# 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

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 $X$.

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

Applying $X$ to this diagram, we get

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.

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 $\tau \colon {\mathbb {S}}^{0, -1} \to {\mathbb {S}}$.

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 $C\tau$.

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.

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:

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

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

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.