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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.
added pointer to:
Probably because the person who started writing the entry thinks that it is not a fully viable product in the present state (for instance, there is not the definition for ordinary cohomology). This is not the same as an article which is satisfying in its present state but can be completed with further material.
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