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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57 errata listed.

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p. 7,
The sentence before equation (17.1.9)
It is more natural to write "$a_1 \otimes a_2 \mapsto a_1a_2$" instead of "$b_1 \otimes b_2 \mapsto b_1b_2$". Javier de la Bodega
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,
Equation (17.1.10)
It should say "$a \mapsto i_2(a)-i_1(a)$". Javier de la Bodega
p. 7,
First sentence after Remark 17.5
It should say "an $R$-derivation $d\colon A \to \Omega^1_{A/R}$"; i.e. the superscript 1 is missing. Javier de la Bodega
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. 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. 42,
Line 7
Replace $h\colon T\to Y$ by $h\colon T\to X$. Yunhao Sun
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
p. 62,
Line 22
Replace "subscheme $U_1$ of $X$ such that $U$ is ..." by "subscheme $U_1$ of $X$ such that $U_1$ is ...". Xiaolong Liu
p. 65,
Exercise 18.11, line 2
Replace $f_{S'}\times X\times_SS'\to S'$ by $f_{S'}: X\times_SS'\to S'$. Xiaolong Liu
p. 66,
Exercise 18.24
Replace `Björn Poonen' by `Bjorn Poonen'. U. Görtz
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. 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. 85,
Line 14
Replace 'local intersection ring' by 'complete intersection ring'. U. Görtz
p. 118,
Remark 20.66 (2)
The argument shows only that the property "finite locally" is stable under base change, under composition, under fpqc descent, and is compatible with cofiltered limits with affine transition maps. To ensure that all these permanency properties also hold for etale covers, one also has to note that the property "etale" is also stable under base change and composition (Remark 18.35), under fpqc descent (Remark 18.46), and compatible with cofiltered limits with affine transition maps (Corollary 18.43). T. Wedhorn
p. 178,
Line 10
Change "namely $i^{-1}$ and $i^!$" into "namely $i^{-1}$ and $i_!$". Xiaolong Liu
p. 178,
Line -9
In the third full paragraph (Then we relate Cech Cohomology to ...), the third sentence should read "... only on $X$ and $\mathscr F$ but not on $\mathcal U$ *by* forming the colimit on all open coverings." where now it says "... on $\mathcal U$ *be* forming the colimit". Gabe O
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. 294,
Exercise 22.26
The hint (resp. remark) refer to (b) (resp. (a)). This should be (2) (resp. (1)). T. Wedhorn
p. 307,
Line -3 (statement of Cor. 23.18)
Replace "finite generated" by "finitely generated". Matthieu Romagny
p. 328,
Remark 23.85
The symbol $k$ is used at the same time for the base field and for the integer. Javier de la Bodega
p. 334,
Line -2
"The hypothesis in (c)" should be "The hypothesis in (d)". Yunhao Sun
p. 349,
Lines 2, 3
Replace $\kappa$ by $\kappa(s)$ (twice). U. Görtz
p. 375,
Exercise 23.45 (1)
Replace $D\otimes_SS'$ by $D \times_S S^\prime$. Haoyang Yuan
p. 408,
Proposition 24.69, line 4
Replace "$\mathrm{Pic}(\mathbb{P}(E))$" to "$\mathrm{Pic}(\mathbb{P}(\mathscr{E}))$". Xiaolong Liu
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. 522,
$\Gamma(C,\mathscr O_C)^\times$ should be $\Gamma(X,\mathscr O_X)^\times$. Christian Dahlhausen
p. 524,
Second paragraph of the proof of Proposition 26.25
It says "the classes $z^1, \dots, z^{-r}$ must be", but it should say "the classes $z^{-1}, \dots, z^{-r}$ must be". Javier de la Bodega
p. 524,
First sentence of the proof of Corollary 26.26
It says "a non-constant morphism $f: X \to \mathbb P_1(\mathbb C)$", but it should say "a non-constant morphism $f: X \to \mathbb P^1(\mathbb C)$"; i.e. the 1 of the projective line should be an upper index, and not a lower index. Javier de la Bodega
p. 540,
Line -9
Replace "This can be done" by "This can be checked". U. Görtz
p. 545,
Line -14
Change $X\to X^{(p)}$ to $X^{(p)}\to X$ (twice). U. Görtz
p. 556,
Last sentence of third paragraph, inside the proof of Theorem 26.98
It should say ${\rm Pic}^0(E \times_k T)/p^*{\rm Pic}(T) \to E(T)$; i.e. the $T$ is missing. Javier de la Bodega
p. 556,
Beginning of last paragraph of the proof of Theorem 26.98
It should say (ii) $\Rightarrow$ (i), not (i) $\Rightarrow$ (ii). Javier de la Bodega
p. 559,
Line -9
Replace $g^*\mathscr O(1)$ by $f^*\mathscr O(1)$. Yingying Wang
p. 559,
Proposition 26.105
The final two entries in the list in (ii) should be $\lambda/(\lambda -1)$ and $(\lambda -1)/\lambda$. U. Görtz
p. 560,
Lines 8, 9, 11
The map $\lambda\mapsto j(\lambda)$ is ramified over $\infty$, $0$ and $1728$, not only over $\infty$. The correct value for the $j$-invariant of the curve $y^2 = x^3+ax+b$ is $2^6 3^3 \frac{4a^3}{4a^3+27b^2}$. U. Görtz
p. 596,
Line 17
Replace `Theorem 25.32' by `Section (25.32)'. U. Görtz
p. 614,
The paragraph before 27.28
Replace "For a functor $F$ on $({\rm Sch}/S)$" by "For a functor $F$ on $({\rm Sch}/S)^{\rm opp}$".
p. 615,
Defn. 27.31
Replace "(AbGrp)" by "(Grp)".
p. 617,
Proof of Lemma 27.36
The phrase "Since the diagonal of $Y$ is representable, so is ${\rm Eq}(h_1,h_2) \to U$ by (9.1.4)." does not make sense. The instance of diagram (9.1.4) one wants to use here has the diagonal of $X \to Y$, i.e., $X\to X\times_YX$, in its right column. This diagonal is representable, see [Stacks] 05L9. Christian Dahlhausen
p. 618,
Remark 27.42
"exist" should be "exists" and "if and only if and for every" without "and". Christian Dahlhausen
p. 618,
Definition 27.38 (2)
It should be "... if and only if $S_i$ has the property $\mathbf P$ for all $i \in I$". Christian Dahlhausen
p. 618,
Definition 27.39
I suggest to add a remark on the compatibility with Definition 8.6 which is ensured by the following statement: [Stacks] 03MJ. Christian Dahlhausen
p. 620,
Defn. 27.46
Add a period at the end of this definition.
p. 621,
Lemma 27.53
What are $X$ and $Y$? According to the proof (use of Lemma 27.52) one might think that they are schemes, but in the proof of Lemma 27.61 the statement is used for algebraic spaces. I suggest to fix the proof of Lemma 27.50 (2) (which I pointed to in another comment) and then state and prove Lemma 27.52 for algebraic spaces. See also [Stacks] 03MJ. Christian Dahlhausen
p. 638,
line 1
Replace ((2)) by (2). T. Wedhorn
p. 649,
Line 2
Delete one of the phrases "for group schemes". T. Wedhorn
p. 747,
Line 14, 15
Replace $u$ by $f$ and $\bar{u}$ by $\bar{f}$. F. Leptien
p. 770,
Line 4
Replace the first 'is' by 'if'.
p. 786,
Line 3
"to to" should be replaced by "to". Torsten Wedhorn