## 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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600 errata listed.

 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. 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. 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, ¶ 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. 7, ¶ Line 9 Replace "guide line" by "guideline". (Mahdi Majidi-Zolbanin) 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, ¶ Theorem 1.8 Add the assumption that $A\ne 0$. (D. Gerigk) 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. 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. 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, ¶ Lemma 1.19 (2) Add the assumption that $X\ne\emptyset$. (Ulrich Görtz) p. 14, ¶ Proof of second part of Lemma 1.19 It would be helpful to explain better why $X$ has only finitely many irreducible components. (Mahdi Majidi-Zolbanin) 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. 14, ¶ Line 10 Replace $\emptyset \ne J \subset I$'' by $\emptyset \ne J \subsetneq I$''. (J. Buck) p. 15, ¶ Line 14 Replace Every open subset'' by Every subspace''. (T. Wedhorn) p. 15, ¶ Line -15 Replace "there existed" with "there would exist". (F. Gispert Sánchez) p. 15, ¶ Line 15 Replace Every closed subset'' by Every subspace''. (T. Wedhorn) p. 16, ¶ Line -16 Replace $T_2-T_1^2$ by $T_1-T_2^2$. (P. Zsifkovits) p. 16, ¶ Line -13 Omit superfluous (. (P. Zsifkovits) p. 19, ¶ Line -15 Replace Proposition 1.20'' by Proposition 1.32''. (P. Zsifkovits) p. 19, ¶ Line 9 In the diagram exchange $m$ and $n$. (J. Buck) 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. 19, ¶ Line 12 Replace "obtain the desired inverse homomorphism" by "obtain the desired inverse map", as Hom(X,Y) is just a set. (Mahdi Majidi-Zolbanin) 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, ¶ Definition 1.39 An exception needs to be made if $U$ is the empty set. (Nick Mertes) p. 21, ¶ Line 8 Remove "of" in the sentence "how to identify elements of $f \in \mathscr O_X(U)$ with ..." (Mahdi Majidi-Zolbanin) p. 21, ¶ Line 12 Require $V$ to be non-empty. (Ulrich Görtz) p. 22, ¶ Line -14 Replace $g\circ f$ with $g\circ f_{|f^{-1}(U)}$. (A.Graf) p. 23, ¶ Line -14 1.47 (i), "opposed category" should be "opposite category". (Peng Du) p. 23, ¶ Line 15 (Def. 1.46 (2)) Replace finite covering'' by finite open covering''. (F. Gispert Sánchez) p. 23, ¶ Cor. 1.47 (i) Exclude the zero $k$-algebra. (Ulrich Görtz) p. 23, ¶ Prop. 1.48 Add the assumption that $X\ne \emptyset$ (or put this into the definition of prevariety?). (Ulrich Görtz) 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. 25, ¶ Prop. 1.54 All the open subsets considered here, i.e., also $U'$ and $U$, $V$ in the final sentence, should be assumed to be non-empty. (Ulrich Görtz) p. 26, ¶ Line -13 Replace $k$ by $R$. (D. Gerigk) p. 26, ¶ Prop. 1.56 Add that the inclusion $Z\to X$ is a morphism of prevarieties. (Ulrich Görtz) p. 27, ¶ Line 5 Replace $R_n$ by $X_n$. (Peng Du) 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 2 Replace "space with function" with "space with functions". (F. Gispert Sánchez) 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 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. 34, ¶ Line -12 (Cor. 1.71) Insert "be" before "quadrics". (Peng Du) 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. 35, ¶ Line 4 $r \gt 2$ (rather than: $r \gt 1$) (P. Barik) 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. 40, ¶ 9 Replace "of functions" by "with functions". (Peng Du) p. 41, ¶ 7 Replace "with the category" by "to the category". (Peng Du) 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 -3 (def. 2.18.Sh2) Replace "by (a)" with "by (Sh1)". (F. Gispert Sánchez) p. 48, ¶ Line 9 Add the category of abelian groups'' as the first example. (A. Kaučikas) 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 $-2$ Replace $s^x$ by ${s^x}_{|V_x}$. (A. Graf) p. 51, ¶ Line 12 Replace if and only of'' by if and only if''. (J. Watterlond) 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, ¶ Line 3 Replace $f(V)$'' by $f(U)$''. (K. Mohri) p. 54, ¶ (2.8.2) and (2.8.3) In the book, (2.8.3) is deduced from (2.8.2). However, it is not clear, how (2.8.2) can be proved without using either (2.8.3) or Proposition 2.27, so either the order should be changed, or further details on the proof of (2.8.2) should be added. Indeed, (2.8.3) can be proved without using (2.8.2) by noting that $(f^{-1}\mathcal{G})_x \cong (f^+ \mathcal{G})_x = \varinjlim_{x \in V} (f^+ \mathcal{G})(V) = \varinjlim_{x \in V} \varinjlim_{f(V) \subseteq U} \mathcal{G}(U) = \varinjlim_{f(x) \in U} \mathcal{G}(U) = \mathcal{G}_{f(x)},$ where the first isomorphism is due to Proposition 2.24(1). Given the above, we can deduce (2.8.2) from (2.8.3): First note that it suffices to prove that $f^{-1}(g^+ \mathcal{H}) \cong f^{-1}(g^{-1} \mathcal{H})$, then observe that the induced stalk maps are isomorphisms by (2.8.3) and Proposition 2.24(1), thus (2.8.2) follows from Proposition 2.23(2). (See also Math StackExchange.) (A. Graf / L. P.) p. 54, ¶ Proof of Prop. 2.27 Introducing the element $t$ could be avoided. (E. Viehmann) p. 54, ¶ Line -9 Replace sheaves'' by presheaves''. (J. Watterlond) p. 54, ¶ Line 2 Replace $X$'' by $Y$''. (K. Mohri) p. 55, ¶ Line 2 (Remark 2.28) The assumption on $\psi$ should be that it is a morphism of presheaves. (L. P.) p. 55, ¶ Line 11 Replace statements'' by statement''. (D. Gerigk) p. 55, ¶ Line -13 Insert "$U$" after "open subset". (F. Ebert) p. 55, ¶ 9 Replace "limit" by "inductive limit". (Peng Du) p. 58, ¶ Line -2 The third sum should be over the index set $I$ instead of $J$. (Sebastian Schlegel Mejia) p. 58, ¶ Proof of Thm. 2.33 In the proof, you write: As $D(f)$ is quasi-compact, we can assume that $I$ is finite.'' Indeed, this works for condition (1); however, it is less trivial for (2). Of course, the strategy of proof (i.e., reducing to the case of finite $I$) is successful, but I do think that this deserves an explanation. The key step is to note that any finite subcovering $D(f) = \bigcup_{i \in I} D(f_i) = \bigcup_{j=1}^k D(f_{i_j})$ (where $I$ is possibly infinite) gives rise to a finite subcovering $D(f_l) = \bigcup_{j=1}^k D(f_{i_j}) \cap D(f_l) = \bigcup_{j=1}^k D(f_{i_j} \cdot f_l)$ for every $l \in I$, and then to apply (1). (L. P.) p. 60, ¶ Line -9 Exclude $f=0$ here. (Philipp Reichenbach) p. 61, ¶ Lines -3, -2 Replace $\mathfrak a$'' by $\mathfrak a_1$'' and $\mathfrak b$'' by $\mathfrak a_2$''. (P. Zsifkovits) p. 62, ¶ Line -18 Insert "is" before "the complement". (Philipp Reichenbach) 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, ¶ Exer. 2.13 (b) The assumption that $X$ be connected can be omitted. (A. Beshenov) 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 "elements of $u \in A$" with "elements $u \in R$". (F. Gispert Sánchez) 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. 69, ¶ 10 Replace "(or more" by ". More". (Peng Du) 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. 71, ¶ 3rd paragraph in the proof $\mathcal{O}_{U_{ij}}(U)$ is not well--defined. Instead, one could write: $\ldots$ then we identify $$\mathcal{O}_{U_i}(\psi_i^{-1}(U)) = \mathcal{O}_{U_{ij}}(\psi_i^{-1}(U)) \cong (\varphi_{ij})_*\mathcal{O}_{U_{ji}}(\psi_i^{-1}(U)) = \mathcal{O}_{U_{ji}}(\varphi_{ij}^{-1}\psi_i^{-1}(U)))$$ $$= \mathcal{O}_{U_{ji}}(\psi_j^{-1}(U)) = \mathcal{O}_{U_j}(\psi_j^{-1}(U))$$ via $\varphi_{ij}$.'' (L. P.) p. 71, ¶ Last line of the proof Replace $X = \bigcup U_i$'' by $X = \bigcup_{i \in I} \psi_i(U_i)$''. (L. P.) 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. 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, ¶ 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. 78, ¶ Prop. 3.29(2) The "$V$" appearing in the statement of (2) is not related to the "$V$" introduced in the proof of (2). (L. P.) p. 78, ¶ Prop. 3.29 (3) Add the assumption that all $U_i$ are non-empty. (D. Gerigk) p. 79, ¶ Line 12 Replace "if follows" with "it follows". (F. Gispert Sánchez) p. 79, ¶ Line -1 (Proof of Prop 3.33) Replace "(1)" with "(i)" , "(2)" with "(ii)" and "(3)" with "(iii)". (R.Ishizuka) p. 79, ¶ Line 14 Replace locally finite type'' by locally of finite type''. (J. Watterlond) p. 79, ¶ Proof of Lemma 3.32, second paragraph Replace "the $f_i$ generated the" by "the $f_i$ generate the". (L. P.) 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. 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). (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. 81, ¶ 10 Replace "opposed category" by "opposite category". (Peng Du) p. 82, ¶ Line 13 Replace points'' by type''. (D. Gerigk) p. 83, ¶ Line -8 Replace "$j_*\mathscr O_Y$" by "$(j_*\mathscr O_Y)_{|U}$". (Peng Du) p. 84, ¶ Def. 3.41 (1) The definition of closed subscheme is not ideal, because it does not become sufficiently clear when two closed subschemes are equal. It would be better to say that a closed subscheme is given by a closed subset $Z\subseteq X$ together with an ideal sheaf $\mathscr J\subseteq \mathscr O_X$ such that certain properties hold. (Ulrich Görtz) p. 85, ¶ Line -17 Replace biggest'' by largest''. (P. Johnson) 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. 86, ¶ Line -2 Replace "Remark 3.45" by "Example 3.45". (Peng Du) p. 88, ¶ 3rd line of the proof of Prop. 3.52 Replace "on" by "in" in the sentence "[...] $Z$ is closed on $U$". () p. 88, ¶ Line -7 Replace "ordered set" by "partially ordered set". (Peng Du) 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, ¶ 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. 90, ¶ Exercise 3.13 $X$ should be nonempty. () p. 92, ¶ Line -5 Replace shows'' by show''. (P. Zsifkovits) p. 93, ¶ Line -9 Insert "=" before $f_m(x)$. (Peng Du) 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, ¶ 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 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, ¶ 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, ¶ 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) Replace affine neighborhood $U'$ of $x'$'' by affine neighborhood $U'$ of $f(x')$''. (B. Heintz) 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 The numbering "Frobenius morphism 4.24" here is a bit inconsistent, and confusing when referred to later - better replace by "Definition 4.24 (The Frobenius morphism)". (Ulrich Görtz) p. 104, ¶ Line -3 Replace be'' by by''. (U. Görtz) p. 104, ¶ Line 13 Add "Let" before "$X$ and $Y$ be". (F. Gispert Sánchez) p. 105, ¶ Line -6 Insert be'' before the first projection''. (P. Zsifkovits) p. 105, ¶ Line 2 ff. Rename the variables as $T_i$ since the symbol $X$ is in use already. (Peng Du) p. 105, ¶ Eqn. (4.7.1) The label of the lower arrow should be $(t, h)_S$. (Peng Du) 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. 111, ¶ Line -7 It might be worthwile to add a few words on the construction of $f$ as a morphism of schemes. (Formally, one has to cover $Z$ by affine charts, and glue the morphisms obtained on these charts. At this point in the book, it might be asking too much from the reader to skip this point.) (Ulrich Görtz) p. 112, ¶ Line -4 Add parentheses around $U_i\times_R W_j$. (Peng Du) p. 117, ¶ Line 15 (Exercise 4.21) The first line "... let $f\colon \mathbb A^1_k \to k$ the structure morphism." should read "... let $f\colon\mathbb A^1_k \to k$ be the structure morphism." (Thomas Brazelton) 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. 124, ¶ Lines -3, -2 (Remark 5.16 (3)) It should be made clear that we are talking about the numbers $h(i)$ of the previous theorem here, for the chain of ideals given by the $Z_i$. (At least, rename the $\mathfrak p_i$ as $\mathfrak a_i$; or state the relationship more explicitly). (Ulrich Görtz) p. 124, ¶ Line -9 Add "the" before "following". (Ulrich Görtz) p. 124, ¶ Line 18 (Proof of Lemma 5.14) Replace "finite-dimension" by "finite-dimensional". (Peng Du) p. 125, ¶ Thm. 5.19 (2) Change (a) to (1). (Peng Du) p. 125, ¶ Line 21 Replace "completed to maximal chain" by "completed to a maximal chain". (Ulrich Görtz) 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. 130, ¶ Line 4 The notion of finite scheme morphism is not defined at this point, one should talk about the corresponding ring homomorphism instead. (Ulrich Görtz) 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. 137, ¶ Cor. 5.56 (2), (3) Insert "then" before $X_K$ (twice). (Peng Du) p. 138, ¶ Line 5 Replace "roughly spoken" by "roughly speaking". (Peng Du) p. 140, ¶ Line 7 Replace $af+bg$ by $af+bg=0$. (Peng Du) p. 142, ¶ Exer. 5.6 The assumption that $f$ is closed is required only for part (a). (Ulrich Görtz) p. 142, ¶ Exercise 5.7 Before the first Show'', add Let $X=\mathop{\rm Spec} A[T]$.'' (A. Steinbach) p. 142, ¶ Exer. 5.8 (b) Replace $\mathop{\rm codim}(Y, X)$ by $\mathop{\rm codim}\nolimits_X(Y)$ etc. (Peng Du) 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. 147, ¶ Line 13 It would be more precise to say "cardinality of any minimal generating set" (rather than "... a minimal ..."). (Ulrich Görtz) 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. 152, ¶ Line 22 Replace "isoomorphism" by "isomorphism". (Ulrich Görtz) p. 153, ¶ Def. 6.14 (1) It might be helpful to add that requiring that $j$ is a morphism of $R$-schemes amounts to saying that the composition of $j$ with the projection to $\mathop{\rm Spec}(R)$ is $f$. (Peng Du) p. 153, ¶ Def. 6.14 (1) Conflict of notation: $j$ is used for the map and as an index. (Ulrich Görtz) p. 153, ¶ Def. 6.14 (2) Replace in all points'' by at all points''. (P. Johnson) p. 154, ¶ Line 8 Replace "for $X$" by "for $f$". (Ulrich Görtz) p. 155, ¶ Part (3) (Line 8) Add the condition that $k$ has characteristic $\ne 2$. (Peng Du) p. 155, ¶ Prop. 6.18 Add that $g$ is monic. (Peng Du) p. 155, ¶ Part (4) (Line 10) Add the condition that $k$ has characteristic $\ne 2$. Also, to see that for $f$ with multiple zeros the given scheme is not smooth, it seems that one wants to invoke (the relevant part of) Theorem 6.28. (Ulrich Görtz) p. 156, ¶ Prop. 6.21, end of proof The second half of the proof (starting from "Replacing $X$ by $U$ ...") should be replaced by the following (in the current "proof", evaluating $g$ at $T_i=0$ ($i=1, \dots, d$) might not produce a separable polynomial; also, after applying Prop. 6.18, since this gives only an isomorphism between dense opens, it is not enough to show that $\Sigma$ is non-empty): As $X$ is geometrically reduced, its function field is a separable extension of $k$ (Proposition 5.49). By Proposition 6.18 we may assume that $X={\rm Spec}(B)$, where $B=k[T_1, \dots, T_{d+1}]/(g)$ for a separable monic irreducible polynomial $g\in k(T_1, \dots, T_d)[T_{d+1}]$ with coefficients in $k[T_1,\dots, T_d]$. We have a finite morphism ${\rm Spec}(B) \to \mathbb A^d_k = {\rm Spec}(k[T_1, \dots, T_d])$. The subset of points $z\in\mathbb A^d_k$ such that the image $\overline{g}$ of $g$ in $\kappa(z)[T_{d+1}]$ is non-separable is the vanishing locus of the discriminant of the polynomial $g$, hence a Zariski closed subset. Since $g$ is separable over $k(T_1, \dots, T_d)$ it does not contain the generic point, so its complement $V$ is open and dense. Whenever $z\in U$, the fiber of the above morphism over $z$ is ${\rm Spec}(\kappa(z)[T_{d+1}]/(\overline{g}))$, a product of separable extensions of $k$ (since $\overline{g}$ might not be irreducible over $\kappa(z)$, we might have more than one factor). (It might be useful to add a few more words about the discriminant, maybe in App. B.) (Peng Du) p. 156, ¶ Proof of Thm. 6.19 The final part of the proof can be simplified: The polynomial $g$ being a $p$-th power is already a contradiction to the irreducibility as a polynomial in $T_{d+1}$. Alternatively, and this seems even simpler, it is enough to consider $\partial g/\partial T_{d+1}$ rather than all the partial derivatives. (Ulrich Görtz) 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. 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. 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, ¶ Line 12 Replace $T_{\mathbb A^n_k, y}$ by $T_{\mathbb A^n_k, y}^*$. (F. Gispert Sánchez) p. 159, ¶ Proof of Lemma 6.27 In the second line of the proof, replace columns'' by rows''. (B. Smithling) 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. 160, ¶ Line 9 Replace "point of $x$" with "point of $X$". (F. Gispert Sánchez) 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. 163, ¶ Line -6 Replace "Exercise 6.19" by Exercise "6.18". (Peng Du) p. 165, ¶ Line 17 (ex. 6.4) Replace "$df_{(e,e)}$" with "$dm_{(e,e)}$". (F. Gispert Sánchez) p. 165, ¶ Line 21 (Exer. 6.4) Replace $df$ by $df_e$. (Jan Willing) p. 166, ¶ Line 7 Insert "be" before "the set". (Peng Du) p. 169, ¶ Lines 7, 14 Wrong quotation marks around "globalizations" and "sheaf version". (Zhaodong Cai) p. 170, ¶ Equation (7.1.1) Add the notation $\mathscr F(x)$ and $s(x)$ to the Index of Symbols in the end. (Philipp Reichenbach) p. 171, ¶ Line 4 Replace "homomorphism" with "homomorphisms". (F. Gispert Sánchez) p. 171, ¶ Line 4 Replace $\mathscr O$ by $\mathscr O_X$. (Peng Du) 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. 175, ¶ Line -5 In addition to $s^{\otimes n}$, the notation $s\otimes t$ should also be defined (a couple of lines above). (Peng Du) p. 176, ¶ Line 4 A more appropriate reference than (7.1) is Section (7.3), specifically the comment after equation (7.3.6). (Philipp Reichenbach) 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 12 Replace "(2.27)" by "Proposition 2.27". (Peng Du) p. 181, ¶ Line -7 Replace "and" by "an". (F. Gispert Sánchez) p. 184, ¶ Line 18 Replace "identity" by "identity morphism of". (Peng Du) 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. 188, ¶ Line -14 Replace "of Dedekind schemes" with "by Dedekind schemes". (Peng Du) p. 189, ¶ Line -16 Replace "$y\in U_x\cap X_0$" by "$y\in (U_x\cap X_0)\setminus\{x\}$". (Peng Du) p. 189, ¶ Line -13 Replace "$\mathscr{L}_{|U_x}$" by "$\mathscr{L}_{D|U_x}$" (F. Gispert Sánchez) p. 190, ¶ Line 23 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. 190, ¶ Line -15 Add "be" before "extended". (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. 191, ¶ Line -1 Replace "generated" by "generate". (F. Gispert Sánchez) p. 194, ¶ Line -9 Replace "analogue" by "analogous". (Peng Du) p. 194, ¶ Line 8 Replace $\mathop{\rm Spec} A$ by $\mathop{\rm Spec} B$. (Longxi Hu) 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. 195, ¶ Remark 7.43 It might be worth adding a reference to arxiv:1011.0038 and/or the Stacks project (058B, 05A5). (Ulrich Görtz) 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. 197, ¶ Line 15 Replace "holds of" by "holds for". (Peng Du) p. 202, ¶ Exercise 7.20 Replace ${\mathscr F}$ by ${\mathscr G}$. (J. Calabrese) p. 202, ¶ Exer. 7.18 Add the condition $\mathscr I \ne 0$. (Peng Du) 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. 206, ¶ Statement of Proposition 8.4 Replace “Then the locus, where $v$ is surjective, is open” with “Then the locus where $v$ is surjective is open”. (Owen Colman) p. 207, ¶ Proof of Prop. 8.4 For part (1), one could add a reference to Exer. 7.2 (c). For part (2), maybe the argument in the last sentence should be expanded a little bit (one could say that $f^*(\tilde{v}) = 0$ if and only if the ideal generated by $\mathscr I$ in $\mathscr O_T$ is zero, which happens if and only if $f$ factors through $V(\mathscr I)$. (Peng Du) p. 209, ¶ Line 15 The second subscript $F$ is missing in the fiber product $F_i \times_F F_j\times F_k$. (Shahram Mohsenipour) p. 209, ¶ Line 10 Change the first "and" to "by". () p. 209, ¶ Line 12 Replace "all $S$" by "all $T$". (Peng Du) p. 209, ¶ Lines -14, -13 Replace all $S$ by $T$. (Peng Du) p. 209, ¶ Equation (8.3.1) Replace $U_{ij}\cap U_{jk}$ by $U_{ij}\cap U_{ik}$. () 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 -19 Replace "therefore" by "therefore we get". (Peng Du) p. 212, ¶ Line -16 Insert and is'' before also''. (D. Gerigk) p. 212, ¶ Line -5 Replace "subvector space of $K^n$" by "subvector space $U$ of $K^n$". (F. Gispert Sánchez) p. 214, ¶ Proof of Proposition 8.17 It would be useful to explain in greater detail how to apply Proposition 8.4 (2). Maybe something along the lines of "Let $h: X \to S$ be an $S$-scheme. Given $X \to {\rm Grass}^e(\mathcal{E}_1)$, let $\mathcal{V}_X$ be the corresponding element of ${\rm Grass}^e(\mathcal{E}_1)(X)$, and apply Proposition 8.4 (2) to the composite $\ker(h^*(v)) \to h^*(\mathscr E_1)/\mathcal{V}_X$." (Owen Colman) p. 214, ¶ Line 11 Replace "$S$ scheme" by "$S$-scheme". (Peng Du) p. 216, ¶ Line 4 Replace "of $\mathscr{O}_S$-module" by "of $\mathscr{O}_S$-modules". (F. Gispert Sánchez) p. 216, ¶ Line -7 Replace "disjoint sum" with "disjoint union". (F. Gispert Sánchez) p. 217, ¶ Line -2 Replace ${\rm Grass}_1$ by ${\rm Grass}^1$. (Peng Du) p. 218, ¶ Line 10 Rephrase as "... we have a bijection, functorial in $S$,". (Peng Du) p. 219, ¶ Line -13 Replace $k$ by $\mathbb C$. (Peng Du) p. 219, ¶ Line -7 Replace "Corollary 5.45" by "Proposition 5.51". (Peng Du) p. 219, ¶ Line -11, Def. 8.25 Insert "$X$" after "$k$-scheme". (D. Gerigk/Peng Du) p. 219, ¶ Line 10 Replace "the kernel of $\bigwedge^e \mathscr E\dots$" by "the kernel of $\bigwedge^e f^*\mathscr E\dots$. (And maybe also, in the same line, $\mathbb P(\bigwedge^e \mathscr E)$ by $\mathbb P(\bigwedge^e \mathscr E)(T)$.) (Peng Du) 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. 221, ¶ Line -16 Replace "(v)" by "(iv)". () p. 221, ¶ Line -13 Add subscript $k$ to $\otimes$. (Peng Du) 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 16, 17 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 (cf. Prop. 10.30, Remark 10.31) 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, ¶ 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. 234, ¶ Lines -3, -1 Replace "$\mathcal R$" by "$\mathcal R(X, Y)$" (twice). (Peng Du) 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. 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. 236, ¶ Line -3 Replace "$K$" with "$K(X)$". (F. Gispert Sánchez) p. 236, ¶ Line 5 Replace "$(\lambda,\mu)$" with "$(\lambda : \mu)$". (F. Gispert Sánchez) p. 237, ¶ Line -16 Replace "tape" with "type". (F. Gispert Sánchez) p. 237, ¶ Line 24 (Remark 9.34) Replace $f\colon - - \rightarrow Y$ by $f\colon X - - \rightarrow Y$. (P. Barik) p. 238, ¶ Line 3 Delete the second $X$. (Peng Du) 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. 241, ¶ Line -16 Replace "among" by "along". (Peng Du) p. 241, ¶ 8 Replace "roughly spoken" by "roughly speaking". (Peng Du) 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, ¶ Line 20 Replace "Its sum" by "The sum". (Peng Du) 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. 254, ¶ Line 3 Replace "Definition 10.5" by "Definition 10.12". (Peng Du) 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. 259, ¶ Line 16 Replace "inductive limit" by "inductive limits". (Peng Du) p. 261, ¶ Line 17 Replace "inverse images under continuous maps" by "inverse images under morphisms". (Peng Du) p. 263, ¶ Line 4 Replace $v_{0\lambda}^{-1}$ by $x_{0\lambda}^{-1}$. (Peng Du) 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. 267, ¶ Line -8 Add "or" after property (1). (Peng Du) p. 268, ¶ Line 22 Replace "morphism" with "morphisms". (F. Gispert Sánchez) 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, ¶ l $-2$ Remove one that'' and replace morphism'' by morphisms''. (U. Hartl) 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 -22 Replace "Frobenius (4.24)" by "Frobenius morphism (Definition 4.24)". (Cf. this erratum.) (Peng Du) 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 particular, 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. 280, ¶ Line 20 It would look better to replace $N>1$ by $N\ge 1$ (as in the following line). (Peter Johnson) p. 281, ¶ Line 9 Replace the final part of the sentence by "and $\kappa(x)$ is a finite extension of $\kappa(f(x))$.". (Peng Du) 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. 284, ¶ Line 20 Replace "generic" by "generically". (Peng Du) p. 286, ¶ Line 5 Delete "Line bundles and". (Peng Du) p. 286, ¶ Line -9 Replace "roughly spoken" by "roughly speaking". () p. 286, ¶ Line 11 Replace "between to ..." by "between two ...". (Peng Du) p. 287, ¶ Line 3 Replace "$S$" by "$X$". (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. 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. 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 18 Replace "straight forward" by "straightforward". (Ulrich Görtz) p. 293, ¶ Line -14 Replace "1-cocycle" with "1-cocycles". (F. Gispert Sánchez) p. 293, ¶ Line 9 Replace "1-cocycle" with "1-cocycles". (F. Gispert Sánchez) 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. 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. 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. 304, ¶ Line 1 Delete the second $U$. (Peng Du) 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. 305, ¶ Line -13 Replace "isomorphism" by "isomorphisms". (Ulrich Görtz) 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, ¶ Line -8 (Lemma 11.33) The notation for the codimension is different from the notation used in chapter 5. (F. Gispert Sánchez) p. 306, ¶ Line -16 Replace "$\mathscr{O}_{X,c}$" with "$\mathscr{O}_{X,C}$". (F. Gispert Sánchez) p. 306, ¶ Line 7 Replace $\mathop{\rm codim}(Z, X)$ by $\mathop{\rm codim}\nolimits_X(Z)$ (twice). (U. Görtz) p. 307, ¶ Line -6 Replace (B.75) by B.75. (Peng Du) 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. 308, ¶ Line 13 Replace $U$ by $U_i$ (twice). (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. 309, ¶ Line 11 Replace $\mathop{\rm codim}(Z, X)$ by $\mathop{\rm codim}\nolimits_X(Z)$. (Peng Du) p. 310, ¶ Line 15 The symbol $D(f)$ is not defined, and should be defined explicitly as the Cartier divisor $(D_+(T_i), f/T_i^d)_i$. (Peng Du) p. 311, ¶ Line 17 Replace "Example 7.13" by "(7.13)". (Peng Du) p. 313, ¶ Line -6 Replace set'' by sets''. (P. Johnson) 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. 315, ¶ Line -13 Replace $d$ by $d_i$ in the definition of $\mathscr E^\lambda$ (twice). (Peng Du) 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 -9 The (finite) covering $(U_i)_i$ of $X$ should be an affine open covering. (F. Gispert Sánchez) p. 323, ¶ Line 12 Replace $X' = \mathop{\rm Spec} B' \otimes_{B} A$ by $X' = \mathop{\rm Spec}(B \otimes_A A')$. (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. 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 -7 Switch the order in all pairs $z/x, \dots$ in this line. (Peng Du) p. 329, ¶ Line 18 Replace "be" with "by". (F. Gispert Sánchez) p. 329, ¶ Line 4 In addition to the notion of degree of $f$, add the notion of rank of $f$ with the same meaning. Nevertheless, we will try to stick to the term degree (change this on p. 331, l. 3, l. 15; p. 479, l. 1). (Peng Du / Ulrich Görtz) p. 330, ¶ Line 2 Switch the order in all pairs $z/x, \dots$ in this line. (Peng Du) 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, ¶ Example 12.29 In several places, replace $t$ by $t'$. (S. Köbele) p. 333, ¶ Line -14 Replace $a_0$ by $a_n$. (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. 339, ¶ Line -13 Replace $L$ by $L_X$. (Peng Du) 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. 348, ¶ Line -14 Replace "Remark 12.10 (2)" by "Remark 12.10 (3)". (Peng Du) p. 351, ¶ Line -9 Replace "$\varphi$" by "$\varphi\colon A\to B$". (Peng Du) 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. 356, ¶ line -8 Replace "this is implies" by "this implies". (Shahram Mohsenipour) 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, ¶ Line 6 Replace "my" by "may". (Peng Du) p. 361, ¶ Ex. 12.6 Replace isomporphism by isomorphism. (T. Keller) p. 363, ¶ Line -22, Exer. 12.21 Insert "be" before "Dedekind rings". (Peng Du) p. 367, ¶ Line 20 Replace $\sum_{d\ge 0}$ by $\sum_d$ (to also cover the case of graded modules). (Peng Du) 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, ¶ Equation (13.1.1) Replace $\mod f-1$ by $\mod (f-1)$ (A. Kaučikas) p. 368, ¶ Line -20 The element $f$ should be homogeneous. (U. Görtz) p. 368, ¶ Line 10 Replace "graded ideal" by "homogeneous ideal". (Peng Du) 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. 370, ¶ Line 10 $f \in rad(g)_+$ should read $g \in rad(f)_+$. (F. Gispert Sánchez/N. Pflueger) p. 371, ¶ Line 4 (Remark 13.7) In the last line of the remark, it should read $A_{(f)}=A'_{(f^{k\delta})}$. (Jesús Martín O.) p. 371, ¶ Line -18 Remove "be" and replace "$R$-algebras" by "$R$-schemes". (Peng Du) p. 371, ¶ Line -19 Insert "be" before "a graded". (Peng Du) p. 375, ¶ Line -3 Replace definition of $\sigma$ by $\sigma(t'):= t / f^n \in \Gamma_*(\mathscr F)_{(f)}$''. (P. Johnson) p. 376, ¶ Line -15 Insert "$M$" after "module". (Peng Du) 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, ¶ 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. 377, ¶ Line 15 Replace $n \ge 0$'' by $n\ge n_0$''. (P. Johnson) p. 378, ¶ Line 12 Insert "the" before "saturation". (Peng Du) p. 378, ¶ line 2 Replace "on" with "one." (N. Pflueger) p. 378, ¶ Line 18 Replace "Corollary 13.24" by "Proposition 13.24". (Peng Du) 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. 379, ¶ Prop. 13.28 In the first line of the statement of the proposition, replace modules'' by module''. (P. Johnson) p. 379, ¶ Lines -14, -13 The notation $-\otimes_{\mathscr O_S}\mathscr O_{S'}$ (used twice) is a bit sloppy. Maybe the $\otimes$ should be replaced by $\boxtimes$, or the tensor product could be replaced by $g'^*$, where $g'$ is the base change of $g$ (which would have to be defined). (Peng Du) 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. 380, ¶ Line 11 Replace "by" with "given by". (Peng Du) p. 380, ¶ Line 17 Add "be" before "the structure morphism". (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. 380, ¶ Line 8 Replace "$\mathscr{O}(n)$" with "$\mathscr{O}_X(n)$". (F. Gispert Sánchez) p. 382, ¶ Line 21 Replace "$\mathscr{O}^{n+1}$" with "$\mathscr{O}^{n+1}_X$". (F. Gispert Sánchez) p. 382, ¶ Lines 15, 18 Replace $\mathbb P^{n+1}$ by $\mathbb P^n$ (three times). (K. Kidwell) 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. 384, ¶ Line -4 Insert the condition $x\ne x'$. (Peng Du) p. 384, ¶ Line 16 Replace "$\mathscr O_S$-module" by "$\mathscr O_S$-module $\mathscr E$". (Peng Du) p. 387, ¶ Line 8 Replace "$f$ is invertible in $x$" by "$f(x) \ne 0$ in the fiber $\mathscr L(x)$". (Ulrich Görtz) p. 388, ¶ Line -6, Line -3 Add that $f$ is homogeneous. (F. Gispert Sánchez) p. 389, ¶ Line -17 Replace "relevant prime ideals ... of $A_+$" by "relevant prime ideals ... of $A$". (Peng Du) 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 -15 Replace "$n\geq n_0+m_0$" with "$n\geq d+m_0$". (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 20 Add that "$S={\rm Spec}\ R$" somewhere ($R$ has not been defined). (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. 397, ¶ Line 10 Replace "be" by "by". (Peng Du) 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 -3 (the diagram) Add label $r$ to the right hand arrow. (Peng Du) p. 404, ¶ Line 19 Replace "by Corollary 13.42" with "by Example 13.69". (F. Gispert Sánchez) p. 405, ¶ Line 22 It should be stated that, in the definition of $H_m$, we set $m = e+1 (={\rm rk}\ \mathscr F)$. (Peng Du) 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. 409, ¶ Prop. 13.91 Replace "let $Z$ be a closed subscheme of $X$" by "let $Z$ be a closed subscheme of $X$ with corresponding ideal sheaf $\mathscr I$". (Peng Du) p. 410, ¶ Line 21 (the 2nd displayed eqn.) The identification $A[I f^{-1}] = A[(T_\alpha)_\alpha] / (f T_\alpha - x_\alpha)_\alpha$ is not true in general and should be replaced by $A[I f^{-1}] = (A[(T_\alpha)_\alpha] / (f T_\alpha - x_\alpha)_\alpha)/(f-{\rm torsion})$, i.e., replace the right hand side by its quotient by the ideal of all elements annihilated by a power of $f$. (Owen Colman) p. 410, ¶ Line 21 (the 2nd displayed eqn.) Replace the second $=$ by $-$. (Peng Du) p. 410, ¶ Lines -5, -3 Replace $=$ by $-$ (one in each line). (Peng Du) p. 410, ¶ Line -6 Delete "of". (Peng Du) p. 411, ¶ Line -2 (Proof of Prop. 13.96) Replace $(I \oplus J)/J$ by $(I+J)/J$. (Peng Du / Matthieu Romagny) p. 412, ¶ Line 20 Replace "roughly spoken" by "roughly speaking". (Peng Du) p. 416, ¶ Line 19 Replace "morphisms" with "morphism". (F. Gispert Sánchez) p. 417, ¶ Line -6 Replace "particular" by "particularly". (Peng Du) p. 417, ¶ Line -20 Switch the exponents $d+1$ and $d$. (Peng Du) 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. 420, ¶ Exer. 13.13 Replace "inductive limits of schemes" by "inductive limits of rings". (Peng Du) p. 422, ¶ Line 17 Insert "be" before "its structure morphism". (Peng Du) 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, ¶ Prop. 14.16 The term "special point" which is used here was not defined before. Define it (and maybe also "special fiber") before the proposition. (Peng Du) p. 427, ¶ Proof of prop. 14.16 In the proof, $X$ is replaced by $\mathop{\rm Spec} \mathscr{O}_{X,x}$ and $f$ is replaced by the composition of $f$ with the canonical morphism $\mathop{\rm Spec} \mathscr{O}_{X,x} \to X$ 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, ¶ Lemma 14.19 Replace such'' by such that''. (U. Görtz) 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. 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 3 Replace "a open immersion" with "an open immersion". (F. Gispert Sánchez) p. 443, ¶ Line -16 Add "a" between "Let $\mathbf{P}$ be" and "property". (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 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/T. Wedhorn) p. 457, ¶ Line 3 Replace "$\gamma(a_\delta)$" with "$\gamma(b_\delta)$". (F. Gispert Sánchez) p. 458, ¶ Line -10 Replace "straight forward" by "straightforward". (Peng Du) 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. 461, ¶ Proof of Prop. 14.90 Replace "Corollary 5.45" by "Corollary 5.54" and "Theorem 6.28" by "Corollary 6.32". (Peng Du) p. 461, ¶ Line 8 Define the symbol $X^{\rm sep}$ as $X\otimes_k k^{\rm sep}$. (Peng Du) p. 462, ¶ Line 9 Replace the second $=$ by $-$. (Peng Du) p. 464, ¶ Line 4 Insert "Proposition" before "5.30 (2)". (Peng Du) 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. 472, ¶ Proof of Cor. 14.127 It is not clear from Prop. 14.107 (1) and Cor. 14.118 why the fibers over the maximal points of $Y$ are non-empty, so it seems better to invoke Cor. 14.116 (and Prop. 14.102) instead. (Peng Du) p. 475, ¶ Proof of Cor. 14.127 It might be helpful to add a reference to Thm. B.54 (4) for the equality of the dimensions of the local rings of $x$ and $y$. (Peng Du) p. 477, ¶ Line 19 Delete the comma after "means". (Peng Du) 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. 490, ¶ Line 20 Replace "integers" by "be integers". (Peng Du) p. 490, ¶ Line 22 Replace "Theorem 15.8" by "Theorem 15.9". (Peng Du) p. 491, ¶ Line 7 Switch ${\rm Spec}\ K$ and ${\rm Spec}\ R$. (Peng Du) p. 495, ¶ Line -9 Replace "two sets" by "three sets". (Peng Du) p. 495, ¶ Line 19 The term "complete" (= proper over $k$) has not been defined. (Peng Du) 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. 499, ¶ Line -12 Replace "over a field" by "over a field $k$". (Peng Du) p. 501, ¶ Line 13 Replace "discrete valuation" by "discrete valuation ring". (Peng Du) p. 501, ¶ Line 9 Replace morpism'' by morphism''. (J. Scarfy) p. 503, ¶ Line 12 Replace "roughly spoken" by "roughly speaking". (Peng Du) p. 506, ¶ Line -6 Replace $\mathbb P^{nm}_R$ by $\mathbb P^{nm-1}_R$. (Peng Du) p. 507, ¶ Line -9 Replace $\times_k$ by $\times$. (Peng Du) p. 507, ¶ Line 6 Replace "geometrically integral" by "is geometrically integral". (Peng Du) p. 507, ¶ Lines 17, -3 Replace $v_n$ by $v_m$ and $w_m$ by $w_n$ (with indices as given, the identifications of Hom spaces and spaces of matrices do not hold as stated). Correspondingly, in Line -3, change $k^n$ to $k^m$. (Peng Du) p. 508, ¶ Line 5 Delete "the same argument and". (Peng Du) p. 508, ¶ Line 3 Change $k^n$ to $k^m$ (cf. the corresponding erratum on page 507). (Peng Du) p. 511, ¶ Line 3 Replace $\otimes_k R$ by $\otimes_R k$. (Peng Du) p. 512, ¶ Line 14 Replace $\underline{\rm Hom}_{\mathscr G, \mathscr H}$ by $\underline{\rm Hom}(\mathscr G, \mathscr H)$ (Peng Du) p. 512, ¶ Line 21 (Proof of Lemma 16.20) Replace $k[T_{ij}]$ by $R[T_{ij}]$. (Peng Du) p. 513, ¶ Line 13 (the diagram), Line -10 Replace $\pi_{\mathscr F}$ by $\pi_{\mathscr V}$. (Peng Du) p. 514, ¶ Line 3 Remove "below". (Peng Du) 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. 516, ¶ Line 6 Delete "an". (Ulrich Görtz) p. 516, ¶ Proof of Lemma 16.28 The definition of $g'$ should be replaced by $g'(e_j') = \sum_{i=1}^n b_{ij}e_i$, $j=1, \dots, n'$. Then $B=(b_{ij})$ is in $M_{n\times n'}(R)$. Then $v(a, b) = (Aa+Bb, b)$, so viewing the direct sums in the source and target of $v$ as column vectors, the matrix $C$ is $C=\left(\begin{array}{cc}A & B \\ 0 & I_{n'}\end{array}\right)$. Finally, in the end we want to consider $(n-r+1)$-minors of $A$ and $(n+n'-r+1)$-minors of $C$. (Peng Du) p. 517, ¶ Line -14 Delete "the spectrum of". (Peng Du) p. 519, ¶ Line 17 Add period at the end of the sentence. (Peng Du) p. 524, ¶ Line -7 Replace $(\beta, \alpha)$ by $(\beta : \alpha)$. (Peng Du) 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. 529, ¶ Caption of Figure 16.2 Add $=0$ in the end. (Peng Du) p. 531, ¶ Line -14 Replace "Remark (12.31)" by "Remark 12.31". (Peng Du) p. 532, ¶ Line 5 To emphasize that the minus signs are not a typo, maybe replace "continued fraction" by "negative regular continued fraction". (Peng Du) p. 534, ¶ Line 20 Replace "Proposition 5.51" by "Proposition 5.49". (Peng Du) p. 534, ¶ Line 5 of the Proof of Proposition 16.54 $p^{-1}(y)$ should be $p_2^{-1}(y)$. () 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 -8 The correct formula is $f(a_1a_2)f(a_2)^{-1}f(a_1)^{-1}$. (Peng Du) p. 534, ¶ Prop. 16.45, statement + Line 18, Proof of Prop. 16.45 In the statement of the proposition, $Y$ must be assumed to be integral rather than just reduced (to ensure that the $V$ in the proof is dense in $Y$, hence $X\times V$ dense in $X\times Y$). In the proof (line 18 of the page), replace "Proposition 12.67" by "Corollary 12.67". This result is applied to the morphism $X\otimes\kappa(y')\to U\otimes\kappa(y')$ whence $Y$ in this line must be replaced by $U\otimes_k\kappa(y')$. (Peng Du) p. 534, ¶ Line -8 Replace $f(a_1 a_2)f(a_1)^{-1}f(a_2)^{-1}$ by $f(a_1 a_2)f(a_2)^{-1}f(a_1)^{-1}$. (Philipp Reichenbach) p. 534, ¶ Line 6 of the Proof of Proposition 16.54 "nieghborhood" should be "neighborhood" () p. 536, ¶ Line 17 Replace "Example 16.18" by "Example (16.18)". (Peng Du) p. 539, ¶ Caption of Figure 16.3 Add $=0$ at the end of the first line. Maybe it would look nicer to replace $0.1$ by $\frac{1}{10}$. (Peng Du) 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. 540, ¶ Line -6 Replace "is no divisor of" by "does not divide". (Peng Du) p. 540, ¶ Line 18 Add space between the comma and the citation. (Ulrich Görtz) p. 545, ¶ Line -14, -7 Better to switch the definition of the zero object with the remark, i.e., define the zero object as an initial and final object (the correct definition fo arbitrary categories), and remark that in an additive category it suffices to check one of the two properties. (Peng Du) 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. 554, ¶ Line -1 Add the assumption that $A$ is a domain here (since this is when $M_{\rm tors}$ was defined). Same for p. 555, Prop. B.28. (Peng Du) p. 554, ¶ Line -2 It might be worth adding a reference to arxiv:1011.0038 and/or the Stacks project (058B, 05A5). (Cf. also the Erratum for Remark 7.43, p. 195). (Ulrich Görtz) p. 558, ¶ Line -16 Replace "$\dots, \mathfrak a_n)$" by "$\dots, a_n)$". (Peng Du) p. 558, ¶ Line -3 Replace "$A$-modules" by "abelian groups". (Peng Du) p. 559, ¶ Line -16 Replace $\mathfrak I^n$ by $\mathfrak I_n$ (and similarly for $n+1$; four changes altogether). (Peng Du) p. 560, ¶ Prop. B.55 Add the hypothesis that $A$ is noetherian (cf. [Atiyah-Macdonald], Prop. 5.17). (Akira Masuoka) p. 560, ¶ Line 2 Replace "form" by "forms". (Ulrich Görtz) 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. 563, ¶ Line 15 Replace "as in (iv)" by "as in (v)". (Peng Du) p. 564, ¶ Prop. B.72 (3) Replace "be a noetherian" by "be noetherian". (Peng Du) 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. 566, ¶ Def./Prop. B.84 Add the equivalent term "Dedekind ring", and also change the index to reflect that "Dedekind ring = Dedekind domain". (Peng Du / Ulrich Görtz) p. 568, ¶ Def. B.95 Since some of its terms are used before, Def. B.95 and the paragraph preceding it should be moved to right after Def. B.88. (Peng Du) p. 569, ¶ Def. B.95 (3) Rephrase as "every extension $K\to \Omega$" with $\Omega$ algebraically closed" to avoid the condition being misread as "$K$ being algebraically closed in $\Omega$". (Peng Du) p. 574+575, ¶ Add (IND) for faithfully flat and surjective (cf. EGA IV 8.10.5 (vi)). (Y. Zaehringer) p. 576, ¶ Line 16 Replace "Converse" by "Conversely". (Peng Du) p. 576, ¶ Line 5 Replace "as usually" by "as usual". (Ulrich Görtz) p. 576, ¶ Line 12 Replace "universal homomorphism" by "universal homeomorphism". (Yun Hao) p. 583, ¶ Line 4 (Reference [AK]) The reference (from p. 478) should instead point to A. Altman, S. Kleiman, Compactifying the Picard scheme, Adv. Math. 35 (1980), 50--112. (Ulrich Görtz) p. 600, ¶ 2nd column, line 8 Replace $P^n(k)$ by $\mathbb P^n(k)$. (Ulrich Görtz) p. 603, ¶ Line 17 Replace Brauer-Severy'' by Brauer-Severi''. (J. Calabrese) p. 612, ¶ 1st column, Line -11, -9 The entries for quasi-finite morphism (of schemes) should be combined. (Ulrich Görtz)