## Errata and Addenda for Algebraic Geometry I (Edition 2)

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.

You can submit errata through the web site (or just send us an email).

49 errata listed.

## Major Errors

PageDescriptionSubmitted byEd.
p. 457,
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 quasi-separated, then it is not possible to find a quasi-compact 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

PageDescriptionSubmitted byEd.
p. 13,
Line 16
Replace "subset" by "open subset". A. Graf 1 2
p. 86,
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 Lima-Filho 1 2
p. 526,
Line -5
In the expression for $g$, in the third parenthesis there should be $(X_2-1)^2$ instead of $(X_2^2-1)^2$. Alejandro Vargas and Tim Seynnaeve 1 2

## Typographical and Trivial Errors

PageDescriptionSubmitted byEd.
p. 8,
16
Replace "field" by "fields". A. Graf 1 2
p. 15,
Line -15
Replace $\mathscr M$ by $\mathcal M$. I. Tselepidis 1 2
p. 33,
Line -13, -12
Replace $\mathbb{P}_k^n$ by $\mathbb{P}^n(k)$. E. Hong 1 2
p. 47,
Line -10
Replace "$R$-module" by "an $R$-module". A. Graf 1 2
p. 62,
Line -20
Replace $i^\flat_x$ by $i^\sharp_x$. A. Graf 1 2
p. 66,
Line -13
Replace "subset" by "open subset". A. Graf 1 2
p. 72,
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,
Line -1
Replace the reference to Section (4.14) by a reference to Section (4.13). N.T. 1 2
p. 84,
Line -21
Add index $i$ to the second intersection. A. Graf 1 2
p. 86,
Definition 3.41 (1), second line near the end
Replace $\mathcal{O}$ by $\mathcal{O}_X$. Andreas Blatter 2
p. 92,
Line 23
Replace "component" by "components". A. Graf 1 2
p. 107,
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,
Line -15
Replace "as $S$-scheme" by "as an $S$-scheme". A. Graf 1 2
p. 131,
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,
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. 142,
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. 143,
Line -13
Replace $\bigoplus$ by $\bigoplus_d$. A. Graf 1 2
p. 153,
Lines -2, -1
Replace the reference to Section (4.14) by a reference to Section (4.13). Dominik Briganti 1 2
p. 155,
15
Replace "$k$-scheme" by "$K$-scheme". Dominik Briganti 1 2
p. 171,
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. 174,
Line 17
Replace "of an $\mathscr O_X$-module" by "of an $\mathscr O_X$-module $\mathscr F$". A. Graf 1 2
p. 177,
Line -1
Add period at the end of the sentence. A. Graf 1 2
p. 185,
Line 15
Replace "with respect" by "with respect to". U. Görtz 1 2
p. 185,
Line -12
Replace "localization in the prime ideal" by "localization at the prime ideal". A. Graf 1 2
p. 195,
Line -3
Replace "as" by "as an". A. Graf 1 2
p. 202,
Exer. 7.4
Add a comma before $s_r(x)$ in both (a) and (b). A. Graf 1 2
p. 204,
Exer. 7.10 (a)
Replace the first occurrence of "polynomials" by "polynomial". A. Graf 1 2
p. 223,
Line -6
Replace $M_{1 \times n}$ by $M_{1\times n}(R)$. Sz-Sheng Wang 1 2
p. 456,
Line -10
Replace $\varphi$ by $\varphi'$ (twice). Peng DU 1 2

## Remarks

PageDescriptionSubmitted byEd.
p. 71,
Prop. 3.8
Add a reference to Exer. 2.18. A. Graf 1 2
p. 72,
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,
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,
Proof of Thm. 3.37
The point that $X(k)$ is connected should be addressed explicitly. A. Graf 1 2
p. 92,
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,
Line -19
Add reference to Cor. 4.7. A. Graf 1 2
p. 129,
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,
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,
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. 137,
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 finite-dimensional $K$-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. 143,
Proof of Lemma 5.62
It would be helpful to add a reference to equation (4.12.4). A. Graf 1 2
p. 178,
Around line -9
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. 189,
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,
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,
Example A.6.(3)
The category of sets should also be in the list. Andreas Blatter 1 2