \tag{64}\] The Jacobian factor is an integral unit congruent to one modulo \(x\), since \(g(x)/x\in1+xR\). Residue substitution remains valid in characteristic \(p\): for monomials, the requisite identities depend on only finitely many coefficients of an invertible formal series and hold integrally, as is seen by verifying them first over characteristic zero. They then specialize to \(k\) and extend to Laurent series.

Formula (64) extends to measures in \(\Delta\). At any fixed coefficient precision, moving the coefficients by \(g^{-1}\) and multiplying by the Jacobian require only a finite level of the group action; inversion is continuous in the group coordinates. Equivalently, integrate the displayed formula on sufficiently fine finite quotients, at each precision. The resulting compatible measures define \(b^\dagger\in\Delta\) and satisfy the adjoint identity. Modulo \(x\), this is group inversion on the completed group algebra. Its associated graded map sends the generator symbols \(A,B,C\) to their negatives. It therefore preserves the two asserted symbol conditions. ◻

Proof. Assume the left side of (65). We first bound the images of the polar representatives belonging to the cokernel part of \(W(Y)\), then use the paired lattice to obtain large killed subspaces of the paired span. Composites into the generator vanish on the cokernel part, so they annihilate the kernel projections of these subspaces. Lemma 39 will force the operators representing those composites into the left ideal \(Sa\).

Proof. Take \(S_0=S\), \(D_0=D=2(p-1)\) and \(\mathscr B_0=\mathscr B\) in Theorem 34. Lemma 38 and the positive even value of \(D\) verify its ring and period assumptions. Use the full thick subcategory \(\mathcal C_{\mathrm{comp}}\) and the exact functor \(\Psi\) defined above. The subcategory contains \(U\) and \(T'\), and \(\Psi(T')\simeq\mathscr B\). Since \(\mathcal C_{\mathrm{comp}}\) is full and thick in \(\mathbf D(\mathcal N)\) and contains \(T'\), the ordinary thick closure of \(T'\) computed in either category is the same. Equation (60) and Lemma 38 verify that \(H(\Psi(U))\) is torsion in every degree and nonzero. Proposition 42 verifies (56) for every map into that thick closure, including every tower tail. Theorem 34 therefore gives \[U\notin\mathop{\mathrm{Thick}}_{\mathcal C_{\mathrm{comp}}}(T') =\mathop{\mathrm{Thick}}_{\mathbf D(\mathcal N)}(T').\] ◻

Proof. Put \[\begin{gathered} B=E_*=\mathbb Z_{(p)}[v_1^{\mathrm{Haz}},(v_2^{\mathrm{Haz}})^{\pm1}], \\ I_0=(0),\quad I_1=(p),\quad I_2=(p,v_1^{\mathrm{Haz}}),\quad Q_h=B/I_h\quad(0\leq h\leq2). \end{gathered}\] The ideals \(I_h\) are invariant under the chart Hopf algebroid, by (4) and the triangular coordinate change. Because the coefficient and cooperation degrees lie in \(D\mathbb Z\), the degree phases split \(\mathscr C_E\) into a product of \(D\) abelian categories. After shifting a phase and reindexing its degrees as weights, flat descent on the chart of Proposition 2 identifies it with \(\operatorname{QCoh}(\mathcal M)\). This is abelian descent, independent of the differential regrading and of topological algebraicity.

Proof of Theorem 44. We now follow the comparison in Pstrągowski v2 [27]. Propositions 4.4 and 4.8, item \((\spadesuit6)\), identify stage-zero reduction with \(E\)-homology on the topological side and differential-comodule homology on the algebraic side. Lemma 5.3 identifies the bottom functor on finitely projective comodules with their inclusion into the standard comodule heart, and its cocontinuous extension gives the heart identification. Lemmas 5.6 and 5.8 identify the heart and the homotopy objects used by the stages. Corollary 6.5 uses the preferred unit/free-algebra map and commutes with reduction to stage zero; Corollary 6.7 restricts that comparison to potential stages.

Lemma 45 supplies the dimension \(d=6\) and the two convergence inputs used by v2 Theorem 4.10. Its stabilization stage is \(\ell=d+k-1\) at homotopy level \(k\), hence \(\ell=7\) at level two. Theorem 6.8, in the stated Corollary 6.9 range \(p>8\), then gives the zigzag of homotopy 2-category equivalences \[h_2\mathrm{Sp}_E\xrightarrow{\sim}h_2\mathcal M_7^{\mathrm{top}} \simeq h_2\mathcal M_7^{\mathrm{alg}} \xleftarrow{\sim}h_2\mathcal D(E_*E).\] Choose an inverse to the final leg. Compose from left to right and then apply the regrading above on homotopy categories. Call the resulting equivalence \(\mathcal F\). The stage-zero compatibility throughout the restriction and assembly gives its natural identification of sheaf homology with \(E_*(-)\).

The recovery of triangles uses Appendix B of v2, with one local correction: the proof of Lemma B.3 overstates the local connectivity of the projection from a stable infinity-category to its homotopy 2-category. Its needed uniqueness follows directly from the cone mapping space. Write \(\mathcal H=h_2\mathscr T\) for a stable infinity-category \(\mathscr T\), and fix a span \(\sigma=(B\leftarrow A\to C)\). The space of cones with vertex \(W\) is \[\operatorname{Cone}_{\sigma}(W)\simeq \operatorname{Map}_{\mathcal H}(B,W) \mathop{\times}^{h}_{\operatorname{Map}_{\mathcal H}(A,W)} \operatorname{Map}_{\mathcal H}(C,W),\] where the homotopy pullback retains the cone’s 2-cell. If \(q_c\) is a cone with vertex \(V\) and \(q\) is any target cone with vertex \(W\), Lemma A.5 identifies the mapping space of cones as \[\operatorname{Map}_{\mathcal H_{\sigma/}}(q_c,q) \simeq\operatorname{hofib}_{q}\bigl( \operatorname{Map}_{\mathcal H}(V,W) \longrightarrow\operatorname{Cone}_{\sigma}(W)\bigr).\] The arrow postcomposes the entire cone \(q_c\) with a vertex map. When \(q_c\) is 1-cocartesian, Definition B.1 says that the displayed comparison is 1-connected. Its fiber over the arbitrary cone \(q\) is therefore nonempty and connected, so it has exactly one \(\pi_0\) class. Existence follows by lifting the span to \(\mathscr T\), forming a pushout, and applying Lemma B.2 to its image. This proves the existence and full uniqueness needed in Lemma B.3. Corollary B.4 then identifies any two such cones. The arbitrary-target form also gives suspension maps and their naturality: for \(f:X\to Y\), precompose the chosen cone under \(0\leftarrow Y\to0\) to obtain a cone under \(0\leftarrow X\to0\) with vertex \(\Sigma Y\), and apply that uniqueness. The two-dimensional lifting in Theorem B.6, together with Lemma B.2 and Corollary B.4, consequently recovers distinguished triangles, their boundary classes, and the natural suspension comparison; Corollary B.7 makes the induced equivalence triangulated.

Apply this recovery to the chosen equivalence of homotopy 2-categories and then use the exact regrading. It equips the same \(\mathcal F\) with its triangulated structure. Iterating its natural suspension isomorphism and its inverse gives the displayed isomorphisms for every integer \(j\). ◻

Proof. There is a relative smash identification \[\mathrm{BP}\wedge b\simeq (\mathrm{BP}\wedge\mathrm{BP})\otimes_{\mathrm{BP}}b.\] As a right \(\mathrm{BP}\)-module, \(\mathrm{BP}\wedge\mathrm{BP}\) is a wedge of free shifts: lift a right-module basis of its homotopy to obtain the equivalence. Thus the flat Künneth calculation gives the asserted tensor product. Under that identification, insertion of units, which defines the coaction, agrees with the cooperation coproduct. Only the homotopy cooperation formulas, not a further multiplicative enhancement, enter this calculation.

Landweber exactness now gives, on the Hazewinkel chart, \[A_{\rm Haz}\otimes_{P,\eta_L}\Gamma_P \otimes_{\eta_R,P}B_2.\] These are the functions on strict arrows with the chosen chart law at the left and a \(\mathcal X\)-law at the right, with grading frames identified. Descent is composition of arrows. This is precisely the pullback of \(f_*\mathcal O_{\mathcal X}\). All degrees lie in \(D\mathbb Z\), so its regraded sheaf is pure. Since \(f\) is affine and \(\mathcal O_{\mathcal X}\) is \(p\)-monic, so is \(T_{\mathbb Z}\).

The same calculation also identifies the completed generator. The completion \(b_p^\wedge\) is the homotopy limit of \(b/p^a\). The tower is surjective on homotopy, so its coefficients are \(B_2\otimes_{\mathbb Z_{(p)}}\mathbb Z_p\), with ordinary completion in each graded degree. The limit is complete by its Postnikov description and has the same multiplication-by-\(p\) cofiber as \(b\). It remains a \(\mathrm{BP}\)-module, and the same free-right-module calculation gives \(T_{\mathbb Z_p}\). Its \(p\)-monicity and reduction follow on the affine chart and then by affine pushforward. ◻

Proof. Connective \(\mathrm{BP}\)-module truncation and reduction give a map \(b\to H\mathbb F_p\). For \(N=b,H\mathbb F_p\), consider the connective Künneth spectral sequence for \[H\mathbb F_p\otimes_{\mathrm{BP}}(\mathrm{BP}\wedge N),\] where \(\mathrm{BP}\) acts on the first factor of \(\mathrm{BP}\wedge N\). The free-right-module calculation in Proposition 46 applies to both choices of \(N\). The map on Tor terms is therefore \[\mathbb F_p[t_1,t_2,\ldots]\otimes \bigwedge(\epsilon_i:i>2) \ \longrightarrow\ \mathbb F_p[t_1,t_2,\ldots]\otimes \bigwedge(\epsilon_i:i\geq0), \qquad |\epsilon_i|=2p^i-1,\] and is the inclusion. To compute it, use the regular Koszul resolutions of the right-base-change quotients of \(\Gamma_P\), on \(\eta_Rv_i\) for \(i>2\), and on \(p,\eta_Rv_i\) for \(i\geq1\), respectively. These resolutions are free over the left base. After tensoring with the left \(\mathbb F_p\), every differential vanishes.

The target spectral sequence collapses by the dimensions in Milnor’s dual Steenrod algebra calculation, recalled in [29]. The injective map and naturality then force the source to collapse page by page. Connectivity gives convergence, and the resulting map on mod-\(p\) homology is injective. Dually \(\mathcal A\) surjects onto \(H^*(b;\mathbb F_p)\). There are no classes in the odd degrees of \(Q_0,Q_1,Q_2\), so these primitives kill the degree-zero generator. The resulting quotient map from \(\mathcal A//E(Q_0,Q_1,Q_2)\) is an isomorphism by the same dimension calculation. ◻

Proof. Shift the chosen residue to degree zero. For the first assertion, take a fibrant representative \(C\) with injective terms and write \(B_j,Z_j\) for its boundaries and cycles. Lifting the isomorphism \(W\to H_0C\) to \(Z_0C\) is obstructed by \(\mathop{\mathrm{Ext}}^1(W,B_0C)\). In the intervening positive degrees the homology vanishes, so the short exact sequences \[0\longrightarrow B_jC\longrightarrow C_j \longrightarrow B_{j-1}C\longrightarrow0\] dimension-shift this obstruction through injective terms. Before reaching the next nonzero homology residue it becomes an Ext group of degree greater than \(a\), and hence vanishes. A lift gives a map from the pure \(W\) into \(C\). Repeat it by the period-twist identifications. It induces the given isomorphism on every homology sheaf and is a quasi-isomorphism.

For the second assertion take an injective resolution of \(W\) in degrees \(0,-1,\ldots,-b\), and periodize it. Map \(C\) into this resolution. The isomorphism on \(Z_0C/B_0C\) extends to \(C_0/B_0C\) with values in the first injective. Its composite with the next differential vanishes on cycles, so it prescribes a map on \(B_{-1}C\); extend that map to \(C_{-1}\) by injectivity. Continue in this way. At every intermediate step, vanishing of homology identifies the cycles with the boundaries on which the chain-map equation has already been prescribed. The finite resolution ends before the next homology residue, so its periodically repeated copies do not interfere. The resulting map is again a quasi-isomorphism. The same arguments begin in any chosen residue, proving the assertion for all shifts. ◻

Proof. The sphere has pure comodule homology \(\mathcal O\), and Proposition 13 gives \(\mathop{\mathrm{Ext}}^{>6}(\mathcal O,-)=0\). Thus the first part of Lemma 49 identifies its image, since \(D>12>6+1\).

For the chart objects we claim injective dimension at most \(5\). Resolve \(\mathcal O_{\mathcal X}\) by its sheaf Čech resolution for the three principal opens, of length \(2\). On each quotient-affine chart the localized regular polynomial ring has global dimension at most \(3\). Hence each term has injective dimension at most \(3\) after pushforward. This assertion applies to arbitrary graded modules: a graded free resolution is also an ungraded free resolution, and in its Hom complex the projection of every homogeneous input to the output of matching degree splits off the grading-preserving Hom complex. Its Ext groups are therefore summands of the ordinary Ext groups. Flat-affine adjunction for \(f\) preserves the injective-dimension bound. Combining the length-\(2\) resolution with the bound \(3\) gives the bound \(5\) for \(T_{\mathbb Z}\).

The same argument applies after tensoring the chart sheaf with \(\mathbb Z_p\): global dimension bounds hold for arbitrary modules over the localized regular rings, so no finite-generation assertion over \(\mathbb Z_{(p)}\) is needed. It gives injective dimension at most \(5\) for \(T_{\mathbb Z_p}\). Proposition 46, homology compatibility, and the second part of Lemma 49 now give the remaining identifications, since \(D>5+1\). The bounds are unchanged by the weight twists, so the sparsity proof applies in every residue. Equivalently, once the degree-zero objects have been identified, the suspension structure of \(\mathcal F\) gives the displayed isomorphisms for every \(j\). ◻

Proof. Equivalences preserve zero objects, finite biproducts, and retracts. Closure under shifts is built into the definition. To prove closure under triangles, consider \[A\longrightarrow B\longrightarrow C\longrightarrow\Sigma A\] and first suppose only that \(F(B),F(\Sigma A)\in\mathcal D_0\). Complete the map \(F(B)\to F(C)\) to a triangle \[F(B)\longrightarrow F(C)\xrightarrow{s}D_1\longrightarrow F(B)[1].\] The map \(s\) kills \(F(B)\to F(C)\). Full faithfulness and essential surjectivity transport the exactness of represented sequences in \(\mathcal C\), so \(s\) factors through the image of the next map: \[s=hF(C\to\Sigma A),\qquad h:F(\Sigma A)\longrightarrow D_1.\] Form a weak homotopy pullback \(P\) of \(s\) along \(h\). Its triangle is \[F(B)\longrightarrow P\longrightarrow F(\Sigma A) \longrightarrow F(B)[1],\] so \(P\in\mathcal D_0\). The pair consisting of the identity of \(F(C)\) and \(F(C\to\Sigma A)\) has equal composites to \(D_1\). The exact sequence defining the weak pullback therefore gives a map \(F(C)\to P\) whose composite with \(P\to F(C)\) is the identity. Thus \(F(C)\) is a retract of \(P\) and belongs to \(\mathcal D_0\). Weak pullbacks here can be constructed from fiber triangles and the octahedral axiom.

Now suppose \(A\) and \(B\) belong to the subcategory in the statement. Apply the preceding argument to each shifted triangle: its two inputs are \(F(\Sigma^jB)\) and \(F(\Sigma^{j+1}A)\), both in \(\mathcal D_0\). It follows that \(F(\Sigma^jC)\in\mathcal D_0\) for every \(j\). Rotating triangles gives the other two-out-of-three implications. ◻

Proof. We first construct enough \(p\)-monic epimorphisms of sheaves. On the whole graded \(\mathrm{BP}\) stack, the cofree-on-the-base comodule \(\Gamma_P\) is a union of graded dualizable subcomodules which are finite free over the left base. Specifically, take the left spans of the arrow monomials of bounded weight. There are finitely many such monomials at each bound. The coproduct preserves these spans because it is homogeneous, all base and arrow coefficients have nonnegative weight, and left scalars remain in the outer factor of the coaction.

For a graded comodule \(M\), the diagonal tensor \(\Gamma_P\otimes_PM\) is cofree on the underlying graded module of \(M\). This is also the projection formula for the affine chart \([\mathop{\mathrm{Spec}}P/\mathbb G_m]\to\mathcal M_{\rm all}\) of the whole stack. A homogeneous element of \(M\) therefore corresponds to a map from a grading shift of the unit into this diagonal tensor. It factors through \(N\otimes_PM\) for one of the finite-free subcomodules \(N\): primitives commute with the filtered union, or equivalently the given coaction expression has finite support. The inclusions of these spans are split on underlying left modules, so tensoring with \(M\) preserves the injections and the factorization respects coactions. Duality turns it into a map from a shifted \(N^\vee\) to \(M\). Evaluation at the counit restricted to \(N\) shows that the original element is in its image. Summing these maps gives an epimorphism from a sum of dualizables to every graded comodule.

To apply this on \(\mathcal M\), extend a quasi-coherent sheaf by pushforward from the height-open into the whole stack. This pushforward is quasi-coherent: on the flat affine charts the open is quasi-compact, and its pushforward commutes with flat base change. Restrict the epimorphism just constructed. The restricted dualizables are still \(p\)-monic, so \(\mathcal M\) has enough \(p\)-monic epimorphisms.

For a twisted periodic complex \(C\), use disk sums on these generators to choose a functorial degreewise epimorphism \(P_C\to C\). Its kernel \(K_C\) is termwise \(p\)-monic, as is \(P_C\). The natural map \[\mathop{\mathrm{cofib}}(K_C\longrightarrow P_C)\longrightarrow C\] is a quasi-isomorphism, and its source is termwise \(p\)-monic. The same construction works within the full subcategory of termwise \(p\)-monic complexes, so that subcategory has the same quasi-isomorphism localization as all complexes. An acyclic complex in the subcategory stays acyclic after reduction: apply the short exact multiplication-by-\(p\) sequence. Since the subcategory is closed under shifts and cones, ordinary reduction on it induces an exact functor on its localized homotopy category. This constructs \(Li^*\).

For \(C=i_*F\), the snake sequence for multiplication by \(p\) is \[0\longrightarrow F\longrightarrow K_C/p \longrightarrow P_C/p\longrightarrow F\longrightarrow0\] as a sequence of complexes over \(\mathcal M_0\). The cone on the middle map is a representative of \(Li^*i_*F\). It extends the cokernel \(F\) by the shifted kernel \(F[1]\), giving (70). The iterated boundary is natural for chain maps and hence descends naturally to the localization. ◻

Proof. Shifts and retracts preserve the stated property. Consider a triangle \(A\to B_1\to C\) and exponents \(a,c\) which work on every shift of \(A,C\), respectively. Naturality makes the composite of the \(c\)-fold power on \(B_1[j]\) with the map to \(C[j+2c]\) vanish. Exactness of represented sequences factors that power through \(A[j+2c]\). Another \(a\) applications of \(\delta\) kill the factorization, again by naturality. Thus \(a+c\) works uniformly on all shifts of \(B_1\).

For the second assertion, the cone of each factor in \(\delta_A^N\) belongs to \(\mathop{\mathrm{Thick}}(B(A))\): exactness identifies \(B(A[j])\) with a shift of \(B(A)\). Repeated octahedra show that the cone of their composite also belongs there. If the composite is zero, that cone is \(A[2N]\oplus A[1]\). Taking a retract and then a shift proves the claim. ◻

Proof. The fiber of \(U\to\widehat U\) is \(t\)-invertible by Proposition 15. Pullback preserves this property, and completion kills such objects. Therefore \[\widehat{r^*U}\simeq\widehat{r^*(\widehat U)}.\] The exact functor obtained by composing \(r^*\) with completion reduces the assertion to the generator. Its value on \(\widehat T'\) is its value on \(T'\), by the same invertible-fiber argument. The finite reductions are \(r^*T'_d\), where \(T'_d=T'/t^d\).

At each level the full coinduction description of Proposition 7 gives an exact sequence \[ 0\longrightarrow r^*T'_d\longrightarrow T_{k,d} \xrightarrow{c-1}T_{k,d}\longrightarrow0, \qquad T_{k,d}=T_k/t^d. \tag{72}\] Here \(c\) is a chosen topological generator of the center \(Z=1+p\mathbb Z_p\). We include the surjectivity check, which uses characteristic \(p\). A locally constant function on \(G_{\rm up}/\Lambda\) factors through a finite quotient. Refine this quotient until each central period has increased by a multiple of \(p\). This is possible because the center embeds in the profinite group and its central \(p\)-power images meet the tame \(\Lambda\)-images trivially. On each refined central cycle, the sum of the repeated values is zero in characteristic \(p\). The function can therefore be integrated as a discrete difference. This proves surjectivity of \(c-1\); its kernel is the rigidified coinduction object.

Take homotopy limits of (72). The result is a triangle expressing \(\widehat{r^*T'}\) using two copies of \(\widehat T_k\). Thus \(\widehat{r^*U}\in\mathop{\mathrm{Thick}}(\widehat T_k)\). This argument uses compatible finite reductions, not commutation of \(r^*\) with arbitrary homotopy limits. Restriction along the finite field extension commutes with the limits and sends \(\widehat T_k\) to a finite sum of copies of \(\widehat T_0\). Hence it gives the claimed membership for \(\widehat F\).

We have already proved nilpotence on the open pieces and on the thick subcategories of the complete generators. Apply it to the fracture triangles \[U\longrightarrow\widehat U\oplus LU \longrightarrow L\widehat U, \qquad F\longrightarrow\widehat F\oplus LF \longrightarrow L\widehat F.\] The open terms have zero obstruction, and the complete terms have nilpotent obstruction. Lemma 54 supplies a uniform exponent over all shifts of \(U\) and \(F\). ◻

Proof. Write \(F=e_*r^*U\). If the displayed membership held, the reduction comparison triangle, Lemma 55, and Lemma 54 would give \(F\in\mathop{\mathrm{Thick}}(T_0)\). Exact scalar extension then gives \[e^*F\in\mathop{\mathrm{Thick}}(e^*T_0)=\mathop{\mathrm{Thick}}(T_k).\] The object \(r^*U\) is a retract of \(e^*e_*r^*U\) by separability of the finite field extension. Hence \(r^*U\in\mathop{\mathrm{Thick}}(T_k)\). Apply the exact functor \(Q\) and use \(QT_k=T'\) from Proposition 14. It follows that \(Qr^*U\in\mathop{\mathrm{Thick}}(T')\). Nilpotence of the central comparison obstruction and Lemma 54 now imply \(U\in\mathop{\mathrm{Thick}}(T')\), contrary to (69). ◻

Proof of Proposition 52. Choose \(Z_{\rm sp}\) as in the statement, using essential surjectivity of Theorem 44. By Proposition 50, its ordinary homotopy groups identify with sections homotopy of \(\widetilde U\): \[\begin{aligned} \pi_i Z_{\rm sp} &\cong[\Sigma^iL_ES^0,Z_{\rm sp}]_{h\mathrm{Sp}_E}\\ &\cong[\mathcal O[i],i_*e_*r^*U]_{\mathbf D(\mathcal M)}\\ &\cong\pi_i Qr^*U. \end{aligned}\] These are identifications of abelian groups, since a triangulated functor is additive. The last identification uses the acyclic resolution and closed-base change of Section 2. For example, \(i_*e_*\) sends termwise injectives to complexes of sections-acyclic objects, and the finite sections bounds justify the computation on periodic complexes. The comparison triangle (71) and the first condition in (69) show that these groups are finite \(p\)-groups in every degree.

Set \(X_{\rm sp}=\tau_{\geq0}Z_{\rm sp}\). It is connective, and its homotopy groups are finite \(p\)-groups. Its finite Postnikov stages are \(p\)-complete, and \(p\)-complete spectra are closed under limits. The Postnikov limit consequently proves that \(X_{\rm sp}\) is \(p\)-complete.

Next, \(H\mathbb F_p\) is \(E(2)\)-acyclic. One can see this directly in the same module calculation as Proposition 46: connective ring truncation gives its \(\mathrm{BP}\)-module structure and \[\mathrm{BP}_*H\mathbb F_p=\Gamma_P\otimes_{\eta_R,P}\mathbb F_p.\] The left positive coefficient actions vanish successively, using \(\eta_Rv_i\equiv v_i\pmod{(p,v_1,\ldots,v_{i-1})}\). Landweber base change to the chart with \(v_2\) invertible therefore gives zero. The negative tail \(\tau_{<0}Z_{\rm sp}\) is a filtered colimit of finite Postnikov ranges with finite \(p\)-group homotopy. A composition series of each finite \(p\)-group expresses its Eilenberg–Mac Lane spectrum as a finite construction from \(H\mathbb F_p\). Consequently each range is a finite construction from shifts of \(H\mathbb F_p\) and is \(E(2)\)-acyclic. Thus the whole tail is acyclic and \[L_{E(2)}X_{\rm sp}\simeq Z_{\rm sp}.\]

Let \(V\) be any finite type-three spectrum. Such spectra exist by the periodicity theorem, taking successive self-map cofibers [30]. The chromatic Bousfield-class calculation [30] \[\langle E(2)\rangle =\langle K(0)\vee K(1)\vee K(2)\rangle\] shows that both \(V\) and its Spanier–Whitehead dual are \(E(2)\)-acyclic. Indeed finite duality over each Morava \(K\)-theory field preserves the vanishing that defines type three. If \(DV\) is the Spanier–Whitehead dual and \(W\) is any spectrum, finite duality gives \[[W,V\wedge Z_{\rm sp}]\cong[DV\wedge W,Z_{\rm sp}]=0,\] because \(DV\wedge W\) is \(E(2)\)-acyclic and \(Z_{\rm sp}\) is local. Hence \(V\wedge Z_{\rm sp}=0\). Consequently \[V\wedge X_{\rm sp} \simeq\Sigma^{-1}\bigl(V\wedge\tau_{<0}Z_{\rm sp}\bigr).\] The left side is bounded below and has finite homotopy in each degree, since \(X_{\rm sp}\) is connective with finite homotopy and \(V\) has finitely many cells. The right side is bounded above, since its tail is bounded above and \(V\) has finitely many cells. It follows that \(\pi_*(V\wedge X_{\rm sp})\) is finite in total. Since \(X_{\rm sp}\) is connective and \(p\)-complete, Mahowald–Rezk’s criterion [22] now proves fp-ness as well as fp-type at most \(2\); fp-ness is not an additional assumption in that criterion. This connective-cover step is also related to their general formulation in Proposition 3.5 and the chromatic construction in Section 7.2(2).

Finally suppose that \(X_{\rm sp}\) belonged to \(\mathop{\mathrm{Thick}}_{\mathrm{Sp}_p^\wedge}(b_p^\wedge)\). The full category of \(p\)-complete spectra is stable, so its finite constructions remain finite constructions of spectra. Exact localization gives \[Z_{\rm sp}\in\mathop{\mathrm{Thick}}(L_Eb_p^\wedge).\] Apply the triangulated equivalence \(\mathcal F\). It preserves shifts, distinguished triangles, and retracts, while Proposition 50 identifies the generator with \(T_{\mathbb Z_p}\). Hence \[\widetilde U\in\mathop{\mathrm{Thick}}(T_{\mathbb Z_p}).\] The pure sheaf \(T_{\mathbb Z_p}\) is \(p\)-monic and has reduction \(T_0\) by Proposition 46. Exactness of \(Li^*\) therefore gives \(Li^*\widetilde U\in\mathop{\mathrm{Thick}}(T_0)\). This contradicts Lemma 56 and completes the proof. ◻

Proof. Suppose that \(X_{\rm sp}\) had fp-type at most one. The height-one theorem of Lee–Pstrągowski [21] would give \[X_{\rm sp}\in\mathop{\mathrm{Thick}}(\mathrm{BP}\langle1\rangle_p^\wedge).\] We check that the generator on the right is already in \(\mathop{\mathrm{Thick}}(b_p^\wedge)\).

In the same monoidal category of \(\mathrm{BP}\)-modules used to construct \(b\), form the further scalar cofiber \[b_1=\mathop{\mathrm{cofib}}\bigl(\Sigma^{2(p^2-1)}b \xrightarrow{v_2}b\bigr).\] Multiplication by \(v_2\) is regular on \(\pi_*b=\mathbb Z_{(p)}[v_1,v_2]\), so \(\pi_*b_1=\mathbb Z_{(p)}[v_1]\). The mod-\(p\) comparison in Proposition 47, with the cutoff changed from two to one, identifies \[H^*(b_1;\mathbb F_p) \cong\mathcal A//E(Q_0,Q_1).\] Indeed the same Koszul calculation has exterior generators \(\epsilon_i\) for \(i>1\); its comparison with the calculation for \(H\mathbb F_p\) is injective, so the spectral sequence collapses and the Milnor-basis dimension comparison identifies the displayed Steenrod module. Here \(E(Q_0,Q_1)\) is the exterior subalgebra generated by the first two Milnor primitives. The quotient is connective and of finite type over \(\mathbb Z_{(p)}\). Lee’s odd-prime uniqueness theorem therefore identifies \((b_1)_p^\wedge\) with \(\mathrm{BP}\langle1\rangle_p^\wedge\) as a spectrum [20]; see also [21].

Since \(p\)-completion is exact, completing the defining cofiber sequence of \(b_1\) gives \[\mathrm{BP}\langle1\rangle_p^\wedge\in\mathop{\mathrm{Thick}}(b_p^\wedge).\] The assumed height-one membership would consequently put \(X_{\rm sp}\) in \(\mathop{\mathrm{Thick}}(b_p^\wedge)\), contradicting the transfer criterion. Its fp-type is at most two and not at most one, as required. ◻

Proof. Put \(N=\tau_{<0}Z_{\rm sp}\), so that \[X_{\rm sp}\longrightarrow Z_{\rm sp}\longrightarrow N\] is a cofiber sequence. The spectrum \(N\) is the filtered colimit of its finite Postnikov ranges \(\tau_{\geq-m}N\). Every homotopy group of each such range is a finite \(p\)-group. Filtering those groups by their composition series expresses each range as a finite extension of shifts of \(H\mathbb F_p\).

For \(j>0\), the ordinary mod-\(p\) homology of the finite spectrum on which a \(v_j\)-self-map is chosen is bounded. Iterating a positive-degree self-map therefore gives the zero telescope on that homology, and hence \(H\mathbb F_p\wedge T(j)=0\). The same conclusion for \(j=0\) is rational acyclicity. It follows that \(N\) is \(T(j)\)-acyclic for \(0\leq j\leq2\). Finite localization is smashing and its acyclics are generated by a finite type-three spectrum [31]. Together with the Boolean algebra theorem [30], this identifies \(\langle L_2^fS^0\rangle\) with \(\langle T(0)\vee T(1)\vee T(2)\rangle\). Thus \(N\) is \(L_2^f\)-acyclic.

Every finite spectrum of type greater than two is \(E(2)\)-acyclic. Thus an \(E(2)\)-local spectrum is also \(L_2^f\)-local: it is right orthogonal to the localizing subcategory generated by those finite spectra. In particular \(Z_{\rm sp}\) is \(L_2^f\)-local, and the preceding cofiber sequence proves \(L_2^fX_{\rm sp}\simeq Z_{\rm sp}\). The equality with \(L_2X_{\rm sp}\) was established in the transfer criterion. This argument is consistent with the finite-localization property of the connective-cover examples in [22].

For the second comparison we may work with \(Z_{\rm sp}\), since \(N\) is \(T(2)\)-acyclic and hence also \(K(2)\)-acyclic. The latter implication follows from the field property: \(K(2)_*T(2)\) is a nonzero graded vector space over \(K(2)_*\), and the Künneth isomorphism shows that a \(T(2)\)-acyclic spectrum has zero \(K(2)\)-homology. The same implication shows that every \(K(2)\)-local spectrum is \(T(2)\)-local.

Consider the fiber \[G=\mathop{\mathrm{fib}}\bigl(Z_{\rm sp}\longrightarrow L_{K(2)}Z_{\rm sp}\bigr).\] It is \(K(2)\)-acyclic and \(E(2)\)-local. Here the target is \(E(2)\)-local because every \(E(2)\)-acyclic spectrum is \(K(2)\)-acyclic. The Hopkins–Ravenel smashing theorem implies that \(T(2)\wedge G\) remains \(E(2)\)-local [30]. Its \(K(0)\)- and \(K(1)\)-homology vanish because they vanish on \(T(2)\), while its \(K(2)\)-homology vanishes because it vanishes on \(G\). The Bousfield-class identity \[\langle E(2)\rangle =\langle K(0)\vee K(1)\vee K(2)\rangle\] therefore makes \(T(2)\wedge G\) also \(E(2)\)-acyclic [30]. Being both local and acyclic, it is zero. Thus the map from \(Z_{\rm sp}\) to its \(K(2)\)-localization is a \(T(2)\)-equivalence with \(T(2)\)-local target. This proves the claimed equivalence of the two localizations, and hence the assertion for \(X_{\rm sp}\). ◻

Proof of Theorem 1. The parameter choices for Proposition 28 are available at every sufficiently large prime. For any such prime, Proposition 32 provides an object \(U\) with finite \(\pi_iU\) in every degree and \(\widehat U\in\mathop{\mathrm{Thick}}(\widehat T')\). Theorem 43 gives \(U\notin\mathop{\mathrm{Thick}}(T')\). Applying Proposition 52 produces the connective, \(p\)-complete spectrum \(X_{\rm sp}\) of fp-type at most two outside \(\mathop{\mathrm{Thick}}(\mathrm{BP}\langle2\rangle_p^\wedge)\). Proposition 57 gives exact fp-type two, and Proposition 58 gives both localization comparisons. ◻

Proof. The exact fp-type from Proposition 57 and the \(T(2)\)-to-\(K(2)\) comparison from Proposition 58 verify the hypotheses of [21]. That lemma gives local fp-ness. The equivalent characterizations in [21] then give thick membership, finite generation over Morava operations, and local reflexivity. ◻

Proof. For a spectrum \(P\), write \(P/p=\mathop{\mathrm{cofib}}(p:P\to P)\). The property that \(\pi_i(P/p)\) is finite for every \(i\) is thick: the multiplication-by-\(p\) cofiber functor is exact, and degreewise finiteness is preserved by shifts, finite sums, cofiber sequences, and retracts. The property holds for \(b_p^\wedge\), since \[\pi_*\bigl(b_p^\wedge/p\bigr)=\mathbb F_p[v_1,v_2]\] has finite-dimensional homogeneous pieces.

On the other hand, \[\pi_0E_2=W(\mathbb F_{p^2})[[u_1]],\] where \(W(\mathbb F_{p^2})\) is the Witt-vector ring. Since \(\tau_{\geq0}E_2\) is connective, its multiplication-by-\(p\) cofiber has \[\pi_0\bigl((\tau_{\geq0}E_2)/p\bigr) \cong\mathbb F_{p^2}[[u_1]],\] which is infinite. It cannot lie in the thick subcategory generated by \(b_p^\wedge\). ◻