Let’s try to do some (baby) examples of the Adams spectral sequence. The notation used will be that of yesterday’s post.

**1. **

So let’s start with a silly example, where the answer is tautological: . We could try to compute the homotopy groups of this using the Adams spectral sequence. At a prime , this means that we should get the -adic completion in degree zero, and nothing elsewhere.

It turns out that we can write down a very explicit Adams resolution for . To start with, we need a map where is a wedge of and shifts, and such that is a monomorphism on -homology. We can take the map

the fact that is a monomorphism on -homology follows because is an *epimorphism* on -cohomology, by Serre’s computation of the cohomology of Eilenberg-MacLane spaces. Serre’s computation tells us, in fact, that the cohomology of is the Steenrod algebra mod the ideal generated by the Bockstein. (more…)