Formal-group interpretation for Lin's theorem?

Question

Background

For compact Lie groups, Atiyah and Segal proved a strong relationship between Borel-equivariant K-theory, defined in terms of the K-theory of $X \times_G EG$, and the equivariant K-theory of X defined in terms of equivariant vector bundles. Roughly, for "nice" spaces X the K-theory of $X \times_G EG$ is a completion of the equivariant K-theory of X, and in particular the K-theory of BG is a completion of the complex representation ring of G.

The Segal conjecture is an analogous result proven in subsequent years (by many authors, with Carlsson completing the proof). It's less well-known outside the subject, and obtained by roughly replacing "vector bundles" with "covering spaces" - the original conjecture is that for a finite group $G$, the abelian group of stable classes of maps $\varinjlim[S^n \wedge BG, S^n]$ has as limit to the Burnside ring of finite $G$-sets. There are further statements describing $\varinjlim [S^{n+k} \wedge BG, S^n]$ in terms of a completion of certain equivariant stable homotopy groups. It's notable for the fact that it's not really a computational result - we describe two objects as being isomorphic, without any knowledge of what the resulting groups on either side really are.

There are a number of steps in this proof, and over the years most of them have been recast and reinterpreted in a number of ways. However, the initial steps in the proof are computational. Lin proved this conjecture for the case $G = \mathbb{Z}/2$, and Gunawardena proved it for the case $G = \mathbb{Z}/p$ for odd primes $p$. Lin's original proof involved some very difficult computations in the Lambda algebra and a simplified proof was ultimately written up by Lin-Davis-Mahowald-Adams. It amounts to a computation of certain Ext or Tor groups over the Steenrod algebra - namely, if $H^* \mathbb{RP}^\infty = \mathbb{Z}/2[x]$ has its standard module structure over the Steenrod algebra, then $Ext^{**}(\mathbb{Z}/2[x^{\pm 1}],\mathbb{Z}/2)$ degenerates down to a single nonzero group.

Bordism theory

A lot of the contemporary work in stable homotopy theory uses the relationship between stable homotopy theory and the moduli of formal groups, rather than the Adams-spectral-sequence calculations that are used in the above proofs. The analogous calculation would be the following.

Let L be the Lazard ring carrying the universal formal group law, with 2-series $[2](t)$ whose zeros are the "2-torsion" of the formal group law. Then there is an L-algebra $$ Q = t^{-1} L[[t]]/[2](t) $$ whose functor of points would be described (up to completion issues) as taking a ring R to the set of formal group laws on R equipped with a nowhere-zero 2-torsion point. This comes equipped with natural descent data for change-of-coordinates on the formal group law, and so it describes a sheaf on the moduli stack of formal group laws $\mathcal{M}$.

A student of Doug Ravenel's (Binhua Mao) proved in his thesis that the analogous Ext-computation is valid in the formal-group setting: namely, if one computes the Ext-groups $$Ext_{\mathcal M}(\mathcal{O}, Q \otimes \omega^{\otimes t})$$ where $\omega$ is the sheaf of invariant 1-forms on $\mathcal{M}$, it converges to a completion of $$Ext_{\mathcal M}({\mathcal O}, \omega^{\otimes t}).$$ (The result was stated in different language, and I am still ignoring completion issues.)

However, as I understand the proof (and I don't claim that I really do!) it essentially uses a reduction to the Adams spectral sequence case by using a filtration that reduces to the group scheme of automorphisms of the additive formal group law, and this is very closely connected to the Steenrod algebra. I would regard it as still being computationally focused, and I don't really have a grip on why one might expect it to be true without carrying the motivation from homotopy theory all the way through.

Question (finally)

Is there is a more conceptual interpretation of this computation in terms of the geometry of the moduli of formal groups?

Answer 1

I can't really answer this. I'll just think out loud for a bit.

Let $R$ be a complete local $\mathbb{F}_p$ algebra. The additive formal group $G$ is a formal scheme $Spf(\mathbb{F}_p[[x]])$. An $R$-point of $G(R)$ is an element of $\mathfrak{m}_R$.

Pick $t\in G(R)=\mathfrak{m}_R$, and consider $f(x)=x(x-t)(x-2t)\cdots (x-(p-1)t)=x^p-tx$. Let $F_t=Spec(R[[x]]/(x^p-tx))=Spec(R[x]/(x^p-tx))$. Then $F_t$ is a finite subgroup scheme of $G$. If we base change $F_t$ to $\tilde{F}_t$ over $R[t^{-1}]$, then $\tilde{F}_t$ becomes an etale group scheme.

The universal example of such an $F_t$ lives over $B=\mathbb{F}_p[[t]]$. The scheme $S$ of automorphisms of $G$ (i.e., the dual Steenrod algebra) acts on $B$, and the action lifts to $B[t^{-1}]=\mathbb{F}_p((t))$. If $\omega$ is the module of invariant differentials on $G$ (isomorphic as a module with $S$-action to $tB/t^2B$, then there's a map $$Res_{t=0}: B[t^{-1}] \otimes_{\mathbb{F}_p} \omega\to \mathbb{F}_p,$$ which is a map of $S$-modules. Lin's theorem asserts that this map induces isomorphisms in $Ext_S^*(\omega^i,{-})$.

So Lin's theorem is something about residues.

You have this residue map is other cases, for instance if we replace $G$ with a Lubin-Tate deformation. Neil Strickland has thought about this: in his Formal Schemes and Formal Groups, he spells out some of the relationship between the residues and the Segal conjecture.