Errata and Addenda for Algebraic Geometry I

Here we post a list of errata and addenda. The name tags refer to the people who found the mistake. We are very grateful to all of them. Further remarks and hints - trivial or not - are very welcome.

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 p. 0 - VII, ¶ Contents of Chapter 16 The page numbers given for the last 3 sections of Chapter 16 should be increased by one. p.528 should be p.529, p.532 should be p.533, p.539 should be p.540. (F. Ebert) p. 0 - V, ¶ Contents of chapter 2 The page's number of section "Excursion: Sheaves" of chapter 2 should be 47, not 46. (Ehsan Shahoseini) p. 1, ¶ Line -16 The term "affine variety" is undefined at this point. (Mahdi Majidi-Zolbanin) p. 1, ¶ Line 10 The condition "If the polynomials f_i are linear" should be replaced by "linear with constant term $0$" (or the solution set is only an affine subspace, in general). (Mahdi Majidi-Zolbanin) p. 2, ¶ Line 6 In "asserts that this equations has no solutions" the word "equations" should be replaced with "equation". (Mahdi Majidi-Zolbanin) p. 2, ¶ Line -8 Replace 1994 by 1995. (J. Hilgert) p. 5, ¶ Line 12 then $X$ inherits many properties of $X'$ (rather than: of $X$) (P. Barik) p. 7, ¶ Line 9 Replace "guide line" by "guideline". (Mahdi Majidi-Zolbanin) p. 7, ¶ Figure 1 Replace $T_2^2-T_1^2(T_1+1)$ by $T_2^2-T_1^2(T_1+1)=0$. (A. B. Nguyen) p. 7, ¶ Example 1.1, first paragraph Exercise 1.8 should be replaced by, or extended by, an example where the real-valued points are not connected with respect to the analytic topology, to illustrate the point "and sometime the visualizations obtained in this way may be deceptive, see Exercise 1.8." (Mahdi Majidi-Zolbanin) p. 10, ¶ Theorem 1.8 Add the assumption that $A\ne 0$. (D. Gerigk) p. 10, ¶ Line -17 Omit We will not use this characterization for the proof of the Nullstellensatz.''. A simple form of it is used in the beginning of the proof of Lemma~1.9. (J. Hilgert) p. 10, ¶ Line -3 Replace $\beta_{n-1}+\beta_{n-2}a+\cdots +a^{n-1}$ by $-(\beta_{n-1}+\beta_{n-2}a+\cdots +\beta_0 a^{n-1})$. (P. Zsifkovits) p. 10, ¶ Line 17 The definition of finite ring homomorphism is messed up. Replace it by: A homomorphism $R \rightarrow R'$ of rings is finite, if it is integral and $R'$ is generated as an $R$-algebra by finitely many elements. (K. Mohri, C. Frank) p. 11, ¶ Line 13 (in the Proof of Theorem 1.7) The sentence "Then A[x^(-1)] is a finitely generated K-Algebra not equal to 0" is true for all nonzero x. The proof then shows that there exists a maximal ideal of A not containing x. So there is no need to use proof by contradiction. (Mahdi Majidi-Zolbanin) p. 13, ¶ Line 15 Replace nonconnected by non-connected. (P. Zsifkovits) p. 13, ¶ Line -7 After hence'', add if $Z \cap U \ne \emptyset''. (U. Görtz) p. 14, ¶ Line 10 Replace $\emptyset \ne J \subset I$'' by $\emptyset \ne J \subsetneq I$''. (J. Buck) p. 14, ¶ Line 14 Replace than'' by then'' and omit be''. (P. Zsifkovits) p. 14, ¶ 3 In order to apply Zorn's Lemma here, one should note that every non-empty topological space$X$indeed contains some irreducible subset, e.g., any singleton. (L. Prader) p. 15, ¶ Line -15 Replace "there existed" with "there would exist". (F. Gispert Sánchez) p. 15, ¶ Line 14 Replace Every open subset'' by Every subspace''. (T. Wedhorn) p. 15, ¶ Line 15 Replace Every closed subset'' by Every subspace''. (T. Wedhorn) p. 16, ¶ Line -13 Omit superfluous (. (P. Zsifkovits) p. 16, ¶ Line -16 Replace$T_2-T_1^2$by$T_1-T_2^2$. (P. Zsifkovits) p. 19, ¶ Line 9 In the diagram exchange$m$and$n$. (J. Buck) p. 19, ¶ Line -15 Replace Proposition 1.20'' by Proposition 1.32''. (P. Zsifkovits) p. 19, ¶ Line 14 Insert $A$'' after $k$-algebra''. (J. Buck) p. 19, ¶ Line 11 In the definition of$f$add the missing bracket at the end. (T. Przezdziecki) p. 20, ¶ Line -15 Replace set morphisms'' by set of morphisms''. (P. Zsifkovits) p. 20, ¶ Line -4 (Proof of Lemma 1.38) The proof can be simplified: The closed subset$V(f_1g_2 - f_2g_1)$contains the dense subset$U$, hence equals$X$, and that implies$f_1g_2 - f_2g_1 = 0$, as desired. (Mahdi Majidi-Zolbanin) p. 21, ¶ Line 8 Remove "of" in the sentence "how to identify elements of f \in O_X(U) with ..." (Mahdi Majidi-Zolbanin) p. 22, ¶ Line -14 Replace$g\circ f$with$g\circ f_{|f^{-1}(U)}$. (A.Graf) p. 23, ¶ Line 15 (Def. 1.46.2) Replace finite covering'' by finite open covering''. (F. Gispert Sánchez) p. 23, ¶ Line -14 1.47 (i), "opposed category" should be "opposite category". (Peng Du) p. 23, ¶ Line 6 Replace manifolds'' by manifold''. (J. Buck) p. 24, ¶ Proof of Lemma 1.50 The reduction step in the first paragraph is superfluous because the rest of the proof shows that$D(f)$and$Y$are isomorphic spaces of functions. (Menachem Dov Mostowicz) p. 26, ¶ Line -13 Replace$k$by$R$. (D. Gerigk) p. 28, ¶ Line 7 Replace "space with function" with "space with functions". (F. Ebert) p. 28, ¶ Displayed equation in prop. 1.59 Replace exist'' by $\exists$''. (P. Zsifkovits) p. 29, ¶ Line 8 Replace is then given by'' by is given abstractly by''. Furthermore, the map$K(U_i) \rightarrow K(U_j)$maps$\frac{X_\ell}{X_i} \mapsto \frac{X_\ell}{X_j}\frac{X_j}{X_i} = \frac{X_\ell}{X_i}$. (I.e., as subfields of$K(X_0, \dots, X_n)$, the$K(U_i)$all coincide, and coincide with$K(\mathbb P^n(k))$, and the isomorphism induced by our identifications is the identity map.) (M. Kaneda) p. 29, ¶ Line 2 Replace "space with function" with "space with functions". (F. Gispert Sánchez) p. 29, ¶ Line 7 Add$g \ne 0$in the description of the function field of$K({\mathbb P}^n(k))$. (T. Wedhorn) p. 33, ¶ Lines 4 and 6 Replace$\mathbb P^n$with$\mathbb P^m$and$\mathbb A^n$with$\mathbb A^m$. (Safak Ozden) p. 35, ¶ Line 4$r \gt 2$(rather than:$r \gt 1$) (P. Barik) p. 35, ¶ Figure 1.2 Replace$X^2+Y^2-1$by$X^2+Y^2=1$and$XY-1$by$XY=1$. (A. B. Nguyen) p. 36, ¶ Exercise 1.8 The statement of the exercise is correct as it stands, but it does not illustrate the phenomenon that connectedness for the Zariski topology does not imply connectedness for the analytic topology. It should be replaced by: Show that the affine algebraic set$V(Y^2-X^3+X)\subset \mathbb A^2(k)$is irreducible and in particular connected. Sketch the set$\{ (x, y)\in\mathbb R^2;\ y^2 = x^3-x\}$and show that it is not connected with respect to the analytic topology on$\mathbb R^2$. (Torsten Wedhorn/Alexey Beshenov) p. 38, ¶ Exercise 1.21 (3) Remove "in" in "The set of closed affine cones in$C\subseteq \mathbb A^{n+1}(k)$". (Menachem D. Mostowicz) p. 44, ¶ Proposition 2.10 Replace Let$A$be a ring.'' by Let$\varphi\colon A\rightarrow B$be a ring homomorphism.''. (U. Görtz) p. 48, ¶ Line 9 Add the category of abelian groups'' as the first example. (A. Kaučikas) p. 48, ¶ Line -3 (def. 2.18.Sh2) Replace "by (a)" with "by (Sh1)". (F. Gispert Sánchez) p. 50, ¶ Prop. 2.20 The basis B has to be closed under finite intersections for the (Sh) condition to be well-defined. (In Lemma 1.31 the notion of basis of topology is "defined", in passing, by requiring also that it is stable under finite intersections; but this appears to be non-standard terminology, so both places should be fixed.) (Florian Ebert) p. 50, ¶ Section 2.6 Line 3 Maybe replace containment'' by reverse containment'' or state explicitly in which direction the order goes. () p. 51, ¶ Line 12 Replace if and only of'' by if and only if''. (J. Watterlond) p. 51, ¶ Line$-2$Replace$s^x$by${s^x}_{|V_x}$. (A. Graf) p. 53, ¶ Line -2 Since$f^{-1}$includes sheafification by definition, the restriction$\mathscr G_{|X}$to an open subspace in this sense coincides with Example 2.19 (1) only if$\mathscr G$is a sheaf. (E. Viehmann) p. 54, ¶ Proof of Prop. 2.27 Introducing the element$t$could be avoided. (E. Viehmann) p. 54, ¶ Line 2 Replace $X$'' by $Y$''. (K. Mohri) p. 54, ¶ Line -9 Replace sheaves'' by presheaves''. (J. Watterlond) p. 54, ¶ Line 3 Replace $f(V)$'' by $f(U)$''. (K. Mohri) p. 55, ¶ Line 11 Replace statements'' by statement''. (D. Gerigk) p. 55, ¶ Line -13 Insert "$U$" after "open subset". (F. Ebert) p. 58, ¶ Line -2 The third sum should be over the index set$I$instead of$J$. (Sebastian Schlegel Mejia) p. 61, ¶ Lines -3, -2 Replace $\mathfrak a$'' by $\mathfrak a_1$'' and $\mathfrak b$'' by $\mathfrak a_2$''. (P. Zsifkovits) p. 62, ¶ Exercise 2.3 line 2 Change "every open subset" to "every non empty open subset" (Vishal Gupta) p. 63, ¶ Exercise 2.13 In property (i), add the requirement that$\mathscr F(\emptyset)$is a set with one element. (Louis Martini) p. 63, ¶ Exercise 2.9 (b), line 1 Make it clearer that the reference points to Section (B.13), not to Def. B.13. (Viktor Tabakov) p. 63, ¶ Line 8 (ex. 2.9 (d)) Replace "$u \in A$" with "$u \in R$". (F. Gispert Sánchez) p. 63, ¶ Exer. 2.13 (b) The assumption that$X$be connected can be omitted. (A. Beshenov) p. 64, ¶ Ex.~2.14 (c) The condition that$Z$is locally connected is not sufficient. It should be replaced by the condition that every point in the closure of$Z$has a fundamental system of open neighborhoods which intersect$Z$in a connected set. (J. Calabrese) p. 64, ¶ Line 19 (Exercise 2.16) Replace "show an analogous results" by "show that an analogous result holds". () p. 67, ¶ Prop.+Def. 3.2 (2) Note that the affine open subschemes are not closed under finite intersections. This is inconsistent with the notion of a basis of a topology "defined" in Lemma 1.31. (F. Ebert) p. 67, ¶ Line -6 Replace constitute homomorphism'' by constitute a homomorphism''. (D. Gerigk) p. 68, ¶ Use consistent terminology: principal open'' or principally open'' subset. (E. Viehmann) p. 71, ¶ Example 3.12 Replace $\prod_{i=1}^n A_i$'' by $\mathop{\rm Spec}\prod_{i=1}^n A_i$''. (P. Zsifkovits) p. 71, ¶ Line 18$U \subseteq \psi_i(U_i) \cap \psi_j(U_j)$(rather than:$U \subseteq U_i \cap U_j$) (P. Barik) p. 72, ¶ Line -9 Exchange the indices$\frac{X_i}{X_j}$and$\frac{X_j}{X_i}$, and reverse the direction of the arrow. (F. Grelak, A. Elashry) p. 73, ¶ 15 Replace subset'' by a subset''. (E. Viehmann) p. 76, ¶ Line 20 Replace ideals'' by ideal''. (D. Gerigk) p. 77, ¶ Line 12 Replace scheme'' by non-empty scheme''. (M. Jarden) p. 78, ¶ Prop. 3.29 (3) Add the assumption that all$U_i$are non-empty. (D. Gerigk) p. 78, ¶ End of proof of Prop. 3.29(3) It might be worthwile to spell out the argument of Example 2.37 more explicitly. (E. Viehmann) p. 79, ¶ Line 12 Replace "if follows" with "it follows". (F. Gispert Sánchez) p. 79, ¶ Proposition 3.33 It might be worthwile to aim for a consistent notation for algebraically closed vs. not necessarily algebraically closed fields (e.g.,$k$versus$K$). (V. Gupta) p. 79, ¶ Line 14 Replace locally finite type'' by locally of finite type''. (J. Watterlond) p. 80, ¶ Corollary 3.36 Make it more explicit that$k=\kappa(x)$means that the natural homomorphism$k\rightarrow\kappa(x)$is an isomorphism (rather than just the existence of any isomorphism between these two fields). (Similarly on page 119, Equation (5.2.1).) (V Gupta) p. 81, ¶ Line 11, line 21 "Definition 1.15" should be replaced with "Definition 1.46". In line 21, replace "is the sense" by "in the sense". (Alexander Isaev) p. 82, ¶ Line 13 Replace points'' by type''. (D. Gerigk) p. 85, ¶ Line 14 Replace "$\varphi(g)|U$" with "$\varphi(g)_{|U}$". (F. Gispert Sánchez) p. 85, ¶ Proof of Thm. 3.42, Condition (2) Omit and$x \not\in U_i$for all$i$'' (this is neither (a priori) possible in general, e.g. if$X$is irreducible and$x$its generic point, nor necessary in the sequel of the proof). (B. Heintz) p. 85, ¶ Line -17 Replace biggest'' by largest''. (P. Johnson) p. 88, ¶ 3rd line of the proof of Prop. 3.52 Replace "on" by "in" in the sentence "[...]$Z$is closed on$U$". () p. 90, ¶ Ex. 3.19 The set$(R^{n+1}\setminus\{0\})/R^\times$must be replaced by the set$M/R^\times$, where$M\subset R^{n+1}$denotes the subset of all tuples which have at least one entry in$R^\times$. (B. Heintz) p. 90, ¶ Exercise 3.13$X$should be nonempty. () p. 90, ¶ Ex. 3.14 Add the assumption that the maximal ideal of$A$is the union of all prime ideals properly contained in it. See Knaf's answer to this question on MathOverflow. (B. Heintz) p. 92, ¶ Line -5 Replace shows'' by show''. (P. Zsifkovits) p. 94, ¶ Line -3 (ex. 4.5) Replace "Equivalent" with "Equivalently". (F. Gispert Sánchez) p. 94, ¶ Example 4.5 Insert: Let$\pi\colon R[T_1,\dots, T_n]\rightarrow R$be the projection mapping each$T_i$to$0$.'' after the definition of the$a_i$. (P. Zsifkovits/F. Gispert Sánchez) p. 94, ¶ Example 4.4 Conflict of notation:$T$. (E. Viehmann) p. 95, ¶ Line 12 Replace "for all objects Y we are given an in$Y$functorial map" by "for all objects$Y$we are given a map [...] functorial in$Y$"? (Lam Pham) p. 95, ¶ Corollary 4.7 Replace "$S$-morphism of schemes" by "morphism of$S$-schemes", the notion of$S$-morphism is only introduced later on. (Or define$S$-morphisms in (3.1).) (Lam Pham) p. 95, ¶ Line 7 Replace "objects" with "object". (F. Gispert Sánchez) p. 97, ¶ Definition 4.10 Add the condition that$f \circ p = g \circ q$. (C. Frei) p. 100, ¶ -9 "are morphisms" should be replaced by "be morphisms". (Kuo Tzu-Ang) p. 101, ¶ Prop. 4.20, Condition (I) Replace affine neighborhood$U'$of$x'$'' by affine neighborhood$U'$of$f(x')$''. (B. Heintz) p. 101, ¶ Line -7 In the statement of Prop. 4.20, maybe it should be made clearer that the assumptions (I) and (II) should refer to each factor of$f$, namely if$f=f_r\cdots f_1$, then each$f_i$should satisfy one of the assumptions (I) and (II) (not$f$itself). (Peng Du) p. 101, ¶ Prop. 4.20, condition (I) The proof of the proposition when condition (I) holds uses the fact that all the assertions can be checked locally. However, to pass to the affine situation, one also needs to know that for every open neighborhood$U''$of$f(x')$contained in$U'$,$f^{-1}(U'')$is also quasi-compact. This is true (Prop. 10.1) but had not been stated at this point. (F. Gispert Sánchez) p. 102, ¶ Line -10 Replace "homomorphisms" by "homomorphism" (... "is injective"). (F. Gispert Sánchez) p. 102, ¶ Proof of Prop. 4.20, Case (II) To see that$(p^{-1}(f(X')), \mathscr O_{Z|p^{-1}(f(X'))})$is a fiber product in the category of schemes, one needs to know that the fiber product of schemes taken in the category of locally ringed spaces exists and is a scheme. This stronger version of Thm. 4.18 should have been stated and proved before. It is proved similarly as Thm. 4.18; for the affine case use Proposition 3.4 in the form given (but not proved) in the book. (O. Körner) p. 104, ¶ Line 13 Add "Let" before "$X$and$Y$be". (F. Gispert Sánchez) p. 104, ¶ Line -3 Replace be'' by by''. (U. Görtz) p. 105, ¶ Line -6 Insert be'' before the first projection''. (P. Zsifkovits) p. 106, ¶ Line -2 Add missing parenthesis. (P. Zsifkovits) p. 106, ¶ Line 4 Replace (applied to$X, S, X' = \mathop{\rm Spec} \kappa(s), Y=S$)'' by (applied to$S, X = S, X' = \mathop{\rm Spec} \kappa(s), Y=X$)''. (P. Zsifkovits) p. 107, ¶ Line 20 (proof of prop. 4.30) Replace "$(f \times_S id_Y)\circ (id_X \times_S g)$" with "$(f \times_S id_Y)\circ (id_{X'} \times_S g)$". (F. Gispert Sánchez) p. 107, ¶ Line -10 (def. 4.31) Replace "morphism of$f\colon X\rightarrow S$of$S$-schemes" by "morphism$f\colon X\rightarrow S$of schemes". (F. Gispert Sánchez) p. 108, ¶ Proof of Prop. 4.32 Replace second sentence by: Proposition 4.20 shows that the properties immersion, open immersion and closed immersion are stable under base change (see the discussion at the beginning of (4.11)).'' (P. Zsifkovits) p. 108, ¶ Line -14 Replace "locally" by "local". (F. Gispert Sánchez) p. 108, ¶ Line -6 (prop. 4.34) Replace "immersion" with "immersions". (F. Gispert Sánchez) p. 109, ¶ -14 Strictly speaking,$f^{–1}(Z)$should be$f^{–1}(i(Z))$. (Jan Willing) p. 111, ¶ Line -10 Replace $g(a)$'' by $g(a)=0$''. (P. Zsifkovits) p. 118, ¶ Line 11 Replace "arbitrary morphism" with "arbitrary morphisms". (F. Gispert Sánchez) p. 120, ¶ l. 9 Replace "Corollary 3.33" by "Proposition 3.33". (M. Pereira) p. 121, ¶ Line 9 Replace$\mathfrak p_0$by$\mathfrak p_0 A[T]$, and correspondingly for$\mathfrak p_r$. (P. Zsifkovits) p. 126, ¶ Theorem 5.22 Replace "of finite type" with "locally of finite type". (This slightly more general statement is used in the proof of Proposition 5.26, for example.) Note that the proof does not use the finiteness of the affine open cover. (F. Gispert Sánchez) p. 126, ¶ Proof of Prop. 5.20 Replace the last sentence by: Corollary 5.17 shows that the structure morphism$X\rightarrow \mathop{\rm Spec} k$corresponds to a finite homomorphism$k\rightarrow A$. (P. Hartwig) p. 127, ¶ Line -3 Replace "$\sup_{Z\in I} (Z\cap U)$" with "$\sup_{Z\in I} \dim(Z\cap U)$" (F. Gispert Sánchez) p. 133, ¶ Cor. 5.45 Add the hypothesis that$Y$be non-empty. () p. 134, ¶ Line 18 Replace $X=\mathop{\rm Spec} B$'' by $Y=\mathop{\rm Spec} B$''. (K. Kidwell) p. 134, ¶ Proof of Cor. 5.47 Note that the first statement in the proof holds by Cor. 5.23. (K. Kidwell) p. 142, ¶ Exercise 5.7 Before the first Show'', add Let$X=\mathop{\rm Spec} A[T]$.'' (A. Steinbach) p. 142, ¶ Line -14 (ex. 5.11) In the definition of the special orthogonal group scheme SO, the condition that the determinant must be +1 is missing. (F. Gispert Sánchez) p. 143, ¶ Line -6 (ex. 5.20) Replace "$x\in X_{\Omega}$" with "$x' \in X_{\Omega}$" (F. Gispert Sánchez) p. 144, ¶ Line 10 (ex. 5.21) Replace "intgral" with "integral". (F. Gispert Sánchez) p. 146, ¶ l.$-8$Replace heuristics'' by heuristic''. (P. Johnson) p. 148, ¶ Line -10$D(g_1 \cdots g_s)$should instead read$D(g_1 \cdots g_r)$. (Nathan Pflueger) p. 149, ¶ Lines -8, -7 Remove "is in$T_x X$". (F. Gispert Sánchez) p. 153, ¶ Def. 6.14 (2) Replace in all points'' by at all points''. (P. Johnson) p. 157, ¶ Proof of Prop. 6.23 Instead of renumbering the$f_i$'', we need to renumber the$T_i$such that the$(r \times r)$minor given by the first$r$columns of$J$does not vanish at$x$. (B. Smithling) p. 157, ¶ Statement of Lemma 6.22 Replace$\partial\varphi(Y_j)$by$\partial\varphi(Y_i)$and$\partial X_i$by$\partial X_j$. (B. Smithling) p. 159, ¶ Line 12 Replace$T_{\mathbb A^n_k, y}$by$T_{\mathbb A^n_k, y}^*$. (F. Gispert Sánchez) p. 159, ¶ Proof of Lemma 6.26 Replace By Example 6.5'' by By the argument in Example 6.5'' and/or add explanation why we obtain the desired linear independence over$\kappa(y)$even though$y$might not be a$k$-valued point. (P. Johnson) p. 159, ¶ Proof of Lemma 6.27 In the second line of the proof, replace columns'' by rows''. (B. Smithling) p. 160, ¶ Line 9 Replace "point of$x$" with "point of$X$". (F. Gispert Sánchez) p. 160, ¶ Corollary 6.29 Replace of finite type'' by locally of finite type''. (Theorem 5.22 requires only locally of finite type'' as noted in a previous erratum.) (Fabian Roll) p. 161, ¶ Lines 4, 5 (cor. 6.31) The two ranks in the statement of the corollary should be of the Jacobian matrix evaluated at$x$: replace "$\partial g_i / \partial T_j$" by "$\frac{\partial g_i}{\partial T_j}(x)$" (twice). (F. Gispert Sánchez) p. 161, ¶ Line 3 of Example 6.34 Change$\cdots Spec(\mathbb{F}_p(T)\otimes_{\mathbb{F}_p[T^p]} \mathbb{F}_p(T))$to$\cdots Spec(\mathbb{F}_p(T)\otimes_{\mathbb{F}_p(T^p)} \mathbb{F}_p(T))$. (Shaopeng Z) p. 162, ¶ Remark 6.37 Replace the final sentence of the remark by: If a point in an arbitrary scheme lies on more than one irreducible component, then its local ring will have more than one minimal prime ideal and hence cannot be an integral domain. In particular, such a point is not normal. (U. Hartl/B. Smithling) p. 163, ¶ Line 17 Replace $\dim {\mathscr O}_{X,x} = 2$'' by $\dim {\mathscr O}_{X,x} \geq 2$''. (T. Wedhorn) p. 165, ¶ Line 17 (ex. 6.4) Replace "$df_{(e,e)}$" with "$dm_{(e,e)}$". (F. Gispert Sánchez) p. 169, ¶ Lines 7, 14 Wrong quotation marks around "globalizations" and "sheaf version". (Zhaodong Cai) p. 171, ¶ Line 4 Replace "homomorphism" with "homomorphisms". (F. Gispert Sánchez) p. 171, ¶ Line -3 Wrong quotation marks around "surjective". (Zhaodong Cai) p. 175, ¶ Line -3 Maybe the tensor symbols$\otimes$should be accompanied by O_X(U) as a subscript. (Victor Zhang) p. 177, ¶ Line -11 Replace "$\mathscr{F}_{|U} \otimes_{\mathscr{O}_U} \mathscr{H}_{|U}$" by "$\mathscr{G}_{|U} \otimes_{\mathscr{O}_U} \mathscr{H}_{|U}$". (F. Gispert Sánchez) p. 178, ¶ Line 19 The morphism "$\iota$" has not been defined; it should be (7.5.7). (F. Gispert Sánchez) p. 181, ¶ Line -7 Replace "and" by "an". (F. Gispert Sánchez) p. 185, ¶ Corollary 7.19. (4) Define$\mathscr{F}$and$\mathscr{G}$to be quasi-coherent$\mathscr{O}_X$-modules. (F. Ebert) p. 186, ¶ Line 18 Replace "(7.4)" by "(7.5)". (Invertible$\mathscr{O}_X$-modules are explained at the end of section (7.5).) (F. Gispert Sánchez) p. 189, ¶ Line -13 Replace "$\mathscr{L}_{|U_x}$" by "$\mathscr{L}_{D|U_x}$" (F. Gispert Sánchez) p. 190, ¶ Line -15 Add "be" before "extended". (F. Gispert Sánchez) p. 190, ¶ Line 22 In general, there are$\mathscr O_X$-modules of finite type that are not of finite presentation even over Noetherian schemes (take a quotient of$\mathscr O_X$by a non-quasi-coherent sheaf of ideals) . (It is true that a quasi-coherent$\mathscr O_X$-modules of finite type over a Noetherian scheme is of finite presentation.) (H. Iriarte) p. 191, ¶ Line -1 Replace "generated" by "generate". (F. Gispert Sánchez) p. 191, ¶ Commutative diagram In the right-most column, replace$n$by$m$in both rows. (P. Carlucci) p. 191, ¶ Line -2 Replace "$X''$" by "$U''$". (F. Gispert Sánchez) p. 194, ¶ (7.18.1) Here$\mathscr{F}$needs to be flat over Y (before, it was only assumed to be$f$-flat in a point$x$). (Longxi Hu) p. 194, ¶ Line 8 Replace$\mathop{\rm Spec} A$by$\mathop{\rm Spec} B$. (Longxi Hu) p. 195, ¶ Proof of Lemma 7.42 The corresponding homomorphism$A'^n \to M'$ should be The corresponding homomorphism$A'^r → M'$; similarly, an isomorphism$A^r_s \to M^r_s$' should be an isomorphism$A^r_s \to M_s$'. (Kannappan Sampath) p. 202, ¶ Exercise 7.20 Replace${\mathscr F}$by${\mathscr G}$. (J. Calabrese) p. 203, ¶ Exercise 7.30 The$\mathcal{O}_X$-modules should be finite locally free, as the determinant was defined in that setting. () p. 204, ¶ Line 4 (ex. 7.32) Replace "send" with "sent". (F. Gispert Sánchez) p. 206, ¶ Prop. 8.4 It might be helpful to add a Remark after the Proposition pointing out that as a formal consequence one gets the following: Let$S$be a scheme, let$X$be an$S$-scheme, and let$v$be a homomorphism of quasi-coherent$\mathscr{O}_X$-modules. Then the functor$F'$on$S$-schemes with$F'(T) = \{ f\in \mathop{\rm Hom}_S(T,X);\ f^*(v)\ \text{surjective} \}$is representable by an open subscheme of$X$. (Apply the original proposition to$X$,$v$to obtain an open subscheme$U$of$X$, and observe that the$S$-scheme$U$represents the functor$F'$. This is how the Proposition is often used later, e.g., in the proof of Lemma 8.13. (Likewise for part (2) of the proposition and other similar statements, like Theorem 11.17.) (F. Gispert Sánchez) p. 209, ¶ Line 10 Change the first "and" to "by". () p. 209, ¶ Line 10 The proof would be more clear if one explained more carefully that a representable functor that is an open immersion (resp. surjective open immersion) is a monomorphism (resp. an isomorphism). (H. Iriarte) p. 211, ¶ Line 9 Replace bijective'' by an isomorphism''. (Ulrich Görtz) p. 211, ¶ Line -7 Add "of" between "homomorphism" and "$\mathscr{O}_S$-modules". (F. Gispert Sánchez) p. 212, ¶ Line -5 Replace "subvector space of$K^n$" by "subvector space$U$of$K^n$". (F. Gispert Sánchez) p. 212, ¶ Line -16 Insert and is'' before also''. (D. Gerigk) p. 216, ¶ Line -7 Replace "disjoint sum" with "disjoint union". (F. Gispert Sánchez) p. 216, ¶ Line 4 Replace "of$\mathscr{O}_S$-module" by "of$\mathscr{O}_S$-modules". (F. Gispert Sánchez) p. 219, ¶ Def. 8.25 Insert$S$before the first such that''. (D. Gerigk) p. 220, ¶ Line 4 Replace "$(v,u)$" by "$(v,s)$". (F. Gispert Sánchez) p. 221, ¶ Line -13 Insert "points" between "$R$-valued" and "of". (F. Gispert Sánchez) p. 224, ¶ Exercise 8.11 Repace${\mathscr G}$by$f^*{\mathscr G}$, and the map$f^*{\mathscr G} \to f^*{\mathscr E}$will not be injective in general. (L. Galinat) p. 226, ¶ Lines 10/12 Replace $\{(x,f(x)) ; x \in X\}$is closed in$X\times Y$'' by $\{(y,f(y)); y \in Y\}$is closed in$Y\times X$'' in (ii), and replace $\{x \in X; f(x) = g(x)\}$is closed in$X$'' by $\{y \in Y; f(y) = g(y)\}$is closed in$Y$'' in (iii). (O. Das) p. 226, ¶ Line 20 Replace separable'' by separated''. (T. Wedhorn) p. 227, ¶ Line 19 Replace "an$S$-objects" by "an$S$-object". (F. Gispert Sánchez) p. 228, ¶ Statement of Prop. 9.3 (3) Replace$f$by$u$everywhere (i.e., in 3 places). (K. Kidwell) p. 229, ¶ Line 13 Replace "$p\colon X\times_S Y$" by "$p\colon X\times_S Y \to X$". (F. Gispert Sánchez) p. 230, ¶ Lines 14, 15 Replace$X\rightarrow X_{\rm red}$by$X_{\rm red}\rightarrow X$, and likewise for$Y$, and replace$f_{\rm red}\circ i_Y$by$i_Y\circ f_{\rm red}$. (P. Barik/U. Hartl) p. 232, ¶ Lines 6-9 Replace "Grass$_{n,n-e}$" with "Grass$_{n-e,n}$" (4 times) to keep the same notation as in chapter 8. (F. Gispert Sánchez) p. 233, ¶ Def. 9.21 Replace $\mathfrak p_x$'' by $\mathfrak m_x$''. (A. Isaev) p. 233, ¶ Remark 9.20, (1) The "conversely" part is clear, but it does not follow from Proposition 9.19 (ii) that "open schematically dense" implies "dense". In fact, it contradicts this example (Stacks project). From here and here (which follows from Lemma 1.25, p. 15, and Definition 10.1 (ii), p.242) it follows that it is the case whenever the ambient scheme X is locally Noetherian. (Laura Brustenga Moncusí) p. 233, ¶ Prop. 9.19 (iii)$f$and$g$should be$S$-morphisms. (K. Kidwell) p. 234, ¶ Lemma 9.23 In (ii) and (iii) the condition of$t$being not a zero divisor is not strong enough, since in general the restriction of a regular global section to some open subset is not regular anymore. Instead,$t$should be regular in each of the stalks (which implies that$t$is regular), in this way the condition becomes stable under restriction. Cf. the erratum to p. 300, l. 3 below. (Louis Martini) p. 234, ¶ Line 28 (Remark 9.25) Replace can checked'' by can be checked''. (L. Galinat) p. 234, ¶ Line 30 (Remark 9.25) Replace$U\subseteq S$by$U\subseteq X$. (P. Barik) p. 235, ¶ Def. 9.26 Our definition of rational map is different from that in other places in the literature. Notably, in EGA I (new ed.) 8.1, it is not required that the open subset$U$be schematically dense. Cf. however the notion of pseudo-morphism introduced in EGA IV, 20.2. (P. Hartwig) p. 235, ¶ Line 9 The claimed injectivity does not hold in general. It does hold if$Y$is separated over$S$(by Prop. 9.19). (K. Kidwell) p. 236, ¶ Line 5 Replace "$(\lambda,\mu)$" with "$(\lambda : \mu)$". (F. Gispert Sánchez) p. 236, ¶ Line -3 Replace "$K$" with "$K(X)$". (F. Gispert Sánchez) p. 237, ¶ Line 24 (Remark 9.34) Replace$f\colon - - \rightarrow Y$by$f\colon X - - \rightarrow Y$. (P. Barik) p. 237, ¶ Line -16 Replace "tape" with "type". (F. Gispert Sánchez) p. 238, ¶ Line 17 (Exercise 9.2) Replace $X \times_S Y \to X \times_S Y$'' by $X \times_S Y \to X \times_T Y$'' (L. Galinat) p. 242, ¶ Rmk. 10.2 (4) At the end, replace if$Y$not separated'' by if$Y$is not separated''. (U. Hartl) p. 248, ¶ Line 9 Replace ad'' by and''. (U. Hartl) p. 248, ¶ Line -14 Replace "on" with "an". (F. Gispert Sánchez) p. 249, ¶ Line 1 Replace "$\mathscr{O}_{Y,f(y)}$" with "$\mathscr{O}_{Y,f(x)}$" (F. Gispert Sánchez) p. 251, ¶ Lines 29, 31 (Proof of Prop. 10.30) Replace$i\colon Z\rightarrow X$by$i\colon Z\rightarrow Y$and$i^\flat$by$i^\flat \colon \mathscr O_Y \rightarrow i^* \mathscr O_Z$. (P. Barik) p. 253, ¶ Remark 10.40 (3) Add the assumption that$X$has a basis of retro-compact open subsets. (A. Gross) p. 255, ¶ Line 4 Replace$Z$by$C$. (P. Barik) p. 256, ¶ Line 18 (Cor. 10.49) In the conclusion of the corollary, add that$\mathscr{F}$is also quasi-coherent. (F. Gispert Sánchez) p. 257, ¶ Line 7 Replace "an" with "a". (F. Gispert Sánchez) p. 264, ¶ Line 2 Replace "1., 3., 5." with "1.-3., 5.". (F. Gispert Sánchez) p. 265, ¶ l.$-2$Replace $R$-scheme'' by of$R$-schemes''. (U. Hartl) p. 265, ¶ Line -15 (cor. 10.67) Add "of" between "morphism" and "$S$-schemes". (F. Gispert Sánchez) p. 267, ¶ Line -10 (prop. 10.75) Replace "$f$" with "$f_0$". (F. Gispert Sánchez) p. 268, ¶ l$-2$Remove one that'' and replace morphism'' by morphisms''. (U. Hartl) p. 268, ¶ Theorem 10.76 It is not true in general that$B$is the inductive limit of its smooth$A$-sub-algebras. All one can say in general is that$B$is isomorphic to a filtered inductive limit of smooth$A$-algebras. See the discussion in Spivakovsky's paper [Sp] (Problem 1.3; Section 10). (M. Hoyois, S. Kelly) p. 268, ¶ Line 22 Replace "morphism" with "morphisms". (F. Gispert Sánchez) p. 268, ¶ l.$-7$Insert if'' after In fact,''. (U. Hartl) p. 270, ¶ Line 17 Remove "it". (F. Gispert Sánchez) p. 272, ¶ Line 12 Replace "subscheme) structure" with "subscheme structure)". (F. Gispert Sánchez) p. 273, ¶ Line -14 Add "is" before "constructible". (F. Gispert Sánchez) p. 274, ¶ Line 4 Replace suffice'' by suffices''. (U. Görtz) p. 274, ¶ Line 9 Replace "$E\subseteq \pi^{-1}(\mathcal{P})$" with "$\overline{E}\supseteq \pi^{-1}(\mathcal{P})$". (If$\pi(\mathfrak{p})=p$with$p\in\mathcal{P}$, then$\pi(\mathfrak{m})=p$for all closed points$\mathfrak{m}$which are specializations of$\mathfrak{p}$, as$\pi$is continuous. Since all such closed points belong to$E$and$R$is a finitely-generated$\mathbb{Z}$-algebra and, in particulalr, Jacobson, we conclude that$\mathfrak{p}\in\overline{E}$.) (F. Gispert Sánchez) p. 275, ¶ Line 21 Replace "Corollary 10.85" with "Theorem 10.84" (we want to apply it to$\mathscr{H}$, not to$\mathscr{O}_X$). (F. Gispert Sánchez) p. 276, ¶ Line 20 Add "of" between "is" and "finite type". (F. Gispert Sánchez) p. 277, ¶ Line 21 Add that$S$is noetherian. (F. Gispert Sánchez) p. 277, ¶ Line 10 Replace "Corollary 5.12" with "Corollary 5.17". (F. Gispert Sánchez) p. 278, ¶ Lines 21-26 (last paragraph of the proof of thm. 10.97) Replace "$X_\xi$" with "$X_\eta$" (three times). Moreover, if$U$is the non-empty open subset of$S$which we are considering, the last equation holds for$v\in V=Y\cap f^{-1}(U)$(not all$Y$). (F. Gispert Sánchez) p. 281, ¶ Line 25 (ex. 10.22) Replace "is" with "be its". (F. Gispert Sánchez) p. 283, ¶ Line -14 (ex. 10.34) Replace "inductive limit" with "inductive system". (F. Gispert Sánchez) p. 287, ¶ Line 3 Replace "$S$" by "$X$". (F. Gispert Sánchez) p. 288, ¶ 2 lines above equation (11.1.6) Replace$f\colon X\rightarrow Y$by$f\colon Y\rightarrow X$. (Zhaodong Cai) p. 288, ¶ Line 2 Replace "$\binom{r+n-1}{r}$" with "$\binom{r+n-1}{n}$" (or "$\binom{r+n-1}{r-1}$"). (F. Gispert Sánchez) p. 288, ¶ Last line of the statement of Proposition 11.1$({\rm Sch})^{\rm opp}$should be$({\rm Sch}/X)^{\rm opp}$. () p. 289, ¶ Prop. 11.3 In the statement of the proposition (and in its proof), one should add parentheses in order to emphasize that the target of the isomorphism is$\Gamma(T, (h^* \mathscr E)^\vee)$. (P. Hartwig) p. 291, ¶ Line 9 Remove the extra parenthesis in "$\mathscr{S}(V/X))$". (F. Gispert Sánchez) p. 292, ¶ -14 Replace "is the group$G$itself" by "is the sheaf of groups$G$itself" (Félix Baril Boudreau) p. 293, ¶ Line 9 Replace "1-cocycle" with "1-cocycles". (F. Gispert Sánchez) p. 293, ¶ Line -14 Replace "1-cocycle" with "1-cocycles". (F. Gispert Sánchez) p. 295, ¶ Lines -2, -3, -5 Replace$\mathscr Isom(\mathscr E, \mathscr O_X^n)$by$\mathscr Isom(\mathscr O_X^n, \mathscr E)$. (A. Schiller) p. 295, ¶ Line 5 Add somewhere that$\varphi$is the map$G'\to G$in the short exact sequence (11.5.5). (F. Gispert Sánchez) p. 296, ¶ Line -3 The discussion in Example 11.42 only considers the noetherian case. (The result is true in general, see e.g. Stacks project 0BCH.) Also, add "ring" after "factorial" (alternatively, remove "a" before). (F. Gispert Sánchez) p. 298, ¶ Line 6 We can only conclude that the immersion is locally of finite presentation. (F. Gispert Sánchez) p. 299, ¶ Line 6 The sheaf$\mathscr{R}_X$has not been defined (it is the sheaf given by$U\mapsto R(U)$), and maybe should not a priori be called a constant sheaf'' here. (F. Gispert Sánchez) p. 299, ¶ 6th line after Definition 11.19 TeX:$\mathop{\rm Div}(X)$should be upright () p. 300, ¶ Line 3 The mapping$U\mapsto {\rm Frac}(\Gamma(U,\mathscr O_X))$is not a presheaf in general (rather, one should consider the localization with respect to elements which are non-zero divisors in every stalk). See Kleiman, Misconceptions about$K_X$. Enseign. Math. (2) 25 (1979), no. 3-4, 203--206 (1980), for a detailed discussion. (P. Hartwig) p. 303, ¶ Line 3 Replace detailed'' by detail''. (J. Calabrese) p. 305, ¶ Line -8 (Lemma 11.33) The notation for the codimension is different from the notation used in chapter 5. (F. Gispert Sánchez) p. 305, ¶ Line -3 Replace for every maximal point of${\rm Supp} D$'' by for every maximal point$\eta$of${\rm Supp} D$''. (T. Wedhorn) p. 305, ¶ Prop. 11.32 Replace the definition of regular'' given in parentheses by i.e., the associated homomorphism$\mathscr O_X \rightarrow \mathscr L$is injective''. (K. Kidwell) p. 305, ¶ Line 27 Replace "$\mathscr{O}$" with "$\mathscr{O}_X$". (F. Gispert Sánchez) p. 305, ¶ Line 19 (displayed bijection) Replace "cartier" with "Cartier". (F. Gispert Sánchez) p. 306, ¶ Line -16 Replace "$\mathscr{O}_{X,c}$" with "$\mathscr{O}_{X,C}$". (F. Gispert Sánchez) p. 308, ¶ Line 13 Replace$U$by$U_i$(twice). (B. Smithling) p. 308, ¶ Prop. 11.40 The proposition is not true as it stands (a counterexample is given by$X$the plane with a fattened origin, and$Z$the origin). It is correct with the additional assumption that$U$be schematically dense in$X$(use the characterization in Lemma 9.23 (ii) to conclude that the exactness at$Z^1(X)$of the corresponding sequence of groups of cycles yields the exactness at${\rm Cl}(X)$of the sequence in the proposition). (B. Smithling) p. 309, ¶ Line -4ff Replace as product$f_1f_2\dots f_r$, where$f_i \in S$are irreducible homogeneous polynomials'' by as product$f_1^{d_1}f_2^{d_2}\dots f_r^{d_r}$, where$f_i \in S$are irreducible homogeneous polynomials and$d_i \in {\mathbb Z}$'' and replace $Z\colon {\mathcal R} \to Z^1({\mathbb P}^n_k), f \mapsto \sum_i [V_+(f_i)]$'' by $Z\colon {\mathcal R} \to Z^1({\mathbb P}^n_k), f \mapsto \sum_i d_i[V_+(f_i)]$''. (T. Wedhorn) p. 313, ¶ Line 1 The morphism$\phi$is not necessarily flat: See mathoverflow.net/questions/65267/global-sections-of-flat-scheme-also-flat. The statement of the proposition is true, though: One should first remark that one can work locally on target and source and hence can assume that everything is affine. (Hernan I.) p. 313, ¶ Line -6 Replace set'' by sets''. (P. Johnson) p. 315, ¶ Line -2 The notion of being "of pure dimension 1" has not been defined. It is defined in brackets in Proposition 15.1. The easiest fix here might be to replace it by equidimensional of dimension 1''. (F. Gispert Sánchez) p. 319, ¶ Line -13 (ex. 11.22 (a)) Replace "an" with "a". (F. Gispert Sánchez) p. 321, ¶ Line -16 Replace "a affine scheme" with "an affine scheme". (F. Gispert Sánchez) p. 321, ¶ Line 24 Replace affine over$X$'' by affine over$Y$''. (P. Barik) p. 323, ¶ Line 17 Replace$\otimes_{(A'\otimes_A B})\otimes$by$\otimes_{(A'\otimes_A B)}$. (P. Barik) p. 323, ¶ Line 8 In the last expression of 12.2.3 replace$Y'$by$X'$. (P. Barik) p. 323, ¶ Line -9 The (finite) covering$(U_i)_i$of$X$should be an affine open covering. (F. Gispert Sánchez) p. 323, ¶ Line 12 Replace$X' = \mathop{\rm Spec} B' \otimes_{\mathop{\rm Spec} B} \mathop{\rm Spec A}$by$X' = \mathop{\rm Spec} B \otimes_{\mathop{\rm Spec} A} \mathop{\rm Spec A'}$. (P. Barik) p. 324, ¶ Line -17 Remove a'' at the end of the line. (K. Kidwell) p. 324, ¶ Line 2 Replace "$\mathscr{F}(X)$" with "$\mathscr{F}(U_i)$" (F. Gispert Sánchez) p. 325, ¶ Line -3 Corollary 5.12'' should be Proposition 5.12''. (U. Hartl) p. 328, ¶ Prop. 12.18 The first formula is false (almost always, e.g. if$Y=\mathop{\rm Spec} \mathbb F_p$,$X=\mathop{\rm Spec} \mathbb F_{p^2}$,$X'=\mathop{\rm Spec} \mathbb F_{p^2}$). (Yehao Zhou) p. 329, ¶ Line 18 Replace "be" with "by". (F. Gispert Sánchez) p. 332, ¶ Lines 2 and 22 Both in the statement (twice) and in the proof (once) of Proposition 12.27, "$\mathscr{O}$" is used to denote "$\mathscr{O}_X$". (F. Gispert Sánchez) p. 332, ¶ Line -15 Replace Corollary 5.12'' by Proposition 5.12''. (S. Köbele) p. 333, ¶ Line -14 Replace$a_0$by$a_n$. (S. Köbele) p. 333, ¶ Example 12.29 In several places, replace$t$by$t'$. (S. Köbele) p. 335, ¶ Equation (12.8.1) The middle term should be$\frac{f_j^{k+\ell}x_{\mathbf i}}{f_{\mathbf i}^k}$instead of$\frac{f_j^{k+\ell}x_{\mathbf i}}{f_{\mathbf i}^r}$. (P. Barik) p. 338, ¶ Line 6 Replace "$H^1(\mathscr{O}_Y,\mathscr{F})$" with "$H^1(Y,\mathscr{F})$". (F. Gispert Sánchez) p. 338, ¶ Line 8 Add "be" between "let$\eta$" and "its generic point". (F. Gispert Sánchez) p. 341, ¶ Line 11 Add "be" before "its normalization". (F. Gispert Sánchez) p. 342, ¶ Line -12 (prop. 12.53) Add that$X$is integral. (F. Gispert Sánchez) p. 344, ¶ Line -11 Replace$f^{-1}(U)\subseteq V$by$f^{-1}(V)\subseteq U$. (P. Barik) p. 346, ¶ Line -4 Replace "$\mathscr{O}_X$-module" with "$\mathscr{O}_X$-modules". (F. Gispert Sánchez) p. 347, ¶ Line 7 Add "is" before "not irreducible". (F. Gispert Sánchez) p. 348, ¶ Line -10 Add "be" before "its Stein factorization". (F. Gispert Sánchez) p. 352, ¶ Line -12 Replace relation'' by relations''. (U. Görtz) p. 352, ¶ Line 18 Replace finite'' by integral'': We do not know whether$A'$is finite over$A$, but it being integral is enough for the following argument. (Yugo Takanashi) p. 353, ¶ Proof of Prop. 12.76 In the beginning of the proof, it would be useful to note that the inclusion$\mathop{\rm Isol}(B/A) \supseteq \mathop{\rm LocIsom}(B/\overline{A})$follows easily from Lemma 12.75 (1) and going up for integral extensions, so that only the opposite inclusion is considered in the following. In the second step, to see that$\mathfrak q'\in \mathop{\rm Isol(A'/A)}$, note that the existence of$U'$shows that$\mathfrak q'$is open in its fiber over$\mathop{\rm Spec} A$. By Lemma 12.72 it is therefore isolated, because the fiber is of finite type over$\kappa(\mathfrak q)$. (F. Gispert Sánchez) p. 356, ¶ Lines 6 to 8 Replace "$W'$" with "$W$" and "$X'$" with "$X$" (three times altogether). Replace$f\mathscr O_Y$by$\mathscr O_Y$(twice). (F. Gispert Sánchez) p. 357, ¶ Proof of Thm. 12.73 In general, it is not true that$f'$as defined in the book is of finite type; hence Cor. 12.78 cannot be applied directly. Instead, one can proceed as follows: Start the proof as in the book: the restriction of$c$to${\rm Isol}(c)$is an open immersion, hence$h$is an open immersion. Now approximate$X'$by$X'_\mu$which are affine of finite type over$Y$(analogously to approximating$Z$by$Z_\lambda$). Using Lemma 12.84, we may assume that$V'\rightarrow X'_\mu$is an open immersion (for some fixed large$\mu$). Now replace$X'$by$X'_\mu$and continue as in the book: Define$Z$, apply 12.78 (which is now justified), observe that$V' \subseteq {\rm Isol}(X'/Z)$, and apply Lemma 12.84 again to approximations of$Z$. (Yehao Zhou) p. 361, ¶ Ex. 12.6 Replace isomporphism by isomorphism. (T. Keller) p. 368, ¶ Prop. 13.2 (3) Replace with all relevant prime ideals'' by with the intersection of all relevant prime ideals''. (K. Kidwell) p. 368, ¶ Line -6 Replace ideal'' by ideals''. (U. Görtz) p. 368, ¶ Statement of Prop. 13.2 (2) We must assume that$0\notin S$and that$S$contains an element of$A_+$. (K. Kidwell) p. 368, ¶ Line -20 The element$f$should be homogeneous. (U. Görtz) p. 368, ¶ Equation (13.1.1) Replace$\mod f-1$by$\mod (f-1)$(A. Kaučikas) p. 370, ¶ Line 10$f \in rad(g)_+$should read$g \in rad(f)_+$. (F. Gispert Sánchez/N. Pflueger) p. 375, ¶ Line -3 Replace definition of$\sigma$by $\sigma(t'):= t / f^n \in \Gamma_*(\mathscr F)_{(f)}$''. (P. Johnson) p. 376, ¶ Line -10 (thm. 13.20, displayed equation, under the arrow) Replace "$\mathscr{F} \mapsfrom \Gamma_{*}(\mathscr{F})$" with "$\Gamma_{*}(\mathscr{F}) \mapsfrom \mathscr{F}$". (F. Gispert Sánchez) p. 377, ¶ Line 15 Replace $n \ge 0$'' by $n\ge n_0$''. (P. Johnson) p. 377, ¶ Proposition 13.22 (statement and proof) The symbol$n$is used for two different things: the number of generators of$A_+$and as an index for$\mathscr{F}(n)$. (F. Gispert Sánchez) p. 378, ¶ line 2 Replace "on" with "one." (N. Pflueger) p. 379, ¶ Prop. 13.28 In the first line of the statement of the proposition, replace modules'' by module''. (P. Johnson) p. 379, ¶ Line -2 Replace "$\varphi$" with "$\varphi_i$". (F. Gispert Sánchez) p. 379, ¶ Line -1 The domain and codomain of$g_i$are swapped: switch$\mathscr{A}_i$and$\mathscr{A}'_i$. (F. Gispert Sánchez) p. 379, ¶ Line -9 The meaning of the symbol$\pi$should be stated again. Both in the statement of proposition 13.28 and at the end of the proof (in the next page), I would write "$g^*_{\mathscr{L}}(\mathscr{O}_X(n)\otimes\pi^*(\mathscr{L}^{\otimes n}))$" with the extra parentheses (or, alternatively, replace "$\pi$" with "$\pi'$"). (F. Gispert Sánchez) p. 380, ¶ Line 17 Add "be" before "the structure morphism". (F. Gispert Sánchez) p. 380, ¶ Line 8 Replace "$\mathscr{O}(n)$" with "$\mathscr{O}_X(n)$". (F. Gispert Sánchez) p. 380, ¶ Line 6 Add "be" before "the structure morphism". (F. Gispert Sánchez) p. 380, ¶ Line 3 Replace "$f_j^{-1}$" with "$f_j^{-n}$". (F. Gispert Sánchez) p. 382, ¶ Lines 15, 18 Replace$\mathbb P^{n+1}$by$\mathbb P^n$(three times). (K. Kidwell) p. 382, ¶ Line 21 Replace "$\mathscr{O}^{n+1}$" with "$\mathscr{O}^{n+1}_X$". (F. Gispert Sánchez) p. 382, ¶ Line -12 Replace$\alpha_i$by$\alpha_j$. (T. Keller) p. 383, ¶ Line 10 Replace "$R$-modules" with "$R$-module". (F. Gispert Sánchez) p. 388, ¶ Line -6, Line -3 Add that$f$is homogeneous. (F. Gispert Sánchez) p. 390, ¶ Line -9 Replace "$(i\times \id_{S'}) \mathscr{O}_{P'}(1)$" with "$(i\times \id_{S'})^* \mathscr{O}_{P'}(1)$". (F. Gispert Sánchez) p. 390, ¶ Line -16 Omit the statement in parentheses. It is not true that the existence of an ample line bundle implies properness. (P. Hartwig) p. 391, ¶ Line 2 Replace "(the globalization) of" with "(the globalization of)". (F. Gispert Sánchez) p. 392, ¶ Line -17 Remove "be". (F. Gispert Sánchez) p. 393, ¶ Line 8 Replace "send" with "sent". (F. Gispert Sánchez) p. 393, ¶ Line -15 Replace "$i'\colon X \hookrightarrow P'$" with "$i'\colon X \to P'$" (as$i'$is not necessarily an immersion, only$i$is). (F. Gispert Sánchez) p. 393, ¶ Lines -8 and -7 Replace "(2)" with "(1)" (twice). (F. Gispert Sánchez) p. 394, ¶ Line 20 Add that "$S=\Spec R$" somewhere ($R$has not been defined). (F. Gispert Sánchez) p. 394/395, ¶ (13.13) In line$-8$, replace non-zero'' by regular''. At the end of that paragraph, add a reference to Prop. 11.32 (in addition to Cor. 11.28). At the beginning of page 395, add the assumption that$X$be integral (to ensure that all non-zero global sections of a line bundle are regular). (K. Kidwell) p. 394, ¶ Line -15 Replace "$n\geq n_0+m_0$" with "$n\geq d+m_0$". (F. Gispert Sánchez) p. 395, ¶ Def. 13.60 Insert$\mathscr L$after$\mathscr O_X$-module. (P. Johnson) p. 396, ¶ Line -6 Replace "$g^{-1}(X_s)$" with "$(g')^{-1}(X_s)$". (F. Gispert Sánchez) p. 396, ¶ Line -1 Add the missing closing parenthesis at the end. (F. Gispert Sánchez) p. 398, ¶ Lines -6 & -2 (def./prop. 13.68) Replace "a quasi-coherent$\mathscr{O}_X$-module$\mathscr{E}$" with "a quasi-coherent$\mathscr{O}_S$-module$\mathscr{E}$" (twice). (F. Gispert Sánchez) p. 398, ¶ Line -9 Replace$Y\backslash \varepsilon(S)$by$C\backslash \varepsilon(S)$. (U. Görtz) p. 404, ¶ Line 19 Replace "by Corollary 13.42" with "by Example 13.69". (F. Gispert Sánchez) p. 409, ¶ Diagram (13.19.1) In the top right corner, replace$\mathop{\rm Bl}\nolimits_X(Z)$by$\mathop{\rm Bl}\nolimits_Z(X)$. (J. Watterlond) p. 416, ¶ Line 19 Replace "morphisms" with "morphism". (F. Gispert Sánchez) p. 418, ¶ Line -6 (ex. 13.2 (a)) Replace "(resp. bijective)" with "(resp. surjective)". (F. Gispert Sánchez) p. 419, ¶ Line 1 (ex. 13.3) Replace "$A$-modules" with "$A$-module". (F. Gispert Sánchez) p. 423, ¶ Line 11 Replace morphisms'' by morphism''. (K. Kidwell) p. 424, ¶ Prop. 14.3. (1) Replace$\mathop{\rm Spec} A \rightarrow \mathop{\rm Spec} B$by$\mathop{\rm Spec} B \rightarrow \mathop{\rm Spec} A$. (F. Ebert) p. 426, ¶ Line -2 Replace $\mathop{\rm Spec} R$'' by $Y$''. (T. Wedhorn) p. 426, ¶ Line 16 (cor. 14.12) Replace "a morphism" with "morphisms". (F. Gispert Sánchez) p. 427, ¶ Proof of prop. 14.16 In the proof,$X$is replaced by$\Spec \mathscr{O}_{X,x}$and$f$is replaced by the composition of$f$with the canonical morphism$\Spec \mathscr{O}_{X,x} \to X$without saying so. It would be useful to state this explicitly. (F. Gispert Sánchez) p. 428, ¶ Proof of prop. 14.17 Here, Proposition B.70 (6) is not enough because$X$is not assumed to be locally noetherian. Thus, we need a slightly different result: If$f$is flat, then it is even faithfully flat (since it is closed and dominant). Now use that if$A\subseteq B$are domains with the same field of fractions and$B$is faithfully flat over$A$, then$A=B$. (Say$\frac as\in B$for$a, s\in A$. Then$a\in sB\cap A = sA$by Matsumura, Commutative Ring Theory, Thm. 7.5 (ii), so$\frac as\in A$.) (F. Gispert Sánchez) p. 428, ¶ Proposition 14.20, Lemma 14.21 The lemma is incorrect as stated, because the reduction to the local case does not work as claimed. It should be replaced by the following: (1) Let$A$be a noetherian ring, let$B$be a noetherian$A$-algebra, and let$M$be a$B$-module of finite type which is flat over$A$. Let$f\in B$such that for every maximal ideal$\mathfrak M\subset B$, multiplication by$f$is an injection$M/(\mathfrak M\cap A) \rightarrow M/(\mathfrak M\cap A)$. Then$M/fM$is flat over$A$. (2) With the techniques of Chapter 10 the hypothesis noetherian'' can be replaced by suitable conditions like finite presentation''. See also MathOverflow and Matsumura, Commutative Ring Theory, Thm. 22.6. To avoid the additional difficulty for non-noetherian rings, in the statement of the proposition the assumption that$S$and$X$be noetherian should be added. Assuming that$S=\mathop{\rm Spec} A$,$X = \mathop{\rm Spec B}$, the injectivity assertion means that for all maximal ideals$\mathfrak m\subset A$,$f\not\in \mathfrak mB$; it follows that$f\not\in (\mathfrak M \cap A)B$for all maximal ideals$\mathfrak M\subset B$, so that the lemma (statement (1) of the previous paragraph) can be applied. In the proof of Theorem 14.22 (in the non-noetherian case) a version of the Lemma as alluded to in (2) above is required. (U. Görtz) p. 428, ¶ Lemma 14.19 Replace such'' by such that''. (U. Görtz) p. 434, ¶ Line -14 Add "the" between "If$y$is not" and "closed point". (F. Gispert Sánchez) p. 435, ¶ Line -17 Replace "morphism" with "morphisms". (F. Gispert Sánchez) p. 443, ¶ Line -16 Add "a" between "Let$\mathbf{P}$be" and "property". (F. Gispert Sánchez) p. 443, ¶ Line 3 Replace "a open immersion" with "an open immersion". (F. Gispert Sánchez) p. 447, ¶ Line 10 Replace "$(\mathscr{G},\psi)$" with "$(\mathscr{G}',\psi)$". (F. Gispert Sánchez) p. 447, ¶ Line 10 Replace$\mathscr G$by$\mathscr G'$. (K. Kidwell) p. 449, ¶ Line 5 Replace "$q=p_1\circ p=p_2\circ p$" with "$q=p\circ p_1=p\circ p_2$". (F. Gispert Sánchez) p. 452, ¶ Line -10 Replace "homomorphism" with "homomorphisms". (F. Gispert Sánchez) p. 453, ¶ Line 14 The reference to Theorem 14.17 is wrong. It should refer to Theorem 14.70. (F. Gispert Sánchez) p. 454, ¶ Line 2 Replace "or fppf-sheaves of sheaves" with "or fppf-sheaves or sheaves". (F. Gispert Sánchez) p. 454, ¶ Line -17 Replace "an morphism" with "a morphism". (F. Gispert Sánchez) p. 455, ¶ Line -10 Replace "$G_{S'}$" with "$G_{|S'}$". (F. Gispert Sánchez) p. 457, ¶ Proof of Thm. 14.83 With the given reference to Morita equivalence in (8.12) the proof covers only the case of finite-dimensional vector spaces. The theorem follows from this because every finite-dimensional$k'$-subvector space of a given$V'$is contained in a$\Gamma$-invariant finite-dimensional$k'$-subvector space of$V'$, so we can write$V'$as the union of its finite-dimensional$\Gamma$-invariant subvector spaces, and get the result for$V'$from the finite-dimensional case. Also it should maybe be stated that for the last assertion of the theorem (taking invariants is a quasi-inverse), it is enough to show - given that we have proved that base change is an equivalence of the two categories - that$(V\otimes k')^\Gamma = V$, which is clear. (U. Görtz) p. 457, ¶ Line 3 Replace "$\gamma(a_\delta)$" with "$\gamma(b_\delta)$". (F. Gispert Sánchez) p. 460, ¶ Diagram (14.22.1), definition of$c(\gamma)$It looks like if we want$c$to be 1-cocycle, then$c(\gamma)$should be$\gamma_Y\circ\gamma^{-1}$instead of$\gamma^{-1}\circ\gamma_Y$as you defined using the diagram. (Han Zhou) p. 468, ¶ Line -11 Replace Lemma 14.106 2'' by Lemma 14.106 (2)''. (U. Görtz) p. 470, ¶ Line 12 The correct formula is$\mathop{\rm dim} X = \mathop{\rm dim} Y + \mathop{\rm dim} f^{-1}(y)$, i.e.,$X$and$Y$must be exchanged. (Yong Hu) p. 471, ¶ Cor. 14.116 In part (1), we only obtain$\mathop{\rm dim} f^{-1}(V) = \mathop{\rm dim} f^{-1}(y) + \mathop{\rm dim}(V)$. (In fact, just take$X=Y=\mathop{\rm Spec} R$, where$R$is a (universally catenary) discrete valuation ring, and$f$the identity morphism. Then the original statement holds for$V=Y$, but is false for$V$consisting only of the generic point of$Y$.) In the proof of part (2), it would be easier to appeal to Lemma 14.109, than to invoke Theorem 14.110. (And note that the statement is void for the empty fibers, anyway.) (Yong Hu) p. 478, ¶ Line -11/Page 583, Line 2 The reference [AK] should point to the following article: Altman, Allen B.; Kleiman, Steven L. Compactifying the Picard scheme. Adv. in Math. 35 (1980), no. 1, 50-112. (P. Hartwig) p. 485, ¶ Prop. 15.1 (ii) Add and none of the$X_i$consists of only one point''. (U. Görtz) p. 487, ¶ Line 11 Replace the title of the segment by Morphisms from spectra of valuation rings to schemes''. (U. Görtz) p. 488, ¶ Prop. 15.7 Rephrase the statement of the Proposition to indicate that the equality$g(\eta) = y$can/should be understood in the schematic sense, i.e., that$g$extends the morphism$\mathop{\rm Spec} K \rightarrow Y$coming from the inclusion$\kappa(y)\subseteq K$. (U. Görtz) p. 488, ¶ Line 11 Replace we can in addition assume that the ring$A$is noetherian'' by then the ring$A$is noetherian''. (U. Görtz) p. 497, ¶ Line 13 Replace If$C$is a separated curve over a field,'' by If$C$is a separated curve over a field and$U$is chosen affine,''. (T. Wedhorn) p. 499, ¶ First line after Definition 15.33 Replace left by right. () p. 501, ¶ Line 9 Replace morpism'' by morphism''. (J. Scarfy) p. 514, ¶ Line -7 Replace $Y = \mathop{\rm Spec} B[1/d_{I,J}]$'' by $Y_{I,J} = \mathop{\rm Spec} B[1/d_{I,J}]$''. (T. Wedhorn) p. 528, ¶ Line 14 Replace $V_+(p)\in\mathbb P^3_k$'' by $V_+(p)\subset\mathbb P^3_k$''. (U. Görtz) p. 528, ¶ Line -18 Replace$T_2T_3^3$by$T_2T_3^2$. (U. Görtz) p. 534, ¶ Line 11 Replace the definition of$g$by $g := f \circ (x \times \mathop{\rm id}\nolimits_Y) \circ p_2$''. (K. Kidwell) p. 534, ¶ Line 6 of the Proof of Proposition 16.54 "nieghborhood" should be "neighborhood" () p. 534, ¶ Line 5 of the Proof of Proposition 16.54$p^{-1}(y)$should be$p_2^{-1}(y)$. () p. 540, ¶ Exercise 16.7 As a connected component$G'$has not necessarily a rational point, it is not geometrically connected in general (and in particular not geometrically irreducible). The exercise should be reformulated as follows: Let$k$be a field and let$G$be a$k$-group schemes locally of finite type. Show that every connected component$G'$of$G$is irreducible and of finite type. Show that the geometric number of connected components of$G'$is equal to the geometric number of irreducible components of$G'$. For$k = {\mathbb Q}$and$G = \mu_p$(Exercise 16.6) for a prime number$p$show that$G$has two connected components and that the geometric number of connected components of$G$is$p$. (B. Conrad) p. 546, ¶ Line -10/-9 Replace left'' by right'' and right'' by left''. (D. Heiss) p. 549, ¶ Line -2 The rank of a free module over the zero ring is not uniquely determined, so this case should be excluded. (P. Hartwig) p. 552, ¶ Line 7 (ex. B.17) Replace "$\mathfrak{m}M\neq M$" with "$\mathfrak{m}M=M$". (F. Gispert Sánchez) p. 560, ¶ Prop. B.55 Add the hypothesis that$A$is noetherian (cf. [Atiyah-Macdonald], Prop. 5.17). (Akira Masuoka) p. 561, ¶ Definition B.58 In the second line of the definition,$r$should be replaced by$n$. (P. Hartwig) p. 563, ¶ Prop. B69 (2) The assumption that the extension be algebraic is not required, see [BouAC] VI § 1.2 Cor. to Thm. 2. (Akira Masuoka) p. 565, ¶ Remark B.75 (2) Replace the reference to B.70 (2) by B.70 (3). (Akira Masuoka) p. 566, ¶ Prop. B.81 (2), Prop. B.82 In both cases$\mathop{\rm depth}\nolimits_A(\mathfrak a)$needs to be replaced by$\mathop{\rm depth}(\mathfrak a, A)$(the length of a maximal$A$-regular sequence of elements in$\mathfrak a\$, and this symbol needs to be defined). (Akira Masuoka) p. 574+575, ¶ Add (IND) for faithfully flat and surjective (cf. EGA IV 8.10.5 (vi)). (Y. Zaehringer) p. 576, ¶ Line 12 Replace "universal homomorphism" by "universal homeomorphism". (Yun Hao) p. 603, ¶ Line 17 Replace Brauer-Severy'' by Brauer-Severi''. (J. Calabrese)