Formalization in Lean of faithfully flat descent of projectivity

Abstract

We formalize in Lean the following foundational result in commutative algebra: Let $R \to S$ be a faithfully flat map of (not necessarily noetherian) commutative rings, and let $P$ be an arbitrary $R$-module. Then $P$ is projective over $R$ if and only if $S\otimes_R P$ is projective over $S$. This formalizes and verifies Perry's fix of a subtle gap in the classical work of Raynaud and Gruson, a result which is a key ingredient in the study of finitistic dimension of commutative noetherian rings.