## Errata and Addenda for Algebraic Geometry I

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

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 p. 102, ¶ Proof of Prop. 4.20, Case (II) To see that $(p^{-1}(f(X')), \mathscr O_{Z|p^{-1}(f(X'))})$ is a fiber product in the category of schemes, one needs to know that the fiber product of schemes taken in the category of locally ringed spaces exists and is a scheme. This stronger version of Thm. 4.18 should have been stated and proved before. It is proved similarly as Thm. 4.18; for the affine case use Proposition 3.4 in the form given (but not proved) in the book. (O. Körner) p. 268, ¶ Theorem 10.76 It is not true in general that $B$ is the inductive limit of its smooth $A$-sub-algebras. All one can say in general is that $B$ is isomorphic to a filtered inductive limit of smooth $A$-algebras. See the discussion in Spivakovsky's paper [Sp] (Problem 1.3; Section 10). (M. Hoyois, S. Kelly) p. 300, ¶ Line 3 The mapping $U\mapsto {\rm Frac}(\Gamma(U,\mathscr O_X))$ is not a presheaf in general (rather, one should consider the localization with respect to elements which are non-zero divisors in every stalk). See Kleiman, Misconceptions about $K_X$. Enseign. Math. (2) 25 (1979), no. 3-4, 203--206 (1980), for a detailed discussion. (P. Hartwig) p. 308, ¶ Prop. 11.40 The proposition is not true as it stands (a counterexample is given by $X$ the plane with a fattened origin, and $Z$ the origin). It is correct with the additional assumption that $U$ be schematically dense in $X$ (use the characterization in Lemma 9.23 (ii) to conclude that the exactness at $Z^1(X)$ of the corresponding sequence of groups of cycles yields the exactness at ${\rm Cl}(X)$ of the sequence in the proposition). (B. Smithling) p. 328, ¶ Prop. 12.18 The first formula is false (almost always, e.g. if $Y=\mathop{\rm Spec} \mathbb F_p$, $X=\mathop{\rm Spec} \mathbb F_{p^2}$, $X'=\mathop{\rm Spec} \mathbb F_{p^2}$). (Yehao Zhou) p. 357, ¶ Proof of Thm. 12.73 In general, it is not true that $f'$ as defined in the book is of finite type; hence Cor. 12.78 cannot be applied directly. Instead, one can proceed as follows: Start the proof as in the book: the restriction of $c$ to ${\rm Isol}(c)$ is an open immersion, hence $h$ is an open immersion. Now approximate $X'$ by $X'_\mu$ which are affine of finite type over $Y$ (analogously to approximating $Z$ by $Z_\lambda$). Using Lemma 12.84, we may assume that $V'\rightarrow X'_\mu$ is an open immersion (for some fixed large $\mu$). Now replace $X'$ by $X'_\mu$ and continue as in the book: Define $Z$, apply 12.78 (which is now justified), observe that $V' \subseteq {\rm Isol}(X'/Z)$, and apply Lemma 12.84 again to approximations of $Z$. (Yehao Zhou) p. 390, ¶ Line -16 Omit the statement in parentheses. It is not true that the existence of an ample line bundle implies properness. (P. Hartwig) p. 428, ¶ Proposition 14.20, Lemma 14.21 The lemma is incorrect as stated, because the reduction to the local case does not work as claimed. It should be replaced by the following: (1) Let $A$ be a noetherian ring, let $B$ be a noetherian $A$-algebra, and let $M$ be a $B$-module of finite type which is flat over $A$. Let $f\in B$ such that for every maximal ideal $\mathfrak M\subset B$, multiplication by $f$ is an injection $M/(\mathfrak M\cap A) \rightarrow M/(\mathfrak M\cap A)$. Then $M/fM$ is flat over $A$. (2) With the techniques of Chapter 10 the hypothesis noetherian'' can be replaced by suitable conditions like finite presentation''. See also MathOverflow and Matsumura, Commutative Ring Theory, Thm. 22.6. To avoid the additional difficulty for non-noetherian rings, in the statement of the proposition the assumption that $S$ and $X$ be noetherian should be added. Assuming that $S=\mathop{\rm Spec} A$, $X = \mathop{\rm Spec B}$, the injectivity assertion means that for all maximal ideals $\mathfrak m\subset A$, $f\not\in \mathfrak mB$; it follows that $f\not\in (\mathfrak M \cap A)B$ for all maximal ideals $\mathfrak M\subset B$, so that the lemma (statement (1) of the previous paragraph) can be applied. In the proof of Theorem 14.22 (in the non-noetherian case) a version of the Lemma as alluded to in (2) above is required. (U. Görtz)