a page or so

sga-1/index.Rmd

@@ -44,7 +44,7 @@ Exposés [I](#I) to [IV](#IV) present the local notions of *étale* and *smooth*
 they hardly ever use the language of schemes, as detailed in Chapter I of the *Éléments*.^[A more complete study is now available in EGA IV 17,18.]
 [Exposé V](#V) presents the axiomatic description of the fundamental group of a scheme, which is useful even in the classical case, where the scheme is simply the spectrum of a field, since we then find a strong and convenient reformulation of the usual Galois theory.
 Exposés [VI](#VI) and [VIII](#VIII) present the *theory of descent*, which has become more and more important in algebraic geometry over the past few years, and which could do the same in analytic geometry and in topology.
-We note that Exposé VII was not transcribed, but its contents can be found incorporated into an article by J. Giraud ("Méthode de la Descente". *Bull. Soc. Math. France* **2** (1964), viii + 150 p.).
+We note that Exposé VII was not transcribed, but its contents can be found incorporated into an article by J. Giraud ("Méthode de la Descente". *Bull. Soc. Math. France* **2** (1964), viii+150 p.).
 In [Exposé IX](#IX), we study more specifically the theory of descent by étale morphisms, obtaining a systematic approach for Van Kampen type theorems for the fundamental group, which appear here as simple translations of theorems of descent.
 It essentially deals with a calculation of the fundamental group of a connected scheme $X$ endowed with a surjective and proper morphism (say, $X'\to X$) in terms of the fundamental groups of the connected components of $X'$ and of the fibre products $X'\times_X X'$, $X'\times_X X'\times_X X'$, and the homomorphisms between these groups induced by the canonical simplicial morphisms between the above schemes.
 [Exposé X](#X) gives the theory of *specialisation of the fundamental group* for a proper and smooth morphism, with the most striking result being the determination (more or less) of the fundamental group of a smooth algebraic curve in characteristic $p>0$, thanks to the known result obtained by transcendental methods in characteristic zero.
@@ -143,7 +143,7 @@ It did not seem useful to make a note of the dates.
 Exposé VII, which is referenced at various points throughout [Exposé VIII](#VIII), has not been written by the speaker, who, in the oral conferences, was limited to outlining the language of descent in general categories, by working from a strictly utilitarian point of view and not entering into the logical difficulties that often arise due to this language.
 It seemed that a proper exposé of this language would go beyond the limits of these current notes, even if only due to length.
 For a proper exposé of the theory of descent, I refer the reader to an article in preparation by Jean Giraud.
-Whilst waiting for its appearance^[It is now published: J. Giraud, "Méthode de la Descente". *Bull. Soc. Math. France* **2** (1964), viii + 150 p.], I think that an attentive reader will have no problems in supplementing, by their own means, the phantom references in [Exposé VIII](#VIII).
+Whilst waiting for its appearance^[It is now published: J. Giraud, "Méthode de la Descente". *Bull. Soc. Math. France* **2** (1964), viii+150 p.], I think that an attentive reader will have no problems in supplementing, by their own means, the phantom references in [Exposé VIII](#VIII).
 
 Other oral exposés, found after [Exposé XI](#XI), and to which there are references in certain places of the text, have also not been written down, and were meant to form the substance of an Exposé XII and an Exposé XIII.
 The first of these oral exposés covered, in the framework of schemes and analytic spaces with nilpotent elements (as introduced in the *Séminaire Cartan 1960/61*), the construction of the analytic space associated to a prescheme of locally finite type over a complete valuation field $k$, GAGA-type theorems in the case where $k$ is the field of complex numbers, and the application to the comparison of the fundamental group defined by transcendental methods and the fundamental group studied in these notes (cf. A. Grothendieck, "Fondements de la Géométrie Algébrique". *Séminaire Bourbaki* **190** (December 1959), page 10).

sga-1/sga-1-i/sga-1-i-10.Rmd

@@ -50,7 +50,7 @@ i. $K$ is unramified over $Y$.
 ii. If $L$ is an extension of $K$ that is unramified over $Y$, and if $Y'$ is a normal prescheme, of field $L$, that dominates $Y$ (e.g. the normalisation of $Y$ in $L$), and $M$ an extension of $L$ that is unramified over $Y'$, then $M/K$ is unramified over $X$ (this is the _transitivity_ property).
 
 iii. Let $Y'$ be a normal integral prescheme that dominates $Y$, of field $K'/K$;
-    if $L$ is an extension of $K$ that is unramified over $Y$, then $L\otimes_K K'$ is an extension of $K'$ that is unramified over $Y'$ (this is the _translation_ property)
+  if $L$ is an extension of $K$ that is unramified over $Y$, then $L\otimes_K K'$ is an extension of $K'$ that is unramified over $Y'$ (this is the _translation_ property)
 :::
 
 Furthermore:

sga-1/sga-1-i/sga-1-i-11.Rmd

@@ -4,26 +4,26 @@ We have already said that a connected étale cover of an integral scheme is not
 Here are two examples of this fact.
 
 a. Let $C$ be an algebraic curve with an ordinary double point $x$, and let $C'$ be its normalisation, with $a$ and $b$ the two points of $C'$ over $x$.
-    Let $C'_1$ and $C'_2$ be copies of $C'$, with $a_i$ (resp. $b_i$) the point of $C'_i$ corresponding to $a$ (resp. $b$).
-    In the curve $C'_1\coprod C'_2$, we identify $a_1$ with $b_2$, and $a_2$ with $b_1$ (we leave making this process of identification precise to the reader; it will be explained in Chapter VI of the multiplodoque, but, in the case of curves over an algebraically closed field, is already covered in Serre's book on algebraic curves).
-    We obtain a curve $C''$ which is _connected_ and _reducible_, and which is a degree-$2$ étale cover of $C$.
-    The reader can verify that, generally, the "Galois" connected étale covers $C''$ of $C$ whose inverse images $C''\times_C C'$ are _trivial_ covers of $C'$ (i.e. isomorphic to the sum of a certain number of copies of $C'$) are "cyclic" of degree $n$, and, conversely, for every integer $n>0$, we can construct a cyclic connected étale cover of degree $n$.
-    In the language of the fundamental group (which will be developed later), this implies that the quotient of $\pi_1(C)$ by the closed invariant subgroup generated by the image of $\pi_1(C')\to\pi_1(C)$ (the homomorphism induced by the projection) is isomorphic to the compactification of $\mathbb{Z}$.
-    More precisely, we should show that the fundamental group of $C$ is isomorphic to the (topological) free product of the fundamental group of $C$ with the compactification of $\mathbb{Z}$.
-    We note that is was questions of this sort that gave birth to the "theory of descent" for schemes.
+  Let $C'_1$ and $C'_2$ be copies of $C'$, with $a_i$ (resp. $b_i$) the point of $C'_i$ corresponding to $a$ (resp. $b$).
+  In the curve $C'_1\coprod C'_2$, we identify $a_1$ with $b_2$, and $a_2$ with $b_1$ (we leave making this process of identification precise to the reader; it will be explained in Chapter VI of the multiplodoque, but, in the case of curves over an algebraically closed field, is already covered in Serre's book on algebraic curves).
+  We obtain a curve $C''$ which is _connected_ and _reducible_, and which is a degree-$2$ étale cover of $C$.
+  The reader can verify that, generally, the "Galois" connected étale covers $C''$ of $C$ whose inverse images $C''\times_C C'$ are _trivial_ covers of $C'$ (i.e. isomorphic to the sum of a certain number of copies of $C'$) are "cyclic" of degree $n$, and, conversely, for every integer $n>0$, we can construct a cyclic connected étale cover of degree $n$.
+  In the language of the fundamental group (which will be developed later), this implies that the quotient of $\pi_1(C)$ by the closed invariant subgroup generated by the image of $\pi_1(C')\to\pi_1(C)$ (the homomorphism induced by the projection) is isomorphic to the compactification of $\mathbb{Z}$.
+  More precisely, we should show that the fundamental group of $C$ is isomorphic to the (topological) free product of the fundamental group of $C$ with the compactification of $\mathbb{Z}$.
+  We note that is was questions of this sort that gave birth to the "theory of descent" for schemes.
 
 b. Let $A$ be a complete integral local ring;
-    we know that its normalisation $A'$ is finite over $A$ (by Nagata), and is thus a complete semi-local ring, and thus local, since it is integral.
-    Suppose that the residue extension $L/k$ that it defines is non-radicial (in the contrary case, we say that $A$ is _geometrically unibranch_; cf. below).
-    This will be the case, for example, for the ring $\mathbb{R}[[s,t]]/(s^2+t^2)\mathbb{R}[[s,t]]$, where $\mathbb{R}$ is the field of real numbers.
-    Then let $k'$ be a finite Galois extension of $k$ such that $L\otimes_k k'$ decomposes;
-    let $B$ be a finite and étale algebra over $A$ corresponding to the residue extension $k'$ (recall that $B$ is essentially unique).
-    Then the residue algebra of $B'=A'\otimes_A B$ over $B$ is $L\otimes_k k'$, which is not local, and so $B'$ is not a local ring, and thus $B$ has zero divisors (since it is complete).
-    Now $B'$ is contained in the total ring of fractions of $B$ (since it is free over $A'$, thus torsion free over $A'$, thus torsion free over $A$, thus contained in $B'\otimes_A K=B'_{(K)}=A'_{(K)}\otimes_K B_{(K)}=B_{(K)}$, since $A'_{(k)}=K$), and so $B$ is not integral.
-    In the case of the ring $\mathbb{R}[s,t]/(s^2+t^2)\mathbb{R}[s,t]$, taking $k'/k=\mathbb{C}/\mathbb{R}$, we see that $B$ is the local ring of two secant lines at their point of intersection.
+  we know that its normalisation $A'$ is finite over $A$ (by Nagata), and is thus a complete semi-local ring, and thus local, since it is integral.
+  Suppose that the residue extension $L/k$ that it defines is non-radicial (in the contrary case, we say that $A$ is _geometrically unibranch_; cf. below).
+  This will be the case, for example, for the ring $\mathbb{R}[[s,t]]/(s^2+t^2)\mathbb{R}[[s,t]]$, where $\mathbb{R}$ is the field of real numbers.
+  Then let $k'$ be a finite Galois extension of $k$ such that $L\otimes_k k'$ decomposes;
+  let $B$ be a finite and étale algebra over $A$ corresponding to the residue extension $k'$ (recall that $B$ is essentially unique).
+  Then the residue algebra of $B'=A'\otimes_A B$ over $B$ is $L\otimes_k k'$, which is not local, and so $B'$ is not a local ring, and thus $B$ has zero divisors (since it is complete).
+  Now $B'$ is contained in the total ring of fractions of $B$ (since it is free over $A'$, thus torsion free over $A'$, thus torsion free over $A$, thus contained in $B'\otimes_A K=B'_{(K)}=A'_{(K)}\otimes_K B_{(K)}=B_{(K)}$, since $A'_{(k)}=K$), and so $B$ is not integral.
+  In the case of the ring $\mathbb{R}[s,t]/(s^2+t^2)\mathbb{R}[s,t]$, taking $k'/k=\mathbb{C}/\mathbb{R}$, we see that $B$ is the local ring of two secant lines at their point of intersection.
 
-    We note also that, if there exists a connected étale cover $X$ of $Y$ that is integral but not irreducible, then every irreducible component of $X$ gives an example of an unramified cover $X'$ of $Y$ that dominates $Y$ but is not étale over $Y$.
-    In the case of example (a), we thus see that $C'$ is unramified over $C$, without being étale at the two points $a$ and $b$ (note that, directly, by inspection of the completions of the local rings at $x$ and $a$, from the "formal" point of view, $C'$ at the point $a$ can be identified with a closed subscheme of $C$ at the point $x$, i.e. one of the two "branches" of $C$ passing through $x$).
+  We note also that, if there exists a connected étale cover $X$ of $Y$ that is integral but not irreducible, then every irreducible component of $X$ gives an example of an unramified cover $X'$ of $Y$ that dominates $Y$ but is not étale over $Y$.
+  In the case of example (a), we thus see that $C'$ is unramified over $C$, without being étale at the two points $a$ and $b$ (note that, directly, by inspection of the completions of the local rings at $x$ and $a$, from the "formal" point of view, $C'$ at the point $a$ can be identified with a closed subscheme of $C$ at the point $x$, i.e. one of the two "branches" of $C$ passing through $x$).
 
 In both (a) and (b), we see that the fact that the conclusions of [9.5 (i) and (ii)](#I.9.5) fail to hold is directly linked with the fact that a point of $Y$ "blows up" at _distinct_ points of the normalisation (in (b), the fact that the residue extension is non-radicial should be interpreted geometrically in this way).
 More precisely, we say that an integral local ring $A$ is _geometrically unibranch_ if its normalisation has only a single maximal ideal, with the corresponding residue extension being radicial;

sga-1/sga-1-i/sga-1-i-3.Rmd

@@ -79,6 +79,6 @@ In order for $B/A$ to be unramified, it is necessary and sufficient that $\hat{B
 
 - In the case where we don't suppose that the residue extension is trivial, we can reduce to the case where it is by taking a suitable finite flat extension of $A$ which destroys the aforementioned extension.
 - Consider the example where $A$ is the local ring of an ordinary double point of a curve, and $B$ a point of its normalisation:
-    then $A\subset B$, $B$ is unramified over $A$ with trivial residue extension, and $\hat{A}\to\hat{B}$ is surjective but _not injective_.
-    We are thus going to strengthen the notion of unramified-ness.
+  then $A\subset B$, $B$ is unramified over $A$ with trivial residue extension, and $\hat{A}\to\hat{B}$ is surjective but _not injective_.
+  We are thus going to strengthen the notion of unramified-ness.
 :::

sga-1/sga-1-i/sga-1-i-4.Rmd

@@ -7,11 +7,11 @@ these facts will be proved later, if there is time^[cf. [IV](#IV).].
 —
 
 a. Let $f\colon X\to Y$ be a morphism of finite type.
-    We say that $f$ is _étale_ at $x$ if $f$ is both flat and unramified at $x$.
-    We say that $f$ is étale if it is étale at all points.
-    We say that $X$ is étale at $x$ over $Y$, or that it is a $Y$-prescheme which is étale at $x$ etc.
+  We say that $f$ is _étale_ at $x$ if $f$ is both flat and unramified at $x$.
+  We say that $f$ is étale if it is étale at all points.
+  We say that $X$ is étale at $x$ over $Y$, or that it is a $Y$-prescheme which is étale at $x$ etc.
 b. Let $f\colon A\to B$ be a local homomorphism.
-    We say that $f$ is étale, or that $B$ is étale over $A$, if $B$ is flat and unramified over $A$.^[cf. regrets in [III 1.2](#III.1.2).]
+  We say that $f$ is étale, or that $B$ is étale over $A$, if $B$ is flat and unramified over $A$.^[cf. regrets in [III 1.2](#III.1.2).]
 :::
 
 ::: {.itenv #I.4.2 title="Proposition 4.2"}
@@ -59,7 +59,7 @@ Indeed, (i) is trivial, and for (ii) and (iii) it suffices to note that it is tr
 
 As a matter of fact, there are also corresponding comments to make about local homomorphisms (without the finiteness condition), which in any case should appear in the multiplodoque (starting with the case of unramified).
 
-_[Trans.] Grothendieck's "multiplodoque d'algèbre homologique" was the final version of his Tohoku paper --- see (2.1) in 'Life and work of Alexander Grothendieck' by Ching-Li Chan and Frans Oort for more information._
+_[Trans.] Grothendieck's "multiplodoque d'algèbre homologique" was the final version of his Tohoku paper --- see (2.1) in "Life and work of Alexander Grothendieck" by Ching-Li Chan and Frans Oort for more information._
 
 ::: {.itenv #I.4.7 title="Corollary 4.7"}
 The cartesian product of two étale morphisms is étale.

sga-1/sga-1-i/sga-1-i-5.Rmd

@@ -11,22 +11,22 @@ The necessity is trivial, and the sufficiency remains to be shown.
 We are going to give two different proofs: the first is shorter, the second is more elementary.
 
 1. A flat morphism is open, and so we can suppose (by replacing $Y$ with $f(X)$) that $f$ is an onto _homeomorphism_.
-    For any base extension, it remains true that $f$ is flat, radicial, surjective, thus a homeomorphism, and a fortiori closed.
-    Thus $f$ is _proper_.
-    Thus $f$ is _finite_ (reference: Chevalley's theorem), defined by a coherent sheaf $\scr{B}$ of algebras.
-    Now $\scr{B}$ is locally free, and further, by hypothesis, of rank $1$ everywhere, and so $X=Y$.
+  For any base extension, it remains true that $f$ is flat, radicial, surjective, thus a homeomorphism, and a fortiori closed.
+  Thus $f$ is _proper_.
+  Thus $f$ is _finite_ (reference: Chevalley's theorem), defined by a coherent sheaf $\scr{B}$ of algebras.
+  Now $\scr{B}$ is locally free, and further, by hypothesis, of rank $1$ everywhere, and so $X=Y$.
 
 2. We can suppose that $Y$ and $X$ are _affine_.
-    We can further easily reduce to proving the following:
-    if $Y=\Spec(A)$, with $A$ local, and if $f^{-1}(y)$ is non-empty (where $y$ is the closed point of $Y$), then $X=Y$ (indeed, this would imply that every $y\in f(X)$ has an open neighbourhood $U$ such that $X|U=U$).
-    We will then have that $X=\Spec(B)$, and wish to prove that $A=B$.
-    But, for this, we can reduce to proving the analogous claim where we replace $A$ by $\hat{A}$, and $B$ by $B\otimes_A\hat{A}$
-    (taking into account the fact that $\hat{A}$ is faithfully flat over $A$).
-    We can thus suppose that $A$ is _complete_.
-    Let $x$ be the point over $y$.
-    By [2.2](#I.2.2), $\cal{O}_x$ is finite over $A$, and is thus (being flat and radicial over $A$) identical to $A$.
-    So $X=Y\coprod X'$ (disjoint sum).
-    But since $X$ is radicial over $Y$, $X'$ is empty.
+  We can further easily reduce to proving the following:
+  if $Y=\Spec(A)$, with $A$ local, and if $f^{-1}(y)$ is non-empty (where $y$ is the closed point of $Y$), then $X=Y$ (indeed, this would imply that every $y\in f(X)$ has an open neighbourhood $U$ such that $X|U=U$).
+  We will then have that $X=\Spec(B)$, and wish to prove that $A=B$.
+  But, for this, we can reduce to proving the analogous claim where we replace $A$ by $\hat{A}$, and $B$ by $B\otimes_A\hat{A}$
+  (taking into account the fact that $\hat{A}$ is faithfully flat over $A$).
+  We can thus suppose that $A$ is _complete_.
+  Let $x$ be the point over $y$.
+  By [2.2](#I.2.2), $\cal{O}_x$ is finite over $A$, and is thus (being flat and radicial over $A$) identical to $A$.
+  So $X=Y\coprod X'$ (disjoint sum).
+  But since $X$ is radicial over $Y$, $X'$ is empty.
 :::
 
 ::: {.itenv #I.5.2 title="Corollary 5.2"}
@@ -140,4 +140,4 @@ For $g$ to be flat (resp. étale) at $x$, it is necessary and sufficient for $g\
 ::: {.proof}
 For "flat", the statement only serves as a reminder, since this is one of the fundamental criteria of flatness^[cf. [IV.5.9](#IV.5.9)].
 For "étale", this follows by taking [5.8](#I.5.8) into account.
-:::
+:::

sga-1/sga-1-i/sga-1-i-6.Rmd

@@ -29,4 +29,4 @@ We can reduce to the case where $A$ is Artinian (by replacing $A$ by $A/\frak{m}
 It remains to prove that, for every finite and separable $k$ algebra (or we can simply say "étale", for brevity) $L$, there exists some $B$ étale over $A$ such that $R(B)$ is isomorphic to $L$.
 We can suppose that $L$ is a separable extension of $k$, and, as such, it admits a generator $x$, i.e. it is isomorphic to an algebra $k[t]/Fk[t]$, where $F\in k[t]$ is a monic polynomial.
 We can lift $F$ to a monic polynomial $F_1$ in $A[t]$, and we take $B=A[t]/F_1A[t]$.
-:::
+:::