aboutsummaryrefslogtreecommitdiffhomepage
diff options
context:
space:
mode:
authorGravatar Tim Hosgood <thosgood@users.noreply.github.com> 2021-07-29 00:39:10 +0100
committerGravatar GitHub <noreply@github.com> 2021-07-29 00:39:10 +0100
commit44565f097e5e12e184cd06a6a64faa2e32c5eb6a (patch)
treeed9478c40bfd303115598b861d086f5b57ff58db
parent21c333e1af020733e4d7af5a636852c5ec0c6f45 (diff)
parent1b06dd70d1310b8bb86d72589943db258de101f3 (diff)
downloadega-44565f097e5e12e184cd06a6a64faa2e32c5eb6a.tar.gz
ega-44565f097e5e12e184cd06a6a64faa2e32c5eb6a.zip
Merge branch 'master' into ega2-2
-rw-r--r--README.md223
-rw-r--r--ega1.tex20
-rw-r--r--ega2.tex16
-rw-r--r--ega2/ega2-1.tex2
-rw-r--r--ega2/ega2-2.tex2
-rw-r--r--ega2/ega2-4.tex2
-rw-r--r--ega3.tex14
-rw-r--r--ega4.tex42
-rw-r--r--ega4/ega4-16.tex10
-rw-r--r--what.tex4
10 files changed, 178 insertions, 157 deletions
diff --git a/README.md b/README.md
index 94c6f1a..886073f 100644
--- a/README.md
+++ b/README.md
@@ -1,118 +1,139 @@
# EGA
Community translation (French to English) of A. Grothendieck's EGA.
-S’il-vous plaît pardonnez-nous, Grothendieck.
+On est désolé, Grothendieck.
-View online [here](https://ega.fppf.site/).
+~~View online [here](https://ega.fppf.site/).~~ (the web version is not currently kept up to date; please refer to the PDF versions linked below)
For discussion regarding this project, visit [#ega:matrix.org](https://matrix.to/#/#ega:matrix.org)!
To compile, `make book`, `make pdfs`, or `make all`.
+
## PDFs
-There is the [**full document**](https://fppf.site/ega/book-auto.pdf), or individual sections can be downloaded separately:
-- [What this is](https://fppf.site/ega/what-auto.pdf)
-- [Introduction](https://fppf.site/ega/intro-auto.pdf)
-- [EGA 0](https://fppf.site/ega/ega0-auto.pdf)
-- [EGA I](https://fppf.site/ega/ega1-auto.pdf)
-- [EGA II](https://fppf.site/ega/ega2-auto.pdf)
-- [EGA III](https://fppf.site/ega/ega3-auto.pdf)
-- [EGA IV](https://fppf.site/ega/ega4-auto.pdf)
-- [References](https://fppf.site/ega/ref-auto.pdf)
+Individual volumes can be downloaded separately:
+
+- [What this is](https://labs.thosgood.com/ega/what-auto.pdf)
+- [Introduction](https://labs.thosgood.com/ega/intro-auto.pdf)
+- [EGA 0](https://labs.thosgood.com/ega/ega0-auto.pdf)
+- [EGA I](https://labs.thosgood.com/ega/ega1-auto.pdf)
+- [EGA II](https://labs.thosgood.com/ega/ega2-auto.pdf)
+- [EGA III](https://labs.thosgood.com/ega/ega3-auto.pdf)
+- [EGA IV](https://labs.thosgood.com/ega/ega4-auto.pdf)
+- [References](https://labs.thosgood.com/ega/ref-auto.pdf)
+
+Alternatively, the full document can be downloaded:
+
+- [Book](https://labs.thosgood.com/ega/book-auto.pdf)
+
+(All the PDFs are auto-compliled every hour if any changes have been made since the last auto-compile, so will always be up to date with the latest commit).
-All the PDFs are auto-compliled every hour if any changes have been made since the last auto-compile, so will always be up to date with the latest commit.
## Current status
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)_ ![introstatus](https://img.shields.io/badge/-5%2F5-brightgreen)
-- **Preliminaries (EGA 0_I)** _(proofread by @thosgood)_ ![EGA0(I)status](https://img.shields.io/badge/-70%2F70-brightgreen)
- + [x] 1. Rings of fractions _(@ryankeleti)_
- + [x] 2. Irreducible spaces. Noetherian spaces _(@ryankeleti)_
- + [x] 3. Supplement on sheaves _(@ryankeleti)_
- + [x] 4. Ringed spaces _(@ryankeleti)_
- + [x] 5. Quasi-coherent sheaves and coherent sheaves _(@ryankeleti)_
- + [x] 6. Flatness _(@ryankeleti)_
- + [x] 7. Adic rings _(@ryankeleti)_
-- **Preliminaries (EGA 0_III)** ![EGA0(III)status](https://img.shields.io/badge/-20%2F75-orange)
- + [x] 8. Representable functors _(@ryankeleti)_
- + [x] 9. Constructible sets _(@ryankeleti)_
- + [x] 10. Supplement on flat modules _(@thosgood)_
- + [ ] 11. Supplement on homological algebra _(@ryankeleti)_
- + [ ] 12. Supplement on sheaf cohomology (~25 pages)
- + [ ] 13. Projective limits in homological algebra (~10 pages)
-- **Preliminaries (EGA 0_IV)** ![EGA0(IV)status](https://img.shields.io/badge/-5%2F215-red)
- + [x] (14-ε). Summary _(@thosgood)_
- + [x] 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)_ ![EGAIstatus](https://img.shields.io/badge/-136%2F136-brightgreen)
- + [x] 0. Summary
- + [x] 1. Affine schemes _(@ryankeleti)_
- + [x] 2. Preschemes and their morphisms _(@thosgood)_
- + [x] 3. Products of preschemes _(@thosgood, @ryankeleti)_
- + [x] 4. Subpreschemes and immersions _(@ryankeleti)_
- + [x] 5. Reduced preschemes; separation condition _(@thosgood)_
- + [x] 6. Finiteness conditions _(@thosgood)_
- + [x] 7. Rational maps _(@thosgood)_
- + [x] 8. Chevalley schemes _(@thosgood)_
- + [x] 9. Supplement on quasi-coherent sheaves _(@thosgood)_
- + [x] 10. Formal schemes _(@thosgood, @ryankeleti)_
-- **Elementary global study of some classes of morphisms (EGA II)** ![EGAIIstatus](https://img.shields.io/badge/-130%2F205-yellow)
- + [x] 0. Summary _(@ryankeleti / proofread by @thosgood)_
- + [x] 1. Affine morphisms _(@ryankeleti)_
- + [x] 2. Homogeneous prime spectra _(@thosgood)_
- + [ ] 3. Homogeneous prime spectrum of a sheaf of graded algebras (~20 pages)
- + [x] 4. Projective bundles; Ample sheaves _(@thosgood)_
- + [x] 5. Quasi-affine morphisms; quasi-projective morphisms; proper morphisms; projective morphisms _(@thosgood)_
- + [ ] 6. Integral morphisms and finite morphisms (~25 pages)
- + [x] 7. Valuative criteria _(@thosgood)_
- + [x] 8. Blowup schemes; based cones; projective closure _(@thosgood)_
- + [x] _Errata and addenda (list 1)_
-- **Cohomological study of coherent sheaves (EGA III)** ![EGAIIIstatus](https://img.shields.io/badge/-20%2F160-red)
- + [x] 0. Summary _(@thosgood / proofread by @thosgood)_
- + [x] 1. Cohomology of affine schemes _(@ryankeleti)_
- + [ ] 2. Cohomological study of projective morphisms (~15 pages)
- + [x] 3. Finiteness theorem for proper morphisms _(@ryankeleti)_
- + [ ] 4. The fundamental theorem of proper morphisms. Applications (~30 pages)
- + [ ] 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)
- + [x] 8. ~~The duality theorem for projective bundles~~
- + [x] 9. ~~Relative cohomology and local cohomology; local duality~~
- + [x] 10. ~~Relations between projective cohomology and local cohomology. Formal completion technique along a divisor~~
- + [x] 11. ~~Global and local Picard groups~~
- + [ ] _Errata and addenda (list 2)_
-- **Local study of schemes and their morphisms (EGA IV)** ![EGAIVstatus](https://img.shields.io/badge/-53%2F825-red)
- + [x] 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)
- + [x] 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)_
+### Introduction (EGA I) _(proofread by @thosgood)_ ![introstatus](https://img.shields.io/badge/-5%2F5-brightgreen)
+
+### Preliminaries (EGA 0_I) _(proofread by @thosgood)_ ![EGA0(I)status](https://img.shields.io/badge/-70%2F70-brightgreen)
+
+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) ![EGA0(III)status](https://img.shields.io/badge/-20%2F75-orange)
+
++ [x] 8. Representable functors _(@ryankeleti)_
++ [x] 9. Constructible sets _(@ryankeleti)_
++ [x] 10. Supplement on flat modules _(@thosgood)_
++ [ ] 11. Supplement on homological algebra _(@ryankeleti)_
++ [ ] 12. Supplement on sheaf cohomology (~25 pages)
++ [ ] 13. Projective limits in homological algebra (~10 pages)
+
+### Preliminaries (EGA 0_IV) ![EGA0(IV)status](https://img.shields.io/badge/-5%2F215-red)
+
++ [x] (14-ε). Summary _(@thosgood)_
++ [x] 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)_ ![EGAIstatus](https://img.shields.io/badge/-136%2F136-brightgreen)
+
+0. Summary
+1. Affine schemes _(@ryankeleti)_
+2. Preschemes and their morphisms _(@thosgood)_
+3. Products of preschemes _(@thosgood, @ryankeleti)_
+4. Subpreschemes and immersions _(@ryankeleti)_
+5. Reduced preschemes; separation condition _(@thosgood)_
+6. Finiteness conditions _(@thosgood)_
+7. Rational maps _(@thosgood)_
+8. Chevalley schemes _(@thosgood)_
+9. Supplement on quasi-coherent sheaves _(@thosgood)_
+10. Formal schemes _(@thosgood, @ryankeleti)_
+
+### Elementary global study of some classes of morphisms (EGA II) ![EGAIIstatus](https://img.shields.io/badge/-130%2F205-yellow)
+
++ [x] 0. Summary _(@ryankeleti / proofread by @thosgood)_
++ [x] 1. Affine morphisms _(@ryankeleti)_
++ [x] 2. Homogeneous prime spectra _(@thosgood)_
++ [ ] 3. Homogeneous prime spectrum of a sheaf of graded algebras (~20 pages)
++ [x] 4. Projective bundles; Ample sheaves _(@thosgood)_
++ [x] 5. Quasi-affine morphisms; quasi-projective morphisms; proper morphisms; projective morphisms _(@thosgood)_
++ [ ] 6. Integral morphisms and finite morphisms (~25 pages)
++ [x] 7. Valuative criteria _(@thosgood)_
++ [x] 8. Blowup schemes; based cones; projective closure _(@thosgood)_
++ [x] _Errata and addenda (list 1)_
+
+### Cohomological study of coherent sheaves (EGA III) ![EGAIIIstatus](https://img.shields.io/badge/-20%2F160-red)
+
++ [x] 0. Summary _(@thosgood / proofread by @thosgood)_
++ [x] 1. Cohomology of affine schemes _(@ryankeleti)_
++ [ ] 2. Cohomological study of projective morphisms (~15 pages)
++ [x] 3. Finiteness theorem for proper morphisms _(@ryankeleti)_
++ [ ] 4. The fundamental theorem of proper morphisms. Applications (~30 pages)
++ [ ] 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)
++ [x] 8. ~~The duality theorem for projective bundles~~
++ [x] 9. ~~Relative cohomology and local cohomology; local duality~~
++ [x] 10. ~~Relations between projective cohomology and local cohomology. Formal completion technique along a divisor~~
++ [x] 11. ~~Global and local Picard groups~~
++ [ ] _Errata and addenda (list 2)_
+
+### Local study of schemes and their morphisms (EGA IV) ![EGAIVstatus](https://img.shields.io/badge/-53%2F825-red)
+
++ [x] 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)
++ [x] 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)_
diff --git a/ega1.tex b/ega1.tex
index d6ab08c..e26ffda 100644
--- a/ega1.tex
+++ b/ega1.tex
@@ -12,16 +12,16 @@
\section*{Summary}
\begin{longtable}{ll}
- \textsection\hyperref[section:I.1]{1}. & Affine schemes.\\
- \textsection\hyperref[section:I.2]{2}. & Preschemes and morphisms of preschemes.\\
- \textsection\hyperref[section:I.3]{3}. & Products of preschemes.\\
- \textsection\hyperref[section:I.4]{4}. & Subpreschemes and immersion morphisms.\\
- \textsection\hyperref[section:I.5]{5}. & Reduced preschemes; the separation condition.\\
- \textsection\hyperref[section:I.6]{6}. & Finiteness conditions.\\
- \textsection\hyperref[section:I.7]{7}. & Rational maps.\\
- \textsection\hyperref[section:I.8]{8}. & Chevalley schemes.\\
- \textsection\hyperref[section:I.9]{9}. & Supplement on quasi-coherent sheaves.\\
- \textsection\hyperref[section:I.10]{10}. & Formal schemes.
+ \hyperref[section:I.1]{\textsection1}. & Affine schemes.\\
+ \hyperref[section:I.2]{\textsection2}. & Preschemes and morphisms of preschemes.\\
+ \hyperref[section:I.3]{\textsection3}. & Products of preschemes.\\
+ \hyperref[section:I.4]{\textsection4}. & Subpreschemes and immersion morphisms.\\
+ \hyperref[section:I.5]{\textsection5}. & Reduced preschemes; the separation condition.\\
+ \hyperref[section:I.6]{\textsection6}. & Finiteness conditions.\\
+ \hyperref[section:I.7]{\textsection7}. & Rational maps.\\
+ \hyperref[section:I.8]{\textsection8}. & Chevalley schemes.\\
+ \hyperref[section:I.9]{\textsection9}. & Supplement on quasi-coherent sheaves.\\
+ \hyperref[section:I.10]{\textsection10}. & Formal schemes.
\end{longtable}
\bigskip
diff --git a/ega2.tex b/ega2.tex
index 1754b29..c55abe1 100644
--- a/ega2.tex
+++ b/ega2.tex
@@ -12,14 +12,14 @@
\section*{Summary}
\begin{longtable}{ll}
- \textsection\hyperref[section:II.1]{1}. & Affine morphisms.\\
- \textsection\hyperref[section:II.2]{2}. & Homogeneous prime spectra.\\
- \textsection\hyperref[section:II.3]{3}. & Homogeneous prime spectrum of a sheaf of graded algebras.\\
- \textsection\hyperref[section:II.4]{4}. & Projective bundles; ample sheaves.\\
- \textsection\hyperref[section:II.5]{5}. & Quasi-affine morphisms; quasi-projective morphisms; proper morphisms; projective morphisms.\\
- \textsection\hyperref[section:II.6]{6}. & Integral morphisms and finite morphisms.\\
- \textsection\hyperref[section:II.7]{7}. & Valuative criteria.\\
- \textsection\hyperref[section:II.8]{8}. & Blowup schemes; projective cones; projective closure.\\
+ \hyperref[section:II.1]{\textsection1}. & Affine morphisms.\\
+ \hyperref[section:II.2]{\textsection2}. & Homogeneous prime spectra.\\
+ \hyperref[section:II.3]{\textsection3}. & Homogeneous prime spectrum of a sheaf of graded algebras.\\
+ \hyperref[section:II.4]{\textsection4}. & Projective bundles; ample sheaves.\\
+ \hyperref[section:II.5]{\textsection5}. & Quasi-affine morphisms; quasi-projective morphisms; proper morphisms; projective morphisms.\\
+ \hyperref[section:II.6]{\textsection6}. & Integral morphisms and finite morphisms.\\
+ \hyperref[section:II.7]{\textsection7}. & Valuative criteria.\\
+ \hyperref[section:II.8]{\textsection8}. & Blowup schemes; projective cones; projective closure.\\
\end{longtable}
\bigskip
diff --git a/ega2/ega2-1.tex b/ega2/ega2-1.tex
index dc1d195..123c4d0 100644
--- a/ega2/ega2-1.tex
+++ b/ega2/ega2-1.tex
@@ -109,7 +109,7 @@ Let $f:X\to S$ and $g:Y\to S$ be the structure morphisms.
First, suppose that $S=\Spec(A)$ and $X=\Spec(B)$ are affine; we must prove that for every homomorphism $\omega:f_*(\sh{O}_X)\to g_*(\sh{O}_Y)$ of $\sh{O}_S$-algebras, there exists a unique $S$-morphism $h:Y\to X$ such that $\sh{A}(h)=\omega$.
By definition, for every open $U\subset S$, $\omega$ defines a homomorphism $\omega_U=\Gamma(U,\omega):\Gamma(f^{-1}(U),\sh{O}_X)\to\Gamma(g^{-1}(U),\sh{O}_Y)$ of $\Gamma(U,\sh{O}_S)$-algebras.
In particular, for $U=S$, this gives a homomorphism $\vphi:\Gamma(X,\sh{O}_X)\to\Gamma(Y,\sh{O}_Y)$ of $\Gamma(S,\sh{O}_S)$-algebras, to which corresponds a well-defined $S$-morphism $h:Y\to X$, since $X$ is affine \sref[I]{I.2.2.4}.
-It remains to prove that $\sh{A}(h)=\omega$, or, in other words, that, for every open set $U$ of a basis for $S$, $\omega_U$ coincides with the homomorphism of algebras $\vphi_U$ corresponding to the $S$-morphism $g^{-1}(U)\to f^{-1}(U)$, a restiction of $h$.
+It remains to prove that $\sh{A}(h)=\omega$, or, in other words, that, for every open set $U$ of a basis for $S$, $\omega_U$ coincides with the homomorphism of algebras $\vphi_U$ corresponding to the $S$-morphism $g^{-1}(U)\to f^{-1}(U)$, a restriction of $h$.
We can reduce to the case where $U=D(\lambda)$, with $\lambda\in S$; then, if $f=({}^a\rho,\widetilde{\rho})$, where $\rho:A\to B$ is a ring homomorphism, we have $f^{-1}(U)=D(\mu)$, where $\mu=\rho(\lambda)$, and $\Gamma(f^{-1}(U),\sh{O}_X)$ is the ring of fractions $B_\mu$; the diagram
\[
\xymatrix{
diff --git a/ega2/ega2-2.tex b/ega2/ega2-2.tex
index 102c75a..147ae60 100644
--- a/ega2/ega2-2.tex
+++ b/ega2/ega2-2.tex
@@ -155,7 +155,7 @@ in particular, if $f\in S_d$ ($d>0$), for all $x\in S_{m-nd}$, then the relation
\label{II.2.1.9}
Let $n_0$ be an integer $>0$;
for all $n\geq n_0$, let $\mathfrak{p}_n$ be a subgroup of $S_n$.
-For there to exist a graded prime ideal $\mathfrak{p}$ of $S$ not containing $S_+$ and such that $\mathfrak{p}\cap S_n=\mathfrak{p}_n$ for all $n\geq n_0$, it is necessary and sufficient for the following coniditions to be satisfied:
+For there to exist a graded prime ideal $\mathfrak{p}$ of $S$ not containing $S_+$ and such that $\mathfrak{p}\cap S_n=\mathfrak{p}_n$ for all $n\geq n_0$, it is necessary and sufficient for the following conditions to be satisfied:
\begin{enumerate}
\item[{\rm(1st)}] $S_m\mathfrak{p}_n\subset\mathfrak{p}_{m+n}$ for all $m\geq 0$ and all $n\geq n_0$.
\item[{\rm(2nd)}] For $m\geq n_0$, $n\geq n_0$, $f\in S_m$, $g\in S_n$, the relation $fg\in\mathfrak{p}_{m+n}$ implies $f\in\mathfrak{p}_m$ or $g\in\mathfrak{p}_n$.
diff --git a/ega2/ega2-4.tex b/ega2/ega2-4.tex
index c226e61..f3ee19a 100644
--- a/ega2/ega2-4.tex
+++ b/ega2/ega2-4.tex
@@ -825,7 +825,7 @@ Then it follows from \sref{II.4.5.7} and \sref{II.4.5.8} that
\quad\text{and}\quad
P^+ - P^+ = P
\]
-or, in other words, $P^+\cup\{0\}$ is the set of \emph{positive} elements in $P$ for a \emph{preorder} structure on $P$ that is compatible with its group structure, and is even \emph{archimedian}, by \sref{II.4.5.8}.
+or, in other words, $P^+\cup\{0\}$ is the set of \emph{positive} elements in $P$ for a \emph{preorder} structure on $P$ that is compatible with its group structure, and is even \emph{archimedean}, by \sref{II.4.5.8}.
This is why we sometimes say ``positive sheaf'' instead of ample sheaf, and ``negative sheaf'' for the inverse of an ample sheaf (but we will not use this terminology).
\end{remark}
diff --git a/ega3.tex b/ega3.tex
index 604326f..e480375 100644
--- a/ega3.tex
+++ b/ega3.tex
@@ -15,13 +15,13 @@ build hack
\section*{Summary}
\begin{longtable}{ll}
- \textsection\hyperref[section:III.1]{1}. & Cohomology of affine schemes.\\
- \textsection\hyperref[section:III.2]{2}. & Cohomological study of projective morphisms.\\
- \textsection\hyperref[section:III.3]{3}. & Finiteness theorem for proper morphisms.\\
- \textsection\hyperref[section:III.4]{4}. & The fundamental theorem of proper morphisms. Applications.\\
- \textsection\hyperref[section:III.5]{5}. & An existence theorem for coherent algebraic sheaves.\\
- \textsection\hyperref[section:III.6]{6}. & Local and global Tor functors; K\"unneth formula.\\
- \textsection\hyperref[section:III.7]{7}. & Base change for homological functors of sheaves of modules.\\
+ \hyperref[section:III.1]{\textsection1}. & Cohomology of affine schemes.\\
+ \hyperref[section:III.2]{\textsection2}. & Cohomological study of projective morphisms.\\
+ \hyperref[section:III.3]{\textsection3}. & Finiteness theorem for proper morphisms.\\
+ \hyperref[section:III.4]{\textsection4}. & The fundamental theorem of proper morphisms. Applications.\\
+ \hyperref[section:III.5]{\textsection5}. & An existence theorem for coherent algebraic sheaves.\\
+ \hyperref[section:III.6]{\textsection6}. & Local and global Tor functors; K\"unneth formula.\\
+ \hyperref[section:III.7]{\textsection7}. & Base change for homological functors of sheaves of modules.\\
\textsection8. & The duality theorem for projective bundles\\
\textsection9. & Relative cohomology and local cohomology; local duality\\
diff --git a/ega4.tex b/ega4.tex
index 963f4e9..cbbeeab 100644
--- a/ega4.tex
+++ b/ega4.tex
@@ -16,27 +16,27 @@ build hack
\oldpage[IV-1]{222}
\begin{longtable}{ll}
- \textsection\hyperref[section:IV.1]{1}. & Relative finiteness conditions. Constructible sets in preschemes.\\
- \textsection\hyperref[section:IV.2]{2}. & Base change and flatness.\\
- \textsection\hyperref[section:IV.3]{3}. & Associated prime cycles and primary decomposition.\\
- \textsection\hyperref[section:IV.4]{4}. & Change of base field for algebraic preschemes.\\
- \textsection\hyperref[section:IV.5]{5}. & Dimension, depth, and regularity for locally Noetherian preschemes.\\
- \textsection\hyperref[section:IV.6]{6}. & Flat morphisms of locally Noetherian preschemes.\\
- \textsection\hyperref[section:IV.7]{7}. & Relations between a local Noetherian ring and its completion. Excellent rings.\\
- \textsection\hyperref[section:IV.8]{8}. & Projective limits of preschemes.\\
- \textsection\hyperref[section:IV.9]{9}. & Constructible properties.\\
- \textsection\hyperref[section:IV.10]{10}. & Jacobson preschemes.\\
- \textsection\hyperref[section:IV.11]{11}.\footnote{The order and content of \textsection\textsection11--21 are given only as an indication of what the titles will be, and will possibly be modified before their publication. \emph{[Trans.] This was indeed the case: many of \textsection\textsection11--21 ended up having entirely different titles or content. See \hyperref[section:what-ega4-sections]{here}.}} & Topological properties of finitely presented flat morphisms. Flatness criteria.\\
- \textsection\hyperref[section:IV.12]{12}. & Study of fibres of finitely presented flat morphisms.\\
- \textsection\hyperref[section:IV.13]{13}. & Equidimensional morphisms.\\
- \textsection\hyperref[section:IV.14]{14}. & Universally open morphisms.\\
- \textsection\hyperref[section:IV.15]{15}. & Study of fibres of a universally open morphism.\\
- \textsection\hyperref[section:IV.16]{16}. & Differential invariants. Differentially smooth morphisms.\\
- \textsection\hyperref[section:IV.17]{17}. & Smooth morphisms, unramified (or net) morphisms, and \'etale morphisms.\\
- \textsection\hyperref[section:IV.18]{18}. & Supplement on \'etale morphisms. Henselian local rings and strictly local rings.\\
- \textsection\hyperref[section:IV.19]{19}. & Regular immersions and normal flatness.\\
- \textsection\hyperref[section:IV.20]{20}. & Meromorphic functions and pseudo-morphisms\\
- \textsection\hyperref[section:IV.21]{21}. & Divisors.
+ \hyperref[section:IV.1]{\textsection1}. & Relative finiteness conditions. Constructible sets in preschemes.\\
+ \hyperref[section:IV.2]{\textsection2}. & Base change and flatness.\\
+ \hyperref[section:IV.3]{\textsection3}. & Associated prime cycles and primary decomposition.\\
+ \hyperref[section:IV.4]{\textsection4}. & Change of base field for algebraic preschemes.\\
+ \hyperref[section:IV.5]{\textsection5}. & Dimension, depth, and regularity for locally Noetherian preschemes.\\
+ \hyperref[section:IV.6]{\textsection6}. & Flat morphisms of locally Noetherian preschemes.\\
+ \hyperref[section:IV.7]{\textsection7}. & Relations between a local Noetherian ring and its completion. Excellent rings.\\
+ \hyperref[section:IV.8]{\textsection8}. & Projective limits of preschemes.\\
+ \hyperref[section:IV.9]{\textsection9}. & Constructible properties.\\
+ \hyperref[section:IV.10]{\textsection10}. & Jacobson preschemes.\\
+ \hyperref[section:IV.11]{\textsection11}.\footnote{The order and content of \textsection\textsection11--21 are given only as an indication of what the titles will be, and will possibly be modified before their publication. \emph{[Trans.] This was indeed the case: many of \textsection\textsection11--21 ended up having entirely different titles or content. See \hyperref[section:what-ega4-sections]{here}.}} & Topological properties of finitely presented flat morphisms. Flatness criteria.\\
+ \hyperref[section:IV.12]{\textsection12}. & Study of fibres of finitely presented flat morphisms.\\
+ \hyperref[section:IV.13]{\textsection13}. & Equidimensional morphisms.\\
+ \hyperref[section:IV.14]{\textsection14}. & Universally open morphisms.\\
+ \hyperref[section:IV.15]{\textsection15}. & Study of fibres of a universally open morphism.\\
+ \hyperref[section:IV.16]{\textsection16}. & Differential invariants. Differentially smooth morphisms.\\
+ \hyperref[section:IV.17]{\textsection17}. & Smooth morphisms, unramified (or net) morphisms, and \'etale morphisms.\\
+ \hyperref[section:IV.18]{\textsection18}. & Supplement on \'etale morphisms. Henselian local rings and strictly local rings.\\
+ \hyperref[section:IV.19]{\textsection19}. & Regular immersions and normal flatness.\\
+ \hyperref[section:IV.20]{\textsection20}. & Meromorphic functions and pseudo-morphisms\\
+ \hyperref[section:IV.21]{\textsection21}. & Divisors.
\end{longtable}
\bigskip
diff --git a/ega4/ega4-16.tex b/ega4/ega4-16.tex
index 2a75e78..9a313b9 100644
--- a/ega4/ega4-16.tex
+++ b/ega4/ega4-16.tex
@@ -35,7 +35,7 @@ is called the sheaf of graded rings \emph{associated to} $f$. The sheaf $\shGr_1
It is clear that the $\sh{O}_{Y^{(n)}} = \psi^*(\sh{O}_X)/\sh{I}_f^{n+1}$ (that we also denote $\sh{O}_{Y_f^{(n)}})$ form a
\oldpage[IV-4]{6}
-projective system of sheaves of rings on $Y$, the transition homomorphism $\vphi_{nm}:\sh{O}_{Y^{(m)}} \to \sh{O}_{Y^{(n)}}$ for $n \leq m$ identifies $\sh{O}_{Y^{(n)}}$ with the quotient of $\sh{O}_{Y^{(m)}}$ by the power $(\sh{I}_f/\sh{I}_f^{n+1} )^m$ of the \emph{agumentation ideal} of $\sh{O}_{Y^{(n)}}$, kernel of $\vphi_{0n}: \sh{O}_{Y^{(n)}} \to \sh{O}_{Y}$.
+projective system of sheaves of rings on $Y$, the transition homomorphism $\varphi_{nm}:\sh{O}_{Y^{(m)}} \to \sh{O}_{Y^{(n)}}$ for $n \leq m$ identifies $\sh{O}_{Y^{(n)}}$ with the quotient of $\sh{O}_{Y^{(m)}}$ by the power $(\sh{I}_f/\sh{I}_f^{n+1} )^m$ of the \emph{augmentation ideal} of $\sh{O}_{Y^{(n)}}$, kernel of $\varphi_{0n}: \sh{O}_{Y^{(n)}} \to \sh{O}_{Y}$.
The $Y^{(n)}$ therefore form a inductive system of ringed spaces, all having underlying space $Y$, and we have canonical morphisms of ringed spaces $h_n: Y^{(n)} \to X$ equal to $(\psi, \theta_n)$, where $\theta^\#_n$ is the canonical morphism $\psi^*(\sh{O}_X) \to \psi^*(\sh{O}_X)/\sh{I}_f^{n+1}$.
It is clear that the sheaf $\shGr_\bullet(f)$ is a sheaf of graded algebras over the sheaf of rings $\sh{O}_Y = \shGr_0(f)$ and the $\shGr_k(f)$ of $\sh{O}_Y$-modules.
@@ -227,7 +227,7 @@ account the exactness of the functor $\rho^*$, we obtain a di-homomorphism of gr
\gr(u): \rho^*(\shGr_\bullet(f)) \to \shGr_\bullet(f')
\tag{16.2.1.3}
\]
-(or, if you like, a $\rho$-morphism \sref[0]{0.3.5.1} $\shGr_\bullet(f) \to \shGr_\bullet(f')$), and in particular a di-homomorphism of conormal sheafs
+(or, if you like, a $\rho$-morphism \sref[0]{0.3.5.1} $\shGr_\bullet(f) \to \shGr_\bullet(f')$), and in particular a di-homomorphism of conormal sheaves
\[
\gr_1(u): \rho^*(\shGr_1(f)) \to \shGr_1(f').
\]
@@ -396,7 +396,7 @@ If the $\shGr_n(f)$ are flat for $n\leq m$, then we first see by induction on $n
\medskip\noindent
\begin{enumerate}
\item[(i)] The reasoning of \sref{IV.16.2.2}[(i)] still applies to \sref{IV.16.2.1.1} when these are morphisms of \emph{locally ringed spaces} \sref[I]{I.1.8.2}.
- \item[(ii)] In \sref{IV.16.2.2}[(ii)], the conclusion is no longer necessairly valid if we only suppose that $v$ and $f$ are morphisms of preschemes ($f$ satisfying the condition of \sref{IV.16.1.1}).
+ \item[(ii)] In \sref{IV.16.2.2}[(ii)], the conclusion is no longer necessarily valid if we only suppose that $v$ and $f$ are morphisms of preschemes ($f$ satisfying the condition of \sref{IV.16.1.1}).
For example (with the notation of the proof of \sref{IV.16.2.2}[(ii)]), it can happen that $\mathfrak{I} = 0$ but the kernel $\mathfrak{I}'$ of $A' \to B' = B \otimes_A A'$ is not zero and that $B' \neq 0$, in which case we have $Y^{(n)} = Y$ for all $n$, but ${Y'}^{(n)} \neq Y'$.
We have an example of this by taking $A = \bb{Z}$, $B = \bb{Q}$, $A' = \prod_{h = 1}^\infty (\bb{Z}/m^h\bb{Z})$ where $m>1$.
\end{enumerate}
@@ -529,7 +529,7 @@ In all notation introduced in \sref{IV.16.3.1} and \sref{IV.16.3.6}, we will som
\begin{env}[16.3.7]
\label{IV.16.3.7}
Suppose in particular that $S = \Spec(A)$ and $X = \Spec(B)$ are affine schemes, $B$ then being an $A$-algebra.
-Then $\Delta_f$ correspeonds to the canonical surjective homomorphism $\pi: B \otimes_A B \to B$ such that $\pi(b\otimes b') = bb'$, with kernel $\mathfrak{I} = \mathfrak{I}_{B/A}$ \sref[0]{0.20.4.1};
+Then $\Delta_f$ corresponds to the canonical surjective homomorphism $\pi: B \otimes_A B \to B$ such that $\pi(b\otimes b') = bb'$, with kernel $\mathfrak{I} = \mathfrak{I}_{B/A}$ \sref[0]{0.20.4.1};
$\sh{P}_{f}^n$ is the structure sheaf of the prescheme $\Spec(P_{B/A}^n)$, where
\[
P_{B/A}^n = (B \otimes_A B)/\mathfrak{I}^{n+1};
@@ -2776,4 +2776,4 @@ For example, if $S = \Spec(k)$, where $k$ is a field of characteristic $p > 0$,
\oldpage[IV-4]{56}
that $\Omega_{X/S}^1$ has rank $1$, and that the morphism $X \to S$ has rank $1$, and that the morphism $X \to S$ is differentially smooth up to order $p - 1$ \sref{IV.16.11.3}[(iii)], but not of order $p$.
However, the proof of \sref{IV.16.12.2} proves that if $\Omega_{X/S}^1$ is locally free, and if $n! 1_{\sh{O}_X}$ is inversible in $\Gamma(X, \sh{O}_X)$, then $X$ is differentially smooth over $S$ up to order $n$.
-\end{env} \ No newline at end of file
+\end{env}
diff --git a/what.tex b/what.tex
index f487775..aae1cf0 100644
--- a/what.tex
+++ b/what.tex
@@ -110,7 +110,7 @@ If the change is minor (e.g. `intersection' replacing `inter-section') then we w
\label{section:what-ega4-sections}
In EGA~IV-1, the summary included a tentative list of section that EGA~IV would contain.
-As EGA~IV was written, \textsection5, \textsection7, and the \textsection\textsection11--21 would contain different sections than initially envisaged.
+As EGA~IV was written, \textsection5, \textsection7, and \textsection\textsection11--21 would contain different sections than initially envisaged.
We include the original listing here:
\begin{longtable}{ll}
\textsection5. & Dimension and depth for preschemes.\\
@@ -130,7 +130,7 @@ We include the original listing here:
\bigskip
\section*{Mathematical warnings}
-EGA uses \emph{prescheme} for what is now usually called a scheme, and \emph{scheme} for what is now usually called a separated scheme.
+EGA uses \emph{prescheme} for what is now usually called a scheme, and \emph{scheme} for what is now usually called a separated scheme --- we have decided to translate ``literally'' or ``historically'', and thus continue to use the word \emph{scheme} to mean separated scheme, and \emph{prescheme} to mean scheme.
In some cases, we (the translators) have changed ``$\to$'' to ``$\mapsto$'' where appropriate.