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some basics at Steenrod algebra
added to the citations at Steenrod algebra a pointer to
Jacob Lurie, 18.917 Topics in Algebraic Topology: The Sullivan Conjecture, Fall 2007. (MIT OpenCourseWare: Massachusetts Institute of Technology), Lecture notes
Lecture 2 Steenrod operations (pdf)
Lecture 3 Basic properties of Steenrod operations (pdf)
Lecture 4 The Adem relations (pdf)
Lecture 5 The Adem relations (cont.) (pdf)
Looks like you’ve written F_2 when you meant F_p in your presentation of the dual Steenrod algebra.
Also maybe worth mentioning that Spec of the dual steenrod algebra is a group scheme that acts on Spf(H^*(RP^{\infty})), i.e. the formal affine line, and it acts faithfully. This induces an isomorphism between Spec of the dual Steenrod algebra and automorphisms of the additive formal group over F_2.
There is a similar but more complicated story for odd primes: you still look at the action of the dual Steenrod algebra on H^*(BC_p) but now you need to remember the odd degree terms which Spec doesn’t see. The usual trick works: say the word “super”. So you can write down what a super formal group is and a super group scheme and you get the same answer as above. I don’t know a reference for this offhand…
Thanks! Fixed now. And thanks for the further comments.
Is there a Kapranov-ian reason for the “super” turning up?
Sure, the $\mathbb{Z}$-graded commutativity of $\pi_\bullet(E)$ for ring spectra $E$ is via the $\mathbb{Z} = \pi_0(\mathbb{S})$. This comes down to the graded commutativity of the smash product of plain spheres in the homotopy category (here).
The fascinating 2-categorical analogue of Steenrod algebra (related also to Adams-Novikov spectral sequence study, and to secondary cohomological operations) has been extensively studied by Hans-Joachim Baues and his collaborators. In additions to many papers, Baues published at least one book on the very subject.
Re #7, and there’s some reason odd degree terms don’t appear for $p = 2$.
Hm, not sure if there is an abstract reason. It comes out of the explicit computation,which gives that only for $p \gt 2$ there are these odd degree generators $\tau_i$ in the dual Steenrod algebra (here).
@David: There are odd degree elements (many!), it’s just that they aren’t nilpotent because p=2, so Spec sees them. The point is just that Spec of a graded ring over a field only sees the even part (since odd^2 = 0) except when the characteristic is 2.
I see. Thanks.
Er, sorry. A glitch due to that “polynomial algebra” versus “exterior algebra” terminology. I should add a note to the entry…
If you want to unify p odd and p=2 : Define P^n to be Sq^{2n} when p=2 and \beta = Sq^1 (it is the Bockstein, after all). Then the dual steenrod algebra is the free (or symmetric) graded algebra over F_p on generators \xi_n and \tau_n defined as in the odd case. All the formulas then look the same for all primes.
Then the difference boils down to the usual dichotomy that a free graded algebra in char 2 is polynomial while a free graded algebra in char p>2 is polynomial on even generators and exterior on odd.
(I realize this isn’t very exciting, but I find it a useful exercise to get rid of the “p=2, p>2” dichotomy wherever one can so as to isolate the places where the difference is real.)
Yes, absolutely. Thanks for amplifying, I am editing the entry now. While I have your attention on these matters: might you have a minute left to look over my derivation at May spectral sequence?
Okay, I have edited accordingly, see here. Is there a citable source that puts it in this nice way?
I made E-Steenrod algebra a redirect to this page here, and added pointer back to Adams spectral sequence – The first page.
Optimally, E-Steenrod algebra should be a page of its own. Maybe later.
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