EGA

Community translation (French to English) of A. Grothendieck and J. Dieudonné’s Éléments de géométrie algébrique. Désolé, Grothendieck.

For discussion regarding this project, visit #ega:matrix.org!

To compile, make book, make pdfs, or make all.

PDFs

The PDFs are automatically generated whenever any changes are made, and you can always find the latest version on the releases page.

Current status

Summary:

• EGA I preliminaries
• EGA I
• EGA II
• EGA III preliminaries
• EGA III
• EGA IV preliminaries
• EGA IV

Details: Here is the current status of the translation, along with who is currently working on/has worked on which sections. (Page counts and percentages are usually just rough estimates).

Introduction (EGA I) (proofread by @thosgood)

Preliminaries (EGA 0I) _(proofread by @thosgood)

1. Rings of fractions (@ryankeleti)
2. Irreducible spaces. Noetherian spaces (@ryankeleti)
3. Supplement on sheaves (@ryankeleti)
4. Ringed spaces (@ryankeleti)
5. Quasi-coherent sheaves and coherent sheaves (@ryankeleti)
6. Flatness (@ryankeleti)
7. Adic rings (@ryankeleti)

Preliminaries (EGA 0_III)

• 8. Representable functors (@ryankeleti)
• 9. Constructible sets (@ryankeleti)
• 10. Supplement on flat modules (@thosgood)
• 11. Supplement on homological algebra (@ryankeleti, @wayneyang108)
• 12. Supplement on sheaf cohomology (~25 pages)
• 13. Projective limits in homological algebra (~10 pages)

Preliminaries (EGA 0_IV)

• (14-ε). Summary (@thosgood)
• 14. Combinatorial dimension of a topological space (@thosgood)
• 15. M-regular and F-regular sequences (~10 pages)
• 16. Dimension and depth of Noetherian local rings (~15 pages)
• 17. Regular rings (~15 pages)
• 18. Supplement on extensions of algebras (~20 pages)
• 19. Formally smooth algebras and Cohen rings (~45 pages)
• 20. Derivations and differentials (~35 pages)
• 21. Differentials in rings of characteristic p (~30 pages)
• 22. Differential criteria for smoothness and regularity (~30 pages)
• 23. Japanese rings (~5 pages)

The language of schemes (EGA I) (proofread by @thosgood)

1. Summary
2. Affine schemes (@ryankeleti)
3. Preschemes and their morphisms (@thosgood)
4. Products of preschemes (@thosgood, @ryankeleti)
5. Subpreschemes and immersions (@ryankeleti)
6. Reduced preschemes; separation condition (@thosgood)
7. Finiteness conditions (@thosgood)
8. Rational maps (@thosgood)
9. Chevalley schemes (@thosgood)
10. Supplement on quasi-coherent sheaves (@thosgood)
11. Formal schemes (@thosgood, @ryankeleti)

Elementary global study of some classes of morphisms (EGA II)

• 0. Summary (@ryankeleti / proofread by @thosgood)
• 1. Affine morphisms (@ryankeleti)
• 2. Homogeneous prime spectra (@thosgood)
• 3. Homogeneous prime spectrum of a sheaf of graded algebras (@thosgood)
• 4. Projective bundles; Ample sheaves (@thosgood)
• 5. Quasi-affine morphisms; quasi-projective morphisms; proper morphisms; projective morphisms (@thosgood)
• 6. Integral morphisms and finite morphisms (@thosgood)
• 7. Valuative criteria (@thosgood)
• 8. Blowup schemes; based cones; projective closure (@thosgood)
• Errata and addenda (list 1) (@thosgood)

Cohomological study of coherent sheaves (EGA III)

• 0. Summary (@thosgood / proofread by @thosgood)
• 1. Cohomology of affine schemes (@ryankeleti)
• 2. Cohomological study of projective morphisms (~15 pages) (@thosgood)
• 3. Finiteness theorem for proper morphisms (@ryankeleti)
• 4. The fundamental theorem of proper morphisms. Applications (~30 pages) (@calebji0)
• 5. An existence theorem for coherent algebraic sheaves (~10 pages)
• 6. Local and global Tor functors; Künneth formula (~40 pages)
• 7. Base change for homological functors of sheaves of modules (~30 pages)
• 8. The duality theorem for projective bundles
• 9. Relative cohomology and local cohomology; local duality
• 10. Relations between projective cohomology and local cohomology. Formal completion technique along a divisor
• 11. Global and local Picard groups
• Errata and addenda (list 2)

Local study of schemes and their morphisms (EGA IV)

• 0. Summary (@thosgood)
• 1. Relative finiteness conditions. Constructible sets of preschemes (~30 pages)
• 2. Base change and flatness (~30 pages)
• 3. Associated prime cycles and primary decomposition (~15 pages)
• 4. Change of base field for algebraic preschemes (~35 pages)
• 5. Dimension, depth, and regularity of locally Noetherian preschemes (~50 pages)
• 6. Flat morphisms of locally Noetherian preschemes (~50 pages)
• 7. Relations between a local Noetherian ring and its completion. Excellent rings (~40 pages)
• 8. Projective limits of preschemes (~50 pages)
• 9. Constructible properties (~40 pages)
• 10. Jacobson preschemes (~20 pages)
• 11. Topological properties of finitely presented flat morphisms. Flatness criteria (~60 pages)
• 12. Fibres of finitely presented flat morphisms (~15 pages)
• 13. Equidimensional morphisms (~15 pages)
• 14. Universally open morphisms (~25 pages)
• 15. Fibres of a universally open morphism (~25 pages)
• 16. Differential invariants. Differentially smooth morphisms (@solov-t) (~50 pages)
• 17. Smooth morphisms, unramified morphisms, and étale morphisms (@tholzschuh) (~55 pages)
• 18. Supplement on étale morphisms. Henselian local rings and strictly local rings (~75 pages)
• 19. Regular immersions and normal flatness (~40 pages)
• 20. Meromorphic functions and pseudo-morphisms (~30 pages)
• 21. Divisors (~75 pages)
• Errata and addenda (list 3)

To-do (incomplete)

• check errata in EGA III and onwards
• more consistent use of Proposition/Theorem/Definition/whatever in front of \srefs
• lists of notation?
• terminology indices?
• hyperref links to equations (e.g. 3.3.9.1)
• check for any \unsure commands in the source
• check that all \hyperref references are found
• when this is finished (?!) should we assign a DOI?
• web version? (work in progress!)
• more detailed style guidelines? (e.g. language, sentence construction, etc.)
• link (M), (G), (T) references