Errata and Addenda for Algebraic Geometry I (Edition 2) Show errata for edition 1
Here we post a list of errata and addenda. The name tags refer to the people who found the mistake. We are very grateful to all of them. Further remarks and hints  trivial or not  are very welcome.
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45 errata listed.
Major Errors
Page  Description  Submitted by  Ed. 

p. 457,
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Prop. 14.67

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. 13,
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Line 16

Replace "subset" by "open subset".  A. Graf  1 2 
p. 86,
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Line 22

Kernels of sheaves have not been defined yet at this point.  A. Graf  2 
p. 93,
¶
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. 526,
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Line 5

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 
Typographical and Trivial Errors
Page  Description  Submitted by  Ed. 

p. 8,
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16

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

Replace $\mathscr M$ by $\mathcal M$.  I. Tselepidis  1 2 
p. 33,
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Line 13, 12

Replace $\mathbb{P}_k^n$ by $\mathbb{P}^n(k)$.  E. Hong  1 2 
p. 47,
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Line 10

Replace "$R$module" by "an $R$module".  A. Graf  1 2 
p. 62,
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Line 20

Replace $i^\flat_x$ by $i^\sharp_x$.  A. Graf  1 2 
p. 66,
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Line 13

Replace "subset" by "open subset".  A. Graf  1 2 
p. 72,
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Line 16

Replace "with an open subscheme of $T_i$" by "with an open subscheme $T_i$ of $T$".  E. Hong  1 2 
p. 75,
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Line 1

Replace the reference to Section (4.14) by a reference to Section (4.13).  N.T.  1 2 
p. 84,
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Line 21

Add index $i$ to the second intersection.  A. Graf  1 2 
p. 86,
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Definition 3.41 (1), second line near the end

Replace $\mathcal{O}$ by $\mathcal{O}_X$.  Andreas Blatter  2 
p. 92,
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Line 23

Replace "component" by "components".  A. Graf  1 2 
p. 107,
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Remark/Def. 4.24

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. 116,
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Line 15

Replace "as $S$scheme" by "as an $S$scheme".  A. Graf  1 2 
p. 131,
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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. 136,
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Line 13

Better: "... where $S$ is the spectrum of a field $k$ and $X$ is of finite type over $k$."  A. Graf  1 2 
p. 153,
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Lines 2, 1

Replace the reference to Section (4.14) by a reference to Section (4.13).  Dominik Briganti  1 2 
p. 155,
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15

Replace "$k$scheme" by "$K$scheme".  Dominik Briganti  1 2 
p. 171,
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The second line of exercise 6.26

Replace ${\rm Spec} R[X](f)$ by ${\rm Spec} R[X]/(f)$. Also, there is a conflict of notation ($X$ is used for the scheme and for the variable).  Han Hu  2 
p. 177,
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Line 1

Add period at the end of the sentence.  A. Graf  1 2 
p. 185,
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Line 12

Replace "localization in the prime ideal" by "localization at the prime ideal".  A. Graf  1 2 
p. 185,
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Line 15

Replace "with respect" by "with respect to".  U. Görtz  1 2 
p. 195,
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Line 3

Replace "as" by "as an".  A. Graf  1 2 
p. 202,
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Exer. 7.4

Add a comma before $s_r(x)$ in both (a) and (b).  A. Graf  1 2 
p. 204,
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Exer. 7.10 (a)

Replace the first occurrence of "polynomials" by "polynomial".  A. Graf  1 2 
p. 223,
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Line 6

Replace $M_{1 \times n}$ by $M_{1\times n}(R)$.  SzSheng Wang  1 2 
p. 456,
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Line 10

Replace $\varphi$ by $\varphi'$ (twice).  Peng DU  1 2 
Remarks
Page  Description  Submitted by  Ed. 

p. 71,
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Prop. 3.8

Add a reference to Exer. 2.18.  A. Graf  1 2 
p. 72,
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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. 81,
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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. 83,
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Proof of Thm. 3.37

The point that $X(k)$ is connected should be addressed explicitly.  A. Graf  1 2 
p. 92,
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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. 116,
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Line 19

Add reference to Cor. 4.7.  A. Graf  1 2 
p. 129,
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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. 133,
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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. 137,
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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. 137,
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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. 143,
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Proof of Lemma 5.62

It would be helpful to add a reference to equation (4.12.4).  A. Graf  1 2 
p. 189,
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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. 209,
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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. 551,
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Example A.6.(3)

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