p. 130, l. 28-29 (precision concerning an example of a
nonnegative ${\cal C}^\infty$ function on ${\Bbb R}$ whose square
root is not ${\cal C}^\infty$, communicated by J-C. Tougeron)
An example of a ${\cal C}^\infty$ function from ${\Bbb R}$ to
${\Bbb R}$, which is positive outside $0$ and infinitely flat at
$0$, and whose square root is not ${\cal C}^2$ at $0$, is given
in:
G. Glaeser: Racine carrée d'une fonction
différentiable. Ann. Inst. Fourier 13 (1963),
203--210.
p. 189, l. 18-19
Corollary 8.7.13 cannot be used at this place since it is not yet
known that $B$ is noetherian. One can use [264] Chap.\ 8, Theorem
3, which says that, for every prime ideal ${\frak p}$ of $B$,
${}^h\! B \otimes_B k({\frak p})$ is an integral $k({\frak
p})$-algebra whose local rings are separable algebraic extensions
of $k({\frak p})$. In particular, if ${\frak q} \subset {\frak
q}'$ are ideals of ${}^h\! B$ such that ${\frak q}\cap B={\frak
q}'\cap B$, then ${\frak q}={\frak q}'$.
p. 191, l. 5-3 from bottom (communicated by Alessandro
Tancredi)
The argument for the fact that ${\frak m}_x$ is finitely generated
is incorrect and should be replaced by the following one:
Let ${\frak n}_x$ be the ideal of $f \in {\cal P}(V)$ vanishing at
$x$. We have ${\frak n}_x {\cal N}(M)= {\frak m}_x$, hence ${\frak
m}_x$ is finitely generated.
p. 203, l. 11
read "where $\pi$ is the projection ..." instead of "where $\Pi$
is the projection ..."
p. 300, l. 15 (statement of Proposition 12.1.3) :
read "Let $\xi$ be a prealgebraic vector bundle" instead of "Let
$\xi$ be an algebraic vector bundle".