Construction of synthetic spectraThe map τ\tau

3 The map τ\tau

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

(Σa,bX)(P)=ΣbX(ΣabP). (\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 Σ1,1\Sigma ^{1, -1}, while νΣ=Σ1,0ν\nu \Sigma = \Sigma ^{1, 0} \nu . We write Sa,b=Σa,bS{\mathbb {S}}^{a, b} = \Sigma ^{a, b} {\mathbb {S}}.

The main theorem is

Theorem 3.1

There is a map τ ⁣:S0,1S\tau \colon {\mathbb {S}}^{0, -1} \to {\mathbb {S}} with the property that

  1. There is a fully faithful embedding ModCτComodEE\operatorname{Mod}_{C\tau } \to \operatorname {Comod}_{E_*E} that sends CτνXC\tau \otimes \nu X to EXE_* X.

  2. There is an equivalence of categories Modτ1Sτ1S\operatorname{Mod}_{\tau ^{-1} {\mathbb {S}}} \cong \tau ^{-1} {\mathbb {S}} that sends τ1νX\tau ^{-1} \nu X to XX.

The τ\tau -Bockstein spectral sequence for νX\nu X then has E2E_2-page given

E2s,t=ExtEEs,t(E,EX) E^{s, t}_2 = \operatorname{Ext}_{E_*E}^{s, t}(E_*, E_*X)

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

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

τ ⁣:Σ0,1XX. \tau \colon \Sigma ^{0, -1} X \to X.

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

    P \ar[r] \ar[d] & * \ar[d] \\
    * \ar[r] & \Sigma P.

Applying XX to this diagram, we get

      X(P) & * \ar[l] \\
      * \ar[u] & X(\Sigma P). \ar[u] \ar[l]

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

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

This is exactly the map τ\tau we seek.

Remark 3.2

One can show that for any XSynEX \in \operatorname{Syn}_E, the map τ ⁣:Σ0,1XX\tau \colon \Sigma ^{0, -1} X \to X is the tensor product of XX with τ ⁣:S0,1S\tau \colon {\mathbb {S}}^{0, -1} \to {\mathbb {S}}.

Example 3.3

Recall that y(X)(P)=τ0F(P,X)y(X)(P) = \tau _{\geq 0} F(P, X). Then

(Σ0,1y(X))(P)=Στ0F(ΣP,X)=τ1F(P,X), (\Sigma ^{0, -1} y(X))(P) = \Sigma \tau _{\geq 0} F(\Sigma P, X) = \tau _{\geq 1} F(P, X),

and τ\tau is the natural covering map. So y(X)/τ=π0F(P,X)y(X) / \tau = \pi _0 F(P, X) while τ1y(X)=Y(X)\tau ^{-1} y(X) = Y(X).

We now quickly look at modules over τ1S\tau ^{-1} {\mathbb {S}} and CτC\tau .

Definition 3.4

A synthetic spectrum XSynEX \in \operatorname{Syn}_E is τ\tau -invertible if it has a (necessarily unique) τ1S\tau ^{-1} {\mathbb {S}}-module structure. Equivalently, if τ ⁣:Σ0,1XX\tau \colon \Sigma ^{0, -1} X \to X is an equivalence.

Example 3.5

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

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

Theorem 3.6

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

Y(X)τ1νX. Y(X) \cong \tau ^{-1} \nu X.

Now consider CτνXνX/τC\tau \otimes \nu X \cong \nu X / \tau . If XX is EE-injective, then νX=y(X)\nu X = y(X). So

[νA,νX/τ]=π0F(A,X)=HomEE(EA,EX). [\nu A, \nu X / \tau ] = \pi _0 F(A, X) = \operatorname{Hom}_{E_*E}(E_*A, E_* X).

Given a general XX, we can resolve it by EE-injectives, and we find that

Lemma 3.7

Let A,XA, X be any spectrum. Then

[Σa,bνA,νX/τ]=ExtEEb,a+b(EA,EX). [\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 ModCτComodEE\operatorname {Mod}_{C\tau } \to \operatorname {Comod}_{E_*E} that sends νX\nu X to EXE_* X. If EE is Landweber exact, then this is essentially surjective.