On Lurie's theorem and applications

Abstract

Lurie's theorem states that there exists a sheaf of ring spectra on the site of formally \'etale Deligne--Mumford stacks over the moduli stack of $p$-divisible groups of height $n$, which agrees with the classical Landweber exact functor theorem (LEFT) on affines. In other words, this theorem is a global, higher categorical refinement of the LEFT. In recent work, Lurie has introduced many of the ingredients one needs to prove this theorem, and in this article, we gather these ingredients together and prove Lurie's theorem. Applications of this theorem to Lubin--Tate theories, topological modular and automorphism forms, and Adams operations are also discussed.