## 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.

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

102 errata listed.

## Minor Errors

PageDescriptionSubmitted byEd.
p. 29,
Exercise 17.1. (2)
The condition should be $[D,D']\in \mathfrak{g}$ for all $D,D'\in X$. Jan Willing
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
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. 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. 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. 756,
Line 11
At this point $\mathcal A$ is only assumed to be additive, hence we can only conclude that the categories of complexes are additive (rather than abelian as stated in the text). Later on, $\mathcal A$ is assumed to be abelian, and it should be added that then these categories obtained from $\mathcal A$ are abelian, too. Jhan-Cyuan Syu

## Typographical and Trivial Errors

PageDescriptionSubmitted byEd.
p. 7,
Line 14
Insert "$M$" after "$A$-modules". Jan Willing
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,
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,
Equation (17.1.10)
It should say "$a \mapsto i_2(a)-i_1(a)$". Javier de la Bodega
p. 7,
Line 2
Insert homomorphism' after $R$-algebra'. U. Görtz
p. 8,
Prop. 17.7
It should be "$\psi_D\colon A \to D_A(M)$" rather than "$\psi_D\colon A \to D_A(N)$". Mathieu Wydra / L. Potter / Jan Willing
p. 8,
line 13
Missing subscript on $u$. Should read $u_D(1 \otimes a - a \otimes 1)$. L. Potter / Jan Willing
p. 11,
Line 9
Replace "$i^1$" by "$i_1$". L. Potter
p. 16,
Line 9
Close bracket of \Hom in the line with "(11.3.3)" over the equal sign. Jan Willing
p. 16,
Line 8
$f$ is overloaded here, as it already was defined to be the structure morphism $f: X \to S$. Better to use some other letter like $g$. L. Potter
p. 18,
Proof of Prop. 17.33, line -3
Should be "is" instead of "ist". L. Potter
p. 22,
Equation (17.7.2)
Replace $h^*(\mathscr{E})$ with $f^*(\mathscr{E})$, since $\mathscr{E}$ lives on $S$, and not $X$. L. Potter
p. 22,
Line 4
It should read "morphism of S-schemes". Jan Willing
p. 23,
Line 4, line 23
Add periods at the end of sentences. Erik Nikolov
p. 24,
Line 12
p. 28,
Line 16
Replace $\mathscr E$ by $\mathscr E''$ in the top entry of the commutative triangle. L. Potter
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. 31,
Line -4
The scheme $T$ should be an $S$-scheme (so the scheme $S$ should be fixed in the beginning). Jhan-Cyuan Syu
p. 42,
Line 7
Replace $h\colon T\to Y$ by $h\colon T\to X$. Yunhao Sun
p. 60,
Line -14
It should be "is" rather than "ist". L. Potter
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. 64,
Line 1
"Show that $K$ if" should be "Show that $K$ is". L. Potter
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. 154,
Equation (21.1.3)
Instead of $j_!j^*\mathscr{F}$ it should be $j_!j^{-1}\mathscr{F}$. Elías Guisado
p. 154,
Remark 21.3
In the displayed short exact sequence, the right term should be $i_*i^{-1}\mathscr{F}$ instead of $i_*i^*\mathscr{F}$. Elías Guisado
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. 178,
Line 10
Change "namely $i^{-1}$ and $i^!$" into "namely $i^{-1}$ and $i_!$". Xiaolong Liu
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 4 of proof of Thm. 25.151
Replace Theorem 25.141' by Corollary 25.141'. U. Görtz
p. 505,
Line 9
Replace 12.3 3' by 12.3 (3)'. U. Görtz
p. 522,
(26.6.2)
$\Gamma(C,\mathscr O_C)^\times$ 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. 528,
Line 22
$\omega_C \simeq \Omega_{C/k}^1$ appears twice.
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. 592,
3rd line after first equation
Instead of $q^\ell(L)$ it should be $q^{\ell(L)}$. Christian Dahlhausen
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. 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. 620,
Defn. 27.46
Add a period at the end of this definition.
p. 637,
Start of proof of 27.95
Should be $s' \in S'$ with $S'$ instead of $S$. Bianca Fürstenau
p. 638,
line 1
Replace ((2)) by (2). T. Wedhorn
p. 640,
Between Corollaries 27.103 and 27.104
In the last line of the paragraph, “denote” should be “denoted”. Bianca Fürstenau
p. 649,
Line 2
Delete one of the phrases "for group schemes". T. Wedhorn
p. 662,
Line 9
In the codomain of $s_1$, the index $S$ in the fibre square of $X$ is missing. Bianca Fürstenau
p. 739,
Line -4
The final ${\rm colim}$ should be replaced by $\lim_{\rightarrow}$. Jhan-Cyuan Syu
p. 740,
Line 6
The functor $\underline{\lim}_{\mathcal I}$ is from $\mathcal C$ to (Sets), rather than from $\mathcal I$ to $\mathcal C$. Jhan-Cyuan Syu
p. 744,
Definition F.21
Replace $\mathcal{F}$ by $\mathcal{C}$. Jenna Nieminen
p. 747,
Line 14, 15
Replace $u$ by $f$ and $\bar{u}$ by $\bar{f}$. F. Leptien
p. 755,
Lines 9, 10
Replace ${\rm C}(\mathcal A)$ by $C(\mathcal A)$. Jhan-Cyuan Syu
p. 756,
Lines 7, 10, 11, 15, 17
Replace ${\rm C}(\mathcal A)$ by $C(\mathcal A)$. Jhan-Cyuan Syu
p. 757,
Definition F.75, end of second line
In the definition, $h^i$ should be a morphism of objects of $\mathcal A$ (not a morphism of complexes as written). Bianca Fürstenau
p. 757,
Line -7
It should read $(C(F))(C_u)$ rather than $F_{C_u}$ (or maybe it would be better to say that we denote $C(F)$ by $F$ again, and then write $F(C_u)$). Jhan-Cyuan Syu
p. 760,
Lines 2, 7
Replace ${\rm C}(\mathcal A)$ by $C(\mathcal A)$. Jhan-Cyuan Syu
p. 767,
Line 13
Replace $E^{p-t, t}_\infty$ by $E^{p+t, -t}_\infty$. Jhan-Cyuan Syu
p. 767,
Line -12
Replace "converging against" by "converging to". Jhan-Cyuan Syu
p. 770,
Line 4
Replace the first 'is' by 'if'.
p. 772,
Line 13
Insert "introduce" before "these notions". Jhan-Cyuan Syu
p. 773,
Diagram in Def. F.116 (2)
Replace the $X[1]$ in the lower right corner by $X'[1]$. Jhan-Cyuan Syu
p. 775,
Remark F.122 (1)
In line 2, replace "exist" by "exists". Delete the closing parenthesis right before the final period. Jhan-Cyuan Syu
p. 777,
Example F.127
Add a period at the end of the displayed line. Jhan-Cyuan Syu
p. 779,
Line 3
Delete opening parenthesis in front of Hom. Jhan-Cyuan Syu
p. 786,
Line 3
"to to" should be replaced by "to". Torsten Wedhorn
p. 795,
Line -2
In the diagram, the label of the left vertical arrow should be $Q_{\mathcal J}$ rather than $Q_{\mathcal I}$. Jhan-Cyuan Syu

## Remarks

PageDescriptionSubmitted byEd.
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,
Line 3
According to the List of Symbols (p. 857) $D_C(N)$ is an called "augmented extension by square zero ideal". This definition should be added here. Jan Willing
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. 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. 666,
Proof of Proposition 27.167
In the first paragraph, replace $\{1, \dots, 3\}$ with the more readable $\{1, 2, 3\}$. Bianca Fürstenau
p. 755,
Line 20
Maybe better to say: A complex is acyclic if it is acyclic at every $i$. (And it should maybe be added that one often speaks of exact complexes (at some index $i$) instead. Jhan-Cyuan Syu
p. 756,
Line 18
Maybe $\pi^n$ rather than $\pi_n$ is more consistent with the other notation. (However, maybe this map does not need a symbol of its own anyway?) Jhan-Cyuan Syu
p. 763,
Line -13
Since $Z^n(E)$ and $B^n(E)$ are assumed to be flat from the beginning, this does not have to be repeated here. Jhan-Cyuan Syu
p. 796,
Proposition F.170
Some localization functors are silently omitted here. Maybe they should be written out explicitly (in Part (1), $\rho(Q_{\mathcal K}(X)) = Q_{\mathcal J}(A_X)$; in Part (2), $RF^{\mathcal J}(Q_{\mathcal J}(A_X)) = RF(Q_{\mathcal K}(X))$), or it should be stated more clearly that they are omitted. Jhan-Cyuan Syu