# Transchromatic twisted character maps

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## Abstract

In this paper we construct a refinement of the transchromatic generalized character maps by taking into account the torus action on the inertia groupoid (also known as the Fix functor). The relationship between this construction and the geometry of p-divisible groups is made precise.

## Keywords

Generalized character theory HKR Morava \(E\)-theory## 1 Introduction

In chromatic homotopy theory, there is a history of trying to understand height \(n\) phenomena in terms of height \(n-1\) phenomena associated to the free loop space. There is an \(S^1\)-action on the free loop space by rotation. This action plays a key role in topological cyclic homology and the redshift conjecture (such as in [3]) and in Witten’s work on the elliptic genus (see [10, 11]). In generalized character theory the \(S^1\)-action has been traditionally ignored. In this work, we describe a generalized character theory in which this \(S^1\)-action is accounted for and we explain the relationship between it and the geometry of \(p\)-divisible groups.

### **Theorem**

We also show how to canonically recover the transchromatic generalized character maps of [8] using the canonical map \(B_t \overset{}{\longrightarrow } \hat{C_t}\). It should be noted that, in personal correspondence, Lurie has described a method for building the transchromatic twisted character maps from the transchromatic generalized character maps of [8].

## 2 Notation and conventions

## 3 Transchromatic geometry

### 3.1 Non-trivial extensions

This construction was inspired by the work of Ando and Morava in [1, Section 5]. The idea behind the construction is that the ring \(B_{t}^{^{\prime }}\) has \(n-t\) canonical points for the formal group \(\mathbb{G }_0 = \mathbb{G }_{L_{K(t)}E_n}\) given by the set \({\bar{q}}\). The elements of \({\bar{A}}\) are invisible to \(\mathbb{G }_0\) in the sense that there are no maps \(L_t[\![x]\!] \overset{}{\longrightarrow } B_{t}^{^{\prime }}\) such that \(x \mapsto A_i\) for any \(i\). We formalize all of this in the next proposition using the language of formal algebraic geometry.

### **Proposition 3.1**

### *Proof*

- 1.We construct the map \(\mathbb{Z }_{p}^{n-t} \overset{}{\longrightarrow } B_{t}^{^{\prime }}\otimes \mathbb{G }_{0}\) as a mapWe have the isomorphism$$\begin{aligned} \mathbb{Z }_{p}^{n-t} \overset{}{\longrightarrow } \mathbb{G }_0(B_{t}^{^{\prime }}). \end{aligned}$$As \(\mathbb{G }_0\) is formal and by Tate’s Lemma \(0\) this is$$\begin{aligned} \mathbb{G }_0(B_{t}^{^{\prime }}) \cong \underset{k}{\lim }\,\underset{j}{{{\mathrm{colim}}}}\hom _{\text{ cont } L_t}(L_t[\![x]\!]/[p^j](x),B_{t}^{^{\prime }}/(I_t+{\bar{q}})^k). \end{aligned}$$which is the largest ideal of definition for \(B_{t}^{^{\prime }}\) with the topology induced by \(I_t + ({\bar{q}})\). Call this ideal \(J\). The elements of \({\bar{q}}\) are elements of \(J\) and can be used to define the required map.$$\begin{aligned} \hom ({{\mathrm{Spf}}}_{I_t+({\bar{q}})}(B_{t}^{^{\prime }}),{{\mathrm{Spf}}}_{I_t+(x)}(L_t[\![x]\!])) \cong \hom _{\text{ cont } L_t}(L_t[\![x]\!],B_{t}^{^{\prime }}), \end{aligned}$$
- 2.We observe that the map defined in part 1 does not extend to \(\mathbb{Q }_{p}^{n-t}\). This is not immediately clear because we don’t know that the elements of \({\bar{A}}\) are not in \(J\). Assume a continuous mapexists. Consider the composite$$\begin{aligned} L_t[\![x]\!] \overset{x \mapsto A_i}{\longrightarrow } B_{t}^{^{\prime }}\end{aligned}$$It factors through$$\begin{aligned} L_t[\![x]\!] \overset{}{\longrightarrow } B_{t}^{^{\prime }}\overset{}{\longrightarrow } B_{t}^{^{\prime }}/({\bar{q}}). \end{aligned}$$for some \(k\). Now, since the map is assumed to be continuous, the power series \([p^k](x)\) must be in the kernel of the map. Thus we get a map$$\begin{aligned} L_t[\![x]\!] \overset{x \mapsto A_i}{\longrightarrow } L_t\otimes _{E_{n}^0} E_{n}^0[\![A_i]\!]/[p^{k}](A_i) \end{aligned}$$But this map is an inverse to the canonical quotient in the other direction and cannot exist for dimension reasons (the ring on the right has higher rank as a free \(L_t\)-module).$$\begin{aligned} L_t[\![x]\!]/[p^k](x) \overset{x \mapsto A_i}{\longrightarrow } L_t\otimes _{E_{n}^0} E_{n}^0[\![A_i]\!]/[p^{k}](A_i). \end{aligned}$$
- 3.The composite \(\mathbb{Z }_{p}^{n-t} \overset{}{\longrightarrow } B_{t}^{^{\prime }}\otimes \mathbb{G }\) does extend to \(\mathbb{Q }_{p}^{n-t}\). Tate’s Lemma \(0\) gives us thatand so$$\begin{aligned} \underset{k}{\lim } \, B_{t}^{^{\prime }}/(I_t+{\bar{q}})^k \cong \underset{k}{\lim } \, B_{t}^{^{\prime }}/(I_{t}^{k}+[p^k]({\bar{q}})) \end{aligned}$$does detect the elements of \({\bar{A}}\). As \(k\) varies we get$$\begin{aligned} \mathbb{G }(B_{t}^{^{\prime }})&\cong \underset{k}{\lim }\,\underset{j}{{{\mathrm{colim}}}} \hom _{\text{ cont } L_t}(L_t \otimes _{E_{n}^0} E_{n}^0[\![x]\!]/[p^j](x),B_{t}^{^{\prime }}/(I_t+{\bar{q}})^k) \\&\cong \underset{k}{\lim }\,\underset{j}{{{\mathrm{colim}}}} \hom _{\text{ cont } L_t}(L_t \otimes _{E_{n}^0} E_{n}^0[\![x]\!]/[p^j](x),B_{t}^{^{\prime }}/(I_{t}^{k}+[p^k]({\bar{q}}))) \end{aligned}$$$$\begin{aligned} \mathbb{Q }_{p}^{n-t} = \lim \big ( \cdots \overset{}{\longrightarrow } (\mathbb{Q }_p/p^2\mathbb{Z }_p)^{n-t} \overset{}{\longrightarrow } (\mathbb{Q }_p/p\mathbb{Z }_p)^{n-t} \overset{}{\longrightarrow } (\mathbb{Q }_p/\mathbb{Z }_p)^{n-t} \big ). \end{aligned}$$
- 4.By universality in the statement of the claim we mean that \({{\mathrm{Spf}}}_{I_t + ({\bar{q}})}(B_{t}^{^{\prime }})\) represents the functor that brings a complete \(L_t\)-algebra \(R\) to the set of commutative squares of the form From the above we see that a continuous \(L_t\)-algebra map \(B_t^{\prime } \overset{}{\longrightarrow } R\) induces a square of this sort. Also, each square of this sort comes from such a map. Let \(R\) be a connected complete \(L_t\)-algebra. Given such a square, the \(R\) points of the square induce a mapwhich is precisely a map \(L_t[\![{\bar{q}}]\!] \overset{}{\longrightarrow } R\). Now the other side of the square implies that this map extends to a map$$\begin{aligned} (\mathbb{Z }_p)^{n-t} \overset{}{\longrightarrow } \mathbb{G }_0(R), \end{aligned}$$$$\begin{aligned} B_t^{\prime } \overset{}{\longrightarrow } R. \end{aligned}$$

### **Proposition 3.2**

### *Proof*

### 3.2 Relation to \(C_t\)

### **Proposition 3.3**

The map above sends the set \(R \subset B_{t}^{^{\prime }}\) bijectively to the set \(S \subset \hat{C_t}^{\prime }\).

### *Proof*

This implies that \(B_t\) is nonzero.

### **Proposition 3.4**

The map \(L_t/I_t \overset{}{\longrightarrow } B_t/(I_t+{\bar{q}})\) is faithfully flat.

### *Proof*

## 4 Transchromatic twisted character maps

The transchromatic twisted character map is defined to be the composition of two maps. The first map is induced by a map of topological spaces and the second one is algebraic in nature.

### 4.1 The topological map

### **Proposition 4.1**

### *Proof*

### **Definition 4.2**

### **Proposition 4.3**

### *Proof*

This is clear. \(\square \)

### *Example 4.4*

### *Example 4.5*

Let \(G\) be a finite group and \(X\) a finite \(G\)-space (equivalent to a finite \(G\)-CW complex).

### **Proposition 4.6**

The action on the fixed point space \(X^{{{\mathrm{im}}}\alpha }\) by \(C({{\mathrm{im}}}\alpha )\) extends to an action by \(T(\alpha )\).

### *Proof*

Recall from [8] that the set of continuous homomorphisms \(\hom (\mathbb{Z }_{p}^{h},G)\) is a \(G\)-set under conjugation.

The following definition is fundamental to our work here:

### **Definition 4.7**

### *Remark 4.8*

### *Example 4.9*

### **Proposition 4.10**

The set \(C(\alpha ,\beta )\) is a right \(\mathbb{Z }_{p}^{h}\)-set through \(\alpha \).

### *Proof*

### **Definition 4.11**

### **Proposition 4.12**

### *Proof*

Now it follows from Corollary 4.8 of [2] that the fat geometric realization of the map \(f\) is an equivalence. The nerves of the topological groupoids are “good” simplicial spaces in the sense of [7] Definition A.4 and this implies that the geometric realization is canonically equivalent to the fat geometric realization by Theorem A.1 of [7]. We conclude that the geometric realization of \(f\) is an equivalence. \(\square \)

When multiple groups are in use, we will write \({{\mathrm{Twist}}}_{h}^{G}(-)\) to make it clear what group \({{\mathrm{Twist}}}_h(-)\) depends on.

A critical property of \({{\mathrm{Twist}}}_{h}(-)\) is the way that it interacts with abelian subgroups of \(G\).

### **Proposition 4.13**

### *Proof*

We note some more properties of \(T(-)\).

### **Proposition 4.14**

### *Proof*

### **Proposition 4.15**

### *Proof*

Finally we give the construction of the topological part of the twisted character map from height \(n\) to height \(t\). Fix a finite group \(G\). We produce it as the realization of a map of topological groupoids.

Let \(k\) be such that every map \(\mathbb{Z }_{p}^{n-t} \overset{}{\longrightarrow } G\) factors through \(\Lambda _k= (\mathbb{Z }/p^{k})^{n-t}\). Let \(\gamma _k:\mathbb{Z }_{p}^{n-t} \overset{}{\longrightarrow } \Lambda _k\) be the canonical quotient.

We begin by extending the morphism space of \({{\mathrm{Twist}}}_{n-t}^G(X)\).

### **Proposition 4.16**

### *Proof*

### *Example 4.17*

### 4.2 The algebraic map

The algebraic part of the twisted character map uses the properties of the ring \(B_t\) constructed and described in Sect. 3.1 to construct an appropriate codomain for the twisted character map.

The discussion regarding gradings in [8] carries over to this situation.

### **Proposition 4.18**

### *Proof*

### *Example 4.19*

### **Proposition 4.20**

The map \(\Upsilon _G\) is independent of the choice of \(k\).

### *Proof*

This follows from the proof of Proposition 3.13 in [8]. \(\square \)

### **Theorem 4.21**

### *Proof*

### 4.3 The isomorphism

### **Proposition 4.22**

The functor \({{\mathrm{Twist}}}_{n-t}(-)\) commutes with pushouts of finite \(G\)-CW complexes.

### *Proof*

### **Proposition 4.23**

The functor \({{\mathrm{Twist}}}_{n-t}(-)\) commutes with geometric realization of simplicial finite \(G\)-CW complexes.

### *Proof*

This follows from [8]. There are no difficulties in extending the result there for the functor \({{\mathrm{Fix}}}_{n-t}(-)\) to \({{\mathrm{Twist}}}_{n-t}(-)\). \(\square \)

Now we follow [8]. Using the Bousfield-Kan spectral sequence the two facts above allow us to reduce the isomorphism for transchromatic generalized character maps to the case of finite \(G\)-CW complexes with abelian stabilizers. Now Mayer-Vietoris reduces this to the case of \(BA\) for \(A\) a finite abelian group. It may not be entirely clear that the cohomology theory in the codomain of the twisted character map above has the Kunneth isomorphism that we need to reduce to cyclic p-groups. We prove that now.

### **Proposition 4.24**

### *Proof*

The above propositions together with the work in [8] establish the main theorem.

### **Theorem 4.25**

### 4.4 Relation to [8]

## Notes

### Acknowledgments

Once again it is a pleasure to thank Charles Rezk for his assistance with this project. Rezk pointed out a version of the \({{\mathrm{Twist}}}(-)\) construction and suggested that it might be related to non-trivial extensions of \(p\)-divisible groups. I would like to thank Jacob Lurie for several illuminating discussions. I’d also like to thank Matt Ando, Mark Behrens, David Carchedi, David Gepner, Tyler Lawson, Haynes Miller, Chris Schommer-Pries, Olga Stroilova, and the referee for their time and helpful remarks. The author was partially supported by NSF grant DMS-0943787.

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