%Insert after
\def\insa#1#2{\par\noindent After\smallskip #1\smallskip
\noindent insert:\smallskip #2\smallskip\hrule
\goodbreak\smallskip}
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\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 $k<n$}

{\bf 434; 12.}\rep{mapping cylinder}{mapping cone}
{\bf 434; 19.}\rep{mapping cylinder}{mapping cone}


{\bf 436; 3.}\rep{$x_r$}{$x_n$}
{\bf 436; 3.}\rep{$\neq$}{$\neq 0$}
{\bf 436; 4.}\rep{$x_r$}{$x_n$}
{\bf 436; 4.}\rep{17.4}{17.14}


{\bf 437; 8.}\rep{$m\otimes1-1\otimes m$}  {$m\otimes1+1\otimes m$}

{\bf 439; -3.}\desc{The two arrows labeled with $\beta$ should point
  downwards.}
{\bf 439; -2.}\rep{$K(x^\ast)$}{$K'(x^\ast)$}

{\bf 440; 5.} \rep {$H_{n-k}(K(x),\delta_{x^\ast})$}
{$H_{n-k}(K'(x^\ast),\delta_{x^\ast})$}

{\bf 443; -18.}\rep{$S/(x_1,\ldots,x_r)$}{$S/(y_1,\ldots,y_r)$}

{\bf 444; 3.}\rep{$e_j$}{$e_J$}

{\bf 446; 9.}\rep{3.16}{13.16}

{\bf 448; 8.}\desc{The $\sigma$ should be situated on the final arrow.}


{\bf 459; -3.}\del{Let $R$ be a Cohen-Macaulay ring.}


{\bf 462; 13.}\rep{11.12}{11.10}
{\bf 462; -16.}\rep{S1'}{S1}

{\bf 466; -10.}\rep{7.17}{7.7}

{\bf 468; 11.}\rep {Fulton [1992]} {Fulton [1993]}

{\bf 471; 10.}\rep{a domain iff $r\ge 3$}{a normal domain iff $r\ge 3$}
{\bf 471; 10.}\del{normal iff $r\geq 4$;}

{\bf 473.}
\insb{the first line of text ((as on p. 423))}
{{\it In this chapter all the rings 
considered are assumed to be Noetherian}} 

{\bf 476; 2}\rep{$R$-module}{module over a regular local ring of
dimension $n$}

{\bf 477; 11.}\rep{$PF_n$}{$PF_{n-1}$}
{\bf 477; 11.}\rep{$i$}{$i\geq 0$}

{\bf 478; -11.}\rep{Corollary 15.13}{Corollary 15.11}
{\bf 478; -10.}\rep{Corollary 19.6}{Corollary 19.7}

{\bf 484; -9 -- -8.}\rep{It turns\dots remark.}
{We begin with a simple observation:}

{\bf 487; 13.}\del{((the end of proof symbol at the end of the line))}
{\bf 487; -6.}\rep{ $R_P$}{$R[x^{-1}]_P$}
{\bf 487; -1.}\ins{((an end of proof symbol at the end of the line))}

{\bf 490; -18.}\rep{${\rm dim}_k(M_d)$}{${\rm dim}_k(M_d)t^d$}

{\bf 492;3.} \insb{the homogeneous}{which is}

{\bf 492; -7.}\rep{polynomial}{family of polynomials $c_t(M)$}

{\bf 492; -4.} \del{Chern and}

{\bf 492; -5 -- -1.} \rep{((the whole last paragraph))}
{Knowing the Hilbert polynomial of a module $M$ is
equivalent to knowing the Chern polynomial
$c_t(M) = 1+c_1(M)t+c_2(M)t^2+\dots$ mod $t^{n+1}$ and the rank
of $M$. We sketch this equivalence:
The Hirzebruch-Riemann-Roch formula allows one
to deduce the Hilbert polynomial from the Chern
classes and the rank as follows:
First form the {\it Chern character,\/} which
is a power series of the form 
${\rm rank}(M)+c_1(M)t+{1\over 2}(c_1(M)^2-2c_2(M))t^2+\dots$ mod $t^{n+1}$.
The Hilbert polynomial $P_M(d)$ is equal, for large $d$,
to the coefficient of $t^n$ in the expression
$$e^{nt}\bigl({t\over 1-e^{-t}}\bigr)ch(M).$$ 
See Hartshorne [1977]
Appendix A4, or Fulton [1984], especially chapter 15.  
Conversely, to compute the
Chern polynomial from the Hilbert polynomial, consider first the
special case 
$M=S_j:=S/(x_1,\dots,x_j)$.  
From the Koszul complex
resolving $M$ and properties i and ii above we see
$$
c_t(S_j)=
{
\prod_{i\ {\rm even}}(1-it)^{n-j\choose i}
\over 
\prod_{i\ {\rm odd}}(1-it)^{n-j\choose i}
}
{\rm mod }t^{n+1}.
$$
Now any Hilbert polynomial can be written (uniquely) in the
form 
$P_M=\sum_{j\geq 0}a_jP_{S_j}$.
It follows that the Chern polynomial of $M$ is
$c_t(M)=\prod_{j\geq 0}a_jc_t(S_j)$.
Of course the rank
of $M$ is $a_0$, completing the argument.
}



{\bf 493.}
\insb{the first line of text ((as on p. 423))}
{{\it In this chapter all the rings 
considered are assumed to be Noetherian}} 

{\bf 497; -15.}\del{((the end of proof symbol at the end of the line))}
{\bf 497; -4.}\rep{$I_j\varphi = I_{j+p}\varphi$}{$I_j\varphi = I_{j+p}\psi$}

{\bf 502; -3.}\del{((the end of proof symbol at the end of the line))}

{\bf 506; -14.}\insa{is exact,}{$F_2$ is free,}
{\bf 506; -10.}\rep{ $I(\varphi_2)$}{ $I_{n-1}(\varphi_2)$}

{\bf 510; -7.}\rep{${\rm Ext}^j(M,R)_n$}{${\rm Ext}^j(M,S)_n$}

{\bf 521; -12.}\rep{((the entire line))}
{${\rm Ext}^j(M,S)_k=0$ for all $j\le n-1$ and $k=-m-j-1$.}

{\bf 529; 14.}\rep{By Proposition 9.2}{By hypothesis}

{\bf 529; -19.}\insb{whose}{$\varphi$}

{\bf 530; 1.}\insa{Zariski}{Gorenstein rings were defined---in exactly
the way done in this book!---in the artinian and graded cases
by Macaulay in his last paper [1934] (his definition in the affine
case differs slightly from the modern one).}

{\bf 530; 7.}\insa{attests.} {There is also a well-developed noncommutative
theory (Frobenius and quasi-Frobenius rings.)}


{\bf 530; -11.}\rep{Proposition 21.2}{Corollary  21.3}

{\bf 531; 4.} \insa{$x_i$}{and $x_i^{-1}$}

{\bf 531; 5.} \rep{the $S$-module\dots $T$}{$T$ with a 
sub-$S$-module of $K(S)/L$}

{\bf 531; 10.} \insa{generated}{nonzero}

{\bf 531; -14.}\rep{$:=$}{$=$}

{\bf 531; -10.}\insa{any}{ nonzero finitely generated}

{\bf 531; -2.}\insa{).} {(Proof: One checks by linear algebra that these
elements generate the part of the annihilator in degrees $\leq 2$, 
even as a vector space. These elements generate all forms
of degree $\geq 3$.)}

{\bf 532; 11.}\insa{$x\in A$}{that is also a nonzerodivisor on $\omega_A$}

{\bf 532; -18.}\rep{21.4d}{21.5d}


{\bf 533; -8.}\insa{condition.}{Note that c implies that the 
annihilator of $W$ is 0, just as in the special case of Prop. 21.3}

{\bf 533; -3.} \rep{..}{. ((after the Proposition number))}

{\bf 539; 14.}\rep{$x$}{$x\in R$}

{\bf 543; -17.}\rep{$M/xM$}{---}

{\bf 544; 13.}\rep{$x_0, x_1$}{$x_1, x_2$} 

{\bf 544; 18.}\rep{Enzykolp\"adie}{Enzyklop\"adie}

{\bf 547; -6.}\rep{$A^n$}{$A^{n+1}$}

{\bf 549; -7.}\insa{such that}{$x$ is a nonzerodivisor on $W$ and}

{\bf 550; first line of Exercise 21.1}\insa{$y^n)$}{, $n\geq 2$}
{\bf 550; -5.}\rep{$\omega_R$}{$R$}

{\bf 551; 1.}\insa{and}{, with the notation introduced just
after Proposition 21.5,}

{\bf 551; first two displays.}\rep{$$\sum_{i_1,\dots,i_r}$$}
{$$\sum_{i_1,\dots,i_r\geq 0}$$((two occurences))}

{\bf 554; -1.}\del{((this entire line))}

{\bf 556; -7.}\rep{degree}{deg ((two occurences))}

{\bf 559; 6.}\rep{$x_1$}{ $x_0$}
{\bf 559; 13.}\rep{$\sum d_i-r$}{$\sum d_i-r-1$}

{\bf 575; 8--12 and 13--19.}\desc{((These two paragraphs
are both definitions, and should be set in italics, with
a skip below the paragraph, just as with the first paragraph on
this page.))}

{\bf 581; 13.}\insb{module}{of rank $n$}
{\bf 581; 15.}\rep{$\lambda^d$}{$\lambda^{n-d}$}

{\bf 609; -20.}\rep{corollary}{theorem}
{\bf 609; -7.}\rep{Corollary}{Theorem}

{\bf 623; 10.}\rep{a variety}{an affine variety}

{\bf 630; -17.}\rep{both $E$}{both $E'$}


{\bf 630; -4.} \rep{b.$^*$}{b.}
{\bf 631; 1.} \rep{c.}{c.$^*$}


{\bf 645; -9.}\del{$0\to$}

{\bf 652; -11.}\rep{$E^1(A,B)$}{$E_R^1(A,B)$}
{\bf 652; -15.}\rep{$0:\ 0\to A\to A\oplus B\to B\to 0$}
{$0:\ 0\to B\to A\oplus B\to A\to 0$}

{\bf 653; 10.}\rep{$A.$}{$B.$}

{\bf 654; -9.}\rep{$A,B$}{$B,A$}
{\bf 654; -7.}\rep{$A,B$}{$B,A$}
{\bf 654; -6.}\rep{$A,B$}{$B,A$}
{\bf 654; -6.}\rep{an injective}{a projective}
{\bf 654; -5.}\rep{a projective}{an injective}

{\bf 655; 5.}\rep{$A$}{$C$}
{\bf 655; 7.}\rep{$C$}{$A$}
{\bf 655; 10.}\rep{$C$}{$A$}
{\bf 655; 10.}\rep{$A$}{$C$}


{\bf 661; -7.}\rep{$F\to G$}{$G\to F$}

{\bf 669; 3,4.}\rep{$F$}{$A$ ((three occurences))}
{\bf 669; 5.}\del{$\bar F$}
{\bf 669; 8.}\rep{$F$}{$A$}
{\bf 669; 10.}\rep{$F$}{$A$ ((two occurences))}


{\bf 671; -3.}\rep{We consider}
{Let $A=\oplus_{p\in {\bf Z}}G^p$, and let $\alpha:\; A\to A$
be the map defined by the inclusions $G^{p+1}\subset G^p$.
We consider}

{\bf 671; -2.}\rep{$H(G)$}{$H(A)$}
{\bf 671; -2.}\rep{$H(G/\alpha G)$}{$H(A/\alpha A)$}

{\bf 672; Figure A3.29.}\rep{$G$}{$A$ ((four occurences))}
{\bf 672; -9.}\rep{horizontal}{vertical}

{\bf 674; 9.}\insa{$G^q)^p$}{$=\oplus_{i+j=q,j\geq p}F^{i,j}$}

{\bf 674; 10.}\insa{then in}{$(G^{q+1})^{p+r}$}

{\bf 674; 11.}\desc{((delete the display, and close up))}

{\bf 688; 11.}\rep{$G\circ P\rightarrow LF\circ P\rightarrow F$}
{$G\circ P\rightarrow LF\circ P\rightarrow P\circ KF$}

{\bf 694; -10.}\rep {$R$-module}{$S$-module}
{\bf 694; -9.}\rep {$R$}{$S$}
{\bf 694; -8.}Replace two occurences of $R$ by $S$. That is,\hfill\break
\rep {${\rm Hom}_R(---,E(R/P))$}{${\rm Hom}_S(---,E(S/P))$}

{\bf 697; 5.}\rep{collections}{collection}

{\bf 728; -9.}\del{(by Lemma 3.3)}

{\bf 728; -9.}\rep{$\in R_i$}{$\in R$}

{\bf 743; 16.}\rep {b.} {c.}

{\bf 746; 13--17.}\rep{((the complete text of the hint for
Exercise 20.20 by))}
{It follows from local duality that if $S=k[x_0,\ldots,x_n]$ is
the homogeneous coordinate ring of ${\bf P}^n$, then
$$
H^j({\bf P}^n; {\cal F}(m-j))^{\vee} \cong
{\rm Ext}^{n-j}(M,S)_{-n-1-(m-j)}
$$ 
for all $j$. This gives a. If $M$
has depth $\ge 2$ we have ${\rm Ext}^{n-j}(M,S)=0$ for $j\le 0$ and
statement b becomes a direct translation of Proposition
20.16 and Theorem 20.17.}
 
{\bf 746; -1.}\desc{((The dots in the right part of the diagram should
run from SE to NW and should connect the two pieces of this part of the
diagram, in analogy with what happens in the left part of the diagram.))}

{\bf 747; 15.}\rep{Theorem 18.4}{Proposition 18.4}

{\bf 747; 18.}\rep{((The displayed sequence))}
{$$
   0\rightarrow {\rm Ext}^d_R(M'',R)
    \rightarrow {\rm Ext}^d_R(M,R) 
    \rightarrow {\rm Ext}^d_R(M',R) 
    \rightarrow  0
$$}


{\bf 751; -19.} \rep{b.}{c.}
{\bf 751; -14 -- -11.} \rep{For this value\dots $Q$ is injective
((three sentences))} {The map $\alpha$ thus factors through
the projection $I\to I/P^dI$, so by Artin-Rees it factors
through the projection $I\to I/(P^e\cap I) \subset R/P^e$ for large $e$.
Since $Q^{(e)}$ is injective over $R/P^e$, we can extend this
to a map $R/P^e\to Q^{(e)}$, and thus to a map $R\to Q$,
proving that $Q$ is injective.}


{\bf 758; 5.}\rep{Th\'eoremes}{Th\'eor\`emes}

{\bf 758; -19 -- -18.}\rep{((italics))}{((roman))}
{\bf 758; -18.}\rep{Preprint.}
{Inst.~Hautes \'Etudes Sci.~Publ.~Math.~86 (1997) 67--114.}

{\bf 759; 9--13.}\rep{entire reference}{Bayer, D., and M.~Stillman
(1982-1990) Macaulay: A system for computation in algebraic geometry
and commutative algebra. Source and object code available for many
computer platforms. See http://www.math.uiuc.edu/Macaulay2/ for a
pointer to this and to the successor program Macaulay2 by D.~Grayson
and M.~Stillman.}

{\bf 759; 22.}\rep{{\it Alg\'ebre}}{{\it Alg\`ebre}}
{\bf 759; 28.}\rep{Alg\`ebre}{{\it Alg\`ebre}}

{\bf 759; -15.}\ins{93 (1994) 211--229. ((at end of line))}

{\bf 760; 6.}\ins{((as new reference))
Buchsbaum, D.~A. (1969).  Lectures on regular local rings.
In {\it Category theory, homology theory and their applications, I. \/}
Battelle Inst. Conf., Seattle, Washington., 1968, Vol. 1, 13--22.
Springer Lect.~Notes in Math. 86}


{\bf 760; -8.}\rep{1944}{1943}

{\bf 761; -9.} \rep{Schopf}{Sch\"opf}

{\bf 762; 6.}\rep{in the}{is the}

{\bf 762; -13 -- -12.}\rep{(in press).}{Duke Math. J. 84 (1995) 1--45.}

{\bf 763; -18 -- -14.}\rep{the whole Gelfand-Manin reference}
{Gelfand, S.~and Y.~I.~Manin (1989--1996). {\it Methods of homological
algebra.\/} Springer-Verlag, Berlin, 1996. English translation of 
the 1989 Russian original.}

{\bf 764; 24.}\rep{Les foncteurs \dots continu ((the whole title))}
{Cat\'egories d\'eriv\'ees et foncteurs d\'eriv\'es}

{\bf 764; -14.}\rep{point}{points}

{\bf 766; 1.}\rep{Der Frage}{Die Frage}

{\bf 766; 15.}\rep{((boldface type))}{((roman))}

{\bf 767; -15.}\rep{Time}{Times}

{\bf 767; -12.}\rep{van}{von}

{\bf 768; 11.}\rep{Birkhauser}{Birkh\"auser}

{\bf 769 Insert as a new referece.}\ins{Macaulay, F.S.~Modern
algebra and polynomial ideals, {\it Proc.~Cam.~Phil.~Soc.\/} 30,
pp. 27--64.}

{\bf 770; -14.}\rep{van}{von}

{\bf 770; -12}\rep{in press}{1996}

{\bf 771; 7.}\rep{Rabinowitch}{Rabinowitsch}

{\bf 772; -12.}\rep{Bemerking}{Bemerkung}
{\bf 772; -12.}\rep{van}{von}

{\bf 773; -18.}\rep{Uber}{\"Uber}
{\bf 773; -16.}\rep{Bezout}{B\'ezout}
{\bf 773; -2.}\rep{{\it Its}}{{\it its}}

{\bf 774; 4.}\rep{1994}{1995}

{\bf 774; 5.}\rep{(preprint)}{J.~Algebraic Combin. 4 (1995) 253--269.}

{\bf 775; -16 -- -15, second column.}
\ins{$\kappa(P)$, residue field, 60}

{\bf 776; -0, second column.}\ins{$E(M)$, injective envelope, 628}

{\bf 785; 20, second column.}\rep{18}{17}

{\bf 789; 10, second column.}\rep{Kozul}{Koszul}

{\bf 790; 8 of second column.}\insa{M-sequence,}{243,248,423ff,}

{\bf 793; -17 of first column.} \rep{167}{308,316}

\end