3 The map
We define a bigrading on by setting
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 , while . We write .
The main theorem is
There is a map with the property that
There is a fully faithful embedding that sends to .
There is an equivalence of categories that sends to .
-Bockstein spectral sequence for
and converges to . Unsurprisingly, this is the Adams spectral sequence for .
We begin by constructing , which is in fact a natural transformation
Fix , and let . In , we have a pushout diagram
Applying to this diagram, we get
There is nothing that requires this to be a pushout diagram, but we get a comparison map
This is exactly the map we seek.
Recall that . Then
and is the natural covering map. So while .
We now quickly look at modules over and .
A synthetic spectrum is -invertible if it has a (necessarily unique) -module structure. Equivalently, if is an equivalence.
For any , the spectral Yoneda embedding is -invertible.
In fact, every -invertible synthetic spectrum is of this form:
The spectral Yoneda embedding is fully faithful with essential image given by -invertible synthetic spectra. Further, there is a natural equivalence
Now consider . If is -injective, then . So
Given a general , we can resolve it by -injectives, and we find that
Let be any spectrum. Then
In fact, it is true that
([3, Theorem 4.46, Proposition 4.53])
There is a fully faithful embedding that sends to . If is Landweber exact, then this is essentially surjective.