# Automorphisms of categories of schemes

## TL;DR

This paper proves that any categorical equivalence between schemes over different bases arises from a unique isomorphism of those bases, removing previous restrictions and fully resolving a longstanding mathematical question.

## Contribution

It establishes a complete characterization of equivalences between categories of schemes, showing they are induced by base isomorphisms without Noetherian or finite type assumptions.

## Key findings

- Equivalence of scheme categories implies base scheme isomorphism.
- Removes Noetherian and finite type restrictions from previous results.
- Answers a question posed by Brandenburg in 2011.

## Abstract

Given two schemes $S$ and $S'$, we prove that every equivalence between $\mathbf{Sch}_S$ and $\mathbf{Sch}_{S'}$ comes from a unique isomorphism between $S$ and $S'$. 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.