@Denis-CharlesCisinski this looks great, thanks!

I'm confused. If I work over k=F_2 and conside P^3_k and the standard line bundle O(1), can I find a global section NOT vanishing at the points [1:0:0:0],[0:1:0:0], [1:1:0:0]? I have talked to the author of a paper I am reading and he says that it can be done, bur I don't see how. Like, if I write a section as ax_0+bx_1+cx_2+dx_3

And evaluare it on the first point I get that a=1, and then on the second point b=1. But on the third I get that it vanishes.

The author claims I can choose c=d=1 and b=a=0, but is this correct?

Does anyone know if the Handbook of Homotopy Theory will ever cost less than $240? I really want to own it in hard copy, but I can't justify spending that much....

A spectral sequence for Hallowe'en

@davik well, there's certainly some sort of action of $HH^\bullet(A)$ on $A$, since after all it is just the endomorphism object. is your question whether this is this is specifically an $E_2$-action? i would guess the answer is yes, considering $A \in Mod_k$ and not in $Alg_k$ (i.e. it's just an action on the underlying module of $A$).

@LiamKeenan oh, thanks for reminding me about that paper! i had forgotten about it, apologies for the omission. what is a "graded" cyclotomic structure, in either/both definitions of cyclotomic spectra?

@Dedalus i agree this is impossible; which paper/claim is this?

$HH^{\bullet}(A)$ acts on QCoh(A) (or Perf(A) if you perfer) and by taking Hochschild homology of everything in sight (using that Hoschild homology is symmetric monoidal from stable categories to spectra) gives an action of $HH_{\bullet}(HH^{\bullet}(A)$ on $HH_{\bullet}(A)$ (as a left module say); then this means that $HH_{\bullet}(A)$ is a $HH^{\bullet}(A)$ E2-module as $HH_{\bullet}(HH^{\bullet}(A)$ is the enveloping algebra)

@AaronMazel-Gee Yeah I'm actually really confused about this. So I actually think the action is just E1 and not E2 (where E2 structure on $HH^{\bullet}(A)$ is the usual one on hochschild cohomology). (I certainly consider actions in Mod_k not Alg_k). So a little background is I learned that Hochschild homology supports a E2 action which is produced as follows.

shoot my two messages got posted out of order, hope the meaning is clear

So ok I don't know but I would be really interested to see a proof that the action is a E2 module structure, because I guess I don't see how the enveloping algebra should act on A