Automorphisms of categories of schemes

Abstract

Given two schemes $S$ and $S^{\prime }$, we prove that every equivalence between ${\mathbf{Sch}}_S$ and ${\mathbf{Sch}}_{S^{\prime }}$ comes from a unique isomorphism between $S$ and $S^{\prime }$. This eliminates all Noetherian and finite type hypotheses from a result of Mochizuki and fully answers a programme set out by Brandenburg in a series of questions on MathOverflow in 2011.

17 pages