On close fields and the local Langlands correspondence

We prove that Fargues-Scholze’s semisimplified local Langlands correspondence (for quasisplit groups) with ${\bar{\mathbb{F}}}_{\mathrm{\ell }}$-coefficients is compatible with Deligne and Kazhdan’s philosophy of close fields. From this, we deduce that the same holds with ${\bar{\mathbb{Q}}}_{\mathrm{\ell }}$-coefficients after restricting to wild inertia, addressing questions of Gan-Harris-Sawin and Scholze. The proof involves constructing a moduli space of nonarchimedean local fields and then extending Fargues-Scholze’s work to this context.