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 link the required proofs in the wider programme and identify providers outside this selected offline 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; two subsections of lesson 7 by Claude Opus 5.5 (Anthropic). All lessons and the collection: CC0 1.0. The Stacks Project is cited for the results it proves. Navigation and locator integration: OpenAI Codex, GPT-6 Astra, Ultra, 3 October 2026.