Chromatic Smith theory: cyclic length and fixed-point loss

Source and verification

This article explains family 314 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Fixed points can lose chromatic information. Family 314 asks for the exact amount lost when one passes from a subgroup’s fixed points to the whole group’s fixed points. Its claimed answer is a discrete invariant of the embedded subgroup: the shortest subnormal chain with cyclic quotients. The answer is independent of chromatic height, including rational height zero. (OpenAI, 2026)

🏷️ Preliminaries: genuine spectra and chromatic tests

A genuine $G$-spectrum is a stable homotopy object with a finite-group action in which suspension by representations, rather than only by trivial-dimensional spheres, is invertible. This structure supports restriction, transfer, and geometric fixed points ${\Phi }^{H}X$. Geometric fixed points remove contributions induced from proper subgroups before taking fixed points. They differ from both ordinary fixed points and homotopy fixed points.

For the suspension spectrum of a finite based $G$-complex, geometric fixed points recover the suspension spectrum of its fixed-point complex. More generally they are exact and preserve finite spectra. A finite genuine spectrum here includes a compact retract of a finite cell spectrum.

At a prime $p$, Morava $K$-theory $K(i)$ tests chromatic height $i>0$; set $K(0)=H\mathbb{Q}$. A spectrum is $K(i)$-acyclic when its $K(i)$-homology vanishes. For a finite nonzero spectrum, its type is the least height with nonzero Morava homology.

Define $r_n(G,H)$ to be the least $r\ge 0$ for which every finite $p$-local genuine $G$-spectrum satisfies

$$\begin{array}{c}K(n+r{)}_{\ast }({\Phi }^{H}X)=0\\ ⟹K(n{)}_{\ast }({\Phi }^{G}X)=0.\end{array}$$

Thus $r$ measures the extra chromatic height one must test at $H$ to force acyclicity at height $n$ for $G$.

🏷️ Cyclic length and the exact loss

A subnormal chain from $H$ to $G$ is a sequence in which each subgroup is normal in the next. Define $\ell (G,H)$ as the smallest number of nontrivial steps in such a chain with cyclic quotients. These chains exist when $G$ is a finite $p$-group; set $\ell (G,G)=0$.

Manuscript's chromatic Smith formula

For every prime $p$, finite $p$-group $G$, subgroup $H\le G$, and $n\ge 0$,

$$r_n(G,H)=\ell (G,H).$$

For every smaller proposed loss, there is a finite genuine spectrum violating the corresponding implication. The witnesses arise by separating tensor-triangular primes. (OpenAI, 2026)

The upper bound follows from the established cyclic chromatic Smith theorem: a nontrivial cyclic $p$-group costs at most one height. Apply it successively along a chain, using transitivity of normal geometric fixed points. A cyclic quotient of order $p^a$ still costs only one, not $a$.

The embedding of $H$ matters. In $G=C_{p^2}\times C_p$, the subgroup $H_1=p{C}_{p^2}\times 0$ has quotient $C_p\times C_p$, while $H_2=0\times C_p$ has quotient $C_{p^2}$. Both subgroups are isomorphic to $C_p$, but their losses are respectively two and one. Normality is automatic in this example; in general $G/H$ need not be a group, so the definition uses an actual subnormal chain.

The difficult direction is sharpness for every embedded subgroup. Knowing only how many generators an elementary abelian quotient has can miss this invariant.

🏷️ A cofree construction detecting height loss

For an ordinary spectrum $D$, let

$$\begin{array}{c}{\mathrm{Bor}}_P(D)=F(E{P}_{+},{\inf }_{P}D),\\ U_P(D)={\Phi }^P{\mathrm{Bor}}_P(D).\end{array}$$

Here $EP$ is a contractible free $P$-space, $F$ denotes the function spectrum, and inflation gives $D$ the trivial genuine action. The first construction is the cofree spectrum attached to the trivial action. The second measures what survives in its full geometric fixed points.

For $l>0$, let $T(l)$ be the telescope of a periodic self-map of a finite type-$l$ spectrum; set $T(0)=H\mathbb{Q}$. On finite spectra, $T(l)$ and $K(l)$ detect the same acyclicity. The proof does not identify their localizations on arbitrary spectra.

The first main intermediate estimate is

$$\begin{array}{c}{U}_P(L_{T(l)}S{)}_{\mathbb{Q}}\ne 0\\ ⟹\ell (P,1)\le l.\end{array}$$

The proof constructs characters indexed by maps from a specially chosen pro-$p$ group $\Gamma$, an inverse limit of finite $p$-groups, with $\Gamma \twoheadrightarrow P$. Every finite quotient of every closed subgroup of $\Gamma$ needs at most $l$ generators. This is stronger than a bound on the generators of $\Gamma$ alone.

A maximizing tower produces $\Gamma$. Enumerate finite target groups and at each stage maximize the number of conjugacy classes of maps to the next target among occurring extensions. The generator bound makes this integer bounded, so a maximum exists. Once a target is maximized, later surjections cannot introduce additional maps. Every fixed finite diagram of groupoids therefore stabilizes at a common stage. This finite-diagram qualification is needed for compatible transfer calculations.

🏷️ Integration weights and subgroup congruences

A finite groupoid is a finite collection of objects and invertible arrows, up to equivalence. Its homotopy cardinality weights an isomorphism class by the reciprocal of its automorphism-group order. In the source’s character coordinates, integration has exactly these weights.

If $a_{\alpha }$ is the size of the automorphism group of a character $\alpha$, the coordinate $\beta$ of a pushforward along $f$ is

$$(f_{!}x{)}_{\beta }=\sum _{[f\alpha ]=[\beta ]}\frac{a_{\beta }}{a_{\alpha }}{x}_{\alpha }.$$

The diagonal transfer has coordinate $a_{\alpha }$ on the $\alpha$ diagonal. Composing the diagonal with projection gives the identity, forcing integration of that coordinate idempotent to equal $1/a_{\alpha }$. The relative formula follows by the projection formula. This argument retains automorphisms in the kernel of the map between automorphism groups.

Apply these weights to finite sets with equivariant labels in a finite continuous $U$-set $X$, where $U$ is an open subgroup of $\Gamma$. If $b_j(X)$ is the weighted count of labeled sets of size $j$, put $B_X(t)={\sum }_j{b}_j(X)t^j$. Decomposition into transitive components gives

$$[t^{p^h}]\log B_X(t)=\frac{1}{p^h}\sum _{[U:V]=p^h}|X^V|.$$

The sum is over actual open subgroups, not conjugacy classes: pointing a transitive set kills its automorphisms and specifies its stabilizer.

The source proves that these weighted coefficients are $p$-integral and agree modulo $p$ with the coefficients for the trivial action on the same underlying set. Hence $B_X/B_{X_{\mathrm{triv}}}=1+pA(t)$ with $p$-integral coefficients. The logarithm still has coefficients divisible by $p$, since $q-{\nu }_p(q)\ge 1$ for every positive integer $q$, also when $p=2$. Comparing the indicated coefficients yields

$$\sum _{[U:V]=p^h}(|X^V|-|X|)\in p^{h+1}{\mathbb{Z}}_{(p)}.$$

Here ${\mathbb{Z}}_{(p)}$ consists of rationals with denominator prime to $p$. Coinduction of labeled groupoids is what supplies this constraint for every open subgroup $U$, rather than only for $\Gamma$ itself. (OpenAI, 2026)

🏷️ From congruences to a rational height bound

Another character calculation gives $|{\mathrm{Hom}}_{\mathrm{cts}}(\Gamma ,F)|/|F|\in {\mathbb{Z}}_{(p)}$ for every finite group $F$. If a nontrivial $\Gamma$ had finite abelianization of order $p^b$, taking $F=C_{p^r}$ for large $r$ would give the nonintegral value $p^{b-r}$. Thus $\Gamma$ has a quotient ${\mathbb{Z}}_p$.

The source’s profinite argument then uses the subgroup congruences to extract further ${\mathbb{Z}}_p$ quotients from successive nontrivial kernels. Shrinking the action makes a direct product with the already extracted factors while preserving the congruences and finite-section bound. After $d$ steps, a nontrivial remaining kernel would give an elementary abelian quotient of rank $d+1$. Therefore extraction stops after at most $l$ steps.

This supplies a closed subnormal series with ${\mathbb{Z}}_p$ factors. Mapping it into the finite quotient $P$ gives a cyclic chain of length at most $l$. The shrinking-action and kernel-counting lemmas are substantial profinite inputs; the finite-section bound alone is not asserted to imply the conclusion. At height zero, rational averaging separately forces $P=1$ whenever the cofree geometric fixed points are nonzero.

🏷️ Raising the estimate to every chromatic height

The case $k=0$ is the preceding rational estimate. The case $l=0$ follows separately from rational averaging. The next target is

$$\begin{array}{c}T(k)\wedge U_P(L_{T(l)}S)\ne 0\\ ⟹k+\ell (P,1)\le l.\end{array}$$

For $k>0$, ring nilpotence first turns telescope detection into Morava detection for the particular unital ring spectrum involved. This is a ring-specific argument, not the telescope conjecture. Base change then gives a nonzero commutative $K(k)$-local Morava $E_k$-algebra $B$.

For $r\ge 1$ and $C=(C_{p^r}{)}^k$, the full-subgroup Burnside projector on $E_k^{BC}$ keeps precisely the surjective characters ${\mathbb{Z}}_p^k\to C$. Its trace is

$$N_{k,r}=p^{k^2(r-1)}\prod _{i=0}^{k-1}(p^k-p^i)>0.$$

Indeed, a matrix over $\mathbb{Z}/p^r$ is surjective exactly when its reduction modulo $p$ is invertible. Every invertible matrix modulo $p$ has $p^{k^2(r-1)}$ lifts.

Local symmetric monoidal base change preserves this trace. A torsion-nilpotence theorem makes the rational unit of nonzero $B$ nonzero, so $N_{k,r}{1}_B$ cannot vanish after rationalization. Thus the full projector survives even without a coefficient-flatness assumption on $B$.

A coherent, unit-preserving comparison

$$U_{P\times C}(D)⟶U_C(U_P(D))$$

now implies rational nonvanishing for $P\times C$. The direction and unitality of this map matter: the nonzero target unit forces the source unit to survive. The rational estimate gives $\ell (P\times (C_{p^r}{)}^k,1)\le l$ for every $r$.

Choose compatible chains by compactness and pass to $P\times {\mathbb{Z}}_p^k$. Exactly $k$ procyclic steps raise the free rank. At each such step the factor is torsion-free ${\mathbb{Z}}_p$, so the finite intersection with $P$ cannot grow. Deleting these repetitions leaves at most $l-k$ cyclic steps for $P$. This proves the height estimate.

🏷️ Sharpness through separation of supports

Define the prime ideal

$${\mathcal{P}}_i(H)=\{X:K(i{)}_{\ast }({\Phi }^{H}X)=0\}$$

in the tensor category of finite genuine spectra. Its index is the tested height; some literature shifts this index by one. The support of an object is the set of primes that do not contain it. Specialization reverses prime inclusion.

The universal implication defining loss is precisely ${\mathcal{P}}_{n+r}(H)\subseteq {\mathcal{P}}_n(G)$. The height estimate, combined with finite chromatic fracture, bounds interactions between the right adjoints to subgroup geometric-fixed-point functors.

Suppose a specialization violates the predicted bound, and choose its least height $m$. Truncate to heights at most $m$. There are now finitely many subgroup–height points. A finite tensor product of separating finite spectra realizes the closure of $(G,n)$ as the support of one compact object $z$.

At the offending subgroup, minimality and the height estimate give vanishing in both directions between its layer and all other layers. Both directions are necessary. Remove the other subgroup layers in increasing order of subgroup size; previously removed layers stay absent. Adjunction then makes the connecting map vanish in every degree, yielding a splitting $z\simeq X\oplus Q$ into compact summands with disjoint supports.

One summand contains the offending point and the other contains $(G,n)$. Both supports are nonempty closed sets. This contradicts irreducibility of the closure of a point. Thus a specialization forces $m\ge n+\ell (G,H)$, proving the lower bound. Failure of every smaller prime inclusion gives the promised finite-spectrum witness, without an explicit cell construction.

The manuscript also combines this result with established conjugacy and $p$-subnormal reductions for arbitrary finite groups. This exposition focuses on the finite $p$-group theorem. The audit covers the displayed character, trace, chain, and support mechanisms; it does not independently verify all profinite lemmas or the external equivariant and chromatic machinery.

🏷️ Related posts

• Chromatic support and tensor ideals introduces another support classification, for dualizable Morava-local objects rather than finite genuine equivariant spectra.
• Smith–Toda complexes discusses constructions of finite spectra with controlled chromatic type, complementing the fixed-point restrictions studied here.

🏷️ Key steps and proof outlines

Ring nilpotence is the precise bridge between two height tests

For $k>0$, the gluing argument initially detects a unital ring spectrum $R$ by $T(k)$, the telescope of a positive-degree periodic self-map $v:{\Sigma }^{d}V\to V$ of a finite type-$k$ spectrum. It needs nonzero $K(k)$-homology to enter the Morava-theoretic character calculation.

Suppose instead that $K(k)\wedge R=0$. In the ring spectrum $\mathrm{End}(V)\wedge R$, consider $v\otimes 1$. At height $k$ it is zero by assumption; at other finite heights its image is nilpotent by the periodic self-map property. Its mod-$p$ Hurewicz image is also nilpotent, because it has positive degree in the bounded graded homology of $\mathrm{End}(V)$ before mapping into the larger ring.

The ring nilpotence theorem makes some power of $v\otimes 1$ zero. Acting on $V\wedge R$ shows that its telescope vanishes, contradicting $T(k)\wedge R\ne 0$. The unital ring structure is the essential extra hypothesis. This proof supplies the required detection implication for the ring in use; it does not compare telescope and Morava localizations on arbitrary spectra. (OpenAI, 2026)

A nonzero trace survives a possibly nonflat base change

For $C=(C_{p^r}{)}^k$, the full-subgroup projector on $E_k^{BC}$ selects the surjective characters ${\mathbb{Z}}_p^k\to C$. Its trace is the positive integer

$$N_{k,r}=p^{k^2(r-1)}\prod _{i=0}^{k-1}(p^k-p^i).$$

The product counts ordered independent columns modulo $p$; the leading factor counts their lifts modulo $p^r$. Because the cochain module is finite free, this is also the categorical trace of the projector after rationalizing morphisms.

Extend scalars in the $K(k)$-local module category to the nonzero commutative algebra $B$ supplied by the preceding construction. The source’s semiadditivity input identifies this local base change with $B^{BC}$ and preserves the finite-groupoid transfers defining the projector. A symmetric monoidal functor preserves the trace of a dualizable object, so the new trace is $N_{k,r}{1}_B$.

Why is it nonzero? The torsion-nilpotence theorem for commutative algebras in this local category says every torsion element of ${\pi }_{0}B$ is nilpotent. If the unit were torsion, it would be nilpotent and therefore zero; unitality would force $B=0$. Thus the unit survives rationalization, as does its nonzero integer multiple. The projector cannot vanish because its trace does not. This proof avoids assuming that the coefficient extension to $B$ is flat.

Isolating one subgroup layer needs vanishing in both directions

The sharpness argument works in a finite height truncation. Write ${\varphi }_J$ for the subgroup-layer functor, $F_J$ for its fully faithful right adjoint, and ${\mathcal{A}}_J=F_J(1)$ for the associated layer algebra. For a dualizable object $z$, the projection formula identifies

$$Q=F_B{\varphi }_{B}z\simeq z\otimes {\mathcal{A}}_B.$$

There is a triangle $X\to z\to Q$. Applying ${\varphi }_B$ shows ${\varphi }_{B}X=0$, but this does not yet split the triangle. The source separately establishes

$$\begin{array}{rlrl}{\varphi }_K(z\otimes {\mathcal{A}}_B)& =0& & (K\nsim B),\\ {\varphi }_B(z\otimes {\mathcal{A}}_J)& =0& & (J\nsim B).\end{array}$$

The first condition confines $Q$ to the chosen layer. The second prevents that layer from reappearing while the fiber is simplified.

Remove the other subgroup layers from $X$ in increasing order of subgroup size. At each step, take the fiber of the adjunction unit to $F_J{\varphi }_J$ of the current object. Full faithfulness removes the $J$-component. A previously removed subgroup cannot contain a conjugate of a later, larger subgroup, so the subgroup-layer formulas preserve its vanishing. The second displayed condition preserves vanishing at $B$: a unital module over ${\mathcal{A}}_J$ is a retract of its tensor product with ${\mathcal{A}}_J$.

At the end, every geometric-fixed-point component is zero, so the remaining object is zero. The first displayed condition and adjunction make the mapping spectrum from $Q$ to each removed layer zero. Inducting backward therefore gives $\mathrm{Map}(Q,X)=0$, in every degree. In particular the connecting map $Q\to \Sigma X$ vanishes, and the triangle splits.

When $z$ is compact, both summands are compact retracts of $z$, and their supports are disjoint. Applied to a compact object whose support is the closure of one point, this would split an irreducible closed set into two nonempty closed pieces. That contradiction rules out the proposed specialization and produces the finite-spectrum witnesses for every loss smaller than the claimed cyclic length.

🏷️ Manuscripts and verification scope

• Cyclic length and chromatic fixed-point loss — pinned manuscript and companion source.

No family scope page was present at lean/docs/314.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. Cyclic length and chromatic fixed-point loss.