4:26 AM
so things like the map S[t] -> S that sends t to 2 is kind of confusing and i feel like, at best, there are a bunch of facts that you can learn to make it less confusing
one is that left multiplication by t and right multiplication by t are both "2" as a maps of spectra S -> S. but: right multiplication by t is not "2" as a map of left S[t]-modules
e.g. you can calculate End_{S[t]} (S), whose coefficient ring is S_*[β]/(β^2) where β is in degree -1. the image of t in this ring is (2 + η β). so even though left & right multiplication by t are the same on the underlying spectrum, if you know one of them then you can see that the other one is different.
alternatively, still thinking of S as a bimodule, there is the following: (t-2)*1 = 0 and 1*(t-2) = 0 in S_*, but these aren't the "same" identity. there's a nullhomotopy of (t-2)*1 and a nullhomotopy of 1*(t-2) but those nullhomotopies aren't "the same".
(this is phrasable as <t-2,1_S,t-2> contains η if you like brackets)
and that's a little weird, but it's indicative of the fact that (t-2) isn't strictly central in S[t] (just like t is, but 2 isn't).