3 The map $\tau$

We define a bigrading on $\operatorname{Syn}_E$ by setting

(\Sigma ^{a, b} X)(P) = \Sigma ^{-b} X(\Sigma ^{-a - b} P).

The precise combinations on the right are chosen for the purpose of agreeing with the Adams grading in the Adams spectral sequence. Under this grading convention, categorical suspension is $\Sigma ^{1, -1}$ , while $\nu \Sigma = \Sigma ^{1, 0} \nu$ . We write ${\mathbb {S}}^{a, b} = \Sigma ^{a, b} {\mathbb {S}}$ .

The main theorem is

Theorem 3.1

There is a map $\tau \colon {\mathbb {S}}^{0, -1} \to {\mathbb {S}}$ with the property that

There is a fully faithful embedding $\operatorname{Mod}_{C\tau } \to \operatorname {Comod}_{E_*E}$ that sends $C\tau \otimes \nu X$ to $E_* X$ .
There is an equivalence of categories $\operatorname{Mod}_{\tau ^{-1} {\mathbb {S}}} \cong \tau ^{-1} {\mathbb {S}}$ that sends $\tau ^{-1} \nu X$ to $X$ .

The

\tau

-Bockstein spectral sequence for

\nu X

then has

E_2

-page given

E^{s, t}_2 = \operatorname{Ext}_{E_*E}^{s, t}(E_*, E_*X)

and converges to $\pi _{t - s} X$ . Unsurprisingly, this is the Adams spectral sequence for $X$ .

We begin by constructing $\tau$ , which is in fact a natural transformation

\tau \colon \Sigma ^{0, -1} X \to X.

Fix $X \in P_\Sigma ^{\mathrm{Sp}}({\mathrm{Sp}}_E^{fp})$ , and let $P \in {\mathrm{Sp}}_E^{fp}$ . In ${\mathrm{Sp}}_E^{fp}$ , we have a pushout diagram

$\begin{useimager} \[ \begin{tikzcd} P \ar[r] \ar[d] & * \ar[d] \\ * \ar[r] & \Sigma P. \end{tikzcd} \] \end{useimager}$

Applying $X$ to this diagram, we get

$\begin{useimager} \[ \begin{tikzcd} X(P) & * \ar[l] \\ * \ar[u] & X(\Sigma P). \ar[u] \ar[l] \end{tikzcd} \] \end{useimager}$

There is nothing that requires this to be a pushout diagram, but we get a comparison map

\Sigma X(\Sigma P) \to X(P).

This is exactly the map $\tau$ we seek.

Remark 3.2

One can show that for any $X \in \operatorname{Syn}_E$ , the map $\tau \colon \Sigma ^{0, -1} X \to X$ is the tensor product of $X$ with $\tau \colon {\mathbb {S}}^{0, -1} \to {\mathbb {S}}$ .

Example 3.3

Recall that $y(X)(P) = \tau _{\geq 0} F(P, X)$ . Then

(\Sigma ^{0, -1} y(X))(P) = \Sigma \tau _{\geq 0} F(\Sigma P, X) = \tau _{\geq 1} F(P, X),

and $\tau$ is the natural covering map. So $y(X) / \tau = \pi _0 F(P, X)$ while $\tau ^{-1} y(X) = Y(X)$ .

We now quickly look at modules over $\tau ^{-1} {\mathbb {S}}$ and $C\tau$ .

Definition 3.4

A synthetic spectrum $X \in \operatorname{Syn}_E$ is $\tau$ -invertible if it has a (necessarily unique) $\tau ^{-1} {\mathbb {S}}$ -module structure. Equivalently, if $\tau \colon \Sigma ^{0, -1} X \to X$ is an equivalence.

Example 3.5

For any $X \in {\mathrm{Sp}}$ , the spectral Yoneda embedding $Y(X)$ is $\tau$ -invertible.

In fact, every $\tau$ -invertible synthetic spectrum is of this form:

Theorem 3.6

The spectral Yoneda embedding $Y \colon {\mathrm{Sp}}\to \operatorname{Syn}_E$ is fully faithful with essential image given by $\tau$ -invertible synthetic spectra. Further, there is a natural equivalence

Y(X) \cong \tau ^{-1} \nu X.

Now consider $C\tau \otimes \nu X \cong \nu X / \tau$ . If $X$ is $E$ -injective, then $\nu X = y(X)$ . So

[\nu A, \nu X / \tau ] = \pi _0 F(A, X) = \operatorname{Hom}_{E_*E}(E_*A, E_* X).

Given a general $X$ , we can resolve it by $E$ -injectives, and we find that

Lemma 3.7

Let $A, X$ be any spectrum. Then

[\Sigma ^{a, b} \nu A, \nu X / \tau ] = \operatorname {Ext}_{E_*E}^{b, a + b}(E_*A, E_*X).

In fact, it is true that

Theorem 3.8 ([3, Theorem 4.46, Proposition 4.53])

There is a fully faithful embedding $\operatorname {Mod}_{C\tau } \to \operatorname {Comod}_{E_*E}$ that sends $\nu X$ to $E_* X$ . If $E$ is Landweber exact, then this is essentially surjective.

3 The map τ\tau τ

3 The map $\tau$