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).
Explanation: Major error Minor error Typo/Trivial Remark Unclassified
102 errata listed.
| Page | Description | Submitted by | |
|---|---|---|---|
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p. 7,
¶
Line 2
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Insert `homomorphism' after `$R$-algebra'. | U. Görtz | |
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p. 7,
¶
After Equation (17.1.9)
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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 | |
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p. 7,
¶
Equation (17.1.10)
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It should say "$a \mapsto i_2(a)-i_1(a)$". | Javier de la Bodega | |
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p. 7,
¶
Line 14
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Insert "$M$" after "$A$-modules". | Jan Willing | |
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p. 7,
¶
Line 3
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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 | |
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p. 7,
¶
First sentence after Remark 17.5
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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 | |
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p. 7,
¶
The sentence before equation (17.1.9)
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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 | |
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p. 8,
¶
line 13
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Missing subscript on $u$. Should read $u_D(1 \otimes a - a \otimes 1)$. | L. Potter / Jan Willing | |
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p. 8,
¶
Prop. 17.7
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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 | |
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p. 11,
¶
Line 9
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Replace "$i^1$" by "$i_1$". | L. Potter | |
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p. 16,
¶
Line 9
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Close bracket of \Hom in the line with "(11.3.3)" over the equal sign. | Jan Willing | |
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p. 16,
¶
Line 8
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$f$ 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 | |
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p. 18,
¶
Proof of Prop. 17.33, line -3
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Should be "is" instead of "ist". | L. Potter | |
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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 | |
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p. 22,
¶
Line 4
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It should read "morphism of S-schemes". | Jan Willing | |
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p. 23,
¶
Line 4, line 23
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Add periods at the end of sentences. | Erik Nikolov | |
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p. 24,
¶
Line 12
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Remove the second "graded" in "graded commutative graded algebra". | Jan Willing | |
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p. 28,
¶
Line 16
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Replace $\mathscr E$ by $\mathscr E''$ in the top entry of the commutative triangle. | L. Potter | |
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p. 29,
¶
Exercise 17.1. (2)
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The condition should be $[D,D']\in \mathfrak{g}$ for all $D,D'\in X$. | Jan Willing | |
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p. 30,
¶
Exercise 17.9
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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 | |
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p. 31,
¶
Line -4
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The scheme $T$ should be an $S$-scheme (so the scheme $S$ should be fixed in the beginning). | Jhan-Cyuan Syu | |
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p. 35,
¶
Proposition 18.11
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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 | |
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p. 42,
¶
Line 7
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Replace $h\colon T\to Y$ by $h\colon T\to X$. | Yunhao Sun | |
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p. 50,
¶
Prop. 18.55
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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 | |
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p. 60,
¶
Line -14
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It should be "is" rather than "ist". | L. Potter | |
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p. 62,
¶
Line 22
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Replace "subscheme $U_1$ of $X$ such that $U$ is ..." by "subscheme $U_1$ of $X$ such that $U_1$ is ...". | Xiaolong Liu | |
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p. 64,
¶
Line 1
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"Show that $K$ if" should be "Show that $K$ is". | L. Potter | |
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p. 65,
¶
Exercise 18.11, line 2
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Replace $f_{S'}\times X\times_SS'\to S'$ by $f_{S'}: X\times_SS'\to S'$. | Xiaolong Liu | |
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p. 66,
¶
Exercise 18.24
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Replace `Björn Poonen' by `Bjorn Poonen'. | U. Görtz | |
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p. 68,
¶
line 7
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B.58 in " a regular sequence (Definition B.58)" should be B.60, and the reference should be made clickable. | Jinyi Xu | |
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p. 68,
¶
line 4
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B.61 in "By Krull’s principal ideal theorem (Corollary B.61)" should B.64 and the reference should be made clickable. | Jinyi Xu | |
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p. 85,
¶
Line 14
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Replace 'local intersection ring' by 'complete intersection ring'. | U. Görtz | |
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p. 118,
¶
Remark 20.66 (2)
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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 | |
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p. 154,
¶
Remark 21.3
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In the displayed short exact sequence, the right term should be $i_*i^{-1}\mathscr{F}$ instead of $i_*i^*\mathscr{F}$. | Elías Guisado | |
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p. 154,
¶
Equation (21.1.3)
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Instead of $j_!j^*\mathscr{F}$ it should be $j_!j^{-1}\mathscr{F}$. | Elías Guisado | |
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p. 178,
¶
Line 10
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Change "namely $i^{-1}$ and $i^!$" into "namely $i^{-1}$ and $i_!$". | Xiaolong Liu | |
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p. 178,
¶
Line -9
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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 | |
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p. 196,
¶
Line 13
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Replace ${\rm Hom}$ by $\mathscr Hom$. | U. Görtz | |
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p. 242,
¶
Theorem 22.22
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In (3) it must say the $R$-submodule instead of $A$-submodule. | T. Wedhorn | |
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p. 278,
¶
Just before proof of Corollary 22.92
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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 | |
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p. 294,
¶
Exercise 22.26
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The hint (resp. remark) refer to (b) (resp. (a)). This should be (2) (resp. (1)). | T. Wedhorn | |
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p. 307,
¶
Line -3 (statement of Cor. 23.18)
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Replace "finite generated" by "finitely generated". | Matthieu Romagny | |
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p. 328,
¶
Remark 23.85
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The symbol $k$ is used at the same time for the base field and for the integer. | Javier de la Bodega | |
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p. 334,
¶
Line -2
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"The hypothesis in (c)" should be "The hypothesis in (d)". | Yunhao Sun | |
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p. 349,
¶
Lines 2, 3
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Replace $\kappa$ by $\kappa(s)$ (twice). | U. Görtz | |
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p. 375,
¶
Exercise 23.45 (1)
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Replace $D\otimes_SS'$ by $D \times_S S^\prime$. | Haoyang Yuan | |
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p. 408,
¶
Proposition 24.69, line 4
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Replace "$\mathrm{Pic}(\mathbb{P}(E))$" to "$\mathrm{Pic}(\mathbb{P}(\mathscr{E}))$". | Xiaolong Liu | |
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p. 476,
¶
Line 5
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Insert `morphism' after `separated' and delete the comma. | U. Görtz | |
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p. 493,
¶
Line -3
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Replace `($S_2$)-module' by `($S_2$)'. | U. Görtz | |
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p. 505,
¶
Line 4 of proof of Thm. 25.151
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Replace `Theorem 25.141' by `Corollary 25.141'. | U. Görtz | |
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p. 505,
¶
Line 9
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Replace `12.3 3' by `12.3 (3)'. | U. Görtz | |
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p. 522,
¶
(26.6.2)
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$\Gamma(C,\mathscr O_C)^\times$ should be $\Gamma(X,\mathscr O_X)^\times$. | Christian Dahlhausen | |
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p. 524,
¶
Second paragraph of the proof of Proposition 26.25
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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 | |
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p. 524,
¶
First sentence of the proof of Corollary 26.26
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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 | |
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p. 528,
¶
Line 22
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$\omega_C \simeq \Omega_{C/k}^1$ appears twice. | ||
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p. 540,
¶
Line -9
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Replace "This can be done" by "This can be checked". | U. Görtz | |
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p. 545,
¶
Line -14
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Change $X\to X^{(p)}$ to $X^{(p)}\to X$ (twice). | U. Görtz | |
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p. 556,
¶
Last sentence of third paragraph, inside the proof of Theorem 26.98
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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 | |
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p. 556,
¶
Beginning of last paragraph of the proof of Theorem 26.98
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It should say (ii) $\Rightarrow$ (i), not (i) $\Rightarrow$ (ii). | Javier de la Bodega | |
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p. 559,
¶
Line -9
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Replace $g^*\mathscr O(1)$ by $f^*\mathscr O(1)$. | Yingying Wang | |
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p. 559,
¶
Proposition 26.105
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The final two entries in the list in (ii) should be $\lambda/(\lambda -1)$ and $(\lambda -1)/\lambda$. | U. Görtz | |
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p. 560,
¶
Lines 8, 9, 11
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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 | |
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p. 592,
¶
3rd line after first equation
|
Instead of $q^\ell(L)$ it should be $q^{\ell(L)}$. | Christian Dahlhausen | |
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p. 596,
¶
Line 17
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Replace `Theorem 25.32' by `Section (25.32)'. | U. Görtz | |
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p. 614,
¶
The paragraph before 27.28
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Replace "For a functor $F$ on $({\rm Sch}/S)$" by "For a functor $F$ on $({\rm Sch}/S)^{\rm opp}$". | ||
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p. 615,
¶
Defn. 27.31
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Replace "(AbGrp)" by "(Grp)". | ||
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p. 617,
¶
Proof of Lemma 27.36
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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 | |
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p. 618,
¶
Definition 27.39
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I suggest to add a remark on the compatibility with Definition 8.6 which is ensured by the following statement: [Stacks] 03MJ. | Christian Dahlhausen | |
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p. 618,
¶
Remark 27.42
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"exist" should be "exists" and "if and only if and for every" without "and". | Christian Dahlhausen | |
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p. 618,
¶
Definition 27.38 (2)
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It should be "... if and only if $S_i$ has the property $\mathbf P$ for all $i \in I$". | Christian Dahlhausen | |
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p. 620,
¶
Defn. 27.46
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Add a period at the end of this definition. | ||
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p. 621,
¶
Lemma 27.53
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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 | |
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p. 637,
¶
Start of proof of 27.95
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Should be $s' \in S'$ with $S'$ instead of $S$. | Bianca Fürstenau | |
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p. 638,
¶
line 1
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Replace ((2)) by (2). | T. Wedhorn | |
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p. 640,
¶
Between Corollaries 27.103 and 27.104
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In the last line of the paragraph, “denote” should be “denoted”. | Bianca Fürstenau | |
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p. 649,
¶
Line 2
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Delete one of the phrases "for group schemes". | T. Wedhorn | |
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p. 662,
¶
Line 9
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In the codomain of $s_1$, the index $S$ in the fibre square of $X$ is missing. | Bianca Fürstenau | |
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p. 666,
¶
Proof of Proposition 27.167
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In the first paragraph, replace $\{1, \dots, 3\}$ with the more readable $\{1, 2, 3\}$. | Bianca Fürstenau | |
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p. 739,
¶
Line -4
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The final ${\rm colim}$ should be replaced by $\lim_{\rightarrow}$. | Jhan-Cyuan Syu | |
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p. 740,
¶
Line 6
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The functor $\underline{\lim}_{\mathcal I}$ is from $\mathcal C$ to (Sets), rather than from $\mathcal I$ to $\mathcal C$. | Jhan-Cyuan Syu | |
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p. 744,
¶
Definition F.21
|
Replace $\mathcal{F}$ by $\mathcal{C}$. | Jenna Nieminen | |
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p. 747,
¶
Line 14, 15
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Replace $u$ by $f$ and $\bar{u}$ by $\bar{f}$. | F. Leptien | |
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p. 755,
¶
Lines 9, 10
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Replace ${\rm C}(\mathcal A)$ by $C(\mathcal A)$. | Jhan-Cyuan Syu | |
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p. 755,
¶
Line 20
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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 | |
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p. 756,
¶
Line 11
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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 | |
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p. 756,
¶
Line 18
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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 | |
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p. 756,
¶
Lines 7, 10, 11, 15, 17
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Replace ${\rm C}(\mathcal A)$ by $C(\mathcal A)$. | Jhan-Cyuan Syu | |
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p. 757,
¶
Line -7
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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 | |
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p. 757,
¶
Definition F.75, end of second line
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In the definition, $h^i$ should be a morphism of objects of $\mathcal A$ (not a morphism of complexes as written). | Bianca Fürstenau | |
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p. 760,
¶
Lines 2, 7
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Replace ${\rm C}(\mathcal A)$ by $C(\mathcal A)$. | Jhan-Cyuan Syu | |
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p. 763,
¶
Line -13
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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 | |
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p. 767,
¶
Line 13
|
Replace $E^{p-t, t}_\infty$ by $E^{p+t, -t}_\infty$. | Jhan-Cyuan Syu | |
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p. 767,
¶
Line -12
|
Replace "converging against" by "converging to". | Jhan-Cyuan Syu | |
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p. 770,
¶
Line 4
|
Replace the first 'is' by 'if'. | ||
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p. 772,
¶
Line 13
|
Insert "introduce" before "these notions". | Jhan-Cyuan Syu | |
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p. 773,
¶
Diagram in Def. F.116 (2)
|
Replace the $X[1]$ in the lower right corner by $X'[1]$. | Jhan-Cyuan Syu | |
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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 | |
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p. 777,
¶
Example F.127
|
Add a period at the end of the displayed line. | Jhan-Cyuan Syu | |
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p. 779,
¶
Line 3
|
Delete opening parenthesis in front of Hom. | Jhan-Cyuan Syu | |
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p. 786,
¶
Line 3
|
"to to" should be replaced by "to". | Torsten Wedhorn | |
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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 | |
|
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 |