Timeline for Solving polynomial equations in spectra?

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Jan 22, 2021 at 16:43	comment	added	Tim Campion		@NicholasKuhn I figured as much. But with just the assumption that $Sq^{2^k}(x) \neq 0$ (even assuming that $Sq^{2^{\geq k+1}}$ acts trivially on $H^\ast(X)$ as well), I couldn't get the Cartan argument to work to show that $Sq^{2^{k+1}}(x \otimes x) \neq 0$. For instance, it might be the case that $Sq^{2^k + 1}(x), Sq^{2^k - 1}(x) \neq 0$ as well, and then there could be cancellation in the Cartan formula. I tried showing that, for $l \geq 1$, $Sq^{2^k + l}$ is in the 2-sided ideal of $\mathcal A^\ast$ generated by $Sq^{2^{k+1}},Sq^{2^{k+2}},\dots$, but I got lost in Adem relations.
Jan 22, 2021 at 16:15	comment	added	Nicholas Kuhn		Also, I was focused on using just the pth power operations as they generate all operations.
Jan 21, 2021 at 22:51	comment	added	Nicholas Kuhn		Yep, I meant a Cartan formula argument.
Jan 21, 2021 at 21:50	history	edited	Tim Campion	CC BY-SA 4.0	added 24 characters in body
S Jan 21, 2021 at 21:43	history	answered	Tim Campion	CC BY-SA 4.0
S Jan 21, 2021 at 21:43	history	made wiki			Post Made Community Wiki by Tim Campion