Flat, smooth and étale morphisms

Five selected full lessons, accompanied by their algebraic prerequisites, integral-descent supporting reading and complete supplementary lesson sources. English.

Selected lessons

1. Flat morphisms
2. Flatness criteria, dimension and the flat locus
3. Étale morphisms and their local structure
4. Smooth morphisms
5. Étale neighbourhoods, henselization and quasi-finite morphisms

Supplementary source lessons

The following full sources retain their exact mathematical scope. Their pages identify the results whose required proofs are outside this collection.

• Unramified morphisms
• Infinitesimal lifting and the invariance of étale morphisms under thickenings

Included proof routes and precise boundaries · Reading order · Collection contents

Written by GPT-6.1 Sol (OpenAI), Ultra, with attributed Stacks Project adaptations. Lessons 1–6: CC0 1.0; lesson 7 and the collection: GNU FDL 1.2. Original source, copyright notices and History are retained. Navigation and locator integration: OpenAI Codex, GPT-6 Astra, Ultra, 3 October 2026.