The Zariski--Nagata purity theorem states that the Galois group of a scheme is invariant under deletion of a closed subscheme of codimension at least \$2\$. A conjecture of Mathew suggests that a similar phenomenon occurs for chromatically-localized \$\mathbb{E}_\infty\$-rings, namely that the map \$L_nR\to L_1R\$ is an equivalence on Galois groups. If \$R\$ is Landweber exact, this follows from the fact that \$L_1\$-localization behaves like restriction to the open substack \$\mathcal{M}_\mathrm{fg}^{\leq1}\hookrightarrow\mathcal{M}_\mathrm{fg}\$, whose compliment, defined by the ideal \$(p,v_1)\$, has codimension \$2\$. The case of the \$E_n\$-local sphere then follows from the fact that the map \$L_n\mathbb{S}\to E_n\$ is descendable.
In this talk, I'll introduce Mathew's theory of Galois groups of \$2\$-rings and descendibility, and discuss some thoughts I've been having about how techniques from the proof of the chromatic Nullstellensatz may apply to this conjecture. This will probably include some high-level discussion of the spherical Witt vector / tilt adjunction, \$T(n)\$-local homotopy, and the process of constructing maps to Lubin--Tate theories which detect nilpotence.