Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: Steenrod algebra

Bottom of Page

1 to 23 of 23

  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2013

    some basics at Steenrod algebra

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2013

    added to the citations at Steenrod algebra a pointer to

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 12th 2016

    I have typed out the generators-and-relations presentation of the general mod pp Steenrod algebra (here) and Milnor’s characterization of its linear dual (here)

    • CommentRowNumber4.
    • CommentAuthorDylan Wilson
    • CommentTimeMay 12th 2016

    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…

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 12th 2016

    Thanks! Fixed now. And thanks for the further comments.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 12th 2016

    Is there a Kapranov-ian reason for the “super” turning up?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 12th 2016

    Sure, the ℤ\mathbb{Z}-graded commutativity of π •(E)\pi_\bullet(E) for ring spectra EE is via the ℤ=π 0(𝕊)\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).

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeMay 12th 2016

    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.

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 13th 2016

    Re #7, and there’s some reason odd degree terms don’t appear for p=2p = 2.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 13th 2016

    Hm, not sure if there is an abstract reason. It comes out of the explicit computation,which gives that only for p>2p \gt 2 there are these odd degree generators τ i\tau_i in the dual Steenrod algebra (here).

    • CommentRowNumber11.
    • CommentAuthorDylan Wilson
    • CommentTimeMay 13th 2016

    @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.

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 13th 2016

    I see. Thanks.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMay 13th 2016

    Er, sorry. A glitch due to that “polynomial algebra” versus “exterior algebra” terminology. I should add a note to the entry…

    • CommentRowNumber14.
    • CommentAuthorDylan Wilson
    • CommentTimeMay 13th 2016

    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.

    • CommentRowNumber15.
    • CommentAuthorDylan Wilson
    • CommentTimeMay 13th 2016

    (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.)

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeMay 13th 2016

    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?

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMay 13th 2016

    Okay, I have edited accordingly, see here. Is there a citable source that puts it in this nice way?

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2020

    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.

    diff, v39, current

  19. The second relation on odd-primary Steenrod operations, those of the form P^i\Beta P^j, holds for all 0<i \leq pj, not 0<i<pj. (See Steenrod–Epstein 1962, Ch6 for eg).

    MPO

    diff, v41, current

  20. The second relation on odd-primary Steenrod operations, those of the form P^i\Beta P^j, holds for all 0<i \leq pj, not 0<i<pj. (See Steenrod–Epstein 1962, Ch6 for eg).

    MPO

    diff, v41, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeDec 16th 2022

    added pointer to:

    diff, v43, current

    • CommentRowNumber22.
    • CommentAuthorAli Caglayan
    • CommentTimeMar 28th 2024
    Why does the page say "under construction"? Isn't that true of most articles on the nlab?
    • CommentRowNumber23.
    • CommentAuthorJ-B Vienney
    • CommentTimeMar 28th 2024

    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.

1 to 23 of 23

  1.