An elementary and constructive proof of Grothendieck's generic freeness lemma

TL;DR

This paper provides a new, direct, and fully constructive proof of Grothendieck's generic freeness lemma, avoiding traditional reduction steps and classical logic assumptions.

Contribution

It introduces a novel, constructive proof method based on topos-theoretic techniques, differing from prior approaches that relied on reduction and classical logic.

Findings

Proof is fully constructive and does not use the law of excluded middle.

Avoids reduction steps common in previous proofs.

Based on a topos-theoretic approach.

Abstract

We present a new and direct proof of Grothendieck's generic freeness lemma in its general form. Unlike the previously published proofs, it does not proceed in a series of reduction steps and is fully constructive, not using the axiom of choice or even the law of excluded middle. It was found by unwinding the result of a general topos-theoretic technique.