small progress

trees/index.tree

@@ -33,6 +33,9 @@
       \li{
         [I. Étale morphisms](sga1-i) ✓
       }
+      \li{
+        [II. Smooth morphisms: generalities, differential properties](sga1-ii) \em{(in progress)}
+      }
     }
   }
   \li{
@@ -46,16 +49,16 @@
 
     \ul{
       \li{
-        [Introduction](sga6-introduction)
+        [Introduction](sga6-introduction) \em{(in progress)}
       }
       \li{
-        [0. Outline of a programme for an intersection theory](sga6-0)
+        [0. Outline of a programme for an intersection theory](sga6-0) \em{(in progress)}
       }
       \li{
-        [0[RRR]. Classes of sheaves and the Riemann–Roch theorem](sga6-0rrr)
+        [0[RRR]. Classes of sheaves and the Riemann–Roch theorem](sga6-0rrr) \em{(in progress)}
       }
       \li{
-        [I. Generalities on finiteness conditions in derived categories](sga6-i)
+        [I. Generalities on finiteness conditions in derived categories](sga6-i) \em{(in progress)}
       }
     }
   }

trees/sga1/sga1-ii/sga1-ii.1.tree

@@ -0,0 +1,9 @@
+\title{Generalities}
+\taxon{section}
+\number{II.1}
+\parent{sga1-ii}
+
+\import{sga1-macros}
+\put\transclude/numbered{false}
+
+

trees/sga1/sga1-ii/sga1-ii.2.tree

@@ -0,0 +1,9 @@
+\title{Some smoothness criteria of a morphism}
+\taxon{section}
+\number{II.2}
+\parent{sga1-ii}
+
+\import{sga1-macros}
+\put\transclude/numbered{false}
+
+

trees/sga1/sga1-ii/sga1-ii.3.tree

@@ -0,0 +1,9 @@
+\title{Permanence properties}
+\taxon{section}
+\number{II.3}
+\parent{sga1-ii}
+
+\import{sga1-macros}
+\put\transclude/numbered{false}
+
+

trees/sga1/sga1-ii/sga1-ii.4.tree

@@ -0,0 +1,9 @@
+\title{Differential properties of smooth morphisms}
+\taxon{section}
+\number{II.4}
+\parent{sga1-ii}
+
+\import{sga1-macros}
+\put\transclude/numbered{false}
+
+

trees/sga1/sga1-ii/sga1-ii.5.tree

@@ -0,0 +1,9 @@
+\title{Case of a base field}
+\taxon{section}
+\number{II.5}
+\parent{sga1-ii}
+
+\import{sga1-macros}
+\put\transclude/numbered{false}
+
+

trees/sga1/sga1-ii/sga1-ii.tree

@@ -0,0 +1,12 @@
+\title{Smooth morphisms: generalities, differential properties}
+\taxon{exposé}
+\number{II}
+\parent{sga1}
+
+\import{sga1-macros}
+
+\transclude{sga1-ii.1}
+\transclude{sga1-ii.2}
+\transclude{sga1-ii.3}
+\transclude{sga1-ii.4}
+\transclude{sga1-ii.5}

trees/sga1/sga1.tree

@@ -7,3 +7,4 @@
 \transclude{sga1-introduction}
 \transclude{sga1-foreword}
 \transclude{sga1-i}
+\transclude{sga1-ii}

trees/sga6/sga6-0/sga6-0.1.tree

@@ -32,7 +32,7 @@
     = \sum_i (-1)^i \cl(\RR^i f_*(\sh{F})).
   }
   As we know, #{\Todd(-)} and #{\ch(-)} are universal polynomials in the Chern classes of the argument with coefficients in #{\QQ}.
-  Since the constant term of #{\Todd(-)} is #{1}, it is an invertible element for any value of the argument, so that Equation (1.1) can be rewritten, after multiplication by #{\Todd(T_Y)^{-1}}, in the form which is more useful for our needs:
+  Since the constant term of #{\Todd(-)} is #{1}, it is an invertible element for any value of the argument, so that \ref{sga6-0.1-equation-1.1} can be rewritten, after multiplication by #{\Todd(T_Y)^{-1}}, in the form which is more useful for our needs:
 
   \subtree[sga6-0.1-equation-1.2]{
     \taxon{equation}
@@ -71,7 +71,7 @@
 }
 
 \p{
-  One of the main goals of this Seminar is to generalise Equation (1.2) simultaneously in two directions:
+  One of the main goals of this Seminar is to generalise \ref{sga6-0.1-equation-1.2} simultaneously in two directions:
 }
 
 \olc{

trees/sga6/sga6-0/sga6-0.2.1.tree

@@ -0,0 +1,18 @@
+\taxon{section}
+\number{2.1}
+\parent{sga6-0.2}
+
+\import{sga6-macros}
+\put\transclude/numbered{false}
+
+\p{
+  \oldpage{3}
+  For
+}
+
+
+
+
+\p{
+  \todo{finish}
+}

trees/sga6/sga6-0/sga6-0.2.tree

@@ -6,5 +6,23 @@
 \put\transclude/numbered{false}
 
 \p{
-  \todo{finish}
+  We now examine generalisation (a) from [[sga6-0.2]], keeping, however, the hypotheses of regularity and of existence of ample invertible modules on #{X} and on #{Y}.
 }
+
+\p{
+  The definition of #{K(X)} and #{K(Y)}, and of the homomorphism #{f_*\colon K(X)\to K(Y)} then gives no new problems, thanks to the fact that #{X} and #{Y} are regular.
+  The most natural route to giving meaning to \ref{sga6-0.1-equation-1.2} thus seems to consist of defining the \em{Chow rings} #{A(X)} and #{A(Y)} and a group homomorphism
+  ##{
+    f_*\colon A(X)\to A(Y)
+  }
+  as well as establishing a theory of Chern classes, providing maps
+  ##{
+    c_i\colon K(X) \to A(X)
+  }
+  (and similarly for #{Y}), and finally giving a description of an virtual relative tangent bundle element
+  ##{
+    T_f\in K(X).
+  }
+}
+
+\transclude{sga6-0.2.1}