A rational obstruction to strong chromatic splitting at height three

Proof. First construct the resolution in profinite modules. A compact p-adic analytic group has a noetherian completed group ring; if it has no p-torsion, then \operatorname{pd}_{\Lambda_G}\Z_p=\operatorname{cd}_p(G)=\dim G. Here projective dimension is in profinite modules. These results for compact groups, including the agreement with abstract finitely generated modules, are recorded in [27, §§1.1–1.2, pp. 275–276]. The noetherianity input for analytic pro-p groups is [19, V.2.2.4]; the compact-group statement uses the finite-index passage recorded by Venjakob. Choose successive finite free presentations. Noetherianity keeps their kernels finitely generated, and the projective-dimension bound makes the last syzygy projective. Their natural compact topologies give the asserted profinite resolution.

To apply the resolution to the geometric coefficient complexes, we pass from continuous actions to solid modules over the completed group ring. Write A=\underline{\Z_p}[\underline G] for the free condensed group ring. A continuous action on a complex means an object of D(A\text{-}\mathrm{Cond}) with derived-solid underlying complex. By [1, Lemma 1.3], the group-ring analytic structure has underived solidification; its normalized ring is

where V runs through open normal subgroups. The ring identification, including multiplication, is [26, §3.6, preceding Proposition 3.21]. Analyticity gives fully faithful embeddings of the unbounded derived category of solid G-modules into both D(A\text{-}\mathrm{Cond}) and D(A^\square\text{-}\mathrm{Cond}), as stated in [1, §1, pp. 3–4]. The solid heart is characterized by solidity of the underlying condensed module. The essential-image criterion of Clausen–Scholze’s analytic theory extends this characterization to unbounded complexes: membership is detected on cohomology objects [22, Proposition 12.4 and Remark 12.5]. Thus its objects are precisely the complexes with derived-solid underlying coefficients. Consequently the external derived Hom complexes satisfy

The first identification is the external derived Hom for the condensed classifying topos BG_{\mathrm{pro\acute et}} [1, §2, pp. 5 and 8]; the last category is that of solid \Z_p[[G]]-modules by [26, Proposition 3.21]. They apply to the given complex K itself, including when its coefficients are neither profinite nor derived p-complete.

The profinite resolution stays exact and projective in solid modules [26, Theorems 3.2 and 3.14(i)]. Exactness can be seen by testing on extremally disconnected profinite sets: maps from such a set lift through a surjection of profinite spaces. Each P_i is a retract of \Lambda_G^{r_i}, and

Evaluation at the point is exact, since the point is extremally disconnected. This proves projectivity against an arbitrary solid coefficient module M, even when M is not profinite. A bounded projective resolution computes derived Hom against any complex, which proves the displayed calculation and the amplitude assertion.

Finally, \Hom_{\Lambda_G}(P_i,H^j(K)) is a retract of a finite sum of H^j(K)(*). The retract commutes with \gamma, because \gamma commutes with the G-action. After inverting p, the entire finite complex computing the jth row therefore has scalar action \lambda_j. Its cohomology has the same action. This uses a scalar identity on point values only; it does not deduce an identity of condensed sheaves from their point values. Boundedness of the resolution gives a finite filtration in each total degree, so exact rationalization preserves the spectral sequence and its abutment.□

Proof. For an inclusion K_1\subset K_2 of compact open subgroups of \GL_n(\Q_p), choose a K_2-stable lattice \Lambda and then r large enough that

This is a uniform subgroup normal in K_2. In the division-algebra case one uses a sufficiently deep congruence subgroup of \cO_{D_n}^{\times}, which is normal in the relevant compact group. For these small groups, Lazard’s comparison is [17, Theorem 3.3.3 and Remark 3.3.4] with trivial \Z_p-coefficients, followed by inversion of p. Its coefficient naturality, rational agreement, and group naturality are [17, Theorem 3.1.1(1)–(3)], with numbering in the arXiv revision specified in the bibliography. Equip the uniform groups with their lower p-series. At these odd primes the proof of Theorem 3.3.3 gives e=1, and the associated graded target is generated in degree one, as required by the group-naturality clause. Inclusions and the block homomorphisms preserve these filtrations. The common subgroup U therefore identifies restriction with the identity on the same ambient Lie algebra cohomology.

Here is why passing back to a compact group is valid even for an index divisible by p. For U\triangleleft K, the augmentation and norm of the finite permutation module \Z[K/U] induce, by continuous Shapiro, the identities

at each finite coefficient level, compatibly with reduction in m. Shapiro here is the derived induction–restriction adjunction: induction for an open subgroup is a direct sum on underlying condensed modules and is exact by (AB4) [26, Theorems 2.2 and 2.4]; restriction is also exact. Their adjunction therefore descends directly to derived categories. Its naturality identifies the permutation maps with the actual restrictions and conjugations in the finite continuous bar complexes. Apply \Rlim_m to these identities before taking cohomology and inverting p. Only then divide by [K:U]. This gives H^*_{\mathrm b}(K)\xrightarrow{\sim}H^*_{\mathrm b}(U)^{K/U} by restriction, with the compact continuous-cochain interpretation of (2.2).

The finite quotient acts trivially in every Lie degree. The Chevalley–Eilenberg complex is a finite-dimensional algebraic \GL_n-representation after a splitting-field extension. Cartan’s formula \mathcal L_X=d\iota_X+\iota_Xd makes the derivative of its action on cohomology zero. In characteristic zero, an algebraic action of the connected group \GL_n with zero derivative is trivial. Thus all inner automorphisms act trivially, in every degree; the same conclusion descends for the division algebra. This proves the compact restriction identification without an all-degree numerical calculation.

After a finite splitting-field extension, either Lie algebra is \mathfrak{gl}_n. The finite Chevalley–Eilenberg calculation in Appendix A gives its asserted groups in degrees one through three, with degree-three representative

or its reduced-trace version for D_n. This is the invariant-form cocycle of [9, §21] and [18, §11]; the appendix records the normalization and restriction used here. These representatives are defined over \Q_p and invariant under inner automorphisms, so the assertions descend from the splitting field. Restriction of the displayed form to the upper-left or lower-right 2\times2 block is the corresponding form for \mathfrak{gl}_2, whose class is nonzero. This proves the compact calculations and their compatibility.

To pass to J_n, let \mathcal B_n be the reduced affine building of \GL_n(\Q_p). It is contractible of dimension n-1. The group J_n preserves vertex types, its face stabilizers K_\sigma are compact open, and a chamber \Delta is a fundamental domain, including its faces. A proof of these building properties by lattice norms is included in Appendix A. Write its bounded augmented cellular resolution as

It is exact also as a complex of condensed modules. Indeed, on a profinite parameter set a continuous section of a discrete cellular chain group has finite image. Contractibility lifts each of the finitely many cycles in that image; choosing those lifts on its clopen fibers gives a continuous lift. The same exact derived induction–restriction adjunction therefore supplies continuous Shapiro for each summand. At coefficient level \Lambda_m, applying derived Hom gives a finite filtration with graded terms \bigoplus_{\dim\sigma=r}C_{\cts}^{\bullet}(K_\sigma,\Lambda_m)[-r] in D(\mathrm{Ab}). A cellular boundary is the literal signed projection of the corresponding coset modules, so naturality of that adjunction makes each face map the signed restriction between stabilizers, with no index multiplier. All of these statements are compatible with coefficient reduction.

Take \Rlim_m of this finite filtration. There are finitely many orbit terms in each of finitely many cellular degrees, so this step uses only finite sums and cones in the cellular direction. Taking cohomology of the resulting finite filtration gives the integral spectral sequence

with precisely those signed restriction faces. It does not interchange an infinite group-cochain totalization with the derived inverse limit. Only after this construction do we invert p; exact rationalization preserves the finite filtration and its abutment.

Under the compact comparison, every restriction between face stabilizers is the identity on the same Lie algebra cohomology; any change of representatives contributes an inner conjugation, which also acts trivially. Thus each rationalized row is the cellular cochain complex of the simplex \Delta with constant coefficients H^s(\mathfrak{gl}_n,\Q_p). It has cohomology only in cellular degree zero. The resulting edge map is restriction to a vertex stabilizer. Restriction further to compact open subgroups, using intersections and the compact comparison, proves the asserted canonical identifications. The block restriction can now be tested on compact open subgroups, where it is the trace calculation above.□

Proof. The kernel of D\to\Z_p^\times\times J_2 is N=\Q_p^2, with its additive topology. Fix m and exhaust it by the compact open subgroups N_r=p^{-r}\Z_p^2 for r\geq0. Their finite-coefficient cohomology, computed by the two-variable Koszul resolution, is

The standard bases identify restriction from N_{r+1} to N_r with multiplication by p^q in degree q: restriction multiplies each degree-one homomorphism by p, and is compatible with cup products. For q>0 this inverse system is pro-zero; for q=0 it is constant with identity transition maps.

This calculation computes the cohomology of N without a missing derived-limit term. Indeed a continuous cochain on N is exactly a compatible family of cochains on the N_r. Restriction of cochains is surjective, since N_r^a is clopen in N_{r+1}^a and a function extends by zero on its complement. Thus the ordinary inverse limit of the cochain complexes realizes their derived inverse limit. The Milnor exact sequence has \varprojlim^1_r=0 in every degree: this holds both for a constant tower with identity maps and for a pro-zero tower. It follows that the constant map is an equivalence

The same proof works with coefficients C_{\cts}(S,\Z/p^m) for every profinite parameter space S. The finite Koszul complex gives finite sums of this module, the positive-degree restrictions are again p^q, and extension by zero is continuous on N_r^a\times S. The condensed standard resolution evaluated on each such parameter identifies this calculation with the internal equivalence

Thus it proves an equivalence of coefficient objects for extension descent. The equivalence is the map of constants, so it is equivariant under the Levi action.

Now take \Rlim_m. In degree zero the parameter modules have surjective transition maps, by lifting locally constant values, and their limit is C_{\cts}(S,\Z_p). Hence the internal coefficient limit satisfies

At each finite level, internal descent for the split extension identifies R\Gamma(D,\underline{\Lambda_m}) with R\Gamma(\Z_p^\times\times J_2, R\Gamma(N,\underline{\Lambda_m})). The Levi-equivariant constant equivalence therefore induces inflation. After point evaluation, the standard-resolution comparison identifies this map with finite continuous-cochain inflation. Taking \Rlim_m and using (2.1) gives the claimed isomorphism integrally before rationalization.

Finally, the compact resolution for \Z_p^\times and the finite building resolution for J_2 give the usual rational product calculation. Since H^*_{\mathrm b}(\Z_p^\times) is exterior on one degree-one generator and H^2_{\mathrm b}(J_2)=0, degree three of the Levi is exactly H^3_{\mathrm b}(J_2). Thus \rho_3^*e_{J_3} is a scalar multiple of \rho_2^*e_{J_2}. Restricting to the section A\mapsto\operatorname{diag}(1,A) shows that this scalar is nonzero, by Proposition 2.2.□

Proof. The imported result is [4, Proposition 2.7.1(1)–(5)], whose hypotheses are h\ge2, a perfect field containing \F_{p^{h(h-1)}}, and the coheight-one Gross–Torii extension with the specified determinant identifications. Here h=3 and k=\F_{p^6} satisfy those hypotheses exactly.

We recall why its formulation gives all surjections and the indicated structure groups. The infinite-level two-tower theorem [4, Theorem 2.6.3], followed by forgetting the étale framing, identifies \Spd\widehat L with exact sequences

whose determinant identification is \iota. Forgetting that framing is the \Z_p^{\times}-torsor in the proof of [4, Proposition 2.7.1(2)]. Given a surjection, its kernel is the line \det\mathcal E_3^0\otimes(\det\mathcal E_2^0)^{-1}; the fixed \iota trivializes this line and recovers the sequence. Thus no additional condition is imposed on the surjection. Descent along the two framing torsors gives exactly the asserted description of Y, also in families.

Quotienting by S_3 instead allows the source bundle to vary while retaining its determinant trivialization. It therefore classifies extensions

with middle term of slope 1/3. By [4, Proposition 2.7.1(5)], these are precisely the extensions that are nonsplit at every geometric point, which form \operatorname{BC}(-1/2)^*.

To specify the joint action, let i_f:\cO\to\mathcal E_3^0 be the kernel injection normalized by \iota. If f'=bfg^{-1}, the determinant isomorphisms for the two exact sequences give

Indeed, g acts on the source determinant by N_3(g) and b acts on the target determinant by N_2(b). The induced diagram of extensions therefore has kernel map N_3(g)N_2(b)^{-1} and quotient map b. Functoriality of extensions by pushout and pullback proves the displayed P_2\times T_3-action. This calculation is an identity of bundle maps on every test object and commutes with base change. For b=1, it is pushout by a=N_3(g). On the extension group, a_*=(aI_2)^*, so this is the action by inverse pullback of the central scalar a^{-1}I_2. The underlying quotient and this T_3 restriction are the parts of [4, Proposition 2.7.1(5)] used below; the joint action includes the target determinant factor just computed. The field assertions are parts (1) and (4) of the same proposition.□

Proof. Apply the underlying cohomology calculation of [4, Proposition 3.6.9] to the quotient in Proposition 3.1, retaining only its T_3-action. The map of constants and the vanishing in the other degrees give a fiber sequence of derived solid k-modules with T_3-action

Here \chi_T denotes the actual character on the line in degree three. This is the restriction of the argument combining quotient descent and cohomology in [4, Theorem 3.6.10] that is needed here. For b=1, the action formula in Proposition 3.1 is pushout by a\in T_3 on the kernel. On the extension group this equals pullback by aI_2, or the action by inverse pullback of a^{-1}I_2. The reduced norm of aI_2 in D_2^\times is a^2. Hence the subgroup

acts on the last term by \alpha\mapsto\alpha^{2} or \alpha\mapsto\alpha^{-2}; either convention gives the same vanishing used below. The displayed exponent is also checked in [4, Lemma 5.2.5]. This is a subgroup of the norm quotient; it must not be replaced by the central \mu_{p-1} in P_3. Since p\ge5, this character is nontrivial. The averaging idempotent for the finite group \mu_{p-1} is defined in characteristic p, so its derived invariants are exact and annihilate k_{\chi_T}. Taking these invariants proves that the underlying map in (3.1) is an equivalence. The map itself is H-equivariant: it is induced by constants from the original G-action on \widehat L, followed by P_3^1-descent, and G fixes k. The forgetful functor from H-equivariant objects detects equivalences, so this gives the required equivalence with its full residual action. No splitting of the fiber sequence or joint character formula is needed.

For completeness, the finite-field hypothesis in this import is substantive. Its proof uses the solid Frobenius comparison [4, Lemma 3.6.3 and Corollary 3.6.8] for \bigl(\bigcup_{r\ge0}k((z^{1/p^r}))\bigr)^{\wedge} over a finite field k. The geometric base change of this point is a punctured perfectoid ball; the proof solves coefficient Frobenius minus identity on its convergent functions and then requires surjectivity in solid modules. For this precise function ring, Lemma B.1 supplies a continuous section and hence proves surjectivity on every profinite parameter space. This justifies the solid enhancement directly, without applying a linear open-mapping theorem to Frobenius minus identity. The resulting natural comparison commutes with the S_3-action. Thus Proposition 3.1 verifies the required hypothesis before finite-field descent is applied.

For the geometric statement, take a quadratic extension F/\Q_p that is a maximal subfield of D_2. Proposition 3.1.2 of [4] identifies

where \mathcal H^1_{F,C} is the one-dimensional Drinfeld upper half-plane for F and the additive group F acts by translations. This identification is equivariant for the central \Q_p^{\times}. It is distinct from the rank-n Drinfeld space \Omega_C^{n-1} used in the modification argument. Proposition 3.5.6 of [4] computes the solid \cO^{\flat}-cohomology of this quotient: it is C^{\flat} in degree zero, a character line in degree [F:\Q_p]+1=3, and zero otherwise. By the b=1 action formula in Proposition 3.1, this norm-quotient \mu_{p-1} acts by kernel scalars. Expressing those as central pullbacks on the target gives reduced-norm character \alpha\mapsto\alpha^2 or its inverse. Both are nontrivial, so the top cohomology has no \mu_{p-1}-invariants; the choice of action or dual convention does not affect the conclusion. The degree-zero identification is the map of constants. Applying the same averaging argument proves (3.2).

Only central equivariance of the Drinfeld description was needed to prove this vanishing. The constant map on X_C itself has the full residual H-equivariance, and an equivariant map is an equivalence when its underlying map is. This proves the full assertion, without assuming a P_2-equivariant splitting or a larger equivariance of that chart. Finally, the actions of P_2 and \overline Z are over the fixed base, so they fix k and C^{\flat} pointwise. All arrows used above are induced by structure maps and group descent; their stated naturality follows.□

Construction; full proof in Appendix C. Fix a parameter set T. Write \mathscr D_\Gamma^{\mathrm{or}} and \mathscr D_C^{\mathrm{or}} for oriented deformations with the specified residue tags of the full and connected reference groups. The connected–étale sequence is uniquely split over the perfect field F. Let Q_T be the base change of its étale quotient, so \Gamma_T=C_T\oplus Q_T. An oriented deformation C_A over a complete adic test ring A has a contractible space of lifts of the tagged étale quotient Q_T. With such a marked lift Q_A, send it to C_A\oplus Q_A, retaining its connected orientation. This constructs \mathfrak s in the left diagram:

Here \mathfrak f takes the connected deformation, and \mathfrak f\mathfrak s\simeq\mathrm{id}. Corepresentability reverses the arrows and gives r_Ti_T\simeq\mathrm{id}. In particular the required map is r_T, supplied by the split section. The forgetful map alone supplies i_T in the other direction.

The tensor-factor identity keeps track of which connected map results. The tensor construction compares the two oriented formal groups through their common Quillen formal group. On the split deformation its comparison is the inverse of the connected inclusion. The resulting connected tag and orientation are therefore exactly those defining c_T, so r_Tb_T\simeq c_T. The appendix verifies this cancellation on the actual tags and proves its compatibility with adic refinement and every continuous group-family map. It then upgrades the identity to the continuous solid maps in the statement. This identifies the right-hand tensor composite with the canonical map used in coefficient descent.□

Proof of Proposition 3.3. The connected endpoint calculation gives the coefficients and ordinary locality of every evaluation. The tensor construction and the oriented section supply the left equivariant ring map; the right map is the canonical connected base change already constructed. The constants calculation and finite Koszul argument prove both fixed-point descent and descent for each coefficient module, with their residual actions and maps from Witt constants. Taking derived H-invariants in the extension P_3^1\to G\to H gives (3.4); \overline Z acts trivially on the E_2 source. The preceding point-evaluation argument proves the ordinary stabilizer-descent identification with its sphere unit.□

Proof. The kernel is a vector bundle: locally at the Cartier divisor, the quotient is a coordinate modulo a local equation of the divisor, and the kernel has a basis obtained by multiplying that coordinate by the local equation. At a geometric point, \rank(\mathcal V_n)=n and \deg(\mathcal V_n)=0. If it had a positive-slope Harder–Narasimhan piece, its maximal destabilizing subbundle \mathcal F would have rank b<n and integral degree d\geq1. The injection \mathcal F\hookrightarrow\mathcal E_n contradicts semistability of \mathcal E_n, because

One may saturate its image in \mathcal E_n without decreasing degree. Hence \mathcal V_n is semistable of degree zero at every geometric point. The relative slope-zero equivalence stated above now applies.□

Proof. If \ell\subset W_C for a proper rational subspace W, modify W\otimes_{\Q_p}\cO along \ell. The resulting rank-\dim W, degree-one bundle embeds as a subbundle of \mathcal E_\ell; the quotient is the unchanged trivial bundle associated to \Q_p^n/W. Its slope is greater than 1/n, so \mathcal E_\ell is not semistable.

For the converse, take a saturated destabilizing subbundle \mathcal F\subset\mathcal E_\ell of rank b<n. Its integral degree d is at least one. Set \mathcal F'=\mathcal F\cap\cO^n. The quotient \mathcal F/\mathcal F' embeds in \mathcal E_\ell/\cO^n and has length \epsilon\leq1. Since \mathcal F' embeds in \cO^n, semistability of the latter gives

Thus d=\epsilon=1 and \deg(\mathcal F')=0. The saturation of \mathcal F' in \cO^n cannot increase its degree: a positive increase would contradict the same semistability inequality. It is therefore already saturated. All its subbundles have nonpositive degree, so it is semistable of slope zero. The classification of bundles at a geometric point identifies it with W\otimes_{\Q_p}\cO for a b-dimensional rational subspace W; here an inclusion between trivial bundles is a matrix over H^0(\mathcal X_C^{\mathrm{FF}},\cO)=\Q_p. The equality \epsilon=1 in the displayed local lattice description then says precisely that \ell\subset W_C.

This proves the criterion at every geometric point. The classification in [13, Theorem II.2.14] gives \cO(1/n) for a semistable rank-n, degree-one bundle, and the relative basic-stratum theorem gives the claimed family of forms.□

Proof. Fixing a quotient line of i_\infty^*\mathcal E_n^0 fixes its kernel \mathcal V_n. By Lemma 4.1 its frames exist pro-étale locally. Requiring a frame to preserve the determinant lattice cuts its \GL_n(\Q_p)-torsor down to a J_n-torsor. Thus \mathcal Z_n/J_n=\mathbb P(i_\infty^*\mathcal E_n^0) as v-sheaves.

In the other direction, a framed lower bundle gives an upper modification \cO^n\hookrightarrow\mathcal E_\ell. By Lemma 4.2 its allowed directions are exactly \Omega_C^{n-1}. Its determinant lattice is induced by the standard one on \cO^n. The isomorphisms \mathcal E_\ell\simeq\mathcal E_n^0 form a D_n^\times-torsor, by the relative basic-stratum theorem. The norm valuation D_n^\times\to\Z is surjective, so this torsor has, v-locally, isomorphisms carrying the induced lattice onto \Lambda_n. Those isomorphisms form precisely a P_n-torsor. It follows that \mathcal Z_n/P_n=\Omega_C^{n-1}.

Both identifications commute with base change and the remaining group action. Taking the further quotients proves Equation (4.1). They are Galois compatible because every operation used—forming the kernel, determinant, lattice condition, and sheaf of frames—is compatible with the descent just constructed. More explicitly, an upper direction is a line in \Q_p^n\otimes_{\Q_p}(\cO(\infty)/\cO)|_\infty. The second factor is common to all coordinates, so its semilinear Galois scalar cancels upon projectivization. The action on \Omega_C^{n-1} is consequently the standard arithmetic action, commuting with J_n. The lower frames form a Galois-equivariant J_n-torsor, which gives the equivariant map to B_CJ_n. Here Galois acts on the \Spd C base of B_CJ_n and trivially on the group J_n; the analogous statement holds for B_CP_n. This assertion does not require the torsor itself to have a Galois-fixed frame.□

Proof. If q:\mathcal E_2\twoheadrightarrow i_{\infty *}\mathcal L is the universal quotient, then \mathcal V_2=\ker(q) and \mathcal V_3=\ker(qf). Thus \mathcal V_3=f^{-1}\mathcal V_2, proving the exact sequence. The two lower modifications are slope zero by Lemma 4.1. The kernel of f has rank one and degree \deg(\mathcal E_3)-\deg(\mathcal E_2)=0, so it too has slope zero.

To check exactness on local systems, pass to a pro-étale cover trivializing the three slope-zero bundles. Full faithfulness of the relative correspondence identifies the maps with matrices of continuous \Q_p-valued functions. The matrices have the indicated ranks on every geometric fiber; on the open loci of invertible minors they give exact, locally split sequences of rational local systems. These loci cover the base, proving exactness before descent.

Transport \Lambda_3 and \Lambda_2 to the determinant local systems of \mathcal V_3 and \mathcal V_2. The determinant isomorphism in Equation (4.2) defines the lattice on the kernel as their tensor quotient. Locally choose a lattice-compatible basis of the kernel and determinant-compatible frames of the quotient, and lift the latter to the middle local system. Such adapted frames form a torsor; their change-of-frame matrices are exactly the displayed matrices in D. The two classifying maps follow by forgetting the filtration or taking its quotient. The construction of m_n retains the original P_n-structured form \mathcal E_n on Y_C and only adds its quotient line. Its P_n-framing torsor is therefore the pullback of the associated framing torsor on Y_C, proving the last assertion as a statement of actual torsors.□

Proof. Define \lambda_m by the powers h_m^j, using adjunction and cup product. It is therefore a global map of derived pushforward objects, compatible with coefficient reduction and all descent data. For arithmetic descent, the linear action on \mathcal W induces descent on the actual line bundle \cO(1). Naturality of the Kummer boundary makes h_m invariant in its twisted coefficient group.

Over \operatorname{Spa}(C,\cO_C), the finite-constant projective-space calculation has basis 1,h_1,\ldots,h_1^r: apply [20, Construction 3.2.1 and Proposition 3.2.2] with \F_p coefficients, since p is invertible on this base. The first Chern class there is the Kummer boundary [20, Variant 3.1.8], so this basis uses our actual classes. To extend the calculation to arbitrary perfectoid test bases, apply the integral Riemann–Hilbert functor \operatorname{RH}_{\Z_p} of [2, Theorem 5.1.7]. Smooth proper primitive comparison [2, Proposition 5.2.2 and Example 5.2.3] applies to \mathbb P_C^r\to\Spd C and the dualizable coefficient object \F_p. Its mod-p target is

by [2, Remark 5.1.9 and the beginning of §5.2]. The symmetric monoidal and pullback compatibility of the functor carries the Kummer basis to its coefficient images. It therefore gives the Frobenius-equivariant Chern-class equivalence with \cO^\flat coefficients over C.

After trivializing \mathcal W on a perfectoid test object, its projective bundle is a base change of \mathbb P_C^r\to\Spd C. This projective morphism is p-bounded. Base change for solid pushforward, [2, Proposition 4.2.7(ii)] with \Lambda=\F_p, transports the equivalence to that test object. The globally defined Chern-class map makes these local equivalences compatible on overlaps, so they descend.

We identify the resulting solid calculation with ordinary v-sheaf pushforward by testing on a perfectoid T\to X. Apply tensor-unit Hom in the solid formalism on T. Adjunction and the tensor-unit cohomology identification [4, Theorem 3.2.1(iii)–(iv)] give

This is natural in T and induced by the same Chern classes. It thus identifies their coefficient-image map with the equivalence of ordinary derived v-sheaf pushforwards

The displayed map is induced by the images of the actual mod-p étale classes h_1^j. It therefore commutes with Frobenius on \cO^\flat, including its descent data. Take the fiber of F-1 on both sides. The Artin–Schreier sequence on the v-site identifies this fiber with \F_p; with twists it identifies the jth fiber with \F_p(-j)[-2j]. Since derived pushforward preserves fibers, the resulting equivalence is exactly \lambda_1. This step uses the Frobenius compatibility of the Chern-class map.

The Kummer classes h_m are compatible under reduction. The coefficient sequences

give compatible exact triangles on the source and target of \lambda_m. Cup product with h_m^j commutes with both coefficient maps. Induction on m therefore proves Equation (5.1). Apply global sections and then the derived inverse limit. The finite direct sum commutes with both operations and gives Equation (5.2). Projection to its zeroth summand is a left inverse to pullback. Exact inversion of p preserves that left inverse.□

Proof. Use the common arithmetic action G_F from Proposition 4.3. The projective presentation is a projective bundle over B_CP_n. Apply Lemma 5.1 first to this relative geometric base. Its two degree-three summands use the finite coefficients \Lambda_m and \Lambda_m(-1) on B_CP_n, respectively, before their derived limits; higher projective summands would have negative cohomological degree and vanish. The compact geometric-point comparison (2.5), applied in the P_n bar degrees and with its Tate fiber, identifies these summands with the group cohomology groups in the G_F-equivariant direct sum

The first summand is the image of the actual constant-cochain pullback, hence contains a nonzero e_{P_n}. The comparison is natural for the Galois action on the \Spd C base. It identifies the untwisted group cochain factors with trivial Galois action because Galois fixes P_n and the untwisted constants. On the second summand it acts by \chi_{\mathrm{cyc}}^{-1}. As \chi_{\mathrm{cyc}}(G_F) is open, this summand has no invariants. Thus

It remains to prove that the pullback of e_{J_n} is nonzero. Choose a compact open uniform subgroup U\subset J_n. Restriction sends e_{J_n} to a nonzero element of H^3_{\mathrm b}(U) by Proposition 2.2. We will detect it after the further pullback to [\Omega_C^{n-1}/U].

Let K_n=\mathscr C_{\Omega_C^{n-1}} be the derived solid integral cohomology complex formed with profinite parameters. Descent, as in Lemma 2.1, gives the first-quadrant spectral sequence

The complex K_n is bounded below, and the group degree has the finite bound supplied by the completed resolution. The sequence therefore has a finite filtration in every degree and may be rationalized exactly. The additional geometric row bound uses only point values: Theorem 2.4 gives H^j(K_n)(*)=0 for j>n-1; the finite Hom calculation below then makes every ordinary point-valued E_2^{s,j} in that row zero. No vanishing assertion for the entire condensed module H^j(K_n) is needed.

Choose the finite projective resolution P_\bullet\to\Z_p over \Lambda=\Z_p[[U]] from Lemma 2.1. The unit \underline{\Z_p}\to K_n factors through H^0(K_n)[0], since K_n is concentrated in cohomological degrees at least zero. The degree-zero calculation and local constants argument following Theorem 2.4 show that this unit is an isomorphism on point values. Each \Hom_\Lambda(P_i,-) is a retract of a finite sum of point values, and the equivariant unit commutes with the idempotent defining that retract. It follows that the unit gives an isomorphism of the integral bottom-row computing complexes

After [1/p], (2.2) and (2.6) identify the restricted group class with the constants bottom-row edge class induced by u_{\Omega_C^{n-1},U,\mathrm b}. This uses only the point-value isomorphism, not an identification of the entire condensed module.

For a fixed j, the rationalized row is computed by

Every term is a direct summand of a finite sum of H^j(K_n)(*)[1/p]. By the integral Drinfeld theorem, [11, Theorem 1.1], a Galois element \gamma acts on the latter by the scalar \chi_{\mathrm{cyc}}(\gamma)^{-j}. It commutes with \Lambda and with the idempotents defining the projective summands, so it has the same scalar action on the entire row and on its cohomology. No identification of a solid module by its point value is involved.

The constants row j=0 has trivial Galois action. Choose \gamma\in G_F with \chi_{\mathrm{cyc}}(\gamma) of infinite order. The only possible incoming differentials to bidegree (3,0) are

The second source is absent when n=2. After inversion of p, the respective source scalars are \chi_{\mathrm{cyc}}(\gamma)^{-1} and \chi_{\mathrm{cyc}}(\gamma)^{-2}, whereas the target scalar is 1. Every differential commutes with \gamma, so both are zero. There are no outgoing differentials from the bottom row. Hence the nonzero bottom-row image of the restriction of e_{J_n} survives in H^3_{\mathrm b}([\Omega_C^{n-1}/U]). Its abutment image is u_{\Omega_C^{n-1},U,\mathrm b}(\operatorname{res}^{J_n}_U e_{J_n}) by the unit calculation above. Naturality (2.4) for U\hookrightarrow J_n identifies it with the pullback of u_{\Omega_C^{n-1},J_n,\mathrm b}(e_{J_n}). Thus the class on [\Omega_C^{n-1}/J_n]=\mathcal M_n is nonzero.

Finally, that class is G_F-invariant. Its classifying map to B_CJ_n is equivariant for the action on the \Spd C base, while G_F fixes J_n and the untwisted finite coefficients. Naturality of u_{B_C,J_n,m} for this base action passes through the derived coefficient limit and rationalization. It therefore sends the invariant group class to an invariant geometric class. Equation (5.5) therefore places both nonzero classes on the same one-dimensional \Q_p-line, proving Equation (5.3).□

Proof. By Lemma 2.3, for the inclusion \rho_3:D\to J_3 and projection \rho_2:D\to J_2 there is a scalar b\in\Q_p^\times with

Apply u_{B_C,D,\mathrm b} to this group-cohomology equality and pull it back along d:\mathcal T\to B_CD. The exact filtration in Proposition 4.4 identifies the two J_n-classifying maps with the composites induced by D\to J_n, and (2.4) identifies their pullbacks with these same group classes. The P_n-framing torsors in that proposition give the first and last equalities below. Consequently

All three scalars are nonzero. The projective morphism \pi:\mathcal T\to Y_C satisfies Lemma 5.1, so \pi^* is injective. Taking a=\kappa_3b/\kappa_2 proves the theorem.□

Proof. For a characteristic-p ring R, restriction of Witt components and Verschiebung give an exact sequence of additive groups

The sequence is natural for ring maps and automorphisms. We use it as a sequence of additive coefficient objects; no assertion of linearity over R is needed. It also gives an exact sequence of the associated solid modules and of the corresponding sheaves on the v-site.

Proposition 3.2 identifies the actual constant maps

as equivalences with their residual H-actions. As proved there, the characteristic-p vanishing uses only the central scalar restrictions of [4, Propositions 3.5.6 and 3.6.9], identified with T_3 by the determinant calculation in Proposition 3.1, and finite norm-unit invariants as in [4, Lemma 5.2.5]. Full H-equivariance comes from the actual maps of constants and quotient descent. Apply (6.3) to the source and target of each map. Induction on m, using the resulting morphisms of exact triangles, gives equivalences

Every map used in this induction is the map induced by constants. In particular the equivalences are H-equivariant, including the \overline Z-action; H fixes the constant rings.

Derived global sections and derived invariants preserve limits. Passing to \Rlim_m therefore gives the same equivalences with full Witt coefficients. Here the ordinary Witt module is also the derived limit of its truncations: in Witt coordinates, restriction has the continuous set-theoretic section obtained by appending a zero. Consequently restriction is surjective on continuous functions from every profinite parameter space. The truncation towers have no higher derived-limit term in the solid coefficient category. The argument applies to k, C^\flat, and \widehat L with their respective topologies. Taking residual H-invariants gives the horizontal isomorphisms of (6.1). Naturality of the constant maps under k\to C^\flat gives its commutativity.

For (6.2), recall \widetilde Y=\Spd\widehat L from Section 3. This affinoid perfectoid space is acyclic for \cO^\flat, also with profinite parameters. Applying (6.3) gives, for every profinite set S, the finite equivalence induced by sections

Here Witt coordinates identify W_m(C_{\cts}(S,\widehat L)) with C_{\cts}(S,W_m(\widehat L)). The equivalence is natural in S and in the G-action. Write u^a_{\widetilde Y,G,m} for the degree-a map of (2.3). The actual finite coefficient units give the commutative square

The left map applies \Lambda_m=W_m(\F_p)\to W_m(\widehat L) pointwise; the right map is induced by the corresponding sheaf unit. Both composites are the same locally constant Witt section. These squares commute with the bar faces, including the action face, and with restriction in m. The same calculation with an extra profinite parameter retains the solid structure. The condensed standard resolution, quotient descent, and then \Rlim_m therefore give (6.2). The full Witt module is the derived limit of its truncations as shown above. Under point evaluation of this identification, the Witt image of u_{\widetilde Y,G,\mathrm{int}} is exactly the coefficient map induced by \Z_p\to W(\widehat L) and (2.2).

It remains to prove injectivity, since an equivalence of the horizontal groups alone would not imply it. The finite-resolution statement of Lemma 2.1 applies to H. Indeed an element of order p in P_2 would have minimal polynomial of degree p-1>2 over \Q_p, whereas the reduced characteristic polynomial in D_2 has degree two. Thus P_2, and hence H=P_2\times\Z_p, has no p-torsion. Choose a bounded resolution P_\bullet\to\Z_p by finitely generated projective \Z_p[[H]]-modules. For a trivial solid coefficient module M, the point-valued cochain complex is

Every Q_j is finite free over \Z_p. Thus this complex is obtained from a fixed bounded finite free \Z_p-complex by scalar extension to the underlying module of M; its differentials are the same \Z_p-linear matrices. This description is natural in M and is valid for the valued Witt topology as well as for the p-adic one, by Lemma 2.1.

Take A=W(k) and B=W(C^\flat). The map A\to B is injective. Since A is a discrete valuation ring with uniformizer p and B is p-torsion-free, B is flat over A. Applying the finite complex just described gives

The field extension A[1/p]\hookrightarrow B[1/p] is faithfully flat, so the natural map from the first cohomology group to its scalar extension is injective. The square now proves the assertion. All infinite limits preceded this finite-complex argument.□

Proof. Let \omega_{Z,m} be the map on cohomology complexes induced by \Lambda_m=W_m(\F_p)\to W_m(\cO^\flat) for Z=Y or Y_C. These maps are compatible with restriction in m. Their derived inverse limit gives a map

Let j:Y_C\to Y denote base change, and put

The finite constant maps and the finite Witt units give

This square commutes in each cosimplicial degree before totalization, and is compatible with q_n:G\to P_n and reduction in m. Take the derived coefficient limit, cohomology, and then invert p. The classifying-pullback convention and (2.4) give j^*(e_{P_n}|_Y)=e_{P_n}|_{Y_C}. Moreover, (6.4) identifies the Witt image of e_{P_n}|_Y with c_n^W under (6.2). Thus the geometric relation says that c_3^W-a c_2^W maps to zero in H_W^3(Y_C)[1/p]. The right vertical map of (6.1) is injective after inverting p, so this difference is already zero on Y. This proves (6.6). Applying the coefficient map W(\widehat L)\to\pi_0\widehat B^{\square} then proves (6.7).

Finally, (3.4) is induced by the actual map E_2^{\square}\to\widehat B^{\square}, by first taking P_3^1-invariants and then H-invariants. Its compatibility with constants is part of Proposition 3.3, whose finite Koszul argument proves the required coefficient instance of the descent construction in [4, Proposition 4.6.12]. The action of \overline Z on the source is trivial. Thus inflation from P_2 followed by this map is exactly the coefficient map defining c_2. This proves the last assertion while retaining the residual action.□

Proof. Set A_k=\pi_0E_2^{\square}(*). The class corresponding to c_2 under (3.4) restricts along P_2\hookrightarrow H to the coefficient image of e_{P_2} in H^3(P_2,A_k)[1/p]. It suffices to prove that this image is nonzero.

Choose an algebraic closure \overline k of k and let A_{\overline k} be the deformation ring of the same height-two formal group after this base extension. Deformation theory gives a natural continuous P_2-equivariant map

In compatible coordinates these rings are W(k)[[u'_1]] and W(\overline k)[[u'_1]], with their (p,u'_1)-adic topologies; equivariance follows from deformation base change, without requiring the coordinate to be fixed. Since P_2 is nonextended, it fixes W(\overline k) pointwise.

The splitting theorem of Barthel–Schlank–Stapleton–Weinstein [5, Proposition 2.5.1 and Corollary 2.5.9] gives a continuous additive equivariant retraction

of the inclusion of Witt constants. Restrict its equivariance to P_2. Additivity and continuity make it \Z_p-linear. After condensation it is a morphism of solid P_2-modules, and hence induces a retraction on the continuous derived invariants used here.

The composite of the coefficient maps

is the usual inclusion of constants. Applying the finite free complex argument from Proposition 6.1 with P_2 in place of H gives

The image of e_{P_2} here is e_{P_2}\otimes1, which is nonzero by Proposition 2.2. If its image in H^3(P_2,A_k)[1/p] vanished, the displayed composite would also send it to zero, a contradiction. Restriction to P_2 therefore proves c_2\ne0, and (6.7) with a\ne0 proves c_3\ne0.□

Proof. Use the ordinary descent identification in Proposition 3.3 to form the composite

The first arrow restricts the group action, the second is inflation along G\to P_3, and the last is induced by the equivariant map (3.3). In the middle term with group G, the action on E_3^{\square} is through P_3. Restriction and inflation are the natural maps of continuous fixed points. The source equivalence and its ordinary unit are the point-valued form of [4, Proposition 4.6.9] proved in Proposition 3.3; the last map uses the local split-section completed map of Lemma 3.4 and that proposition.

For every profinite set T, the evaluation \widehat B^{\square}(T) is the ordinary height-two Lubin–Tate theory described in Proposition 3.3, with the Noetherian witness verified there; hence it is K(2)-local [4, Proposition 4.4.2]. The bar construction gives

Ordinary local spectra are closed under limits, so \mathcal D is K(2)-local. This also follows from the pointwise criterion for solid locality [4, Definition 4.5.7 and the following paragraph]. The universal property of ordinary K(2)-localization now gives (7.2).□

Proof. Let F(X)=(X^{hA})(*). This is an exact functor preserving limits and upper homotopy bounds. For the latter assertion, coconnective objects remain coconnective under evaluation and limits. Although X need not be a module spectrum over H\Z_p, each Postnikov layer HM is one. The forgetful functor from module spectra preserves limits, so its continuous fixed points agree with the derived module invariants. On this Eilenberg–Mac Lane object, the finite resolution in Lemma 2.1 gives

The solid spectra are used with the hypersheaf convention of [4, Section 4.5, especially Proposition 4.5.5]. Truncations remain solid, since they retain or kill each solid homotopy module. Evaluation on an extremally disconnected profinite set is t-exact: homotopy sheafification evaluated on this projective object agrees with pointwise homotopy. It therefore commutes with truncation. These evaluations detect equivalences, so Postnikov completeness of ordinary spectra gives X\simeq\varprojlim_n\tau_{\leq n}X.

Put T_n=F(\tau_{\leq n}X). Exactness identifies the successive fiber as

Consequently \pi_qT_{n+1}\to\pi_qT_n is an isomorphism for n\geq q+d; the analogous tower in degree q+1 is also eventually constant. The Milnor exact sequence therefore has zero derived-limit term, and gives

The infinitely negative tail causes no further issue: since F(\tau_{\leq q-1}X) is concentrated in degrees at most q-1, the fiber sequence for this lower cutoff gives

Hence each abutment degree is computed by a finite Postnikov interval.

The exact couple of this Postnikov construction has the E_2-term in (7.4). Its filtration in degree q is

It starts with the whole group, ends in zero by (7.5), and has associated graded E_\infty^{s,q+s}. This proves the asserted strong convergence and its naturality. Tensoring the exact couple and these finite filtrations with \Q is exact. It therefore gives the rational spectral sequence and its stated abutment; no exchange of rationalization with an infinite totalization is involved. The groups here are \Z_p-modules, so this tensor operation agrees with [1/p]. No uniform torsion exponent as t varies is required: each fixed homotopy degree has the finite filtration just constructed.□

Proof. Odd homotopy modules vanish. For t=2m, scalar automorphisms a\in1+p\Z_p lift canonically to scalar automorphisms of the universal deformation. They fix its deformation ring, and act by a^m on its mth invariant-differential line \pi_{2m}E_n^{\square}. This is the central character used in [5, Lemma 2.6.1]; replacing the periodicity convention by its dual replaces m by -m and has the same consequence.

Choose b=1+p. It is central in G_3, since the finite Galois action fixes these scalars. A central group element acting on a coefficient module induces the identity on its group cohomology. This follows already from the bar resolution: its coefficient action is chain homotopic to the action by conjugation, which is the identity for a central element. Consequently (b^m-1) annihilates H^s(G_3,\pi_{2m}E_3^{\square}). For m\ne0 this is a nonzero p-adic scalar, and is invertible after p is inverted. This proves the first vanishing.

For the target, use the natural coefficient descent (3.4) to identify its cohomology with H^s(H,\pi_tE_2^{\square}). The factor \overline Z acts trivially on these coefficients, and the central scalars in P_2 act by the same character b^m. The preceding argument applies to H, and proves the second vanishing.□

Proof. First extend the source primitive to the extended stabilizer. By Proposition 2.2, e_{P_3} is represented under rational Lie comparison by a nonzero multiple of

In the standard presentation of the degree-three division algebra, take its maximal unramified subfield \Q_{p^3} and an element \Pi such that

where \sigma is arithmetic Frobenius of the unramified cubic extension \Q_{p^3}/\Q_p, inducing z\mapsto z^p on the residue field. Its extension fixing \Pi equals conjugation y\mapsto\Pi y\Pi^{-1}, as is checked on the generators \Q_{p^3} and \Pi. It therefore preserves the reduced-trace form. Its action on P_3 factors through the order-three quotient of \Delta; the order-two kernel also fixes the \Z_p-coefficients. Naturality of Lie comparison therefore makes e_{P_3} invariant under \Delta. Since 6 is a unit in \Z_p, finite-group averaging and extension descent give

Let \widetilde e_3 be the class restricting to e_{P_3}, and let z_3 be its image under the unit-coefficient map \Z_p\to\pi_0E_3^{\square}.

Under (2.2), restriction from G_3 to P_3 and inflation along G\to P_3 on the constant classes are the derived limits of their finite continuous-cochain pullbacks. These are the group maps used in (2.4). All coefficient maps in the following square are the maps induced by (7.3):

On bar cochains, the right vertical map is restriction, pullback along G\to P_3, and the coefficient map of E_3^{\square}\to\widehat B^{\square}, in that order. Its composite with the source coefficient unit is the target unit, which is also \Z_p\to W(\widehat L)\to\pi_0\widehat B^{\square}. Here the first map is the integral limit of the finite Witt units in (6.4); the map from full Witt vectors to deformation coefficients is applied only after that limit. The source descent identification used here is the canonical one in Proposition 3.3. In particular the image of z_3 is exactly the class called c_3 in Proposition 6.2: it is obtained from e_{P_3} by projection and the actual constant-coefficient inclusion. That proposition and Lemma 6.3 give

Thus the map between the rational descent E_2-terms is nonzero in bidegree (s,t)=(3,0).

Apply Lemma 7.2 to the source and target of g. By Lemma 7.3, only t=0 survives after rationalization. The differential has bidegree (r,r-1), so every rational differential vanishes. For total homotopy degree -3, the finite filtration has just one possibly nonzero graded piece, namely (s,t)=(3,0). The nonzero map on this piece is therefore a nonzero map on the abutments. These abutments are the ordinary rational homotopy groups asserted in the proposition.□

Proof of Theorem 1.1. Rationalize (7.2). If the canonical map L_0L_{K(3)}S\to L_0L_{K(2)}L_{K(3)}S were zero on \pi_{-3}, its composite L_0g would also be zero there. This contradicts Proposition 7.4. Every step above applies to each prime p\geq5, proving the theorem.□

Proof. Localization is exact and preserves finite sums, so L_{K(2)}(A\vee B)\simeq L_{K(2)}A. Naturality with the summand inclusions identifies \tau_{A\vee B} with \tau_A on L_0A and the zero map on L_0B. It vanishes in degree d. Naturality with an arbitrary equivalence A\vee B\simeq Z transports this conclusion to Z.□

Proof of Corollary 1.2. In (1.1), let A be the first line and B the sum of the last two lines. Smashing for L_1 and the Bousfield-class description of L_1S give L_{K(2)}B=0 [16, Theorem 5.1 and Proposition 5.3].

The p-complete sphere is connective. Each term of the Moore tower S_p^\wedge=\varprojlim_m S/p^m is connective, and the degree-zero transition maps are surjective; the possible negative Milnor term therefore vanishes. Rationalization preserves connectivity. Since L_0L_2\simeq L_0, the two summands of L_0A have no homotopy in degree -3. Thus Lemma 8.1 gives \pi_{-3}\tau_{\mathcal W_3}=0.

Put T=L_{K(3)}S and X=L_2T. The cofiber of the localization map T\to X is E(2)-acyclic. The identity

makes it both rationally and K(2)-acyclic. The naturality square

therefore has vertical equivalences. Any equivalence X\simeq\mathcal W_3 would force its lower horizontal map to vanish on \pi_{-3}, and hence force the same for \tau_T. This contradicts Theorem 1.1. No step specifies the action of the hypothetical equivalence on individual summands.□

Proof. For each coset of the open additive subgroup C^{\circ\circ} in C, choose a representative a and a root b_a of b_a^q-b_a=a. Choose a=b_a=0 on the zero coset. Define

The sum converges, and s(x)^q-s(x)=x. If |x-y|<1, the points lie in the same coset and the first term of the difference series dominates, so |s(x)-s(y)|=|x-y|. If |x-y|\ge1, the equation for s(x)-s(y) gives norm |x-y|^{1/q} (including norm one at the boundary). Thus s is continuous, s(0)=0, and

Define S coefficientwise. For every 0<r<1 this estimate gives

It also gives the corresponding estimate on every coefficient tail. Since r^q is again in (0,1), S preserves convergence and is continuous for the Fréchet topology. It is a section of \phi_q-1. For any profinite space T, postcomposition with S lifts every continuous map T\to R. This proves the asserted surjectivity of condensed groups, and therefore of solid modules.

The kernel consists of the same convergent series with coefficients in k. Its nonzero coefficients all have norm one. The convergence condition says that the exponents in its support form a left-finite subset of \Z[1/p]: below any bound there are only finitely many. Truncation therefore identifies it with the completed perfection E. On this kernel every Gauss seminorm is r^{v_t(f)}; hence its induced topology is exactly the valued-field topology of E. The two-term complex [R\xrightarrow{\phi_q-1}R] now has kernel E and zero cokernel in solid modules. It computes homotopy fixed points for the infinite cyclic action, proving the last assertion.□

Proof of Lemma 3.4. The forgetful transformation in [21, Proposition 6.2.2 and Remark 6.2.3] goes from deformations of a full group to deformations of the left term of a tagged exact sequence with étale quotient. Corepresentability therefore gives a map from the connected deformation ring to the full deformation ring. We construct the reverse oriented map by a section of that transformation.

The field F is perfect of characteristic p, and its inverse Frobenius is continuous. Hence pointwise inverse Frobenius preserves C_{\cts}(T,F), so R_T is perfect. For nonempty T, the map F\to R_T is flat, since F is a field, and relatively perfect, since both Frobenius maps are isomorphisms. The ring R_T need not be a field or Noetherian. The field F is Noetherian and F-finite. More generally, for any perfect \F_p-algebra, every p-divisible group is nonstationary and the absolute cotangent complex is almost perfect, by [21, Remark 3.4.2]. In particular the full group over F, without a connectedness assumption, is an eligible Noetherian LKN witness for (R_T,\Gamma_T), in the sense of [4, Definition 4.3.7 and Example 4.6.4]. The connected pair has the height-two witness (k,\Gamma_2). At the empty parameter set we assign the terminal presheaf value; no LKN assertion for the zero ring is needed.

Over the perfect field F, take the connected–étale sequence and its unique splitting:

Here Q_F is étale of height one. Existence and uniqueness of the splitting are [21, Proposition 2.5.20 and Remark 2.5.24]. We apply the splitting remark only over the perfect field F. Write \iota_F:C_F\to\Gamma_F for the inclusion, and let

be exactly the tautological formal isomorphism of the isomorphism torsor in [4, Example 4.6.3]. Over the discrete field F, the hypotheses of [21, Theorem 2.3.12 and Corollary 2.3.13] hold because p=0. Their fully faithful identity-component functor gives a unique isomorphism \lambda_F:(\Gamma_2)_F\xrightarrow{\sim}C_F, characterized by

Thus our \lambda_F has the direction from \Gamma_2 to the connected summand. Define \sigma_T, its splitting, \iota_T, and \lambda_T by base change to R_T. In particular, C_T denotes a p-divisible group, whereas the identity component of \Gamma_T denotes its formal group.

Naturality of the split reference pair. We check the family naturality of this splitting. If M is a finite projective R_T-module, then

is injective: embed M as a direct summand of R_T^N and use R_T\hookrightarrow\prod_tF. It follows that maps between finite flat group schemes over R_T are detected by these fibers, by applying the same observation to the target of the corresponding coordinate-algebra maps. It detects maps of p-divisible groups on each p^m-torsion level. These are actual equalities of maps: finite flat algebras over the ordinary ring R_T are ordinary finite projective algebras, and their mapping spaces are discrete. The finite-flat functor description recalled in the proof of [21, Proposition 2.5.20] identifies the compatible torsion-level maps with maps of p-divisible groups.

Let \rho:R_T\to R_U be a parameter pullback or a map furnished by a continuous action family, and let \gamma:\rho^*\Gamma_T\xrightarrow{\sim}\Gamma_U be the given pair isomorphism. Both induced sequences are connected–étale: connectedness is preserved by base change by the locally nilpotent augmentation criterion of [21, Remark 2.3.8], and étaleness is preserved by base change. The discrete characteristic-p ring R_U is complete with ideal of definition zero. Thus [21, Theorem 2.5.13] gives the unique isomorphism of these sequences extending \gamma. The two resulting sections Q_U\to\Gamma_U agree at each u\in U, by uniqueness of the splitting over the perfect field F; the preceding detection argument makes them equal over R_U. Applying this with U=T\times G^a, for every a\geq0, proves compatibility with all family, multiplication, and unit maps. The same holds for \lambda_T by the fully faithful identity-component functor. The input is the continuous action on the full completed pair and its tautological formal trivialization from [4, Examples 4.6.3–4.6.4], restricted to the nonextended P_3\times P_2, which fixes k. This argument uses the pulled-back field splitting; it does not assert a splitting theorem for arbitrary perfect rings.

The section on oriented deformations. We now construct the section on oriented deformations. Let A be any complete adic E_\infty-ring, without a Noetherian or connectivity assumption. For nonconnective A, the notation \mathrm{BT}_p(A) means \mathrm{BT}_p(\tau_{\geq0}A). For a finitely generated ideal of definition I, put B_I=\pi_0A/I, write H(B_I) for its discrete Eilenberg–Mac Lane E_\infty-ring, and use the canonical reduction

A subscript I below denotes this reduction or the corresponding ordinary base change of a reference group. No map A\to H(B_I) is asserted. Orientations remain over the full ring A: their Bott map is an equivalence of A-modules. They are not orientations over its connective cover; see [21, Warning 4.3.12].

Represent a connected oriented deformation by (C_A,I,\mu,\alpha_C,e), where I\subset\pi_0A is a finitely generated ideal of definition,

and e orients C_A^\circ over A. Tags are identified only after passage to a common larger finitely generated ideal of definition on which both the residue maps and group isomorphisms agree, as in [21, Definitions 3.1.1 and 3.1.4]. Since R_T has characteristic p, the tag forces p\in I. Completeness for I therefore implies (p)-completeness, and p is topologically nilpotent, by [21, Remarks 0.0.10–0.0.12]. The tag makes C_A formally connected. These facts meet the hypotheses of the identity-component and connected–étale results used below; they do not assert completeness of \tau_{\geq0}A.

The reduction equivalence

of [21, Corollary 2.5.10] lifts \mu^*Q_T to an étale group Q_A, with a specified residual isomorphism \alpha_Q. The space of such lifts with this isomorphism is contractible; the unmarked object Q_A itself may have automorphisms. More precisely, if \mathscr D_{C,I}^{\mathrm{or}}(A) denotes the fixed-I stage, use the homotopy pullback

where the first map sends a tag to \mu^*Q_T and the second is the typed reduction above. Its projection to \mathscr D_{C,I}^{\mathrm{or}}(A) is an equivalence by Corollary 2.5.10; its fiber includes the marking \alpha_Q.

These projections and the split construction form natural diagrams on the category of compatible data (A,I,\mu), including adic maps, residue maps after ideal refinement, and compatible maps of split base sequences. For an adic map f:A\to A', start with a target ideal of definition J_0. Continuity gives f(I^n)\subseteq J_0 for some n, so J_0+(f(I)) is again a finitely generated ideal of definition and contains f(I). Such choices admit common larger ideals. For I\subseteq J, the same Q_A with the reduced marking is used.

The fixed-I tagged categories for complete A are groupoidal by the proof of [21, Lemma 3.1.10]: a tagged morphism is an equivalence modulo I, and equivalence is detected on its finite flat p-torsion level. Orientations are pulled back from \mathrm{BT}_p(A)^\simeq, independently of I; filtered colimits of spaces commute with this pullback. Hence the filtered colimit of the fixed stages is precisely the oriented tagging anima of Definition 3.1.4. The auxiliary projections are an objectwise equivalence of diagrams; take their filtered colimits and invert that equivalence in the functor category. This supplies all lift, refinement, and base-map coherences. We do not replace that colimit by reduction to the nilradical, which [21, Warning 3.1.9] excludes in this generality.

Form the split sequence

It is an exact sequence of p-divisible groups by [21, Proposition 2.4.8], applied to the split pushout square. Tag the entire residual sequence using

This retains the original full tag, including \mu, and chooses no basis or trivialization of the étale factor. Since A is (p)-complete, Q_A^\circ=0 by [21, Proposition 2.5.8]. The identity-component fiber sequence, as in the proof of [21, Proposition 2.5.17], gives C_A^\circ\xrightarrow{\sim}(C_A\oplus Q_A)^\circ. Transport the same orientation e through this equivalence over A.

Write \mathscr D_\sigma^{\mathrm{or}}, \mathscr D_\Gamma^{\mathrm{or}}, and \mathscr D_C^{\mathrm{or}} for the resulting oriented deformation functors. Let m:\mathscr D_\sigma^{\mathrm{or}}\to \mathscr D_\Gamma^{\mathrm{or}} and q:\mathscr D_\sigma^{\mathrm{or}}\to\mathscr D_C^{\mathrm{or}} forget to the middle and left terms. Proposition 6.2.2 of [21] makes the underlying m an equivalence. It remains an equivalence with orientations: a tagged quotient is étale by Remark 2.5.11 of that paper, and the left and middle identity formal groups are naturally equivalent. Set \mathfrak f=qm^{-1}, taking the inverse in the functor category. The split construction gives \widetilde{\mathfrak s}:\mathscr D_C^{\mathrm{or}}\to \mathscr D_\sigma^{\mathrm{or}}, whose left projection is the identity on (I,\mu,\alpha_C,e). Therefore

naturally with all the preceding refinements and base maps.

Corepresentability and the reverse ring map. We justify the oriented mapping property at the possibly non-Noetherian R_T. The oriented mapping property in [21, Remark 6.0.7] is stated in the Noetherian setting of its Theorem 6.0.3; the general statement of BMR Remark 4.1.22 needs the following argument here. For either pair (R_T,\Gamma_T) or (R_T,C_T), the characteristic is p, and [21, Remark 3.4.2] supplies nonstationarity and an almost perfect absolute cotangent complex. Theorem 3.4.1 of that paper therefore gives a complete connective universal spectral deformation ring R^{\mathrm{un}}, a surjection \pi_0R^{\mathrm{un}}\to R_T, and a finitely generated kernel J which is an ideal of definition.

Let E be the orientation classifier of its universal identity formal group, as in [4, Construction 4.1.17 and Definition 4.1.20]. The LKN witnesses and [4, Definition 4.2.1 and Theorem 4.3.8] supply balancedness. By Remark 4.1.16 and Definitions 4.1.19–4.1.20 there, the map \pi_0R^{\mathrm{un}}\to\pi_0E is an isomorphism, the odd homotopy groups of E vanish, and every even homotopy group is an invertible \pi_0E-module. Here completeness of modules means derived completeness. The homotopy-group completeness argument in the proof of [21, Corollary 6.0.5] now applies without a Noetherian step. Specifically, the homotopy-group criterion cited there as SAG Theorem 7.3.4.1 makes \pi_0R^{\mathrm{un}} J-complete because R^{\mathrm{un}} is J-complete. An invertible module over this ring is a direct summand of a finite free module, hence is also J-complete. The same criterion then makes E J-complete.

Endow \pi_0E with the topology transported from \pi_0R^{\mathrm{un}}. An E_\infty-map E\to A is continuous exactly when its restriction from R^{\mathrm{un}} is continuous, because these \pi_0 rings and topologies agree. The universal deformation mapping property of Theorem 3.4.1, followed by the orientation-classifier mapping property, therefore identifies the adic mapping anima from E with the triples consisting of a deformation, its strict tagging class, and an orientation over full A. This locally proves the use of BMR Remark 4.1.22 at R_T. It turns \mathfrak f and \mathfrak s into continuous adic maps

They preserve the SW(k)-structure because they retain the restriction of the same tag \mu to k. The construction is natural in split base sequences. The family argument above therefore makes the r_T continuous G-natural maps.

For precision, this last assertion also survives condensation. Take the category of split LKN sequences with both full and connected pairs LKN. The preceding construction is a natural transformation from its full E-functor to its connected E-functor. Extend these functors and the transformation from the presheaf category and tensor with condensed anima as in [4, Construction 4.5.2]. Hypersheafifying the raw split parameter diagram gives a condensed object with its continuous G-action; its full and connected images are the original pair diagrams. The resulting enriched transformation is the pointwise r_T followed by hypersheafification, giving r^{\mathrm{cond}}.

The connected tensor factor. It remains to identify the E_2 factor. For each T, let b_T:E_2\to E(R_T,\Gamma_T) be the ordinary tensor-factor map followed by completion, and let c_T:E_2\to E(R_T,C_T) be the canonical connected map using \lambda_T. The pointwise tensor construction uses the Noetherian perfect reference pair (k,\Gamma_2), with connected height-two group, and the localized height-two LKN pair as its other input. These meet [4, Construction 4.4.7]. The proof of Theorem 4.4.8 there identifies its mapping spaces on K(2)-local complete adic SW(k)-algebra tests. Choose a common representative ideal I for the two k-linear tag classes, using the strict refinement relation. If e_1,e_2 orient the formal groups of the two deformations H_1,H_2, let

be the isomorphisms from the same Quillen formal group and their comparison. This is the natural orientation correspondence of [4, Remark 4.1.18], using Lurie’s Propositions 4.3.21 and 4.3.23. The orientations are over full A, while j is a formal isomorphism over \tau_{\geq0}A. Only j, not an orientation, is reduced along \tau_{\geq0}A\to H(B_I).

Let \alpha:\Gamma_{T,I}\xrightarrow{\sim}(H_1)_I be the full tag after completion, and let \lambda_{0,I} be the base change of the same torsor isomorphism \lambda_0 to B_I. The tuple repacking of Theorem 4.4.8 first goes from the reference \Gamma_2^\circ through \lambda_{0,I}^{-1} to the reference full formal group, then through \alpha^\circ to (H_1)_I^\circ, and finally through j_I. Thus the formal tag of the second tensor factor is exactly

For the split section take

where i:C_A\to C_A\oplus Q_A is the inclusion. Naturality of the Quillen correspondence gives q_{e_1}=i^\circ q_e, so j=(i^\circ)^{-1}. In our direction for \lambda_T, put

The defining relation for \lambda_F gives \lambda_{0,I}^{-1}=\iota_{T,I}^\circ\lambda_{T,I}^\circ, and the definition of the full split tag gives \alpha^\circ\iota_{T,I}^\circ=i_I^\circ\alpha_C^\circ. Substitution therefore cancels the precise torsor and orientation isomorphisms:

Theorem 2.3.12 and Corollary 2.3.13 of [21], over A and over the discrete characteristic-p ring B_I, identify the formally connected output with C_A and the tag itself with \beta. Their hypotheses were checked above. The output orientation is e, since pushforward through i^\circ is undone through its inverse. The output ring tag is exactly k\to R_T\xrightarrow{\mu}B_I. Completion only precomposes \mu and base changes the same \lambda_0, so the calculation respects the SW(k)-linear tagging condition. There is no additional Bott unit or automorphism normalization. It is natural under strict ideal refinement and the parameter and family maps already treated. Hence, on the stated local test category,

The connected endpoint is K(2)-local by [4, Proposition 4.4.2] for its height-two witness, and it is complete by the argument above. The same statements hold for E_2. Thus both are K(2)-local complete adic SW(k)-algebras. Yoneda on the stated test category, equivalently evaluation on the universal connected deformation at A=E(R_T,C_T), gives r_Tb_T\simeq c_T. This preserves the full connected tag (I,\mu,\alpha_C), its restricted reference-pair tag \beta:(\Gamma_2)_{B_I}\xrightarrow{\sim}(C_A)_I along k\to R_T\xrightarrow{\mu}B_I, and the orientation e; it uses no locality of E(R_T,\Gamma_T).

The continuous solid identity. We identify the actual E_2 tensor composite with c^\square. Let b^\square:E_2^\square\to E^{\mathrm{cond}}(\widehat L,\Gamma^{\mathrm{un}}_{3,\widehat L}) be the tensor-factor map followed by completion. There are two points in passing from the pointwise identity to solid theories. First, the pointwise presheaf in the proof of [4, Proposition 4.5.12] uses the constant ordinary E_2, not E_2^\square(T) as the reference input of Theorem 4.4.8. The preceding identity, natural in T, therefore identifies the two maps after precomposition from E_2^\delta and hypersheafification. By [4, Lemma 4.5.8],

is the localization unit. The connected target is L^\square_{K(2)}-local by its profinite evaluations, so its localization mapping property identifies the two underlying maps from E_2^\square.

Second, this does not give E_2^\delta a continuous P_2-action. Put X=E_2^\square, Y=\widehat B^\square, and write Y^S for the profinite cotensor, so Y^S(T)=Y(S\times T). For the underlying solid spectra, the space of continuous G-equivariant maps is the bar totalization with degree a

Every target cotensor is local. Restriction along \ell is therefore an equivalence of mapping spaces in each degree; transport the bar cofaces through these equivalences. This requires no action on E_2^\delta.

To identify the transported source coface, let g:S\to P_2 be a continuous family. Since k is finite, compactness gives C_{\cts}(S\times T,k)=C_{\cts}(T,C_{\cts}(S,k)_{\mathrm{disc}}). Thus the cotensor of k^\delta by S is (k^\delta(S))^\delta, where k^\delta(S)=C_{\cts}(S,k). The family gives a pair map

at every torsion level its coefficients are locally constant, as required by continuity. The naturality of Lemma 4.5.8 and the cotensor comparison of [4, Remark 4.5.6] give the commuting square

Here a_{X,g} is the given family action and \ell_S is the corresponding localization unit under the cotensor identification. Consequently the source coface after restriction is precomposition of the connected tag by \gamma_g. Target cofaces and interior cofaces are the full-pair family maps and parameter pullbacks. The tuple identification and \mathfrak f\mathfrak s\simeq\mathrm{id} are natural for exactly these operations on every R_{G^a\times T}. They give compatible homotopies in all bar degrees, hence

as continuous G-equivariant maps of underlying solid spectra. Both sides are independently maps of condensed E_\infty-rings, and their source and target are solid. In particular they induce the same coefficient and even-line maps. This uses the localization mapping property degree by degree, not an interchange of localization with totalization or fixed points.□