Errata and Addenda for Algebraic Geometry II

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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21 errata listed.

Minor Errors

PageDescriptionSubmitted byEd.
p. 35,
Proposition 18.11
In general, no such global lift $b$ exists. Consider e.g. the case where $f$ is formally smooth but not formally étale: For two distinct global lifts $b_{1}$ and $b_{2}$ of $a_{0}$ take the open cover $U_{1} = U_{2} = T$.
Instead, the conclusion that is actually proved (and used in the sequel) is that under the given assumptions there exists a lift $b\colon T\to X$ (i.e., $b$ makes the diagram (18.0.1) commutative). As explained in the preceding discussion, this means that one can change each $b_i$ by a derivation in $\mathscr G(U_i)$ (with notation as in Lemma 18.9) to obtain a family $b_i'$ of lifts $U_i\to X$ that can be glued.
M. Herbers, J. K. Hessel
p. 50,
Prop. 18.55
The statement of the proposition is correct, but the proof is incomplete. In fact, the proof of (i) $\Rightarrow$ (ii) only explains the conclusion for $I$ and $B$ as in the definition of a smooth morphism (cf. the beginning of Section (18.10)).
To prove the statement in general, one should use that the morphism ${\rm Spec}(A)\to {\rm Spec}(R)$, being smooth at $\mathfrak p$, is formally smooth in a neighborhood of $\mathfrak p$ (Theorem 18.56). Then one can invoke Proposition 18.20.
U. Görtz

Typographical and Trivial Errors

PageDescriptionSubmitted byEd.
p. 7,
After Equation (17.1.9)
In the text, it says "Then $\Omega_{R/A}^1$ is the kernel of the $R$-algebra...", but it should be $\Omega_{A/R}^1$ instead. Cynthia
p. 7,
Line 2
Insert `homomorphism' after `$R$-algebra'. U. Görtz
p. 23,
Line 4, line 23
Add periods at the end of sentences. Erik Nikolov
p. 30,
Exercise 17.9
The curve should be defined by the homogeneous equation $Y^2Z-X^3-aXZ^2-bZ^3$. The definition of $\omega$ should read $\omega = d(x)/y$ where $x = \frac XZ, y=\frac YZ\in K(E)$. U. Görtz
p. 66,
Exercise 18.24
Replace `Björn Poonen' by `Bjorn Poonen'. U. Görtz
p. 68,
line 7
B.58 in " a regular sequence (Definition B.58)" should be B.60, and the reference should be made clickable. Jinyi Xu
p. 68,
line 4
B.61 in "By Krull’s principal ideal theorem (Corollary B.61)" should B.64 and the reference should be made clickable. Jinyi Xu
p. 85,
Line 14
Replace 'local intersection ring' by 'complete intersection ring'. U. Görtz
p. 196,
Line 13
Replace ${\rm Hom}$ by $\mathscr Hom$. U. Görtz
p. 242,
Theorem 22.22
In (3) it must say the $R$-submodule instead of $A$-submodule. T. Wedhorn
p. 278,
Just before proof of Corollary 22.92
Replace the fact that being “affine” is a stable under fpqc-descent by the fact that being “affine” is a property stable under fpqc-descent Matthieu Romagny
p. 307,
Line -3 (statement of Cor. 23.18)
Replace "finite generated" by "finitely generated". Matthieu Romagny
p. 476,
Line 5
Insert `morphism' after `separated' and delete the comma. U. Görtz
p. 493,
Line -3
Replace `($S_2$)-module' by `($S_2$)'. U. Görtz
p. 505,
Line 9
Replace `12.3 3' by `12.3 (3)'. U. Görtz
p. 505,
Line 4 of proof of Thm. 25.151
Replace `Theorem 25.141' by `Corollary 25.141'. U. Görtz
p. 596,
Line 17
Replace `Theorem 25.32' by `Section (25.32)'. U. Görtz
p. 747,
Line 14, 15
Replace $u$ by $f$ and $\bar{u}$ by $\bar{f}$. F. Leptien
p. 786,
Line 3
"to to" should be replaced by "to". Torsten Wedhorn