%Insert after \def\insa#1#2{\par\noindent After\smallskip #1\smallskip \noindent insert:\smallskip #2\smallskip\hrule \goodbreak\smallskip} %Insert before \def\insb#1#2{\par\noindent Before\smallskip #1\smallskip \noindent insert:\smallskip #2\smallskip\hrule \goodbreak\smallskip} %Insert \def\ins#1{\par\noindent Insert:\smallskip #1\smallskip\hrule \goodbreak\smallskip} %delete \def\del#1{\par\noindent Delete:\smallskip #1\smallskip\hrule \goodbreak\smallskip} %replace \def\rep#1#2{\par\noindent Replace:\smallskip #1\smallskip \noindent by:\smallskip #2\smallskip\hrule \goodbreak\smallskip} %describe \def\desc#1{\par\noindent #1\smallskip\hrule \goodbreak\smallskip} \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