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I wanted to collect some of the stuff recently added to a bunch of chromatic entries in a way that forms an at least semi-coherent story, so I made an entry
This is built mostly from copy-and-pasting stuff that I had added to dedicated entries, equipped with a bit of glue to make it stick together and form a story.
(Special thanks to Marc Hoyois for general discussion and in particular for working on the text on the Lurie spectral sequence.)
I want to further fine-tune this. But not tonight.
Am brushing up the section Canonical cosimplicial resolution of E-infinity algebra.
It used to say there that it is the $E_2$-page of the spectral sequence of a simplicial stable homotopy type which is given by the Moore complexes of the underlying simplicial objects of homotopy groups. I originally took this from Dylan Wilson’s prop. 1.2.3 But it’s instead the $E_1$-page, isn’t it.
What you have should be the E_2-page. The E_1 page is pi_\astTot(cosk_\astY) with a differential coming from alternating sum. Taking homology gives the E_2 page, as stated.
Also it probably says this elsewhere, but convergence of said spectral sequence is delicate in this “cosimplicial” case as opposed to the simplicial case.
Oh, sure. I am being stupid, ignored the $H_p$.
I plan to write up a more detailed discussion of the cosimplicial spectral sequence by next Tuesday. So maybe I will come back to you again with more stupid questions.
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