\magnification=\magstep1
\baselineskip=13pt
\parskip=5pt
\def \Box {\hbox{}\nobreak \vrule width 1.6mm height 1.6mm
depth 0mm \par \goodbreak \smallskip}
\input amssym.def
\rightline {Updated: July 11, 1996}
\smallskip
\centerline{Corrections to}
\centerline{\bf Commutative Algebra with a
View Towards Algebraic Geometry}
\medskip
\centerline{by}
\smallskip
\centerline{David Eisenbud}
\bigskip
\beginsection Introduction
\item{p. 6} line 2* and p. 8, line 2*
Macdonald is misspelled (the D should
be d).
\item{p. 7} line 17: replace `` is'' (at end of line) by ``are''
\item{p. 7} line 3*: replace 1.11 by 1.9
\beginsection Chapter 0.
\item{p. 11} line 12*: Replace this line by:
\smallskip
\noindent A {\bf ring} is an abelian group $R$ with
an operation $(a,b)\mapsto ab$ called {\it multiplication}
\medskip
\item{p. 11} lines 6*-4*: Replace the sentence beginning
``Nearly\dots by `` In this book we use the word ring to denote
a commutative ring, with a very few exceptions that will be
explicitly noted.''
\item{p. 11} line 1*: after ``in which'' insert $1 \neq 0$ and
\item{p. 12} line 8: replace $R$ by $I$
\item{p. 15} line 21: change ``$I=(i), J=(ij)$, and $i$ is a
nonzerodivisor'' to
``$I=(ij), J=(j)$, and $j$ is a nonzerodivisor''
\item{p. 15} line 17*: Add to this paragraph the sentence:
``For example, we say that a nonzero element $r\in R$ is
a {\bf nonzerodivisor on $M$} if $(0:_{M}r) = 0$; that is,
if $r$ annihilates no nonzero element of $M$.''
\item{p. 16} line 11: ``Corollary 4.5'' should be ``Corollary 4.4''
\beginsection Chapter 1.
\item{p. 16} line 2*: ``generated'' should be ``generate''
\item{p. 22}
\itemitem line 3*: change ``unfortunately'' to ``but''
\itemitem line 2*: delete the words ``after all''
\itemitem line 1*: replace ``spawned.'' with ``spawned, including a large
chunk of commutative algebra. The amazing recent
proof of Fermat's last theorem by Wiles and Taylor continued
this tradition: a small but significant step in the proof involves
a novel argument about Gorenstein rings.)
\item{p. 23} line 5: after [1881] insert (see also Edwards [1990])
\item{p. 24} line 14*: change [1992] to [1993]
\item{p. 24} line 3*: replace $R$ by $k$
\item{p. 26} line 3 of the footnote: after ``celebrated''
insert (see for example Browder [1976] or Kaplansky [1977])
\item{p. 26} last line of the footnote: change [1986] to [1990]
\item{p. 27} line 19: before the word ``collection'' insert
``nonempty''
\item{p. 28} line 15*: before the word ``collection'' insert
``nonempty''
\item{p. 29} lines 20*-18*: Change the long sentence
beginning ``If $G$...'' to:
If $G$ is $SL_n(k)$ or $GL_n(k)$ we suppose further that
the matrices $g\in G$ act on $k^r$ as matrices whose
entries are rational functions of the entries of $g$;
such a representation of $G$ is called {\it rational\ }.
\item{p.30} line 7: ``for $j$'' should be ``for $j >>0$''.
\item{p.31} line 11: before the period, insert ``is a
homogeneous polynomial''.
\item{p.31} line 20. Replace the last sentence of the proof
by: Thus any homogeneous element of $R$ is in $R'$.
But if $f\in R$ is any element, then applying $\phi$ to
each homogeneous component of $f$ we see that $f$ is
a sum of homogeneous elements of $R$, so we are done.
\item{p.32} line 10: ``called an affine'' should be
``called affine''
\item{p.32} line 21*. insert ``nonempty'' at the beginning of
the line. Near the end of the line,
change ``to two'' to ``of two nonempty''.
\item{p.33} line 4*. Change ${\bf A}^n_k$ to ${\bf A}^n(k)$
\item{p. 37} line 1*. Change $fg$ to $f\circ g$
\item{p. 41} line 15*. Before ``algebraic'' insert
``projective''
\item{p. 42}
\itemitem{line 13*} Change $x_1$ to $x_0$
\itemitem{line 12*} Change $r-1$ to $r$
\item{p.43} line 12: change ``integer $s$'' to ``integer $s\geq 0$''
\item{p.43} line 14, 15: change `` $H(s)$. For all $s$ we have''
to ``$H(s)$, and''
\item{p.46} line 2: the last ``$s+d$" should
be $s+d+1$
\item{p.47} line 6*: change $\phi$ to $\pi$
\item{p.49} line 11: change ``imples'' to ``implies''
\item{p.49} line 16: change ``$Z(X)$'' to ``$I(X)$''
\item{p.55} line 9: before the period at the end of the line
insert ``and the empty set''.
\beginsection Chapter 2.
\item{p. 58} line 10*. Change ${\bf A}^r_k$ to ${\bf A}^r(k)$
\item{p. 61} line 4*. change ``$S$ is a domain'' to ``$S/I$
is a domain''
\item{p. 63} line 3. change ${\rm Hom}($ to ${\rm Hom_{R}}($
\item{p. 63} line 4. change ${\rm Hom}($ to ${\rm Hom_{R}}($
\item{p. 63} line 21. change $M/A$ to $B/A$
\item{p. 73} line 19: ``then clearly $M \supset M_i$." should be
``the claim follows since $M_i=0$."
\item{p. 73} line 24: change $M' \cap M_{i} =M' \cap M_{i}$
to $M' \cap M_{i+1} =M' \cap M_{i}$
\item{p. 79}
\item\item {line 5 of problem 2.3} Change
``sums'' to ``inclusions''.
\item\item {line 9 of problem 2.3.} Delete the first
superscript $\infty$, and delete from ``where'' to
the end of the sentence. Thus sentence should end:
``have $R\cap JR[U^{-1}]=\sum_{f\in U}(J:f)$.
\item{p.82} line 13: replace ``if $x$'' by ``if $x\in R$
\item{p.85} Exercise 2.23. Add a $^*$ to the exercise
number (indicating that there is a hint, as added below).
\item{p.85} line 7*: delete the $^{*}$
\beginsection Chapter 3.
\item{p.90}{line 3} The subscript $u$ in the display should be
a subscript $U$
\item{p. 90} After line 8 insert the following paragraph,
set off by smallskips.
\smallskip
Like maximal ideals, primes minimal over a given ideal
$I$ exist in {\it any} ring. To see this, note first that
if a set of prime ideals in a ring $R$ is totally ordered by
inclusion, then the intersection of these primes is again
prime. By Zorn's Lemma we may find a maximal totally ordered
subset of the primes containing $I$, and the intersection of
the primes in this subset is necessarily minimal over $I$.
By Theorem 3.1a, the set of primes minimal over $I$ is
finite if $R$ is Noetherian. This
result generalizes the statement that a nonzero polynomial in
one variable can have only finitely many roots.
\smallskip
\item{p.90} lines 19-22. Replace these lines (Cor. 3.2 and
the sentence before) by:
Theorem 3.1 yields a surprising dichotomy:
\proclaim Corollary 3.2. Let $R$ be a Noetherian ring, let
$M$ be a finitely generated, nonzero $R$-module, and let
$I$ be an ideal of $R$. Either $I$ contains a nonzerodivisor
on $M$ or $I$ annihilates an element of $M$.
\item{p.91} line 8*: Change ``an $R$-module'' to `` a
nonzero $R$-module''
\item{p.95} line 15: Change ``some ideal $I$'' to
``some ideal $I\neq 0$''
\item{p.95} line 7*: Change ``primary submodules'' to
``finitely many primary submodules''
\item{p.96} line 3: Change ``of $M'$.'' to ``of 0 in $M'$.''
\item{p.96} before line 10: After the statement of the
theorem, and before the proof, add the following paragraph:
``The language in part c is often stretched, and
the module
$M_i$ is referred to as the $P_i$-primary component of $M$
in the given decomposition (it may depend on the decomposition).
\item{p.96} line 14: Change ``irreducible submodules'' to
``finitely many irreducible submodules''
\item{p.100} line 20: change ``closed'' to
``intersections of ${\rm Ass\ }M$ with the open''
\item{p.101} line 16*: $P$ should be $P_i$
\item{p.111} line 6: replace [in press] by [1995]
\item{p.112} line 10*: before ``show'' insert:
``and given a minimal primary decomposition $0 = \cap M_i$
as in Theorem 3.10,''
\item{p.112} line 9*: replace ``primary components of $0$
in $M$'' by ``$M_i$''
\beginsection Chapter 4.
\item{p.118} line 2: Replace ``basis'' by ``free basis''
\item{p.119} line 15: Replace ``${\bf Q}$'' by ``${\bf Q}(\sqrt 5)$''
\item{p.121} line 3: Replace $(x)$ by $(t)$
\item{p.121} line 20: change $p$ to $P$
\item{p.123} line 5*: ``Proposition 4.5" should be ``Corollary 4.5"
\item{p.130} line 13: after ``that of $R$'' add ``(in the sense
that every element of the big field is the root of a nontrivial
polynomial
with coefficients in the small field)''
\item{p.131} line 21: change ``3.10'' to ``13.10''
\item{p.134} line 11: change the subscript $_{n}$ to $_{r}$
\item{p.134} line 4*: Delete from the sentence ``By Theorem...''
to the end of the page. Replace this text by:
\noindent By Theorem 4.19 the ring $S := k[x_1,\ldots,x_n]$ is
a Jacobson ring, so every prime ideal of $S$, and in
particular every prime ideal that contains
$I$, is an intersection of maximal ideals. Thus $I(Z(I))$
is equal to the intersection of all the prime ideals containing $I$.
By Corollary 2.12, this is ${\rm rad}(I)$, proving the
formula.
The equality $Z(I(X)) = X$ follows directly from the definition
of an algebraic set. The formula just proved shows
that if $I$ is a radical ideal then
$I(Z(I)) = I$. Thus
the functions $Z$ and $I$ are inverse bijections between
algebraic sets and radical ideals as claimed.\hfill\Box
\beginsection Chapter 5.
\item{p.146} line 4: change ``large $i$'' to ``large $n$''
\item{p.146} line 13: change ``later in this chapter" to
``after Proposition 5.3".
\item{p.148} line 4: Change ``an $R$-module'' to
``a finitely-generated $R$-module''
\item{p.148} line 30 = line 8*: the $g_{j}$ at the end of the line
should be $g_{j}t$
\item{p.149} line 18: change $B_{\cal I}$ to $B_{I}$
\beginsection Chapter 6.
\item{p.156} line 16: change ``maximal'' to ``prime''
\item{p.156} line 19*: delete ``, $U\cong \phi(U)$''
\item{p.157} line 2: change first character of the line,
``$X$'', to ``$x$''
\item{p.157} line 12: Append ``We have chosen cases where (most
of) the fibers are finite sets.''
\item{p.158} line 13*: replace ``in not flat'' by ``is not flat''
\item{p.158} line 7*: $X$ should be $x$
\item{p.158} line 5*: replace ``$=R[x]$'' (at the end ofthe line)
by ``$=k[x]$
\item{p.159} line 13*: Delete the sentence beginning
``So that the reader may judge the merits\dots''
\item{p.159} line 8*-5*: Replace these four lines by
\smallskip
\item {1.} If $M$ and $N$ are $R$-modules, and
$\ldots \to F_{i+1}\to F_{i}\to F_{i-1}\to
\ldots \to F_{0} \to M\to 0$
is a free resolutios of
$M$ as an $R$-module, then
${\rm Tor}^{R}_{i}(M,N)$
is the homology at $F_{i}\otimes N$ of the complex
$F_{i+1}\otimes N\to F_{i}\otimes N\to F_{i-1}\otimes N$;
that is, it is the kernel of
$F_{i}\otimes N\to F_{i-1}\otimes N$
modulo the image of
$F_{i+1}\otimes N\to F_{i}\otimes N$.
\item{p. 162}
The elementary ``proof'' offered is garbage --- the
``it follows'' just in the middle of the page is too
optimistic. Therefore:
Delete the last 2 lines of page 161 and all but the
last 3 lines of p. 162.
\item{p. 163} line 1*: in the display, before ``$for\ all\ i$''
insert ``$\ in\ M\ $''
\item{p. 165} line 9*: change ``$m_{j}$'' at the beginning of the
line to $m_{i}$
\item{p. 166} line 19: change ``$1+st$'' to ``$1-ts$''
\item{p. 172}, Exercise 6.5. Change each $P$ (four occurences)
to $\frak m$, and each $Q$ (two occurences) to $\frak n$
\item{p. 173}, line 2*. delete ``should be $((x)\cap(x,t)^{2})$''
\item{p. 175}, lines 5,6. Replace the sentence
beginning ``We may think of'' by
For each prime ideal $P\subset R_0$ with residue
field $\kappa(P) = K(R_0/P)$ we have a graded module
$\kappa(P)\otimes M$ over the ring $\kappa(P)\otimes R$;
Thus $M$ gives rise to a family of graded modules, parametrized
by ${\rm Spec}(R_0)$.
\item{p. 175}, line 12: replace ``Hartshome'' by ``Hartshorne''
\beginsection Chapter 7.
\item{p. 180} fig 7.1: in the figure, replace ``$k[x,y]_{(x,y+1)}$''
by ``$k[x,y]$''
\item{p. 181}line 19: replace ``$\hat m$'' with
``$\hat {\frak m}$''
\item{p. 181}line 20: replace ``$\hat m_{1}$'' with
``$\hat {\frak m}_{1}$''
\item{p. 181}line 24: \itemitem{} replace ``$\hat m$'' with
``$\hat {\frak m}$''
\itemitem{} replace ``$\hat R/\hat {\frak m}\hat R_{\hat {\frak m}}$''
by
``$\hat R_{\hat {\frak m}}/\hat {\frak m}\hat R_{\hat {\frak m}}$''
\item{p. 187} para. 3 line 3: replace (f,g) by (g,h).
\item{p. 188} para. 5 last line: the first $e_n$ should have
a bar over it like the others.
\item{p. 188} para. 7: replace ``by" with ``be".
\item{p. 192}line 13: replace
``$\hat {\frak m}_{n}\subset (\hat {\frak m}_{1})^{n}$''
by
``$(\hat {\frak m}_{1})^{n}\subset \hat {\frak m}_{n}$''
\item{p. 192}line 18: replace ``$/{\frak m}_{n}$''
by ``$/\hat {\frak m}_{n}$''
\item{p. 194} line 7* and 6*: after the second ``Noetherian''
insert `'and ${\frak m}/{\frak m}^{2}$ is a finitely generated
$R/{\frak m}$-module.'' Replace the following
sentence, `` The ring\dots `' up to is Noetherian.''
by : ``The ring ${\rm gr}_{\frak m} R$ is generated
as an $R/{\frak m}$-algebra by any set of generators for the
module ${\frak m}/{\frak m}^{2}$. Thus by the Hilbert Basis
Theorem (Theorem 1.2) ${\rm gr}_{\frak m} R$ is Noetherian.''
\beginsection Chapter 8.
\item{p.214} line 15*: change [1935] to [1985]
\beginsection Chapter 9.
\item{p.226} line 5*: replace ``$M=$'' by ``$(M\cup L) =$''
\item{p.226} line 2*:replace ``$M$'' by ``$M\cup L$''
\item{p. 228} Replace Exercise 9.1 with the following:
\itemitem {a)} Let $R$ be a Noetherian local ring. If the
maximal ideal of $R$ is principal, show that every ideal of $R$
is principal, and any nonzero ideal of $R$ is a power of
the maximal ideal.
\itemitem{b)} Deduce from part a) that if each maximal ideal
of a Noetherian ring is principal, then the ring has
dimension $\leq 1$.
\item{p. 228} 18*. Replace ``is monic in $y$'' by ``is a scalar times a
monic polynomial in $y$''
\item{p. 229} line 2*. replace ``every element'' by ``every nonzero
element''
\beginsection Chapter 10.
\item{p. 234} line 12*: In the section title, replace ``Parameter Ideals''
by ``Ideals of Finite Colength''
\item{p. 235} line 3 of text: replace {\bf parameter ideal for $R$}
by {\bf ideal of finite colength}
\item{p. 235} line 13 of text: replace `` a parameter ideal for''
by ``an ideal of finite colength on''
\item{p. 235} line 11*, line 10* and line 7* (three occurences):
replace ``is a parameter ideal for'' with ``has finite colength on''
\item{p. 235} line 5*: replace ``a parameter ideal for'' with
``an ideal of finite colength on''
\item{p. 237} Theorem 10.10: replace ``map of local rings" by
``map of local rings sending $\frak m$ into $\frak n$''
\item{p. 246} Add the following exercise:
\noindent {\bf Exercise 10.14:} Let $S$ be a Noetherian ring
of dimensionsion $d$,
and let $I$ be a radical ideal of $S$, that is, $I = {\rm rad}(I)$.
We consider the problem of finding the smallest number of
elements $f_{1},\ldots, f_{e}$
that {\it generate $I$ up to radical\/} in the sense
that ${\rm rad}(f_{1},\ldots, f_{e}) = I$. In the case where
$S = k[x_{1},\ldots,x_{d}]$, and $k$ is an algebraically
closed field, the ideal $I$ corresponds to an algebraic
set $X$, and the problem is equivalent, by the Nullstellensatz,
to the problem of determining the
minimal number of hypersurfaces that intersect precisely in $X$.
(In the non-algebraically closed case, the problems are quite
different; show that every algebraic set in ${\bf R}^{d}$ is
a hypersurface, that is, may be defined by a single equation.)
\item{a.} Show that if ${\rm codim\ }I = c$
then $I$ cannot be generated up to radical by fewer than
$c$ elements. Thus there are ideals in $S$ which cannot be generated up
to radical by fewer than $d$ elements.
\item{b.} Show that $I$ is generated up to
radical by at most $d+1$ elements, as follows. If
$I$ is contained in all the minimal primes of $S$, then
$I$ is nilpotent, so the empty set generates $I$ up to radical.
Otherwise,
choose as first generator $f_1$ an element
in $I$ but not in any of the minimal primes of $S$ that do not
contain $I$. If ${\rm codim\ } I >0$, factor
out $(f_1)$ and do induction on the maximum of the dimensions of minimal
primes of $S$ that do not contain $I$.
\item{c.} Suppose that $S$ can be written as a polynomial ring
in one variable over a smaller ring, say $S = R[x]$. Show
by induction on $d-1 = {\rm dim\ }R$ that $I$ can be generated up
to radical by just $d$ elements, perhaps using the following
outline. Conclude that every algebraic set in affine $d$-space
is the intersection of $d$ hypersurfaces. The corresponding
theorem is also true in projective space, and can be proved
by a modification of the argument given below (see Eisenbud--Evans
[1973] or Kunz [1985, Ch.~5] for an
account).\footnote{$^{1}$}
{These results have a controversial history.
Part b was first proved by Kronecker [1881], using difficult
arguments from elimination theory; the argument suggested here
is due to van der Waerden, about 1941. In 1891 Kurt
Vahlen announced an example (a rational quintic curve in
projective 3-space) that he claimed was not the intersection
of 3 hypersurfaces. The subsequent history was once
explained to me by Alfred Brauer:
According to him, Vahlen abandoned mathematics and became,
after the first world war, a
University Rektor (President), and
a prominent Nazi. Perhaps because of this, Oscar Perron
was moved to re-examine the example, Vahlen's only significant
mathematical contribution, and showed that it was wrong!
By hard computation Perron [1941]
exhibited 3 hypersurfaces that intersect in Vahlen's curve.
Later, Kneser [1960] showed that any algebraic set in projective
3-space is the intersection of 3 hypersurfaces. The proof
for affine $d$-space outlined below is from
Storch [1972] and Eisenbud--Evans [1973]. The question of whether
curves in projective 3-space can be expressed as the interesection
of just two hypersurfaces remains tantalizingly open, even in
simple concrete cases; see the discussion at the end of Ch.~15.}
%
\itemitem{i.} Show that it suffices to treat the case where
$S$ (equivalently $R$) is reduced.
Note that if $R$ is reduced and of dimension 0
then $R$ is a product of
fields. Now do the case $d=1$.
%
\itemitem{ii.} Assuming $R$ is reduced, let $U$ be the set of
nonzerodivisors of $R$. Show $R[U^{-1}]$ is a product of
fields, and that you can choose $f_{1}\in I$ so that
$f_{1}R[U^{-1}] = IR[U^{-1}]$.
%
\itemitem{iii.} Show that there is an element $r\in R$ not in any
minimal prime of $R$ such that $rI\subset (f_{1})$. Factor
out $r$ and use the induction hypothesis to find
$d-1$ elements that generate $I(R/(r)[x])$ up to radical.
Lift these elements to elements $f_{2},\ldots,f_{d}\in I$.
Show that together with $f_{1}$ these generate $I$ up to
radical.
\beginsection Chapter 11.
\item{p. 258} line 10*-7*. Replace the two sentences ``This is an uncountable
... . Thus Pic(R).. .'' by
``This is an uncountable divisible group. An easy argument show that if $R$ is
the coordinate ring of an affine open subset of such a curve then
${\rm Pic\ }R$ is a quotient of this torus by a finitely generated subgroup
which maps onto ${\bf Z}$. Thus (still assuming $g>0$) ${\rm Pic\ }R$ is
an uncountable divisible group.
\beginsection Chapter 12.
\item{p. 271} line 13*: bad line break.
\item{p. 272} line 6:replace ``parameter ideal ${\frak q}$ for $M$''
by ``ideal of finite colength on $M$''
\item{p. 272} line 7-8: replace ``parameter ideal ${\frak q}$ of $M$''
by ``ideal ${\frak q}$ of finite colength on $M$''
\item{p. 272} line 20-21: replace ``a parameter ideal for $M$'' by
``an ideal of finite colength on $M$''
\item{p. 275} line 14*-13*: replace ``parameter ideals for'' by
``ideals of finite colength on''
\item{p. 275} line 7*: replace ``a parameter ideal''
by ``an ideal of finite colength''
\item{p. 275} line 3*: replace ``parameter ideal'' by ''ideal of finite colength''
\item{p. 276} line 7: replace ``parameter ideal'' by ''ideal of finite colength''
\item{p. 276} line 6*: replace ``a parameter ideal for ''
by ''an ideal of finite colength on''
\item{p. 277} line 5: delete ``the'' (third word in the line).
\item{p. 277} line 13: replace ``a parameter ideal for ''
by ''an ideal of finite colength on''
\item{p. 278} line 2-3: replace ``a parameter ideal of ''
by ''an ideal of finite colength on''
\item{p. 279} line 17*: replace ``any parameter ideal for ''
by ''any ideal of finite colength on''
\item{p. 279} line 2*: replace ``a parameter ideal for ''
by ''an ideal of finite colength on''
\beginsection Chapter 13.
\item{p. 283} line 8*: change $d_m > 0$ to $d_m\geq 0$
\item{p. 283} line 6*: change $x_{d_{j+1}}$ to $x_{{d_j}+1}$
\beginsection Chapter 14.
\beginsection Chapter 15.
\item{p. 317} line 1: replace ``Restkalssen-'' with ``Restklassen-''
\item{p. 319} line 11: Delete the ' after ``Sturmfels''
\item{p. 320} line 20*: after ``largest monomial in $F$''
insert ``, in the sense of divisibility,''
\item{p. 322}, 1st line of paragraph before last display:
\itemitem{} ``We may now assume that
$\sigma = \sum a_v n_v ^\epsilon (\ker \phi)_n$''
should be
``We may now assume that
$\sigma = \sum a_v n_v \epsilon_v\in (\ker \phi)_n$''
\item{p. 322-323} Delete the last two lines of p. 322
and the first 12 lines of p. 323 (Lemma 15.1 bis and its proof).
\item{p. 325} line 8*: change $m_i$ to $u_im_i$
\item{p. 332} line 15: replace $(g_1,\ldots, g_t)\in F$
by $(g_1,\ldots, g_t)\subset F$
\item{p. 332} line 14* to p. 333, line 4: replace these
paragraphs by\smallskip
choose an element
$$
f = \sum_u f_u\epsilon_u
\qquad{\rm with}\qquad
{\rm in}(\phi(f))\notin ({\rm in}(g_1),\ldots,{\rm in}(g_t)).
$$
Let $m$ be the maximal monomial that occurs among the
terms ${\rm in}(f_ug_u)$. We may assume that the expression
for $f$ is chosen so that $m$ is minimal, and so that
the number of times $m$ occurs among the ${\rm in}(f_ug_u)$
is also minimal. Since
${\rm in}(\phi(f))$ is not divisible by an ${\rm in}(g_i)$, the
terms of the $f_ug_u$ that involve the monomial $m$ must
cancel; in particular, there must be at least two such terms,
and renumbering the $g_u$ we may assume for simplicity that
${\rm in}(f_1g_1)$ and ${\rm in}(f_2g_2)$ are among them. We may
write these terms in the form $n_1m_1$ and $n_2m_2$, where
$n_i$ is a term of $f_i$.
Since $n_1m_1$ and $n_2m_2$ differ by only a scalar,
$n_1$ is divisible by $m_2/{\rm GCD}(m_1,m_2)$,
and thus there is a term
$n\in S$ such that $nm_{2,1} = n_1$.
Consider the element
$$
f' = f - n(\sigma_{1,2} - \sum f_u^{1,2}\epsilon_u),
$$
and write it in the form $f' = \sum f'_u\epsilon_u$.
By our hypothesis $h_{ij}=0$ we see that $\phi(f')=\phi(f)$.
The terms of $\phi(nf_u^{ij}\epsilon_u)$ are all $< m$.
The term $n_1\epsilon_1$ in $f$ cancels with the
term $nm_{2,1}\epsilon_1$ of $n\sigma_{1,2}$, so this
term is missing from $f'$. The
other term $nm_{1,2}\epsilon_2$ of $n\sigma_{1,2}$
combines with the term $n_2\epsilon_2$ of $f$ so that the
number of occurences of $m$ among the ${\rm in}(f'_ug_u)$
is strictly less than among the ${\rm in}(f_ug_u)$.
This contradicts the minimality property of
$f$, so $g_1,\ldots, g_t$ is a Gr\"obner basis after all.
\Box
\item{p 333} line 15*ff: Delete from the paragraph that begins ``There is a
fairly sharp...'' through the end of the page.
\item{p 334} line 1: replace ``This estimate is so large as to suggest'' by
``Worst-case analysis of Gr\"obner bases shows that the degrees of the
elements in a Gr\"obner basis may be extremely large,
suggesting''
\item{p. 337} Footnote: replace
``was a postdoctoral student'' by
``was a young visitor''
\item{p. 339} line 6: delete the word ``graded''
\item{p. 339} line 8*-7*: replace ``inequality'' by
``inclusion''
\item{p. 340} line 11*: replace ``monomials'' by ``terms''
\item{p. 342} line 5*: The first $>$ should be $>_{\lambda}$
\item{p. 347} line 8*: replace ``term of the equation is
$y^2$'' by ``term is $-y^2$''
\item{p. 351} line 2*: Replace the $(1+\delta)$ with
$(1+\gamma)$
\item{p. 352} line 6: replace $\delta_n$ by $\delta_r$
\item{p. 353} display on line 13*: After $t\choose s$
insert $c^{s}$
\item{p. 359} line 19*: replace $p'$ by $g'$ and $p$ by $g$
\item{p. 360} line 8: replace $I'$ by $I$. Replace
``the ${\rm in}(g_i') = {\rm in}(g_i)$.'' by
``the ${\rm in}(g_i)$.
\item{p. 372} Exercise 15.39, replace ``$I'$ is any''
by ``$I'$ is a homogeneous''.
\beginsection Chapter 16.
\item{p. 392} 5th line from the bottom: The direct sum symbol
$\oplus_R$ should be $\otimes_R$.
\item{p. 393} lines 9 and 12: the two occurences of $\otimes_R$
should both be $\otimes_{S_i}$
\item{p. 394} Corollary 16.6, displayed equation:
add two pairs of parentheses, to make it
$$
\Omega_{T/R} \cong
(T\otimes_S\Omega_{S/R})
\oplus
(\oplus_i Tdx_i).
$$
\item{p. 394} Corollary 16.6, line 2 of the proof: add two
pairs of parentheses to the expression on the right hand
side of the equation, to make it
$(T\otimes_S\Omega_{S/R})
\oplus
(T\otimes_{T'}\Omega_{T'/R})
$
\item{p. 394} Corollary 16.6, line 3 of the proof: add a pair
of parentheses to make
$=T\otimes_{T'}\oplus_iT'dx_i=$ into
$=T\otimes_{T'}(\oplus_iT'dx_i)=$
\item{p. 394} Theorem 16.8, line 2: the roman ``B'' should be
a caligraphic B, as on the first line.
\item{p. 394} Theorem 16.8, line 3: the map at the end should
be $\phi: S'\to S$.
\item{p. 402} Theorem 16.19, line 3: Change $K(R/P)$ to
$K(S/P)$.
\beginsection Part III
\item{p. 417} line 11*: [1858] should be [1848]
\beginsection Chapter 17.
\item{p.424} line 4: insert the missing rightarrow
in the display
between $\wedge^n N$ and $\wedge^{n+1} N$; should
be
$$
\cdots \to \wedge^{n-1} N\to \wedge^{n} N
\to \wedge^{n+1} N \to 0.
$$
\item{p. 439} line 8. after `` is a subset'' insert `` of length $s$ of''
\item{p.439} line 17: $\sum_J c_{I,J}$ should be
$\sum_J c_{I,J} e_J$
\beginsection Chapter 18.
\item{p. 466} second line of Exc. 18.11: Insert the word
``primitive'' before ``polynomial''.
\item{p. 477} In Corollary 19.11, delete the phrase
``, and let $F$ \dots monomial order''.
Change the period at the end of the display to a comma, and.
after the display add the phrase
``where the generic initial ideal is taken with respect to
reverse lexicographic order.''
\item{p. 482} line 11: after the word ``respectively'',
and before the semicolon, insert:
(exept that in the case of the Cayley numbers it is
non-associative)
\beginsection Chapter 19.
\item{p. 487} Exercise 19.18. Delete the second sentence
of the exercise, beginnning ``Suppose the degrees\dots''.
\beginsection Chapter 20.
\item{p. 494} First word of the second line of the Proof of
Prop. 20.6: ``localiztion" should be ``localization"
is misspelled in the second line of the proof
of Prop. 20.6.
\item{p. 506} line 4*: $M$ should be $M'$
\beginsection Chapter 21.
\item{p. 522} line 10 replace ``$A,P$'' by ``$(A,P)$''
\item{p. 522} line 3*: replace $\omega_{A}$ by $D(A)$
\item{p. 541} Theorem 21.23: Delete the first occurence of
``$J=(0:_AI)$".
\item{p.550} Exercise 21.15 should have the hypothesis
that $dim(R) = c$, or more generally that $R/J$
is Cohen-Macaulay.
\item{p. 551} Exercise 21.19: Add a ``)" at the end of the first
line.
\beginsection Appendix A1.
\beginsection Appendix A2.
\item{p. 576} end of first paragraph: There should be a space
between ``2" and ``(bi)".
\item{p. 577}: The right margin is ragged.
\item{p. 596} Fig. A2.7: in the middle of the middle row,
the $\wedge^{q}-gG$ should be $\wedge^{q-g}G$. Also,
there is an extra $=$ floating below the symbol $\sum_{p+q-k}$
which should be deleted.
\item{p. 597} Lemma A2.11: The numerals
(and perhaps the ``A''? in ``Theorem A2.10" should
be typeset in roman. (Compare with similar situations in Lemma
A2.5 and Proposition A3.17.)
\beginsection Appendix A3.
\item{p. 614} Footnote: Add
``(In later editions of the book Lang takes a more
moderate position; see Lang [1993].)''
\item{p. 620} lines 4 and 6: The small
italic ``o" in the statement of
Proposition A3.5 and in the first line of the proof
should be the number ``0" (zero).
\item{p. 629} line 12 (last line before the second display):
two pairs of () should be removed, and one ( added, so
that the final equation should be
$\alpha x -\beta x = \partial (h(x)) - h(\partial(x))$.
\item{p. 629} The caption of Figure A3.2 needs to be fixed:
In the third line of the caption $\delta$'s should be
$\partial$'s. The last line of the caption should be
$\alpha x -\beta x = \partial (h(x)) - h(\partial(x))$
\item{p. 630} line 9*: ``A3.,'' should be ``A3.13,''
\item{p. 632} line 1*: in the display replace $F''$ by
$F'' \to 0$
\item{p. 650} line 9*: ``differential of $F$'' should be
``differential of $F[-1]$''
\item{p. 678} line 8: Insert a comma after ``Chapter 1]''
and after ``[1986],'' add ``Gelfand-Manin [1989],''
\item{p. 679} line 8: $H_{i}$ should be $H_{n}$
\item{p. 681} lines 16 AND 18: ``$\to KF$'' should be
``$\to P \circ KF$''
\beginsection Appendix A4.
\item{p.683} line 10*: replace ``zeroeth'' by '
``$0^{\rm th}$''
\item{p.687} line 8: after $Q\neq P$ insert before the
period:
``and that
all the minimal primes of $R$ have the same dimension''
\item{p.687} line 9*-6*: replace these lines (first four
lines of ex A4.3 by
\smallskip
\noindent{\bf Exercise A4.3:} Let $(R,P)$ be a Noetherian local
ring, and suppose that $x_{1},\ldots,x_{n}$ generate $P$.
For each $i\geq 0$ there is a natural map
$H^{i}(K(x_{1},\ldots,x_{n})) \to {\rm H}^{i}_{P}(R)$
from the cohomology of the Koszul complex to the
local cohomology. For
$i=0$ this map is injective, but in general it is neither
injective nor surjective. We say that $R$ is
{\bf Buchsbaum} if these maps are all surjective for
$i< {\rm dim\ }R$. It turns out that this somewhat unappetizing
definition leads to a rich and surprizing theory, initiated in
Vogel [1973]; the definitive exposition
is given in St\"uckrad and Vogel [1986]. Show
\item{p.687} line 5*: before the word ``projective'' insert
``locally Cohen-Macaulay''
\beginsection Appendix A5.
\item{p.690} lines 2-4: Replace the sentence beginning
``All the methods\dots'' by:
``There are difficult foundational issues, but they do not
seem to threaten the use of categorical language in
common situations. For the best-known way out,
see Grothendieck [1972, Ch. I]; for a review, see
Feferman [1969]. We shall take a naive approach, and simply
ignore the problem.''
\item{p.691} lines 8: Replace $M\to N$ by $M\to M$
\item{p.692} line 3: replace $A\to A'$ by $\phi:\ A\to A'$
and replace $B\to B'$ by $\psi:\ B\to B'$
\item{p.692} diagram, symbols next to the left hand vertical arrow:
replace $G(\phi)$ by $G(\psi)$.
\item{p.692} diagram, bottom row left hand side: replace $A'$ by $A$
\beginsection Appendix A6.
\beginsection Appendix A7.
\item{p. 710} line 1: replace ``algebra" with ``algebraic''.
\item{p. 710} line 16: replace ``Theory'' with ``theory''
\beginsection Hints and Solutions for Selected Exercises.
\item{p.714} Before the hint for Exercise 2.27, add a hint
for exercise 2.23, as follows:
\noindent{\bf Exercise 2.23:\ } Check and use the fact that
if an ideal $I$ is contained in a principal ideal $(a)$
then $I = a(I:a)$ to show that if every ideal
containing an ideal $I$ is principal, then $I$ is
either principal or prime.
\item{p. 730} line 4 ``J. Sally'' should be ``S. Wiegand''.
\item{p. 733} line 3 of hint for 18.11: after ``because''
insert ``$f$ is primitive we must have $d,e \geq 1$,
and because''
\item{p. 741} line 4 of hint for A3.13: in the displayed
diagram the subscript ``F'' should be ``$F$''
\beginsection References.
\item{p. 747} The reference to Bass appears out of order. It
should come before the Bayer references.
\item{p. 747} Insert the reference:
Browder, F. (1976). {\it Mathematical developments arising from
Hilbert's problems.\ } Proc. of Symp. in Pure Math. 28.
Amer. Math. Soc., Providence, RI.
\item{p. 749} before line 4* Insert the reference:
\itemitem{} Edwards, H. M. (1990). {\it Divisor Theory.\/}
Birkh\"auser, Boston.
\item{p. 750} Insert new reference:
Eisenbud, D.~and Evans, E.~G.~ (1973). Every algebraic set in
$n$-space is the intersection of $n$ hypersurfaces.
Inv. Math. 19, pp. 107-112.
\item{p. 750} line 11: add ``75, pp. 339--352.''
\item{p. 750} line 16: add ``To be published in revised
form by Springer-Verlag, under the title {\it Why Schemes?}.''
\item{p. 750} line 17*: add ``109, pp. 168--187.''
\item{p. 750} line 13*: Change ``1994''
to ``1996'', and add at the end of the line
``Duke J.~Math. (in press).''
\item{p. 750} before line 7*: Add the reference:
\itemitem Feferman, S. (1969). Set-Theoretical Foundations
of Category Theory. in {\it Reports of the Midwest Category
Theory Seminar}, Ed. S. Maclane, Springer Lect. Notes in Math.
106, pp. 201--247, Springer-Verlag, New York.
\item{p. 752} line 19: change 593 to 523
\item{p. 753} The paper of Heinzer, Ratliffe, and Shah has
now appeared; the reference is
Houston J. Math, 21, 29-52 (1995)
\item{p. 754} After the references to David Hilbert's
papers, insert
(Hilbert's papers above are noe available in an English
translation in {\bf Hilbert's Invariant Theory Papers},
transl. M. Ackermann. Lie Groups: History, Frontiers,
and Applications Volume VIII, Math Sci Press, Boston Mass, 1978.)
\item{p. 758} line 5*, reference to Peskine's book: remove the
accent from the word ``Algebraic".
\item{p. 755} Insert new reference:
\itemitem{} Kaplansky, I. (1977). Hilbert's problems, Lecture
Notes in Mathematics, Univ. Chicago, Chicago, IL.
\item{p. 755} Insert new reference: Kneser, M. (1960).
\"Uber die Darstellung algebraisher Raumkurven als
Durchschnitte von Fl\"achen. Arch.~Math.~11, pp.157-158.
\item{p. 758} Insert new reference:
Perron, O. (1941). \"Uber das
Vahlensche Beispiel zu einem Satz von
Kronecker. Math.~Z.~47, pp. 318-324.
\item{p. 761} Insert new reference:
Storch, U. (1972). Bemerkung zu einem Satz von M.~Kneser.
Arch.~Math.~23, pp. 403-404.
\item{p. 761} line 5: replace ``In Press'' by 1995
\item{p. 761} line 6: add 301, pp. 417--432.
\item{p. 761} before line 5*: Insert new reference:
Vogel, W.(1973). Uber eine Vermutung von D.~A.~Buchsbaum.
{\it J. Alg.} 25, pp. 106--112.
\beginsection Index.
\item{p. 769} line 5, left column: 502 should be 501
\item{p. 769} line 55, left column: the references to
Vogel should be:
7, 111, 278, 301, 337, 363, 687.
\item{p. 769} line 2, left column:
add the page number 15 to the references for the
word ``nonzerodivisor''
\item{p. 776} add an index item in the list after ``ideal'', after
line 15 in the left column:
of finite colength, 234--236,272,275--279
\item{p. 780} remove all the references to ``parameter ideal''
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\beginsection: added 6/16/96
Ch. 10
\item{p. 235} line 17: replace ``parameter ideals for'' by ``ideals
of finite colength on''.
\item{p. 235} line 7*: replace ``is a parameter ideal for''
by ``has finite colength on''. (this is the second occurence
on line 7*!)
\item{p. 236} line 8: replace ``is a parameter ideal for''
by ``has finite colength on''.
\item{p. 236} line 8: replace ``then it is for''
by ``then it has for''.
\item{p. 236} line 11*-10* : replace ``is a parameter ideal for''
by ``has finite colength on''.
\item{p. 236} line 9*-8* : replace ``is a parameter ideal for''
by ``has finite colength on''.
\
Ch. 12
\item{p. 273} line 15*-14* : replace ``a parameter ideal for''
by ``an ideal of finite colength on''.
\item{p. 274} line 4-5 : replace ``a parameter ideal for''
by ``an ideal of finite colength on''.
\item{p. 274} line 8* and line 6* (two occurrences) : replace
``parameter ideal''
by ``ideal of finite colength''.
\item{p. 275} line 2 : replace
``parameter ideal''
by ``ideal of finite colength''.
\item{p. 276} line 17 : replace ``parameter ideal''
by ``ideal of finite colength''.
\item{p. 276} line 9* : replace ``parameter ideal''
by ``ideal of finite colength''.
\item{p. 277} line 2*, 1* (two occurrences!) :
replace ``a parameter ideal for''
by ``an ideal of finite colength on''.
\item{p. 279} line 2 :
replace ``a parameter ideal''
by ``an ideal of finite colength''.
\item{p. 279} line 3*:
replace ``a parameter ideal for''
by ``an ideal of finite colength on''.
\
Appendix A2:
\item{p. 594} line 17* (display) change $\phi_{1}$ to $\phi_{2}$
and change $\phi_{0}$ to $\phi_{1}$.
\item{p. 599} line 13 (display as on p. 594) change $\phi_{1}$ to $\phi_{2}$
and change $\phi_{0}$ to $\phi_{1}$
\
Appendix A3:
\item{p. 677} line 3*. replace ``has just been'' by ``is about to be''.
\item{p. 677} line 1*. after ``1963'' add ``, quoted in the
introduction to Hartshorne [1966b].''
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\beginsection: added 7/11/96
\item{p.33}{line 11}. $f^n$ should be$f^d$
\item{p.43}{line 4*}: "algebra" should be "algebraic".
\item{p.48}{line 9*}: ``$1992$'' should be``$1993$''
\item{p.52}{line 3}: "coincident" should be "incident"
\item{p. 58}{line 3}: ${\bf A}^{r+1}_k$ should be ${\bf A}^{r+1}(k)$
\item{p. 58}{line 13}: ``algebra'' should be ``algebraic''
\item{p. 58}{line 16}: ${\bf A}^r_k$ should be ${\bf A}^r(k)$
\item{p. 58} In the figure $z$ and $x-1$ should have the same sign.
\noindent The branches live in the first and third-quadrant.
\item{p. 70}{line 8} : "left" should be "right"
\item{p.90}{line 3} : more space around the vertical bar in the set definition
\item{p.114}{line 7, Ex. 3.18a} : After ``Show that'' insert ``, if $k$ is
an infinite field, then''
\item{p.129}{line 3} : $x-1$ should be $x+1$
\item{p.136} in Exercise 4.11 the reference to A3.3 should be to A3.2.
\item{p.156}{line 2*} : ";" should be "."
\item{p.158}{line 1*}: At the end there should be a parenthesis ")"
\item{p.251}{line 8} : "ring" should be "reduced ring"
\item{p.252}{line 12*}: $R_{P}$ should be $R_{P_{j}}$
\item{p.303}{line 3}: $\otimes_P$ should be $\otimes_R$
\item{p.304}{line 5}: $f_{i}(x_{1},\ldots,x_{n},y_{0},\ldots,y_{m})$ should
be $f_{i}(x_{0},\ldots,x_{m},y_{0},\ldots,y_{n})$
\item{p.304}{line 6}: $m+1$ should be $n+1$
\item{p.305}{line 17*}: "polymials" should be "polynomials"
\item{p. 325} line 15*. line should end with a colon
\item{p. 330} line 15 (mid of Prop). In the expression that ends with )),
the second one should also be in math mode (not so tilted)
\item{p. 333} line 7. comma before last word should be period
\item{p. 367} line 13 ``ldots" should be ``...'' (missing backslash in the tex
\item{p.384}{line 11} Diagram: $\exists ! e$ should be $\exists ! e'$
\item{p.389}{line 8} $f$ should be $\phi$
\item{p.389}{line 11} $\phi^{*}$ should be $f^{*}$
\item{p.391}{line 13*} $R/S$ should be $S/R$
\item{p.405}{line 15*} $R/P$ should be $S/P$
\item{p.405}{line 6*} $R/P$ should be $S/P$
\item{p.410}{line 2*} after ``as a'' insert ``deformation''
\item{p.431}{line 9*} 17.4 should be 17.14
\item{p.539} line 4*: ``first'' should be ``middle''
\item{p.539} line 3*: should be `` $f_2 = x_1x_3-x_2^2$
\item{p.539} line 1*: change ``$x_0=x_1=0$'' to ``$x_1=x_2=0$''
\item{p. 754} line 8 (Herzog-Kunz reference)
Should be ``Herzog, J., and ''
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\beginsection Not corrected in the Oct 96 printing
In the following ``first'' means first printing, ``sec'' means
second printing (Fall 96).
\beginsection Chapter 1
\item{}{first 152 sec.154 Figure 5.2}
There should not be any space between "in" and "($y^2$..." otherwise
it takes time (for me) to figure out that you actually mean the
initial term of ($y^2-$... .
\item {p. 251} (of first printing). First line of the pf of Cor 11.4:
Prop. 11.3, which is referenced, requires the ring to be reduced.
\item{}first 256 sec.260 Corollary 11.7a and its proof: At the end
of the statement of part a of the Cor, before the period, add:
``modulo the group of units of $R$''. In the fourth line of
the proof of part a, replace ``, so it'' by:
``. 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''
\item {p. 327} (of first printing). There should be an end-of-proof
sign following the statement of Prop 15.4
\item {p. 332} (of first printing). The large insert
fixing the proof of Thm 15.8 was put in the wrong place:
The part from
``choose and expression'' to ``possible. Now''
should be deleted.
\item {}{first 528 sec 532} first paragraph after definition.
Towards the end of the paragraph there is a reference to a
proposition. It should be "Proposition 21.5d" not 21.4d.
\item {p. 546} (of first printing). Exercise 21.6 should refer
explicitly to the notation introduced just after Prop. 21.5.
\item {p. 609} (of second printing) First line after the proof.
``Corollary'' should be ``Theorem''. Same in EXC A2.15
\item {}{first 645 sec 652} exact sequence labeled as 0:
It should be $0\to B\to A+B\to A\to 0$ and not $0\to A\to $ etc.
\item {}{first 645 sec 652 last paragraph of Exc A3.26a}
$ E^1(A,B)$ should be and $E^1_R(A,B)$
\item {}{first 645 sec 652 last paragraph of Exc A3.26c}
should end with $B$
\item {}{first 778,sec 790}
Reference page for ``$M$-sequence'' should be 419ff in the first
printing and 423ff in the second printing
\beginsection Thanks!
Thanks to all those who have helped me compile these
latest corrections -- in particular, to:
\item{$\bullet$}Craig Huneke
\item{$\bullet$}Kenji Matsuki
\item{$\bullet$}Jan Strooker
\item{$\bullet$}Wolfgang Vogel
\item{$\bullet$}Karen Smith
\item{$\bullet$}Hara Charalambous
\item{$\bullet$}Sinan Sertoz
\item{$\bullet$}Roger Wiegand
\item{$\bullet$}Faheem Mitha
\item{$\bullet$}Bernd Sturmfels
\item{$\bullet$}Keith Pardue
\item{$\bullet$}Will Traves
\item{$\bullet$}Leslie Roberts
\item{$\bullet$}Ezra Miller
\item{$\bullet$}CW
\item{$\bullet$}Bill Adams
\item{$\bullet$}Aron Simis
\item{$\bullet$}Richard Belshoff
\item{$\bullet$}Kishor Shah
\item{$\bullet$}Udo Vetter
\item{$\bullet$}Marie Vitulli
\item{$\bullet$}Martin Lorenz
\item{$\bullet$}Allan Adler
\item{$\bullet$}Ebrahim Jahangard
\item{$\bullet$}Pietro de Poi
\item{$\bullet$}Vivek Pawale
\item{$\bullet$}Daniel Mall
\item{$\bullet$}Harm Derksen
\item{$\bullet$}Gerhard Quarg
\item{$\bullet$}Sorin Popescu
\item{$\bullet$}Juan Migliore
\item{$\bullet$}Chris Peterson
\item{$\bullet$}Colin McLarty
\item{$\bullet$}Vesselin Gasharov
\item{$\bullet$}Paolo Oliverio
\item{$\bullet$}Claude Quitte
\bigskip
\noindent{Dept. of Math., Brandeis Univ., Waltham MA 02254 \par
\noindent eisenbud@math.brandeis.edu}
\vfil \eject
\bye