@ArunDebray For the record, the group is Z/4. What you have to show is that if you take the class η hitting the top cell and multiply by two, you get η² on the bottom cell. If we write ι: S¹ → RP² for the inclusion of the bottom cell, then the class η on the top cell is the Toda bracket <η, 2, ι>. So twice this is 2 <η, 2, ι> = <2, η, 2> ι = η² ι, as desired. The other extensions follow from Bott periodicity. You can perform this calculation either in Ext or in homotopy.

Here is a computer-calculated Adams chart: spectralsequences.github.io/rust_webserver/…

Hello. I'm looking at Milnes Etale Cohomology, and I'm slightly confused about his notion of twisted forms. He defines a twisted form $Y'\to X$ of $Y\to X$ for, say, an etale cover $\{U_i\to X\}$ to be $Y'\to X$ such that there are isomorphisms $\varphi_i:Y\times_X U_i\to Y'\times_X U_i$ for every $i\in I$. This is fine. He then observes that this induces an automorphism $\varphi_j^{-1}\circ\varphi_i$ of $Y\times_X U_i\times_X U_j$ where (him and) I have abused notation very slightly by...

writing $\varphi_i$ for the induced morphism on $Y\times_X U_i\times_X U_j$. Then it's clear that such a twisted form induces a $1$-cocycle for the sheaf associated to the presheaf $Aut(Y):U\mapsto Aut_U(Y\times_X U)$. Now the way he writes this section, he makes it seem like this goes through the same for twisted forms of, say, $\mathcal{O}_X$-modules on $X$.

pseudo-redacted

Wait, sorry, that last paragraph (now pseudo-redacted, i.e. still viable by history) isn't quite what I mean to write

What sheaf should play the part of $Aut(Y)$ in the case that we want to consider twisted forms of a sheaf of $\mathcal{O}_X$-modules on $X$?

I'm guessing we would ask for a twisted form $\mathcal{F}'$ of $\mathcal{F}$ that there is a family of isomorphisms $\varphi_i^*\mathcal{F}\cong \varphi_i^*\mathcal{F}'$.

Btw, the relevant page is 134

@GaloisintheField Well, if $\mathcal{F}$ is the sheaf of $\mathcal{O}_X$-modules, the relevant sheaf of groups is $\mathrm{Aut}_{\mathcal{O}_X}(\mathcal{F})$ (sending every $f:U\to X$ to the group of automorphisms of $f^*\mathcal{F}$ as an $\mathcal{O}_U$-module)

In general this works for any stack, although if you're just starting studying étale cohomology odds are that you haven't seen the definition of stack yet

What works for any stack?

This procedure: you can always twist an element $x$ of any stack $\mathfrak{X}$ by torsors over the automorphism sheaf $\mathrm{Aut}_{\mathfrak{X}}(x)$

Sorry, but what does 'an element of a stack' mean? If I have a stack $\mathfrak{X}:F\to C/X$, ($(C/X)_E$ a site), is an element of $\mathfrak{X}$ an object in some $\mathfrak{X}(U\to X)$ (denoting the category over $U\to X$)?

(I just haven't seen that terminology, sorry, and can't find it on google)

Yes, exactly

Okay, thank you, I'll think about this.