\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