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6. Hopkins’ Chromatic Splitting Conjecture


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      Problem 6.1.

      What are the iterated localizations $$L_{K(t_0)}L_{K(t_1)} \dots L_{K(t_n)} \mathbf{S}$$ where $t_0 < t_1 < \dots < t_n$?
          The chromatic splitting predicts an answer. For example is it true that $$L_{K(n-1)} L_{K(n)} \mathbf{S} \cong L_{K(n-1)}(\mathbf{S} \oplus \Sigma^{-1}\mathbf{S})?$$ Is there a way to understand this using the two tower isomorphism? More specifically by upgrading Drinfeld symmetric space with a “Drinfeld upper half Grassmannian”, i.e. a general flag variety cut out its $\mathbf{Q}_p$-rational hyperplanes $\mathscr{H}$ with the action of a parabolic subgroup on $\mathscr{H}$. What is this in the $n=2$ case? May be related to repeated deformations of $p$-divisible groups with prescribed etale parts.


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          Problem 6.2.

          Is $\pi_0(L_{K(n-1)} L_{K(n)} E_n)$ a Huber ring? (Probably not.) Is it a solid ring?


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              Problem 6.3.

              Is it true that $$L_{K(n-1)} (E^{h\mathbf{G}_n^1}) \cong L_{K(n-1)} \mathbf{S}?$$


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                  Problem 6.4.

                  In char $p$, $$H^*(\mathbf{G}_n^1, E_{n, *}/(p, u_1, \dots, u_{n-2}))[u_{n-1}^{\pm 1}] \cong H^*(\mathbf{G}_{n-1}, k[u^{\pm 1}]).$$
                    1. Remark. There is work by Takeshi Torii on this topic, for example, https://link.springer.com/article/10.1007/s00209-009-0605-9 https://people.math.rochester.edu/faculty/doug/otherpapers/torii1.pdf

                          Cite this as: AimPL: Chromatic homotopy theory and p-adic geometry, available at http://aimpl.org/chromhomotopy.