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\centerline{Corrections to the second printing of}
\centerline{\bf Commutative Algebra with a View Toward Algebraic
Geometry}
\bigskip
This file contains all the corrections to the second printing
that I knew of as of 9/7/98.
References are of the form
{\bf n;m.} where n is a page number in the second printing,
m a line number. Descriptive
matter (that is, things not actually appearing in the text)
is surrounded by double parentheses ((like this)).
\bigskip
\hskip 1truein ---David Eisenbud
\bigskip
\hrule\smallskip
{\bf on title page, or somewhere else prominent}\ins
{Third corrected printing}
{\bf 22; 15.} \rep{is an}{is a primitive}
{\bf 35; 8.}\del{reduced}
{\bf 36; -7.}\insa{category of}{reduced}
{\bf 43; -13.} \rep{this is}{we make the convention that this is}
{\bf 52; 8-15.}\rep{((the labels a-h))}{((the numbers 1-8))}
{\bf 52; -8.}\rep{a and b}{1 and 2}
{\bf 52; -7.}\rep{g and h}{7 and 8}
{\bf 52; -4.}\rep{c,d and h}{3,4 and 8}
{\bf 57; -2.}\rep{3.5}{I.3.6}
{\bf 67; 9.}\insa{ideals}{$P$}
{\bf 83; 1--2.}\rep{$x$}{$f$ ((two occurences))}
{\bf 90; -7.}\rep{union} {finite union}
{\bf 111; 12.}\rep{in press}{1995}
{\bf 118; -8.}\rep{algebra}{algebraic}
{\bf 118; -5.}\insa{).}
{Assuming that $X$ and $Y$ are
affine, so is $Y'$, and its coordinate ring is the
normalization of the image of $A(Y)$ in $A(X)$.}
{\bf 124; 4.}\rep{Lemma}{Theorem}
{\bf 129; -14.}\rep{Hartshome}{Hartshorne}
{\bf 130; fig 4.4.}\rep{((the upside down U))}{$\cap$}
{\bf 139; -10 .}\insb{Let}{((as a new part a.))
\item{a.} Show that the quotient field of $k[\Gamma]$ is
$k[G(\Gamma)]$.}
{\bf 139; -7.}\rep{its quotient field}{$k[x_1,\dots,x_n]$}
{\bf 139; -6.}\rep{a.}{b.}
{\bf 140; 3.}\rep{b.}{c.}
{\bf 140; 5--10.}\del{the whole of part d.}
{\bf 140; 12.}\insa{$\{$}{(}
{\bf 149; -20.}\insa{ideal}{such}
{\bf 154; Figure 5.2, first line under the left-hand picture.}
\rep{in $(y^2$}{in$(y^2$}
{\bf 159; -8.}\rep{$a\neq 0$}{$0\neq a$}
{\bf 187; -8.}\rep{equation}{expression}
{\bf 187 ;-2.}\rep{ $5/32$}{ $5/128$}
{\bf 189; 5.}\rep{$e_j$}{$\sum_{j\neq i}e_j$}
{\bf 189; 6.}\insa{$=0$}{for each $j\neq i$}
{\bf 189; 6.}\rep{$m=e_j(n')$}{$m=\sum_{j\neq i}e_j(n_j')$}
{\bf 189; 7.}\rep{$n_j\in M$}{$n_j'\in M$}
{\bf 189; 7.}\rep{$e_j(m)=$\dots$=m$}
{$\sum_{j\neq i}e_j(m)=\sum_{j\neq i}e_j(\sum_{j\neq i}e_j(n_j'))=
\sum_{j\neq i}e_j(n_j')=m$}
{\bf 189; 8.}\rep{$e_j(M)$}{$\sum_{j\neq i}e_j(M)$}
{\bf 189; -18.}\del{$[x]$ ((three occurences))}
{\bf 189; -15.}\insa{(commutative)}{local}
{\bf 189; -4.}\rep{$\bar e_1$}{$e_1$}
{\bf 190; 18.}\insa{((end of line))}{Also, the hypothesis
``local'' is unnecessary: see Proposition 7.10.}
{\bf 194; 4.}\rep{((the first subscript)) $_j$}{((the subscript)) $_n$}
{\bf 195; 4.}\rep{{\it m}}{((fraktur)) $m$}
{\bf 195; 15.}\rep{$1+x$}{$1-x$}
{\bf 195; 17.}\rep{$1+a$}{$1-a$}
{\bf 195; 19.}\rep{$1-a+a^2-\dots$}{$1+a+a^2\dots$}
{\bf 195; 20.}\rep{$1+a$}{$1-a$}
{\bf 195; 21.}\rep{((the display))}{$(1-a)+(1-a)a+(1-a)a^2\dots$}
{\bf 195; 22.}\rep{$1+a^i$}{$1-a^i$}
{\bf 200; -15.}\insa{for each $i$}{and taking convergent
sequences to convergent sequences}
{\bf 201; 17.}\rep{$)^{i+j}$}{$)^{i+j-1}$}
{\bf 203; 19.}\rep {A1.3c}{A1.4c}
{\bf 204; 9.}\del{$\tilde K\subset$}
{\bf 204; 14.}\rep{$\tilde a \in \dots =\tilde K$}{$\tilde a \in R$}
{\bf 204; 16.}\rep{$\tilde K=\varphi(K)$}{$\tilde K\subseteq \varphi(K)$}
{\bf 204; 19--20.}\rep{((entire lines 19-20))}{so $\varphi$ is a
homomorphism and $\varphi(K)$ is a coefficient field containing $\tilde
K$. The previous paragraph shows that $\varphi(K)=\tilde K$}
{\bf 204; -11.}\insb{Since}{We may assume that $\bar u'_w$ and $\bar r_w$
are nonzero.}
{\bf 204; -10.}\rep{$k^q$}{$k$}
{\bf 217; 10.}\rep{1.15c}{1.15b}
{\bf 227; -4.}
\rep{Equivalently. it}{Equivalently, it}
{\bf 230; 18.}\rep{dimenion}{dimension}
{\bf 237; -18.}\del{and using Nakayama's lemma,}
{\bf 238; -11 -- -10.}\rep{parameter ideal}{ideal of finite
colength on}
{\bf 241; Figure 10.4.}\desc{The $X$ at the upper right should be $Y$;
the $Y$ at the lower right should be $X$}
{\bf 242; 4.} \rep{$R_P/PR_P$}{$R/P$}
{\bf 242; 6--8.}\rep{$R_P$}{$R$ ((three occurences))}
{\bf 244; 3.}
\rep {the maximal ideal is generated by $x$}
{the maximal ideal is generated by $y$}
{\bf 244; 4.}\rep{$k[x]_{(x)}$}{$k[y]_{(y)}$}
{\bf 244; 5.}\rep{$k(x)$}{$k(y)$}
{\bf 248; 20.}\rep{dimensionsion}{dimension}
{\bf 253; -12.}\rep{$ar=bs$}{$r^n\in (s)$}
{\bf 253; -12 -- -11.}\rep{a zerodivisor\dots of $s$.}
{nilpotent modulo $(s)$ and is contained in the minimal primes
of $(s)$.}
{\bf 253; -11 -- -11.}\rep{this}{each}
{\bf 253; -10.}\rep {associated}{minimal}
{\bf 254; -21}\del{the end of proof sign at the end of the line}
{\bf 254; -14.} \rep{Continuing}{To complete}
{\bf 254; -14.} \del{next}
{\bf 255; 2.}\ins{the end of proof sign at the end of the line}
{\bf 258; 17.}\rep{$R$}{$R_P$ ((two occurences))}
{\bf 258; 19.}\rep{Since\dots =0}
{Since ${\rm ker} (\varphi_i)_P\otimes R_P\varphi_i$ maps to
$(\varphi_i)_P{\rm ker} (\varphi_i)_P=0$}
{\bf 260; 4.}\insa{$K(R)$}{modulo the units of $R$} {\bf 260;
10.}\rep{so it}{. We have $Ru=Rv$ iff $u$ and $v$ differ by a unit of
$R$, so we may identify the group of principal divisors, under
multiplication, with the group $K(R)^*/R^*$. If $I$ is any invertible
divisor and $Ru$ is a principal divisor, then $(Ru)I = uI$. Thus it}
{\bf 276; -15 -- -13.}\rep{Suppose\dots . ((whole sentence))}
{Suppose that $q\subset R$ is an ideal of finite colength on $M$.
(($q$ should be fraktur))}
{\bf 276; -1.}\rep{$M/x_1,M$ ((part of the subscript in the middle))}
{$M/x_1M$ ((that is, delete the comma))}
{\bf 277; -14 -- -13.}\rep{parameter ideal}{ideal of finite colength
on}
{\bf 277; -11.}\rep{where\dots with}{where the polynomial $F$
has}
{\bf 277; -11.} \rep{whose degree is}{degree}
{\bf 278; 2.}\rep{((comma at the end of the display))}{((period))}
{\bf 278; 3,4.}\rep{((the entire two lines))}
{The equality shows that $F$ has positive leading term,
while the inequality gives the desired degree bound.}
{\bf 282; 12.}\rep{$(n)$ ((second occurence only!!))}{$(i)$}
{\bf 287; -3.}\rep{In fact, if}{If}
{\bf 288; 2.}\rep{$A$}{$R$}
{\bf 289; -3.}\rep{$x_1'-a_1x_e',\dots,x_{e-1}'-a_{e-1}x_e'$
((beginning of the displayed list))}
{$x_1'',\dots,x_{e-1}''$}
{\bf 291; -21.}\insa{a field}{, R is generated by $R_0$ over $R_1$,}
{\bf 291; -10.}\insa{is a field}{and $Q_0=0$}
{\bf 291; -7.}\del{$Q_0\oplus$}
{\bf 296; -3.}\rep{Let}{If $f$ is a unit the assertion is obvious.
Otherwise, let}
{\bf 298; 2,3.}\rep{$L$}{$L'$ ((two occurences))}
{\bf 301; 17.}\rep{Theorem 13.7} {Theorem 13.17}
{\bf 303; 8,9,10.}\rep{$S_I$}{$B_I$ ((Three occurences))}
{\bf 303; -1.}\insb{((the period))}{with equality if $R$ is
universally catenary}
{\bf 308; 1--3.}\rep{((the first paragraph))}
{We will prove Theorem 4.1 as the special case $e=0$ of
Corollary 14.9 to the much stronger Theorem 14.8. For
a direct proof see Exercise 14.1.}
{\bf 308; after 3, as a new paragraph.}\ins
{In general, a morphism $\varphi:\; Y\to X$ of algebraic
varieties is called {\it projective\/} if $\varphi$ can
be factored as $Y\to X\times {\bf P}^n\to X$ with the first
map a closed embedding and the second map the projection.
In these geometric terms, Theorem 14.1 says that a projective
morphism is {\it closed\/} in the sense that it takes
closed sets onto closed sets.}
{\bf 308; -21.}\rep{kernal}{kernel}
{\bf 310; -3.}\rep{Andre}{Andr\'e}
{\bf 316; after line 10; just after theorem 14.8.}
\ins {Let us restate
Theorem 14.8 (or rather its consequence for reduced affine algebras
over an algebraically closed field) in geometric terms: Suppose that
$R\to S$ corresponds to a morphism of varieties $\varphi:\;Y\to
X$. Set $F_e=\{x\in X\mid {\rm dim\ }\varphi^{-1}(x)\geq e\}$ and let
$G_e$ be the set of all points of $y$ so that the fiber
$\varphi^{-1}(\varphi(y))$ has dimension $\geq e$ locally at
$y$. That is, $G_e$ is the union of the large components of the
preimages of points of $F_e$. Theorem 14.8 says that $G_e$ is defined
by the ideal $I_e$ and is thus closed. If the morphism $\varphi$ is
projective then $F_e$ is defined by $J_e$, and is closed as well.
Note that $F_e$ is the image of $G_e$, so we could deduce part b of
Theorem 14.8 from part a together with Theorem 14.1---if it weren't
that we will only prove Theorem 14.1 by using part b.}
{\bf 319; 2.}\rep{principal}{principle}
{\bf 330; -17.}\rep{((the boldace type))}{((roman type))}
{\bf 331; 13.}\insa{((end of line))}{((an ``end of proof'' sign))}
{\bf 332; -18.}\rep{with basis $F$}{$F$ with basis}
{\bf 341; 4.}\rep{irreducible}{irreductible}
{\bf 342; -19 -- -18.}\del{refines the order by total degree and}
{\bf 342; -15.}\rep{Equivalently, as the reader may check, a}
{A}
{\bf 342; -15.}\rep{is ((last word))}{may be}
{\bf 342; -14.}\del{either\dots same and}
{\bf 374; 14 (first line of Exercise 15.33).}\rep{$x=$}{$X=$}
{\bf 411; 16.}\rep{an algebra map}{a surjective algebra map}
{\bf 417; -13.}\rep{Let}{Suppose
that $R$ contains a field of characteristic 0, and let}
{\bf 417; -8.}\insa{field}{of characteristic 0}
{\bf 430; 4.}\rep{$M$}{$M\neq 0$}
{\bf 430; 5.}\rep {some $k$}{some $k