Errata and Addenda for Algebraic Geometry I (Edition 1) Show errata for edition 2
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Major Errors
Page  Description  Submitted by  Ed. 

p. 102,
¶
Proof of Prop. 4.20, Case (II)

To see that $(p^{1}(f(X')), \mathscr O_{Zp^{1}(f(X'))})$ 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  1 
p. 268,
¶
Theorem 10.76

It is not true in general that $B$ is the inductive limit of its smooth $A$subalgebras. 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  1 
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 nonzero divisors in every stalk). See Kleiman, Misconceptions about $K_X$. Enseign. Math. (2) 25 (1979), no. 34, 203206 (1980), for a detailed discussion.  P. Hartwig  1 
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  1 
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  1 
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  1 
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  1 
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 nonnoetherian 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 nonnoetherian case) a version of the Lemma as alluded to in (2) above is required. 
U. Görtz  1 
p. 450,
¶
Prop. 14.65

The lower right triangle of the diagram constructed at the end of the proof is not commutative in general, therefore the proof is not valid as given here. Furthermore, in Step (i) of the proof of Theorem 14.66 (ed. 1) / Theorem 14.68 (ed. 2), a morphism $f': T'\rightarrow S'$ is considered where $T'$ is a finite disjoint union of affine open subschemes of $S'$ covering $S'$. If $S'$ is not quasiseparated, then it is not possible to find a quasicompact such $f'$. Therefore, it seems better to handle individually the two cases of this proposition that are needed in the proof of the theorem: The case where $T'$ is a finite disjoint union of open subschemes of $S'$, and the case where $f' : T'=S \rightarrow S'$ is a section of $p: S' \rightarrow S$. Both cases are easy to deal with (the first one because it is clear that morphisms of this kind satisfy descent). A posteriori, the theorem implies that the proposition is actually true in the form stated.  M. Bruneaux, P. Godfard  1 2 
Minor Errors
Page  Description  Submitted by  Ed. 

p. 10,
¶
Line 3

Replace $\beta_{n1}+\beta_{n2}a+\cdots +a^{n1}$ by $(\beta_{n1}+\beta_{n2}a+\cdots +\beta_0 a^{n1})$.  P. Zsifkovits  1 
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  1 
p. 10,
¶
Theorem 1.8

Add the assumption that $A\ne 0$.  D. Gerigk  1 
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  1 
p. 13,
¶
Line 16

Replace "subset" by "open subset".  A. Graf  1 2 
p. 13,
¶
Line 7

After ``hence'', add ``if $Z \cap U \ne \emptyset$''.  U. Görtz  1 
p. 15,
¶
Line 14

Replace ``Every open subset'' by ``Every subspace''.  T. Wedhorn  1 
p. 15,
¶
Line 15

Replace ``Every closed subset'' by ``Every subspace''.  T. Wedhorn  1 
p. 21,
¶
Definition 1.39

An exception needs to be made if $U$ is the empty set.  Nick Mertes  1 
p. 23,
¶
Line 15 (Def. 1.46 (2))

Replace ``finite covering'' by ``finite open covering''.  F. Gispert Sánchez  1 
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  1 
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  1 
p. 29,
¶
Line 7

Add $g \ne 0$ in the description of the function field of $K({\mathbb P}^n(k))$.  T. Wedhorn  1 
p. 35,
¶
Line 4

$r \gt 2$ (rather than: $r \gt 1$)  P. Barik  1 
p. 48,
¶
Line 3 (def. 2.18.Sh2)

Replace "by (a)" with "by (Sh1)".  F. Gispert Sánchez  1 
p. 48,
¶
Line 9

Add the ``category of abelian groups'' as the first example.  A. Kaučikas  1 
p. 50,
¶
Prop. 2.20

The basis B has to be closed under finite intersections for the (Sh) condition to be welldefined. (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 nonstandard terminology, so both places should be fixed.)  Florian Ebert  1 
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  1 
p. 54,
¶
Line 9

Replace ``sheaves'' by ``presheaves''.  J. Watterlond  1 
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.  1 
p. 62,
¶
Exercise 2.3 line 2

Change "every open subset" to "every non empty open subset"  Vishal Gupta  1 
p. 63,
¶
Exercise 2.13

In property (i), add the requirement that $\mathscr F(\emptyset)$ is a set with one element.  Louis Martini  1 
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  1 
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  1 
p. 77,
¶
Line 12

Replace ``scheme'' by ``nonempty scheme''.  M. Jarden  1 
p. 78,
¶
Prop. 3.29 (3)

Add the assumption that all $U_i$ are nonempty.  D. Gerigk  1 
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  1 
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  1 
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  1 
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  1 
p. 91,
¶
Exercise 3.26(a)

The statement is true for $V = X$, but not in general. (E.g., take $Y= \mathbb A^1_k$, $U=\mathbb A^1_k\setminus \{0\}$, $I=\{1, 2\}$, $i=1$, $V=U_2\subseteq X$.)  Paulo LimaFilho  1 2 
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  1 
p. 97,
¶
Definition 4.10

Add the condition that $f \circ p = g \circ q$.  C. Frei  1 
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 quasicompact. This is true (Prop. 10.1) but had not been stated at this point.  F. Gispert Sánchez  1 
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  1 
p. 121,
¶
Line 9

Replace $\mathfrak p_0$ by $\mathfrak p_0 A[T]$, and correspondingly for $\mathfrak p_r$.  P. Zsifkovits  1 
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  1 
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  1 
p. 133,
¶
Cor. 5.45

Add the hypothesis that $Y$ be nonempty.  1  
p. 134,
¶
Proof of Cor. 5.47

Note that the first statement in the proof holds by Cor. 5.23.  K. Kidwell  1 
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  1 
p. 155,
¶
Prop. 6.18

Add that $g$ is monic.  Peng Du  1 
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 nonempty): 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 nonseparable 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  1 
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  1 
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  1 
p. 159,
¶
Line 12

Replace $T_{\mathbb A^n_k, y}$ by $T_{\mathbb A^n_k, y}^*$.  F. Gispert Sánchez  1 
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  1 
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  1 
p. 163,
¶
Line 17

Replace ``$\dim {\mathscr O}_{X,x} = 2$'' by ``$\dim {\mathscr O}_{X,x} \geq 2$''.  T. Wedhorn  1 
p. 166,
¶
Exercise 6.12 (b) (iii)

Condition (iii) should be changed to: " X \cong \coprod_i \Spec K_i, where K_i are finite separable field extensions of k." (Alternatively, add the assumption that $X$ is quasicompact in the beginning.)  Rafael M. Saavedra  1 2 
p. 178,
¶
Line 19

The morphism "$\iota$" has not been defined; it should be (7.5.7).  F. Gispert Sánchez  1 
p. 185,
¶
Corollary 7.19. (4)

Define $\mathscr{F}$ and $\mathscr{G}$ to be quasicoherent $\mathscr{O}_X$modules.  F. Ebert  1 
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 nonquasicoherent sheaf of ideals) . (It is true that a quasicoherent $\mathscr O_X$modules of finite type over a Noetherian scheme is of finite presentation.)  H. Iriarte  1 
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  1 
p. 203,
¶
Exercise 7.30

The $\mathcal{O}_X$modules should be finite locally free, as the determinant was defined in that setting.  1  
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  1 
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  1 
p. 233,
¶
Prop. 9.19 (iii)

$f$ and $g$ should be $S$morphisms.  K. Kidwell  1 
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  1 
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 pseudomorphism introduced in EGA IV, 20.2.  P. Hartwig  1 
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  1 
p. 253,
¶
Remark 10.40 (3)

Add the assumption that $X$ has a basis of retrocompact open subsets.  A. Gross  1 
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 finitelygenerated $\mathbb{Z}$algebra and, in particular, Jacobson, we conclude that $\mathfrak{p}\in\overline{E}$.)  F. Gispert Sánchez  1 
p. 277,
¶
Line 21

Add that $S$ is noetherian.  F. Gispert Sánchez  1 
p. 278,
¶
Lines 2126 (last paragraph of the proof of thm. 10.97)

Replace "$X_\xi$" with "$X_\eta$" (three times). Moreover, if $U$ is the nonempty 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  1 
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  1 
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  1 
p. 298,
¶
Line 6

We can only conclude that the immersion is locally of finite presentation.  F. Gispert Sánchez  1 
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  1 
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  1 
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  1 
p. 313,
¶
Line 1

The morphism $\phi$ is not necessarily flat: See mathoverflow.net/questions/65267/globalsectionsofflatschemealsoflat. 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.  1 
p. 318,
¶
Exercise 11.18 (a)

Replace '$Z^1({\rm Spec}\ A)\cong\mathbb{Z}^r$' by '$Z^1({\rm Spec}\ A)\cong\mathbb{Z}$'.  Xiaolong Liu  1 2 
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  1 
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  1 
p. 369,
¶
Line 21

We defined $V_+(\mathfrak a)$ only for homogeneous ideals contained in $A_+$. Therefore $V(\mathfrak a^h)$ should be replaced by $V(\mathfrak a^h\cap A_+)$. Note that for $\mathfrak p\in {\rm Proj}(A)$, we have \(\mathfrak a\subseteq \mathfrak p \Leftrightarrow \mathfrak a^h\subseteq\mathfrak p\Leftrightarrow \mathfrak a^h\cap A_+\subseteq\mathfrak p.\) (Alternatively we could define $V_+()$ for all homogeneous ideals (or even all subsets of $A$) and explain why this doesn't make a difference.)  Dominik Briganti  1 2 
p. 375,
¶
Line 3

Replace definition of $\sigma$ by ``$\sigma(t'):= t / f^n \in \Gamma_*(\mathscr F)_{(f)}$''.  P. Johnson  1 
p. 394, 395,
¶
(13.13)

In line $8$, replace ``nonzero'' 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 nonzero global sections of a line bundle are regular).  K. Kidwell  1 
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  1 
p. 418,
¶
Line 6 (ex. 13.2 (a))

Replace "(resp. bijective)" with "(resp. surjective)".  F. Gispert Sánchez  1 
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  1 
p. 460,
¶
Diagram (14.22.1), definition of $c(\gamma)$

It looks like if we want $c$ to be 1cocycle, 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  1 
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  1 
p. 472,
¶
Proof of Cor. 14.119

It is not clear from Prop. 14.107 (1) and Cor. 14.118 why the fibers over the maximal points of $Y$ are nonempty, so it seems better to invoke Cor. 14.116 (and Prop. 14.102) instead.  Peng Du / G. Dill  1 
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, 50112.  P. Hartwig  1 
p. 485,
¶
Prop. 15.1 (ii)

Add ``and none of the $X_i$ consists of only one point''.  U. Görtz  1 
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  1 
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  1 
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 $(nr+1)$minors of $A$ and $(n+n'r+1)$minors of $C$.  Peng Du  1 
p. 521,
¶
Line 8

In the expression for $g$, in the third parenthesis there should be $(X_21)^2$ instead of $(X_2^21)^2$.  Alejandro Vargas and Tim Seynnaeve  1 2 
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  1 
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  1 
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  1 
p. 560,
¶
Prop. B.55

Add the hypothesis that $A$ is noetherian (cf. [AtiyahMacdonald], Prop. 5.17).  Akira Masuoka  1 
p. 564,
¶
Proposition B.70 (6)

What is true (and proved in Matsumura's book) is: If $B$ is normal, then $A$ is normal. If $A$ and $B\otimes_A\kappa(\mathfrak p)$ are normal for all $\mathfrak p\in {\rm Spec}(A)$, then $B$ is normal.
It is not true that normality of $B$ implies normality of the fibers. Consider for instance a discrete valuation ring $A$ with uniformizer $t$ and $B=A[X,Y]/(XYt)$. Then $B$ is regular and in particular normal, but the special fiber $(A/t)[X, Y]/(XY)$ is not normal. 
M. Kerz  1 2 
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  1 
p. 574, 575,
¶

Add (IND) for faithfully flat and surjective (cf. EGA IV 8.10.5 (vi)).  Y. Zaehringer  1 
Typographical and Trivial Errors
Page  Description  Submitted by  Ed. 

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  1 
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  1 
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 MajidiZolbanin  1 
p. 2,
¶
Line 6

In "asserts that this equations has no solutions" the word "equations" should be replaced with "equation".  Mahdi MajidiZolbanin  1 
p. 2,
¶
Line 8

Replace 1994 by 1995.  J. Hilgert  1 
p. 5,
¶
Line 12

then $X$ inherits many properties of $X'$ (rather than: of $X$)  P. Barik  1 
p. 7,
¶
Figure 1

Replace $T_2^2T_1^2(T_1+1)$ by $T_2^2T_1^2(T_1+1)=0$.  A. B. Nguyen  1 
p. 7,
¶
Line 9

Replace "guide line" by "guideline".  Mahdi MajidiZolbanin  1 
p. 8,
¶
16

Replace "field" by "fields".  A. Graf  1 2 
p. 13,
¶
Line 15

Replace nonconnected by nonconnected.  P. Zsifkovits  1 
p. 14,
¶
Lemma 1.19 (2)

Add the assumption that $X\ne\emptyset$.  Ulrich Görtz  1 
p. 14,
¶
Line 10

Replace ``$\emptyset \ne J \subset I$'' by ``$\emptyset \ne J \subsetneq I$''.  J. Buck  1 
p. 14,
¶
Line 14

Replace ``than'' by ``then'' and omit ``be''.  P. Zsifkovits  1 
p. 15,
¶
Line 15

Replace $\mathscr M$ by $\mathcal M$.  I. Tselepidis  1 2 
p. 15,
¶
Line 15

Replace "there existed" with "there would exist".  F. Gispert Sánchez  1 
p. 16,
¶
Line 16

Replace $T_2T_1^2$ by $T_1T_2^2$.  P. Zsifkovits  1 
p. 16,
¶
Line 13

Omit superfluous (.  P. Zsifkovits  1 
p. 17,
¶
Line 20

Replace "function" by "functions".  Ehsan Shahoseini  1 2 
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 MajidiZolbanin  1 
p. 19,
¶
Line 11

In the definition of $f$ add the missing bracket at the end.  T. Przezdziecki  1 
p. 19,
¶
Line 14

Insert ``$A$'' after ``$k$algebra''.  J. Buck  1 
p. 19,
¶
Line 9

In the diagram exchange $m$ and $n$.  J. Buck  1 
p. 19,
¶
Line 15

Replace ``Proposition 1.20'' by ``Proposition 1.32''.  P. Zsifkovits  1 
p. 20,
¶
Line 15

Replace ``set morphisms'' by ``set of morphisms''.  P. Zsifkovits  1 
p. 21,
¶
Line 12

Require $V$ to be nonempty.  Ulrich Görtz  1 
p. 21,
¶
Line 8

Remove "of" in the sentence "how to identify elements of $f \in \mathscr O_X(U)$ with ..."  Mahdi MajidiZolbanin  1 
p. 22,
¶
Line 14

Replace $g\circ f$ with $g\circ f_{f^{1}(U)}$.  A.Graf  1 
p. 23,
¶
Cor. 1.47 (i)

Exclude the zero $k$algebra.  Ulrich Görtz  1 
p. 23,
¶
Line 6

Replace ``manifolds'' by ``manifold''.  J. Buck  1 
p. 23,
¶
Line 14

1.47 (i), "opposed category" should be "opposite category".  Peng Du  1 
p. 23,
¶
Prop. 1.48

Add the assumption that $X\ne \emptyset$ (or put this into the definition of prevariety?).  Ulrich Görtz  1 
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 nonempty.  Ulrich Görtz  1 
p. 26,
¶
Line 13

Replace $k$ by $R$.  D. Gerigk  1 
p. 27,
¶
Line 5

Replace $R_n$ by $X_n$.  Peng Du  1 
p. 28,
¶
Line 7

Replace "space with function" with "space with functions".  F. Ebert  1 
p. 28,
¶
Displayed equation in prop. 1.59

Replace ``exist'' by ``$\exists$''.  P. Zsifkovits  1 
p. 29,
¶
Line 2

Replace "space with function" with "space with functions".  F. Gispert Sánchez  1 
p. 33,
¶
Lines 4 and 6

Replace $\mathbb P^n$ with $\mathbb P^m$ and $\mathbb A^n$ with $\mathbb A^m$.  Safak Ozden  1 
p. 33,
¶
Line 21 and 22

Replace $\mathbb{P}_k^n$ by $\mathbb{P}^n(k)$.  E. Hong  1 2 
p. 34,
¶
Line 12 (Cor. 1.71)

Insert "be" before "quadrics".  Peng Du  1 
p. 34,
¶
Line 5

Replace "function field" with "function fields".  Ehsan Shahoseini  1 2 
p. 35,
¶
Figure 1.2

Replace $X^2+Y^21$ by $X^2+Y^2=1$ and $XY1$ by $XY=1$.  A. B. Nguyen  1 
p. 35,
¶
Line 12

Replace "it" by ", the quadric of rank 3".  Ehsan Shahoseini  1 2 
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  1 
p. 40,
¶
9

Replace "of functions" by "with functions".  Peng Du  1 
p. 41,
¶
7

Replace "with the category" by "to the category".  Peng Du  1 
p. 44,
¶
Proposition 2.10

Replace ``Let $A$ be a ring.'' by ``Let $\varphi\colon A\rightarrow B$ be a ring homomorphism.''.  U. Görtz  1 
p. 46,
¶
Line 7

Replace "$R$module" by "an $R$module".  A. Graf  1 2 
p. 51,
¶
Line $2$

Replace $s^x$ by ${s^x}_{V_x}$.  A. Graf  1 
p. 51,
¶
Line 8

$\mathscr O(U)$ should be $\mathscr O_{\mathbb C}(U)$  Ehsan Shahoseini  1 2 
p. 51,
¶
Line 12

Replace ``if and only of'' by ``if and only if''.  J. Watterlond  1 
p. 54,
¶
Line 2

Replace ``$X$'' by ``$Y$''.  K. Mohri  1 
p. 54,
¶
Line 3

Replace ``$f(V)$'' by ``$f(U)$''.  K. Mohri  1 
p. 55,
¶
9

Replace "limit" by "inductive limit".  Peng Du  1 
p. 55,
¶
Line 11

Replace ``statements'' by ``statement''.  D. Gerigk  1 
p. 55,
¶
Line 13

Insert "$U$" after "open subset".  F. Ebert  1 
p. 55,
¶
Line 2 (Remark 2.28)

The assumption on $\psi$ should be that it is a morphism of presheaves.  L. P.  1 
p. 58,
¶
Line 2

The third sum should be over the index set $I$ instead of $J$.  Sebastian Schlegel Mejia  1 
p. 60,
¶
Line 9

Exclude $f=0$ here.  Philipp Reichenbach  1 2 
p. 61,
¶
Lines 3, 2

Replace ``$\mathfrak a$'' by ``$\mathfrak a_1$'' and ``$\mathfrak b$'' by ``$\mathfrak a_2$''.  P. Zsifkovits  1 
p. 61,
¶
Line 8

Replace $i^\flat_x$ by $i^\sharp_x$.  A. Graf  1 2 
p. 62,
¶
Line 18

Insert "is" before "the complement".  Philipp Reichenbach  1 2 
p. 63,
¶
Line 8 (ex. 2.9 (d))

Replace "elements of $u \in A$" with "elements $u \in R$".  F. Gispert Sánchez  1 
p. 63,
¶
Exer. 2.13 (b)

The assumption that $X$ be connected can be omitted.  A. Beshenov  1 
p. 64,
¶
Line 19 (Exercise 2.16)

Replace "show an analogous results" by "show that an analogous result holds".  1  
p. 65,
¶
Line 14

Replace "subset" by "open subset".  A. Graf  1 2 
p. 67,
¶
Line 6

Replace ``constitute homomorphism'' by ``constitute a homomorphism''.  D. Gerigk  1 
p. 69,
¶
10

Replace "(or more" by ". More".  Peng Du  1 
p. 70,
¶
Line 5

Replace "with an open subscheme of $T_i$" by "with an open subscheme $T_i$ of $T$".  E. Hong  1 2 
p. 71,
¶
Last line of the proof

Replace ``$X = \bigcup U_i$'' by ``$X = \bigcup_{i \in I} \psi_i(U_i)$''.  L. P.  1 
p. 71,
¶
3rd paragraph in the proof

$\mathcal{O}_{U_{ij}}(U)$ is not welldefined. 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.  1 
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  1 
p. 71,
¶
Example 3.12

Replace ``$\prod_{i=1}^n A_i$'' by ``$\mathop{\rm Spec}\prod_{i=1}^n A_i$''.  P. Zsifkovits  1 
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  1 
p. 73,
¶
15

Replace ``subset'' by ``a subset''.  E. Viehmann  1 
p. 74,
¶
Line 7

Replace the reference to Section (4.14) by a reference to Section (4.13).  N.T.  1 2 
p. 76,
¶
Line 20

Replace ``ideals'' by ``ideal''.  D. Gerigk  1 
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.  1 
p. 79,
¶
Line 12

Replace "if follows" with "it follows".  F. Gispert Sánchez  1 
p. 79,
¶
Line 14

Replace ``locally finite type'' by ``locally of finite type''.  J. Watterlond  1 
p. 79,
¶
Proof of Lemma 3.32, second paragraph

Replace "the $f_i$ generated the" by "the $f_i$ generate the".  L. P.  1 
p. 79,
¶
Line 1 (Proof of Prop 3.33)

Replace "(1)" with "(i)" , "(2)" with "(ii)" and "(3)" with "(iii)".  R.Ishizuka  1 
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  1 
p. 81,
¶
10

Replace "opposed category" by "opposite category".  Peng Du  1 
p. 82,
¶
Line 13

Replace ``points'' by ``type''.  D. Gerigk  1 
p. 82,
¶
Line 21

Add index $i$ to the second intersection.  A. Graf  1 2 
p. 83,
¶
Line 8

Replace "$j_*\mathscr O_Y$" by "$(j_*\mathscr O_Y)_{U}$".  Peng Du  1 
p. 85,
¶
Line 14

Replace "$\varphi(g)U$" with "$\varphi(g)_{U}$".  F. Gispert Sánchez  1 
p. 85,
¶
Line 17

Replace ``biggest'' by ``largest''.  P. Johnson  1 
p. 86,
¶
Line 2

Replace "Remark 3.45" by "Example 3.45".  Peng Du  1 
p. 88,
¶
Line 7

Replace "ordered set" by "partially ordered set".  Peng Du  1 
p. 88,
¶
3rd line of the proof of Prop. 3.52

Replace "on" by "in" in the sentence "[...] $Z$ is closed on $U$".  1  
p. 90,
¶
Exercise 3.13

$X$ should be nonempty.  1  
p. 90,
¶
Line 21

Replace "component" by "components".  A. Graf  1 2 
p. 92,
¶
Line 5

Replace ``shows'' by ``show''.  P. Zsifkovits  1 
p. 93,
¶
Line 9

Insert "=" before $f_m(x)$.  Peng Du  1 
p. 94,
¶
Example 4.4

Conflict of notation: $T$.  E. Viehmann  1 
p. 94,
¶
Line 3 (ex. 4.5)

Replace "Equivalent" with "Equivalently".  F. Gispert Sánchez  1 
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  1 
p. 95,
¶
Line 7

Replace "objects" with "object".  F. Gispert Sánchez  1 
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  1 
p. 100,
¶
9

"are morphisms" should be replaced by "be morphisms".  Kuo TzuAng  1 
p. 101,
¶
Prop. 4.20, Condition (I)

Replace ``affine neighborhood $U'$ of $x'$'' by ``affine neighborhood $U'$ of $f(x')$''.  B. Heintz  1 
p. 102,
¶
Line 10

Replace "homomorphisms" by "homomorphism" (... "is injective").  F. Gispert Sánchez  1 
p. 104,
¶
Line 3

Replace ``be'' by ``by''.  U. Görtz  1 
p. 104,
¶
Line 13

Add "Let" before "$X$ and $Y$ be".  F. Gispert Sánchez  1 
p. 104,
¶
4.24 (Frobenius morphism)

Conflict of notation: There are too many $f$'s here, $f\in \Gamma(U, \mathscr O_S)$ in line 2, the morphism $f\colon X\to S$, the polynomial $f$ in (1).  U. Görtz  1 2 
p. 105,
¶
Line 6

Insert ``be'' before ``the first projection''.  P. Zsifkovits  1 
p. 105,
¶
Eqn. (4.7.1)

The label of the lower arrow should be $(t, h)_S$.  Peng Du  1 
p. 105,
¶
Line 2 ff.

Rename the variables as $T_i$ since the symbol $X$ is in use already.  Peng Du  1 
p. 106,
¶
Line 2

Add missing parenthesis.  P. Zsifkovits  1 
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  1 
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  1 
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  1 
p. 108,
¶
Line 14

Replace "locally" by "local".  F. Gispert Sánchez  1 
p. 108,
¶
Line 6 (prop. 4.34)

Replace "immersion" with "immersions".  F. Gispert Sánchez  1 
p. 109,
¶
14

Strictly speaking, $f^{–1}(Z)$ should be $f^{–1}(i(Z))$.  Jan Willing  1 
p. 111,
¶
Line 10

Replace ``$g(a)$'' by ``$g(a)=0$''.  P. Zsifkovits  1 
p. 112,
¶
Line 4

Add parentheses around $U_i\times_R W_j$.  Peng Du  1 
p. 113,
¶
Line 4

Replace "as $S$scheme" by "as an $S$scheme".  A. Graf  1 2 
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  1 
p. 118,
¶
Line 11

Replace "arbitrary morphism" with "arbitrary morphisms".  F. Gispert Sánchez  1 
p. 120,
¶
l. 9

Replace "Corollary 3.33" by "Proposition 3.33".  M. Pereira  1 
p. 124,
¶
Line 18 (Proof of Lemma 5.14)

Replace "finitedimension" by "finitedimensional".  Peng Du  1 
p. 124,
¶
Line 9

Add "the" before "following".  Ulrich Görtz  1 
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  1 
p. 125,
¶
Thm. 5.19 (2)

Change (a) to (1).  Peng Du  1 
p. 125,
¶
Line 21

Replace "completed to maximal chain" by "completed to a maximal chain".  Ulrich Görtz  1 
p. 127,
¶
Line 3

Replace "$\sup_{Z\in I} (Z\cap U)$" with "$\sup_{Z\in I} \dim(Z\cap U)$"  F. Gispert Sánchez  1 
p. 128,
¶
Line 12

Since $d$ is never refered to, its definition in the statement of Prop. 5.30 should be removed.  A. Graf  1 2 
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  1 
p. 133,
¶
Line 15

Better: "... where $S$ is the spectrum of a field $k$ and $X$ is of finite type over $k$."  A. Graf  1 2 
p. 134,
¶
Line 18

Replace ``$X=\mathop{\rm Spec} B$'' by ``$Y=\mathop{\rm Spec} B$''.  K. Kidwell  1 
p. 137,
¶
Cor. 5.56 (2), (3)

Insert "then" before $X_K$ (twice).  Peng Du  1 
p. 138,
¶
Line 5

Replace "roughly spoken" by "roughly speaking".  Peng Du  1 
p. 140,
¶
Line 15

It is clearer to write $A\otimes_kK = \Gamma(C_K\cap D_K, \mathscr O_{C_K\cap D_K})$ (so to make use of the calculation in the previous line).  A. Graf  1 2 
p. 140,
¶
Line 7

Replace $af+bg$ by $af+bg=0$.  Peng Du  1 
p. 140,
¶
Line 13

Replace $\bigoplus$ by $\bigoplus_d$.  A. Graf  1 2 
p. 142,
¶
Exer. 5.8 (b)

Replace $\mathop{\rm codim}(Y, X)$ by $\mathop{\rm codim}\nolimits_X(Y)$ etc.  Peng Du  1 
p. 142,
¶
Exercise 5.7

Before the first ``Show'', add ``Let $X=\mathop{\rm Spec} A[T]$.''  A. Steinbach  1 
p. 143,
¶
Line 6 (ex. 5.20)

Replace "$x\in X_{\Omega}$" with "$x' \in X_{\Omega}$"  F. Gispert Sánchez  1 
p. 144,
¶
Line 10 (ex. 5.21)

Replace "intgral" with "integral".  F. Gispert Sánchez  1 
p. 146,
¶
l. $8$

Replace ``heuristics'' by ``heuristic''.  P. Johnson  1 
p. 147,
¶
Line 4

Add subscript $k$ (twice).  A. Graf  1 2 
p. 148,
¶
Line 10

$D(g_1 \cdots g_s)$ should instead read $D(g_1 \cdots g_r)$.  Nathan Pflueger  1 
p. 149,
¶
Lines 8, 7

Remove "is in $T_x X$".  F. Gispert Sánchez  1 
p. 150,
¶
Line 1

Replace the reference to Section (4.14) by a reference to Section (4.13).  Dominik Briganti  1 2 
p. 152,
¶
Line 22

Replace "isoomorphism" by "isomorphism".  Ulrich Görtz  1 
p. 152,
¶
Line 15

Replace "$k$scheme" by "$K$scheme".  Dominik Briganti  1 2 
p. 153,
¶
Def. 6.14 (1)

Conflict of notation: $j$ is used for the map and as an index.  Ulrich Görtz  1 
p. 153,
¶
Def. 6.14 (2)

Replace ``in all points'' by ``at all points''.  P. Johnson  1 
p. 154,
¶
Line 17

Replace "a $r\times r$ minor" by "an $(nd)\times (nd)$ minor" as $r$ is not defined in Definition 6.14.  Jan Willing  1 2 
p. 154,
¶
Line 8

Replace "for $X$" by "for $f$".  Ulrich Görtz  1 
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  1 
p. 155,
¶
Part (3) (Line 8)

Add the condition that $k$ has characteristic $\ne 2$.  Peng Du  1 
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  1 
p. 159,
¶
Proof of Lemma 6.27

In the second line of the proof, replace ``columns'' by ``rows''.  B. Smithling  1 
p. 160,
¶
Line 9

Replace "point of $x$" with "point of $X$".  F. Gispert Sánchez  1 
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  1 
p. 163,
¶
Line 6

Replace "Exercise 6.19" by Exercise "6.18".  Peng Du  1 
p. 165,
¶
Line 21 (Exer. 6.4)

Replace $df$ by $df_e$.  Jan Willing  1 
p. 165,
¶
Line 17 (ex. 6.4)

Replace "$df_{(e,e)}$" with "$dm_{(e,e)}$".  F. Gispert Sánchez  1 
p. 166,
¶
Line 7

Insert "be" before "the set".  Peng Du  1 
p. 169,
¶
Lines 7, 14

Wrong quotation marks around "globalizations" and "sheaf version".  Zhaodong Cai  1 
p. 171,
¶
Line 4

Replace $\mathscr O$ by $\mathscr O_X$.  Peng Du  1 
p. 171,
¶
Line 3

Wrong quotation marks around "surjective".  Zhaodong Cai  1 
p. 171,
¶
Line 4

Replace "homomorphism" with "homomorphisms".  F. Gispert Sánchez  1 
p. 171,
¶
Line 14

Replace "of an $\mathscr O_X$module" by "of an $\mathscr O_X$module $\mathscr F$".  A. Graf  1 2 
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  1 
p. 175,
¶
Line 3

Maybe the tensor symbols $\otimes$ should be accompanied by O_X(U) as a subscript.  Victor Zhang  1 
p. 175,
¶
Line 3

Add period at the end of the sentence.  A. Graf  1 2 
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  1 2 
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  1 
p. 179,
¶
Line 6

Add "for all open $U\subseteq X$" at the end of the line.  A. Graf  1 2 
p. 181,
¶
Line 7

Replace "and" by "an".  F. Gispert Sánchez  1 
p. 181,
¶
Line 12

Replace "(2.27)" by "Proposition 2.27".  Peng Du  1 
p. 182,
¶
Line 16

Replace "localization in the prime ideal" by "localization at the prime ideal".  A. Graf  1 2 
p. 182,
¶
Line 6

Replace "with respect" by "with respect to".  U. Görtz  1 2 
p. 184,
¶
Line 18

Replace "identity" by "identity morphism of".  Peng Du  1 
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  1 
p. 187,
¶
Theorem 7.22

Replace $X_s$ by $X_s(\mathscr L)$ in the statement (and in the proof before the reduction to $\mathscr L \cong \mathscr O$.  A. Graf  1 2 
p. 188,
¶
Line 14

Replace "of Dedekind schemes" with "by Dedekind schemes".  Peng Du  1 
p. 189,
¶
Line 13

Replace "$\mathscr{L}_{U_x}$" by "$\mathscr{L}_{DU_x}$"  F. Gispert Sánchez  1 
p. 189,
¶
Line 16

Replace "$y\in U_x\cap X_0$" by "$y\in (U_x\cap X_0)\setminus\{x\}$".  Peng Du  1 
p. 190,
¶
Line 15

Add "be" before "extended".  F. Gispert Sánchez  1 
p. 191,
¶
Line 1

Replace "generated" by "generate".  F. Gispert Sánchez  1 
p. 191,
¶
Commutative diagram

In the rightmost column, replace $n$ by $m$ in both rows.  P. Carlucci  1 
p. 191,
¶
Line 2

Replace "$X''$" by "$U''$".  F. Gispert Sánchez  1 
p. 192,
¶
Line 3

Replace "as" by "as an".  A. Graf  1 2 
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  1 
p. 194,
¶
Line 9

Replace "analogue" by "analogous".  Peng Du  1 
p. 194,
¶
Line 8

Replace $\mathop{\rm Spec} A$ by $\mathop{\rm Spec} B$.  Longxi Hu  1 
p. 197,
¶
Line 15

Replace "holds of" by "holds for".  Peng Du  1 
p. 199,
¶
Exer. 7.4

Add a comma before $s_r(x)$ in both (a) and (b).  A. Graf  1 2 
p. 200,
¶
Exer. 7.10 (a)

Replace the first occurrence of "polynomials" by "polynomial".  A. Graf  1 2 
p. 201,
¶
Exercise 7.16

Replace $\mathscr F$ by $\mathscr E$ (four times).  Jan Willing  1 2 
p. 202,
¶
Exer. 7.18

Add the condition $\mathscr I \ne 0$.  Peng Du  1 
p. 202,
¶
Exercise 7.20

Replace ${\mathscr F}$ by ${\mathscr G}$.  J. Calabrese  1 
p. 204,
¶
Line 4 (ex. 7.32)

Replace "send" with "sent".  F. Gispert Sánchez  1 
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  1 
p. 209,
¶
Equation (8.3.1)

Replace $U_{ij}\cap U_{jk}$ by $U_{ij}\cap U_{ik}$.  1  
p. 209,
¶
Line 12

Replace "all $S$" by "all $T$".  Peng Du  1 
p. 209,
¶
Line 10

Change the first "and" to "by".  1  
p. 209,
¶
Lines 14, 13

Replace all $S$ by $T$.  Peng Du  1 
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  1 
p. 211,
¶
Line 7

Add "of" between "homomorphism" and "$\mathscr{O}_S$modules".  F. Gispert Sánchez  1 
p. 211,
¶
Line 9

Replace ``bijective'' by ``an isomorphism''.  Ulrich Görtz  1 
p. 212,
¶
Line 16

Insert ``and is'' before ``also''.  D. Gerigk  1 
p. 212,
¶
Line 19

Replace "therefore" by "therefore we get".  Peng Du  1 
p. 212,
¶
Line 5

Replace "subvector space of $K^n$" by "subvector space $U$ of $K^n$".  F. Gispert Sánchez  1 
p. 214,
¶
Line 11

Replace "$S$ scheme" by "$S$scheme".  Peng Du  1 
p. 216,
¶
Line 4

Replace "of $\mathscr{O}_S$module" by "of $\mathscr{O}_S$modules".  F. Gispert Sánchez  1 
p. 216,
¶
Line 7

Replace "disjoint sum" with "disjoint union".  F. Gispert Sánchez  1 
p. 216,
¶
Line 6

Replace $q'$ by $q''$.  Xiaolong Liu  1 2 
p. 217,
¶
Line 2

Replace $Grass^1(\bigwedge^d\mathscr{O}_{Spec(\mathbb{Z})})$ by $Grass^{\binom{n}{d}1}(\bigwedge^d\mathscr{O}_{Spec(\mathbb{Z})})$.  Xiaolong Liu  1 2 
p. 217,
¶
Line 2

Replace ${\rm Grass}_1$ by ${\rm Grass}^1$.  Peng Du  1 
p. 218,
¶
Line 10

Rephrase as "... we have a bijection, functorial in $S$,".  Peng Du  1 
p. 219,
¶
Line 11, Def. 8.25

Insert "$X$" after "$k$scheme".  D. Gerigk/Peng Du  1 
p. 219,
¶
Line 7

Replace "Corollary 5.45" by "Proposition 5.51".  Peng Du  1 
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  1 
p. 219,
¶
Line 13

Replace $k$ by $\mathbb C$.  Peng Du  1 
p. 220,
¶
Line 4

Replace "$(v,u)$" by "$(v,s)$".  F. Gispert Sánchez  1 
p. 220,
¶
Line 19 (Proof of Proposition 8.26)

Replace $M_{1 \times n}$ by $M_{1\times n}(R)$.  SzSheng Wang  1 2 
p. 221,
¶
Line 13

Add subscript $k$ to $\otimes$.  Peng Du  1 
p. 221,
¶
Line 13

Insert "points" between "$R$valued" and "of".  F. Gispert Sánchez  1 
p. 221,
¶
Line 16

Replace "(v)" by "(iv)".  1  
p. 226,
¶
Line 20

Replace ``separable'' by ``separated''.  T. Wedhorn  1 
p. 227,
¶
Line 19

Replace "an $S$objects" by "an $S$object".  F. Gispert Sánchez  1 
p. 228,
¶
Statement of Prop. 9.3 (3)

Replace $f$ by $u$ everywhere (i.e., in 3 places).  K. Kidwell  1 
p. 229,
¶
Line 13

Replace "$p\colon X\times_S Y$" by "$p\colon X\times_S Y \to X$".  F. Gispert Sánchez  1 
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  1 
p. 232,
¶
Lines 69

Replace "Grass$_{n,ne}$" with "Grass$_{ne,n}$" (4 times) to keep the same notation as in chapter 8.  F. Gispert Sánchez  1 
p. 233,
¶
Def. 9.21

Replace ``$\mathfrak p_x$'' by ``$\mathfrak m_x$''.  A. Isaev  1 
p. 234,
¶
Lines 3, 1

Replace "$\mathcal R$" by "$\mathcal R(X, Y)$" (twice).  Peng Du  1 
p. 234,
¶
Line 30 (Remark 9.25)

Replace $U\subseteq S$ by $U\subseteq X$.  P. Barik  1 
p. 234,
¶
Line 28 (Remark 9.25)

Replace ``can checked'' by ``can be checked''.  L. Galinat  1 
p. 236,
¶
Line 5

Replace "$(\lambda,\mu)$" with "$(\lambda : \mu)$".  F. Gispert Sánchez  1 
p. 236,
¶
Line 3

Replace "$K$" with "$K(X)$".  F. Gispert Sánchez  1 
p. 237,
¶
Line 16

Replace "tape" with "type".  F. Gispert Sánchez  1 
p. 237,
¶
Line 24 (Remark 9.34)

Replace $f\colon   \rightarrow Y$ by $f\colon X   \rightarrow Y$.  P. Barik  1 
p. 238,
¶
Line 3

Delete the second $X$.  Peng Du  1 
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  1 
p. 241,
¶
8

Replace "roughly spoken" by "roughly speaking".  Peng Du  1 
p. 241,
¶
Line 16

Replace "among" by "along".  Peng Du  1 
p. 242,
¶
Rmk. 10.2 (4)

At the end, replace ``if $Y$ not separated'' by ``if $Y$ is not separated''.  U. Hartl  1 
p. 248,
¶
Line 9

Replace ``ad'' by ``and''.  U. Hartl  1 
p. 248,
¶
Line 14

Replace "on" with "an".  F. Gispert Sánchez  1 
p. 249,
¶
Line 1

Replace "$\mathscr{O}_{Y,f(y)}$" with "$\mathscr{O}_{Y,f(x)}$"  F. Gispert Sánchez  1 
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  1 
p. 251,
¶
Line 20

Replace "Its sum" by "The sum".  Peng Du  1 
p. 254,
¶
Line 12, 11

Replace 'constructible' by 'globally constructible' twice.  Xiaolong Liu  1 2 
p. 254,
¶
Line 3

Replace "Definition 10.5" by "Definition 10.12".  Peng Du  1 
p. 255,
¶
Line 4

Replace $Z$ by $C$.  P. Barik  1 
p. 256,
¶
Line 18 (Cor. 10.49)

In the conclusion of the corollary, add that $\mathscr{F}$ is also quasicoherent.  F. Gispert Sánchez  1 
p. 257,
¶
Line 7

Replace "an" with "a".  F. Gispert Sánchez  1 
p. 259,
¶
Line 16

Replace "inductive limit" by "inductive limits".  Peng Du  1 
p. 261,
¶
Line 17

Replace "inverse images under continuous maps" by "inverse images under morphisms".  Peng Du  1 
p. 263,
¶
Line 4

Replace $v_{0\lambda}^{1}$ by $x_{0\lambda}^{1}$.  Peng Du  1 
p. 264,
¶
Line 2

Replace "1., 3., 5." with "1.3., 5.".  F. Gispert Sánchez  1 
p. 265,
¶
Line 15 (cor. 10.67)

Add "of" between "morphism" and "$S$schemes".  F. Gispert Sánchez  1 
p. 265,
¶
l. $2$

Replace ``$R$scheme'' by ``of $R$schemes''.  U. Hartl  1 
p. 267,
¶
Line 10 (prop. 10.75)

Replace "$f$" with "$f_0$".  F. Gispert Sánchez  1 
p. 267,
¶
Line 8

Add "or" after property (1).  Peng Du  1 
p. 268,
¶
Line 22

Replace "morphism" with "morphisms".  F. Gispert Sánchez  1 
p. 268,
¶
l $2$

Remove one ``that'' and replace ``morphism'' by ``morphisms''.  U. Hartl  1 
p. 268,
¶
l. $7$

Insert ``if'' after ``In fact,''.  U. Hartl  1 
p. 270,
¶
Line 17

Remove "it".  F. Gispert Sánchez  1 
p. 272,
¶
Line 12

Replace "subscheme) structure" with "subscheme structure)".  F. Gispert Sánchez  1 
p. 273,
¶
Line 14

Add "is" before "constructible".  F. Gispert Sánchez  1 
p. 273,
¶
Line 22

Replace "Frobenius (4.24)" by "Frobenius morphism (Definition 4.24)". (Cf. this erratum.)  Peng Du  1 
p. 274,
¶
Line 4

Replace ``suffice'' by ``suffices''.  U. Görtz  1 
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  1 
p. 276,
¶
Line 20

Add "of" between "is" and "finite type".  F. Gispert Sánchez  1 
p. 277,
¶
Line 10

Replace "Corollary 5.12" with "Corollary 5.17".  F. Gispert Sánchez  1 
p. 280,
¶
Line 20

It would look better to replace $N>1$ by $N\ge 1$ (as in the following line).  Peter Johnson  1 
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  1 
p. 281,
¶
Line 25 (ex. 10.22)

Replace "is" with "be its".  F. Gispert Sánchez  1 
p. 283,
¶
Line 14 (ex. 10.34)

Replace "inductive limit" with "inductive system".  F. Gispert Sánchez  1 
p. 284,
¶
Line 20

Replace "generic" by "generically".  Peng Du  1 
p. 286,
¶
Line 9

Replace "roughly spoken" by "roughly speaking".  1  
p. 286,
¶
Line 11

Replace "between to ..." by "between two ...".  Peng Du  1 
p. 286,
¶
Line 5

Delete "Line bundles and".  Peng Du  1 
p. 287,
¶
Line 3

Replace "$S$" by "$X$".  F. Gispert Sánchez  1 
p. 288,
¶
Line 2

Replace "$\binom{r+n1}{r}$" with "$\binom{r+n1}{n}$" (or "$\binom{r+n1}{r1}$").  F. Gispert Sánchez  1 
p. 288,
¶
2 lines above equation (11.1.6)

Replace $f\colon X\rightarrow Y$ by $f\colon Y\rightarrow X$.  Zhaodong Cai  1 
p. 288,
¶
Last line of the statement of Proposition 11.1

$({\rm Sch})^{\rm opp}$ should be $({\rm Sch}/X)^{\rm opp}$.  1  
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  1 
p. 291,
¶
Line 9

Remove the extra parenthesis in "$\mathscr{S}(V/X))$".  F. Gispert Sánchez  1 
p. 292,
¶
14

Replace "is the group $G$ itself" by "is the sheaf of groups $G$ itself"  Félix Baril Boudreau  1 
p. 293,
¶
Lines 57

Define $\theta$ and $\theta'$ as $\theta := (g_{ij})_[i,j}$ and $\theta' := (g_{ij}')_[i, j}$, respectively. Maybe replace "for all $i$" with "for all $i \in I$". Mention explicitly that the equation (in line 7) holds on $U_i \cap U_j$.  N. T.  1 2 
p. 293,
¶
Line 14

Replace "1cocycle" with "1cocycles".  F. Gispert Sánchez  1 
p. 293,
¶
Line 9

Replace "1cocycle" with "1cocycles".  F. Gispert Sánchez  1 
p. 293,
¶
Line 18

Replace "straight forward" by "straightforward".  Ulrich Görtz  1 
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  1 
p. 299,
¶
6th line after Definition 11.19

TeX: $\mathop{\rm Div}(X)$ should be upright  1  
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  1 
p. 303,
¶
Line 3

Replace ``detailed'' by ``detail''.  J. Calabrese  1 
p. 304,
¶
Line 1

Delete the second $U$.  Peng Du  1 
p. 305,
¶
Line 8 (Lemma 11.33)

The notation for the codimension is different from the notation used in chapter 5.  F. Gispert Sánchez  1 
p. 305,
¶
Line 19 (displayed bijection)

Replace "cartier" with "Cartier".  F. Gispert Sánchez  1 
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  1 
p. 305,
¶
Line 27

Replace "$\mathscr{O}$" with "$\mathscr{O}_X$".  F. Gispert Sánchez  1 
p. 305,
¶
Line 13

Replace "isomorphism" by "isomorphisms".  Ulrich Görtz  1 
p. 306,
¶
Line 16

Replace "$\mathscr{O}_{X,c}$" with "$\mathscr{O}_{X,C}$".  F. Gispert Sánchez  1 
p. 306,
¶
Line 7

Replace $\mathop{\rm codim}(Z, X)$ by $\mathop{\rm codim}\nolimits_X(Z)$ (twice).  U. Görtz  1 
p. 307,
¶
Line 6

Replace (B.75) by B.75.  Peng Du  1 
p. 308,
¶
Line 13

Replace $U$ by $U_i$ (twice).  B. Smithling  1 
p. 309,
¶
Line 11

Replace $\mathop{\rm codim}(Z, X)$ by $\mathop{\rm codim}\nolimits_X(Z)$.  Peng Du  1 
p. 309,
¶
Lines 4, 1

The expression for $f$ in line 8 (ed. 1: line 4) ignores that there may be denominators. A homogeneous polynomial may be expressed as stated and for those $f$ the expression in line 11 (ed. 1: line 1) is correct. In general, we express an element of $\mathcal R$ as a fraction and extend the map $Z$ in the obvious way so that it becomes a group homomorphism.  J.C. Syu  1 2 
p. 311,
¶
Line 17

Replace "Example 7.13" by "(7.13)".  Peng Du  1 
p. 313,
¶
Line 6

Replace ``set'' by ``sets''.  P. Johnson  1 
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  1 
p. 315,
¶
Line 13

Replace $d$ by $d_i$ in the definition of $\mathscr E^\lambda$ (twice).  Peng Du  1 
p. 319,
¶
Line 13 (ex. 11.22 (a))

Replace "an" with "a".  F. Gispert Sánchez  1 
p. 321,
¶
Line 16

Replace "a affine scheme" with "an affine scheme".  F. Gispert Sánchez  1 
p. 321,
¶
Line 24

Replace ``affine over $X$'' by ``affine over $Y$''.  P. Barik  1 
p. 323,
¶
Line 17

Replace $\otimes_{(A'\otimes_A B})\otimes$ by $\otimes_{(A'\otimes_A B)}$.  P. Barik  1 
p. 323,
¶
Line 12

Replace $X' = \mathop{\rm Spec} B' \otimes_{B} A$ by $X' = \mathop{\rm Spec}(B \otimes_A A')$.  P. Barik  1 
p. 323,
¶
Line 8

In the last expression of 12.2.3 replace $Y'$ by $X'$.  P. Barik  1 
p. 323,
¶
Line 9

The (finite) covering $(U_i)_i$ of $X$ should be an affine open covering.  F. Gispert Sánchez  1 
p. 324,
¶
Line 2

Replace "$\mathscr{F}(X)$" with "$\mathscr{F}(U_i)$"  F. Gispert Sánchez  1 
p. 324,
¶
Line 17

Remove ``a'' at the end of the line.  K. Kidwell  1 
p. 325,
¶
Line 3

``Corollary 5.12'' should be ``Proposition 5.12''.  U. Hartl  1 
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  1 
p. 329,
¶
Line 18

Replace "be" with "by".  F. Gispert Sánchez  1 
p. 329,
¶
Line 7

Switch the order in all pairs $z/x, \dots$ in this line.  Peng Du  1 
p. 330,
¶
Line 2

Switch the order in all pairs $z/x, \dots$ in this line.  Peng Du  1 
p. 332,
¶
Line 15

Replace ``Corollary 5.12'' by ``Proposition 5.12''.  S. Köbele  1 
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  1 
p. 333,
¶
Line 14

Replace $a_0$ by $a_n$.  S. Köbele  1 
p. 333,
¶
Example 12.29

In several places, replace $t$ by $t'$.  S. Köbele  1 
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  1 
p. 338,
¶
Line 6

Replace "$H^1(\mathscr{O}_Y,\mathscr{F})$" with "$H^1(Y,\mathscr{F})$".  F. Gispert Sánchez  1 
p. 338,
¶
Line 8

Add "be" between "let $\eta$" and "its generic point".  F. Gispert Sánchez  1 
p. 339,
¶
Line 13

Replace $L$ by $L_X$.  Peng Du  1 
p. 341,
¶
Line 11

Add "be" before "its normalization".  F. Gispert Sánchez  1 
p. 342,
¶
Line 12 (prop. 12.53)

Add that $X$ is integral.  F. Gispert Sánchez  1 
p. 344,
¶
Line 11

Replace $f^{1}(U)\subseteq V$ by $f^{1}(V)\subseteq U$.  P. Barik  1 
p. 346,
¶
Line 4

Replace "$\mathscr{O}_X$module" with "$\mathscr{O}_X$modules".  F. Gispert Sánchez  1 
p. 347,
¶
Line 7

Add "is" before "not irreducible".  F. Gispert Sánchez  1 
p. 348,
¶
Line 14

Replace "Remark 12.10 (2)" by "Remark 12.10 (3)".  Peng Du  1 
p. 348,
¶
Line 10

Add "be" before "its Stein factorization".  F. Gispert Sánchez  1 
p. 351,
¶
Line 9

Replace "$\varphi$" by "$\varphi\colon A\to B$".  Peng Du  1 
p. 352,
¶
Line 12

Replace ``relation'' by ``relations''.  U. Görtz  1 
p. 356,
¶
line 8

Replace "this is implies" by "this implies".  Shahram Mohsenipour  1 
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  1 
p. 361,
¶
Ex. 12.6

Replace isomporphism by isomorphism.  T. Keller  1 
p. 361,
¶
Line 6

Replace "my" by "may".  Peng Du  1 
p. 363,
¶
Line 22, Exer. 12.21

Insert "be" before "Dedekind rings".  Peng Du  1 
p. 367,
¶
Line 20

Replace $\sum_{d\ge 0}$ by $\sum_d$ (to also cover the case of graded modules).  Peng Du  1 
p. 368,
¶
Equation (13.1.1)

Replace $\mod f1$ by $\mod (f1)$  A. Kaučikas  1 
p. 368,
¶
Line 20

The element $f$ should be homogeneous.  U. Görtz  1 
p. 368,
¶
Prop. 13.2 (3)

Replace ``with all relevant prime ideals'' by ``with the intersection of all relevant prime ideals''.  K. Kidwell  1 
p. 368,
¶
Line 10

Replace "graded ideal" by "homogeneous ideal".  Peng Du  1 
p. 368,
¶
Line 6

Replace ``ideal'' by ``ideals''.  U. Görtz  1 
p. 370,
¶
Line 10

$f \in {\rm rad}(g)_+$ should read $g \in {\rm rad}(f)_+$.  F. Gispert Sánchez/N. Pflueger  1 
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.  1 
p. 371,
¶
Line 18

Remove "be" and replace "$R$algebras" by "$R$schemes".  Peng Du  1 
p. 371,
¶
Line 19

Insert "be" before "a graded".  Peng Du  1 
p. 376,
¶
Line 10 (thm. 13.20, displayed equation, under the arrow)

Replace "$\mathscr{F}\leftarrow\!\shortmid \Gamma_{*}(\mathscr{F})$" with "$\Gamma_{*}(\mathscr{F}) \leftarrow\!\shortmid\mathscr{F}$".  F. Gispert Sánchez  1 
p. 376,
¶
Line 15

Insert "$M$" after "module".  Peng Du  1 
p. 376,
¶
Line 5

Replace '$(M_{(f)})_n$' by '$(M_f)_n$'.  Xiaolong Liu  1 2 
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  1 
p. 377,
¶
Line 15

Replace ``$n \ge 0$'' by ``$n\ge n_0$''.  P. Johnson  1 
p. 378,
¶
Line 12

Insert "the" before "saturation".  Peng Du  1 
p. 378,
¶
Line 18

Replace "Corollary 13.24" by "Proposition 13.24".  Peng Du  1 
p. 378,
¶
line 2

Replace "on" with "one."  N. Pflueger  1 
p. 379,
¶
Prop. 13.28

In the first line of the statement of the proposition, replace ``modules'' by ``module''.  P. Johnson  1 
p. 379,
¶
Line 1

The domain and codomain of $g_i$ are swapped: switch $\mathscr{A}_i$ and $\mathscr{A}'_i$.  F. Gispert Sánchez  1 
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  1 
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  1 
p. 379,
¶
Line 2

Replace "$\varphi$" with "$\varphi_i$".  F. Gispert Sánchez  1 
p. 380,
¶
Line 3

Replace "$f_j^{1}$" with "$f_j^{n}$".  F. Gispert Sánchez  1 
p. 380,
¶
Line 11

Replace "by" with "given by".  Peng Du  1 
p. 380,
¶
Line 6

Add "be" before "the structure morphism".  F. Gispert Sánchez  1 
p. 380,
¶
Line 17

Add "be" before "the structure morphism".  F. Gispert Sánchez  1 
p. 380,
¶
Line 8

Replace "$\mathscr{O}(n)$" with "$\mathscr{O}_X(n)$".  F. Gispert Sánchez  1 
p. 382,
¶
Line 12

Replace $\alpha_i$ by $\alpha_j$.  T. Keller  1 
p. 382,
¶
Lines 15, 18

Replace $\mathbb P^{n+1}$ by $\mathbb P^n$ (three times).  K. Kidwell  1 
p. 382,
¶
Line 18

Replace '$\mathbb{P}^{n+1}=\mathbb{P}(\mathscr{O}_S^{n+1})$' by '$\mathbb{P}_S^{n}=\mathbb{P}(\mathscr{O}_S^{n+1})$'.  Xiaolong Liu  1 2 
p. 382,
¶
Line 21

Replace "$\mathscr{O}^{n+1}$" with "$\mathscr{O}^{n+1}_X$".  F. Gispert Sánchez  1 
p. 383,
¶
Line 10

Replace "$R$modules" with "$R$module".  F. Gispert Sánchez  1 
p. 384,
¶
Line 4

Insert the condition $x\ne x'$.  Peng Du  1 
p. 384,
¶
Line 16

Replace "$\mathscr O_S$module" by "$\mathscr O_S$module $\mathscr E$".  Peng Du  1 
p. 384,
¶
line 16

Replace "Corollary 13.24" with "Proposition 13.24".  N. T.  1 
p. 387,
¶
Line 8

Replace "$f$ is invertible in $x$" by "$f(x) \ne 0$ in the fiber $\mathscr L(x)$".  Ulrich Görtz  1 
p. 388,
¶
Line 6, Line 3

Add that $f$ is homogeneous.  F. Gispert Sánchez  1 
p. 389,
¶
Line 17

Replace "relevant prime ideals ... of $A_+$" by "relevant prime ideals ... of $A$".  Peng Du  1 
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  1 
p. 391,
¶
Line 2

Replace "(the globalization) of" with "(the globalization of)".  F. Gispert Sánchez  1 
p. 392,
¶
Line 17

Remove "be".  F. Gispert Sánchez  1 
p. 393,
¶
Lines 8 and 7

Replace "(2)" with "(1)" (twice).  F. Gispert Sánchez  1 
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  1 
p. 393,
¶
Line 8

Replace "send" with "sent".  F. Gispert Sánchez  1 
p. 394,
¶
Line 20

Add that "$S={\rm Spec}\ R$" somewhere ($R$ has not been defined).  F. Gispert Sánchez  1 
p. 394,
¶
Line 15

Replace "$n\geq n_0+m_0$" with "$n\geq d+m_0$".  F. Gispert Sánchez  1 
p. 395,
¶
Def. 13.60

Insert $\mathscr L$ after $\mathscr O_X$module.  P. Johnson  1 
p. 396,
¶
Line 6

Replace "$g^{1}(X_s)$" with "$(g')^{1}(X_s)$".  F. Gispert Sánchez  1 
p. 396,
¶
Line 1

Add the missing closing parenthesis at the end.  F. Gispert Sánchez  1 
p. 397,
¶
Line 10

Replace "be" by "by".  Peng Du  1 
p. 398,
¶
Lines 6 & 2 (def./prop. 13.68)

Replace "a quasicoherent $\mathscr{O}_X$module $\mathscr{E}$" with "a quasicoherent $\mathscr{O}_S$module $\mathscr{E}$" (twice).  F. Gispert Sánchez  1 
p. 398,
¶
Line 9

Replace $Y\backslash \varepsilon(S)$ by $C\backslash \varepsilon(S)$.  U. Görtz  1 
p. 404,
¶
Line 19

Replace "by Corollary 13.42" with "by Example 13.69".  F. Gispert Sánchez  1 
p. 404,
¶
Line 3 (the diagram)

Add label $r$ to the right hand arrow.  Peng Du  1 
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  1 
p. 405,
¶
Line 19

Replace '$X$' by '$Z$'.  Xiaolong Liu  1 2 
p. 408,
¶
Line 11

Replace '$A^x=k[x,y,\mu]/(\mu x=y)$' by '$A^x=k[x,y,\mu]/(\mu xy)$'.  Xiaolong Liu  1 2 
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  1 
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  1 
p. 410,
¶
Lines 5, 3

Replace $=$ by $$ (one in each line).  Peng Du  1 
p. 410,
¶
Line 6

Delete "of".  Peng Du  1 
p. 410,
¶
Line 21 (the 2nd displayed eqn.)

Replace the second $=$ by $$.  Peng Du  1 
p. 411,
¶
Line 2 (Proof of Prop. 13.96)

Replace $(I \oplus J)/J$ by $(I+J)/J$.  Peng Du / Matthieu Romagny  1 
p. 412,
¶
Line 20

Replace "roughly spoken" by "roughly speaking".  Peng Du  1 
p. 416,
¶
Line 19

Replace "morphisms" with "morphism".  F. Gispert Sánchez  1 
p. 417,
¶
Line 20

Switch the exponents $d+1$ and $d$.  Peng Du  1 
p. 417,
¶
Line 6

Replace "particular" by "particularly".  Peng Du  1 
p. 419,
¶
Line 1 (ex. 13.3)

Replace "$A$modules" with "$A$module".  F. Gispert Sánchez  1 
p. 420,
¶
Exer. 13.13

Replace "inductive limits of schemes" by "inductive limits of rings".  Peng Du  1 
p. 422,
¶
Line 17

Insert "be" before "its structure morphism".  Peng Du  1 
p. 423,
¶
Line 11

Replace ``morphisms'' by ``morphism''.  K. Kidwell  1 
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  1 
p. 426,
¶
Line 16 (cor. 14.12)

Replace "a morphism" with "morphisms".  F. Gispert Sánchez  1 
p. 426,
¶
Line 2

Replace ``$\mathop{\rm Spec} R$'' by ``$Y$''.  T. Wedhorn  1 
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  1 
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  1 
p. 428,
¶
Lemma 14.19

Replace ``such'' by ``such that''.  U. Görtz  1 
p. 430,
¶
Line 8

Replace '$h=g\circ f$' by '$g=h\circ f$'.  Xiaolong Liu  1 2 
p. 434,
¶
Line 14

Add "the" between "If $y$ is not" and "closed point".  F. Gispert Sánchez  1 
p. 435,
¶
Line 17

Replace "morphism" with "morphisms".  F. Gispert Sánchez  1 
p. 443,
¶
Line 3

Replace "a open immersion" with "an open immersion".  F. Gispert Sánchez  1 
p. 443,
¶
Line 16

Add "a" between "Let $\mathbf{P}$ be" and "property".  F. Gispert Sánchez  1 
p. 447,
¶
Line 10

Replace $\mathscr G$ by $\mathscr G'$.  K. Kidwell  1 
p. 447,
¶
Line 10

Replace "$(\mathscr{G},\psi)$" with "$(\mathscr{G}',\psi)$".  F. Gispert Sánchez  1 
p. 449,
¶
Line 10

Replace $\varphi$ by $\varphi'$ (twice).  Peng DU  1 2 
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  1 
p. 452,
¶
Line 10

Replace "homomorphism" with "homomorphisms".  F. Gispert Sánchez  1 
p. 453,
¶
Line 14

The reference to Theorem 14.17 is wrong. It should refer to Theorem 14.70.  F. Gispert Sánchez  1 
p. 454,
¶
Line 2

Replace "or fppfsheaves of sheaves" with "or fppfsheaves or sheaves".  F. Gispert Sánchez  1 
p. 454,
¶
Line 17

Replace "an morphism" with "a morphism".  F. Gispert Sánchez  1 
p. 455,
¶
Line 10

Replace "$G_{S'}$" with "$G_{S'}$".  F. Gispert Sánchez  1 
p. 457,
¶
Line 3

Replace "$\gamma(a_\delta)$" with "$\gamma(b_\delta)$".  F. Gispert Sánchez  1 
p. 458,
¶
Line 10

Replace "straight forward" by "straightforward".  Peng Du  1 
p. 461,
¶
Line 8

Define the symbol $X^{\rm sep}$ as $X\otimes_k k^{\rm sep}$.  Peng Du  1 
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  1 
p. 462,
¶
Line 9

Replace the second $=$ by $$.  Peng Du  1 
p. 464,
¶
Line 4

Insert "Proposition" before "5.30 (2)".  Peng Du  1 
p. 468,
¶
Line 11

Replace ``Lemma 14.106 2'' by ``Lemma 14.106 (2)''.  U. Görtz  1 
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  1 
p. 477,
¶
Line 19

Delete the comma after "means".  Peng Du  1 
p. 481,
¶
Exercise 14.8

Replace $X$ by $Y'$.  1 2  
p. 487,
¶
Line 11

Replace the title of the segment by ``Morphisms from spectra of valuation rings to schemes''.  U. Görtz  1 
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  1 
p. 490,
¶
Line 20

Replace "integers" by "be integers".  Peng Du  1 
p. 490,
¶
Line 22

Replace "Theorem 15.8" by "Theorem 15.9".  Peng Du  1 
p. 491,
¶
Line 7

Switch ${\rm Spec}\ K$ and ${\rm Spec}\ R$.  Peng Du  1 
p. 495,
¶
Line 9

Replace "two sets" by "three sets".  Peng Du  1 
p. 495,
¶
Line 19

The term "complete" (= proper over $k$) has not been defined.  Peng Du  1 
p. 499,
¶
First line after Definition 15.33

Replace left by right.  1  
p. 499,
¶
Line 12

Replace "over a field" by "over a field $k$".  Peng Du  1 
p. 501,
¶
Line 9

Replace ``morpism'' by ``morphism''.  J. Scarfy  1 
p. 501,
¶
Line 13

Replace "discrete valuation" by "discrete valuation ring".  Peng Du  1 
p. 503,
¶
Line 12

Replace "roughly spoken" by "roughly speaking".  Peng Du  1 
p. 506,
¶
Line 6

Replace $\mathbb P^{nm}_R$ by $\mathbb P^{nm1}_R$.  Peng Du  1 
p. 507,
¶
Line 6

Replace "geometrically integral" by "is geometrically integral".  Peng Du  1 
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  1 
p. 507,
¶
Line 9

Replace $\times_k$ by $\times$.  Peng Du  1 
p. 508,
¶
Line 5

Delete "the same argument and".  Peng Du  1 
p. 508,
¶
Second line in the proof of Prop. 16.11

Add "open subset" after dense.  Andreas Blatter  1 2 
p. 508,
¶
Line 3

Change $k^n$ to $k^m$ (cf. the corresponding erratum on page 507).  Peng Du  1 
p. 511,
¶
Line 3

Replace $\otimes_k R$ by $\otimes_R k$.  Peng Du  1 
p. 512,
¶
Line 21 (Proof of Lemma 16.20)

Replace $k[T_{ij}]$ by $R[T_{ij}]$.  Peng Du  1 
p. 512,
¶
Line 14

Replace $\underline{\rm Hom}_{\mathscr G, \mathscr H}$ by $\underline{\rm Hom}(\mathscr G, \mathscr H)$  Peng Du  1 
p. 513,
¶
Line 13 (the diagram), Line 10

Replace $\pi_{\mathscr F}$ by $\pi_{\mathscr V}$.  Peng Du  1 
p. 514,
¶
Line 3

Remove "below".  Peng Du  1 
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  1 
p. 516,
¶
Line 6

Delete "an".  Ulrich Görtz  1 
p. 517,
¶
Line 14

Delete "the spectrum of".  Peng Du  1 
p. 519,
¶
Line 17

Add period at the end of the sentence.  Peng Du  1 
p. 524,
¶
Line 7

Replace $(\beta, \alpha)$ by $(\beta : \alpha)$.  Peng Du  1 
p. 528,
¶
Line 18

Replace $T_2T_3^3$ by $T_2T_3^2$.  U. Görtz  1 
p. 528,
¶
Line 14

Replace ``$V_+(p)\in\mathbb P^3_k$'' by ``$V_+(p)\subset\mathbb P^3_k$''.  U. Görtz  1 
p. 529,
¶
Caption of Figure 16.2

Add $=0$ in the end.  Peng Du  1 
p. 531,
¶
Line 14

Replace "Remark (12.31)" by "Remark 12.31".  Peng Du  1 
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  1 
p. 534,
¶
Line 5 of the Proof of Proposition 16.54

$p^{1}(y)$ should be $p_2^{1}(y)$.  1  
p. 534,
¶
Line 8

The correct formula is $f(a_1a_2)f(a_2)^{1}f(a_1)^{1}$.  Peng Du  1 
p. 534,
¶
Line 6 of the Proof of Proposition 16.54

"nieghborhood" should be "neighborhood"  1  
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  1 
p. 534,
¶
Line 20

Replace "Proposition 5.51" by "Proposition 5.49".  Peng Du  1 
p. 536,
¶
Line 17

Replace "Example 16.18" by "Example (16.18)".  Peng Du  1 
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  1 
p. 540,
¶
Line 18

Add space between the comma and the citation.  Ulrich Görtz  1 
p. 540,
¶
Line 6

Replace "is no divisor of" by "does not divide".  Peng Du  1 
p. 546,
¶
Line 10/9

Replace ``left'' by ``right'' and ``right'' by ``left''.  D. Heiss  1 
p. 552,
¶
Line 7 (ex. B.17)

Replace "$\mathfrak{m}M\neq M$" with "$\mathfrak{m}M=M$".  F. Gispert Sánchez  1 
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  1 
p. 558,
¶
Line 16

Replace "$\dots, \mathfrak a_n)$" by "$\dots, a_n)$".  Peng Du  1 
p. 558,
¶
Line 3

Replace "$A$modules" by "abelian groups".  Peng Du  1 
p. 559,
¶
Line 16

Replace $\mathfrak I^n$ by $\mathfrak I_n$ (and similarly for $n+1$; four changes altogether).  Peng Du  1 
p. 560,
¶
Line 2

Replace "form" by "forms".  Ulrich Görtz  1 
p. 561,
¶
Definition B.58

In the second line of the definition, $r$ should be replaced by $n$.  P. Hartwig  1 
p. 563,
¶
Line 15

Replace "as in (iv)" by "as in (v)".  Peng Du  1 
p. 564,
¶
Prop. B.72 (3)

Replace "be a noetherian" by "be noetherian".  Peng Du  1 
p. 565,
¶
Remark B.75 (2)

Replace the reference to B.70 (2) by B.70 (3).  Akira Masuoka  1 
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  1 
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  1 
p. 576,
¶
Line 5

Replace "as usually" by "as usual".  Ulrich Görtz  1 
p. 576,
¶
Line 16

Replace "Converse" by "Conversely".  Peng Du  1 
p. 576,
¶
Line 12

Replace "universal homomorphism" by "universal homeomorphism".  Yun Hao  1 
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), 50112.  Ulrich Görtz  1 
p. 598,
¶
Left col., lines 3, 4

The symbols for direct sum and product should be exchanged.  J.C. Syu  1 2 
p. 600,
¶
2nd column, line 8

Replace $P^n(k)$ by $\mathbb P^n(k)$.  Ulrich Görtz  1 
p. 603,
¶
Line 17

Replace ``BrauerSevery'' by ``BrauerSeveri''.  J. Calabrese  1 
p. 612,
¶
1st column, Line 11, 9

The entries for quasifinite morphism (of schemes) should be combined.  Ulrich Görtz  1 
Remarks
Page  Description  Submitted by  Ed. 

p. 1,
¶
Line 16

The term "affine variety" is undefined at this point.  Mahdi MajidiZolbanin  1 
p. 7,
¶
Example 1.1, first paragraph

Exercise 1.8 should be replaced by, or extended by, an example where the realvalued 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 MajidiZolbanin  1 
p. 11,
¶
Line 13 (in the Proof of Theorem 1.7)

The sentence "Then A[x^(1)] is a finitely generated KAlgebra 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 MajidiZolbanin  1 
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 MajidiZolbanin  1 
p. 14,
¶
3

In order to apply Zorn's Lemma here, one should note that every nonempty topological space $X$ indeed contains some irreducible subset, e.g., any singleton.  L. Prader  1 
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 MajidiZolbanin  1 
p. 26,
¶
Prop. 1.56

Add that the inclusion $Z\to X$ is a morphism of prevarieties.  Ulrich Görtz  1 
p. 29,
¶
Proof of Corollary 1.60

For the proof of the second assertion, i.e. \mathbb{P}^n is a prevariety, one need to show the connectedness of \mathbb{P}^n. But, the book use part (2) of the Lemma 1.19 (to show irreduciblity) which itself needs the connectedness! If connectedness is trivial, the second assertion of the Corollary 1.60 is proved and there is no need to speak about Lemma 1.19 and irreduciblity; and if connectedness is not trivial, then one can not use the Lemma 1.19. Make the reference to Lemma 1.19 more precise: Use part (1) to show that $\mathbb P^n(k)$ is connected (from the construction and previous discussion it is clear that any two of the standard charts have nonempty intersection.). Then irreducibility follows from part (2) (which needs the connectedness), but that was already noted as a general fact in Proposition 1.48.  Ehsan Shahoseini  1 2 
p. 30,
¶
Line 15

After "are again in a", add reference to Exercise 1.20 (b).  Ehsan Shahoseini  1 2 
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 $V(Y^2X^3+X)\subset \mathbb A^2(k)$ is irreducible and in particular connected. Sketch the set $\{ (x, y)\in\mathbb R^2;\ y^2 = x^3x\}$ and show that it is not connected with respect to the analytic topology on $\mathbb R^2$. 
Torsten Wedhorn/Alexey Beshenov  1 
p. 50,
¶
Section 2.6 Line 3

Maybe replace ``containment'' by ``reverse containment'' or state explicitly in which direction the order goes.  1  
p. 54,
¶
Proof of Prop. 2.27

Introducing the element $t$ could be avoided.  E. Viehmann  1 
p. 58,
¶
Proof of Thm. 2.33

In the proof, you write: ``As $D(f)$ is quasicompact, 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.  1 
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  1 
p. 68,
¶

Use consistent terminology: ``principal open'' or ``principally open'' subset.  E. Viehmann  1 
p. 70,
¶
Prop. 3.8

Add a reference to Exer. 2.18.  A. Graf  1 2 
p. 70,
¶
Prop. 3.10

Add a reference to Exer. 2.16. (And/or odd a reference to Prop. 3.10 to the exercise.)  A. Graf  1 2 
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  1 
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  1 
p. 79,
¶
Proof of Prop. 3.33

Replace the beginning of the first sentence by "If $x\in X$ and if $U={\rm Spec} A$ is an affine open neighborhood of $x$, then $x$ is closed in $U$ and corresponds to ..."  A. Graf  1 2 
p. 81,
¶
Proof of Thm. 3.37

The point that $X(k)$ is connected should be addressed explicitly.  A. Graf  1 2 
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  1 
p. 90,
¶
Exercise 3.19

Replace ${\rm Hom}({\rm Spec}(R), \mathbb P^n_R)$ by ${\rm Hom}_{R}({\rm Spec}(R), \mathbb P^n_R)$ to make explicit that we only consider $R$morphisms here.  A. Graf  1 2 
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  1 
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  1 
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  1 
p. 113,
¶
Line 8

Add reference to Cor. 4.7.  A. Graf  1 2 
p. 126,
¶
Prop. 5.22 (4)

It would be enough to assume that $X$ and $Y$ are locally of finite type over $k$.  A. Graf  1 2 
p. 130,
¶
Proof of Cor. 5.33

Since the formulation "vanishes identically" has not been formally defined, it would be clearer to replace "in which case" by "i.e.,".  A. Graf  1 2 
p. 134,
¶
Line 4

It should be explained why $\bar{x}$ is closed in $X_K$. The point $\bar{x}$ is a point of the fiber $p^{1}(x) = \Spec \kappa(x)\otimes_k K$. We now use Proposition 3.33. Since the extension $\kappa(x)/k$ is finite, $\kappa(x)\otimes_k K$ is a finitedimensional $K$vector space. This implies that the residue class field of $\bar{x}$ (which is a quotient of this tensor product) is finite over $K$. Hence $\bar{x}$ is a closed point of $X_K$. (A variant of the argument: Since $\kappa(x)\otimes_k K$ is finite over $K$, it is an Artin ring, hence the fiber $p^{1}(x)$ has dimension $0$. Therefore $\bar{x}$ is a closed point of the fiber. Since $x$ is closed in $X$, $p^{1}(x)$ is closed in $X_K$. Altogether we see that $\bar{x}$ is closed in $X_K$.)  A. Graf  1 2 
p. 134,
¶
Line 15

It might be clearer to move this remark up so that it comes directly after the proof of Corollary 5.45.  A. Graf  1 2 
p. 140,
¶
Proof of Lemma 5.62

It would be helpful to add a reference to equation (4.12.4).  A. Graf  1 2 
p. 142,
¶
Exer. 5.6

The assumption that $f$ is closed is required only for part (a).  Ulrich Görtz  1 
p. 147,
¶
Example 6.3 (3)

Maybe explain that the finiteness conditions are imposed to ensure that the construction (and passing to the dual vector space, resp.) is compatible with the tensor product.  A. Graf  1 2 
p. 147,
¶
Line 13

It would be more precise to say "cardinality of any minimal generating set" (rather than "... a minimal ...").  Ulrich Görtz  1 
p. 148,
¶
Line 10

Replace "homomorphisms" by "local homomorphisms".  A. Graf  1 2 
p. 148,
¶
Line 14

Label the final arrow with $df_x$.  A. Graf  1 2 
p. 149,
¶
Line 13

Replace "projection $X(k[\varepsilon])\to X(k)$" by "map $X(k[\varepsilon])\to X(k)$ induced by the projection $k[\varepsilon]\to k$, $\varepsilon\mapsto 0$.  A. Graf  1 2 
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  1 
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  1 
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  1 
p. 170,
¶
Line 12

Somewhere (maybe after Example 7.2) add a sentence like "If $\mathscr A$ is a presheaf of rings and $\mathscr F$ is a presheaf of abelian groups with an $\mathscr A$module structure (as presheaves), then the sheafification of $\mathscr F$ is a sheaf of modules over the sheafification of $\mathscr A$, because sheafification is compatible with products.  A. Graf  1 2 
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  1 2 
p. 175,
¶
Around line 7

Add that the tensor product is associative (up to canonical isomorphism) and that one can similarly form tensor products of more than two factors (with the same result as taking successive tensor products).  A. Graf  1 2 
p. 185,
¶
Proof of corollary 7.19 (4)

Maybe replace the final sentence by the following, to make this clearer: These isomorphisms show that the presheaf $\mathscr H$ on the basis of the topology given by the $D(f)$ is actually a sheaf, and therefore yield the desired isomorphism (7.10.3).  Caiyong Qiu  1 2 
p. 188,
¶
11

Add reference to Corollary 3.22 (open subschemes of noetherian schemes are noetherian).  A. Graf  1 2 
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  1 
p. 196,
¶
Line 2

Use a different letter for the submodule $N$, since the letter $N$ is used before and afterwards for stating the universal property.  Jan Willing  1 
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 quasicoherent $\mathscr{O}_X$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  1 
p. 206,
¶
Prop. 8.4

It would be helpful to point out explicitly that for an $S$Scheme $T$ the set ${\rm Hom}_S(T, S)$ is a singleton set (and that therefore in both (1) and (2) of Prop. 8.4 the set $F(T)$ has at most one element).  A. Graf  1 2 
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  1 
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  1 
p. 214,
¶
Line 4, (8.7.1)

Maybe replace $u^*\mathrm{Grass}^e(\mathscr{E})$ by $\mathrm{Grass}^e(\mathscr{E})\times_SS'$ here to emphasize that the Yoneda lemma is used in order to obtain an isomorphism of schemes. The corresponding identification of functors is already obtained in the displayed line above.  Xiaolong Liu  1 2 
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  1 
p. 457,
¶
Proof of Thm. 14.83

It should maybe be stated that for the last assertion of the theorem (taking invariants is a quasiinverse), 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  1 
p. 461,
¶
Proof of Theorem 14.91

For $n=1$ the proof needs to be modified slightly (this case is easier, but the exception should be stated explicitly). Furthermore, in the second part of the proof it might be appropriate to give a few more details, see here. 
J.C. Syu / U. Görtz  1 2 
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  1 
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  1 
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  1 
p. 551,
¶
Example A.5.(3)

The category of sets should also be in the list.  Andreas Blatter  1 2 
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  1 
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  1 
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  1 