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Dedalus
Dedalus
7:03 AM
@dhy The paper is arxiv.org/abs/math/0303382 and it is Lemma 4. I can fix this, so it is not that terrible, but I don't see the proof now as correct. The fix I have is to base change to the algebraic closure, where you easily can find a section not vanishing (since a vector space over an infinite field is not a finite union of proper subspaces). Then, we take this section s and the product of all its Galois conjugates and then descend
 
Dedalus
Dedalus
7:17 AM
I guess this is kind of like Noether normalization for finite fields. If you are over an infinite field, you can project to something linear to get a finite map, but for finite fields you add non-linear terms.
 
 
7 hours later…
dhy
dhy
2:33 PM
@Dedalus Ah OK, I agree the claim about deg(Z)+1 linearly independent sections sufficing is nonsense. Another way to fix this is to take a large enough so that H^1(L^a tensor I)=0, with I the ideal sheaf of Z
 
 
1 hour later…
Liam Keenan
Liam Keenan
3:36 PM
@AaronMazel-Gee You're very welcome! Thanks for putting the TR-trace paper back on my radar; I've been meaning to take a closer look at that for some time.
If you'll permit me to work along the lines of Nikolaus-Scholze, a "cyclotomic graded spectrum" (so as not to refer to a graded object in CycSp) is a graded spectrum with (naive) S^1-action equipped with S^1-equivariant maps $X_{\bullet} \rightarrow X_{p \bullet}^{tC_{p}}$ for all primes p. The trivial square-zero example comes from the cyclic bar construction (using Day convolution) of A+M, where A sits in degree 0 and M sits in degree 1.
 
Dedalus
Dedalus
4:06 PM
@dhy yes, I agree that one can use ampleness to make the evaluation surjective (i.e. twisting by a power of L). Thanks for confirming!
This it is easy to fix, but it is still surprising it slipped through the refereeing process...
 
Jonathan Beardsley
Jonathan Beardsley
5:01 PM
@davik I'm thinking maybe I misunderstood. Were you asking whether HH*(A) is an 𝔼₂-module over A, or whether A is an 𝔼₂-module over HH*(A)? Anyhow, looks like maybe you've got it sorted out with other people.
Rereading, I see that your question is about whether or not A is an 𝔼₂-module over HH*(A).
 
davik
davik
5:16 PM
The latter, but I don't think it's sorted out? At least I still have no idea... I think Aaron thinks it is, but I would love to see a proof of this
 
Jonathan Beardsley
Jonathan Beardsley
Right, yeah I see that now. Good question! I don't think I know.
Sorry for the unhelpful distraction :(
 
Dylan Wilson
Dylan Wilson
5:28 PM
@davik is this even true "one level down"? Like, if A is some pointed module, then End(A) acts on A... but I don't think A is a bimodule over End(A), right? That makes me feel like the 'one level higher' thing also won't be true
(by 'one level down' I mean to take E_0-centers of E_0-algebras and ask about E_1-modules; so that translates to taking endomorphism objects and asking about bimodules)
 
davik
davik
6:02 PM
thanks @DylanWilson! yeah it doesn't happen a level down so I guess it shouldn't be true here too (this is kind of what I expected but I wanted to make sure)
 

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