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).
Jump to: Major error Minor error Typo/Trivial Remark Unclassified
237 errata listed.
Minor Errors
| Page | Description | Submitted by | Ed. |
|---|---|---|---|
|
p. 13,
¶
Proof of Proposition 17.17
|
The implication that $I/I^2=0$ implies $I=0$ does not follow directly from Nakayama's lemma (in its standard form), because even though, as explained, $I$ is in the Jacobson ideal of $A/I$, it will not in general be contained in the Jacobson ideal of $A$. In fact, looking at a projection $A\times B\to A$ of rings, we see that $I$ may be $\ne 0$. Instead we should apply Nakayama's lemma in the form of [GW1] Prop. B.3 (1) (= [Atiyah-Macdonald, Corollary 2.5]) to find $s\in A$ such that $s \equiv 1 \mod I$ and $sI=0$. Localizing by $s$ we find an open neighborhood of $y$ where $I$ vanishes. |
Branislav Sobot | |
|
p. 18,
¶
Right after proof of Corollary 17.32
|
In “Let $S$ be a scheme and let $i : Y \to X$ be a closed immersion of $S$-schemes,” I think we should relax the hypothesis to $i$ just being an immersion: Proposition 18.20 only assumes "immersion", but invokes the sequence (17.5.10) / Proposition 17.33 (and the latter is indeed still true under the weaker assumption of $i$ being an immersion, see Tag 01UZ). | Elías Guisado | |
|
p. 26,
¶
Proof of Proposition 17.54
|
Since our definition of $d$ is $d(a)=1\otimes a-a\otimes 1$ (see (17.1.10)), the formula that (17.10.4) implies is actually $d^1(bd^0(a))=d^0 a\wedge d^0 b$. Hence one needs to do some modification. For example, one can define $u(a\otimes b)=d^0(a)\wedge d^0(b)$, i.e., the negative of the definition in the book. | Yuhao Cheng | |
|
p. 29,
¶
Exercise 17.1. (2)
|
The condition should be $[D,D']\in \mathfrak{g}$ for all $D,D'\in X$. | Jan Willing | |
|
p. 29,
¶
Exercise 17.7
|
Add the assumption that $f$ is not a zero divisor. If $f$ is allowed to be a zero divisor, the statement is false in general. For example, take $R=k[\varepsilon]/(\varepsilon^2)$, $P=\mathbb P^1_R$, and $f=\varepsilon x_0$. On $D_+(x_1)$. with $t=x_0/x_1$ , the ideal is $(\varepsilon t)$, so $\mathscr I/\mathscr I^2\cong k[t]$, whereas $i^*\mathscr O(−1)≅R[t]/(\varepsilon t)$, which is not isomorphic to $k[t]$. | Yuhao Cheng | |
|
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. 175,
¶
Line 10
|
There is no reason that $Z$ should be irreducible in general, so the induction hypothesis does not apply directly. Thus the argument discussed below about reducing to irreducible spaces also needs to be used in the lemma. | LY | |
|
p. 191,
¶
Statement of Proposition 21.94
|
To “there exists a K-flat complex $\mathscr P_{\mathscr F}$”, it should be added “with flat terms.” This strengthening follows from the existing proof: Corollary F.197 implies that $\mathscr P_{\mathscr F}^i$ is a colimit of flat modules; thus, it is flat.
(To derive the equalities appearing in (21.20.4) from (F.25.2) we actually need our K-flat resolutions to have flat terms.) |
Elías Guisado | |
|
p. 213,
¶
Proposition 21.139
|
In the statement, we should add the hypothesis that $\mathcal{D}$ is stable under isomorphisms in $D(X)$ (without this assumption, the correct conclusion is just that “every strictly perfect complex is isomorphic to some object of $\mathcal{D}$”). | Elías Guisado | |
|
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 | |
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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. 582,
¶
Lemma 26.158
|
The Lemma states as a condition: "Suppose that the lowest slope of the HN polygon of E is strictly bigger than the smallest slope of the HN polygon of F." But it should say: "Suppose that the lowest slope of the HN polygon of E is strictly bigger than the highest slope of the HN polygon of F." | Michelle Klemt | |
|
p. 610,
¶
Remark/Def. 27.17
|
The identification $(e^*\Omega^1_{G/S})^\vee = e^*\mathscr T_{G/S}$ does not hold in general. It does hold, if $\Omega^1_{G/S}$ is locally free of finite rank, e.g., if $G$ is smooth over $S$. | ||
|
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. 722-723,
¶
Remark 27.283
|
The claim that, for every integer $n > 1$, every line bundle $\mathscr L$ on $X$ is fppf-locally of the form $\mathscr M^{\otimes n}$ is too strong. For instance, already for an elliptic curve over an algebraically closed field, a line bundle of non-zero degree $d$ cannot be of this form if $n$ does not divide $d$. In the proof given in the reference [DePa], there is a crucial additional hypothesis that $\varphi_{\mathscr L}$ is zero on the $n$-torsion of $X$, and this must be added as an assumption in Remark 27.283. (In the later applications $\varphi_{\mathscr L} = 2\lambda$ is $=0$ on the $2$-torsion, i.e., the assumption is satisfied.) |
Gabriel Dill | |
|
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 | |
|
p. 802,
¶
Lemma F.194
|
It is claimed that the functorial map $X \to I_X$ can be chosen to be “a monomorphism in each degree,” but this is only true if $X$ is itself bounded below ($X^i=0$ for $i < n$), whereas the proof is done for $X$ cohomologically bounded below ($H^i(X)=0$ for $i < n$), and we lose the degree-wise injectivity at the first step of the proof when consider the qis $X \to \tau^{\ge n} X$ and replace $X$ by $\tau^{\ge n} X$. The proof of (2) constructs a map $X \to I_X$ which is functorial in complexes $X$ such that $H^i(X)=0$ for $i < n$, where $n$ is fixed. Something slightly stronger is true: the map $X \to I_X$ is actually functorial in complexes $X$ such that there is some $n$ with $H^i(X)=0$ for all $i < n$. The latter is actually the way in which Lemma F.194 is used e.g. in p. 164, Remark 21.31. Here is the proof of the enhanced version: by assumption for $A$ is concentrated in degree zero then the functor $A \mapsto I_A$ is additive. Now define, for a general complex $X$ bounded below, the double complex $\tilde{I}_X$ to be $\cdots\to I_{X^{i-1}} \to I_{X^i} \to I_{X^{i+1}} \to\cdots$ (i.e., we do not use a fixed cut-off value on the left). Then $\tilde{I}_X$ is functorial in $X$ and is bounded below, because $I_0=0$ by the additivity. I don't know if this is any useful, but one could add to the statement the fact that $X \mapsto I_X$ is additive. This is because it is the composition of the additive functor $X \mapsto \tilde{I}_X$ and the additive functor Tot. |
Elías Guisado |
Typographical and Trivial Errors
| Page | Description | Submitted by | Ed. |
|---|---|---|---|
|
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,
¶
Second sentence of Remark 17.5
|
It should say "Let $\psi: A \to C$ be an $R$-algebra homomorphism". | Javier de la Bodega | |
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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 | |
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p. 7,
¶
Equation (17.1.10)
|
It should say "$a \mapsto i_2(a)-i_1(a)$". | Javier de la Bodega | |
|
p. 7,
¶
Line 14
|
Insert "$M$" after "$A$-modules". | Jan Willing | |
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p. 8,
¶
line 13
|
Missing subscript on $u$. Should read $u_D(1 \otimes a - a \otimes 1)$. | L. Potter / Jan Willing | |
|
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. 9,
¶
Remark 17.11(1)
|
The algebra structure of $E\oplus^M M’$ is not defined. It should be $(e_1,m_1’)(e_2,m_2’)=(e_1e_2,\pi(e_1)m_2’+\pi(e_2)m_1’)$. | Yuhao Cheng | |
|
p. 11,
¶
Line 9
|
Replace "$i^1$" by "$i_1$". | L. Potter | |
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p. 12,
¶
Line -12
|
Replace $X$ by $X’$. | Yuhao Cheng | |
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p. 16,
¶
Line 9
|
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
|
$f$ 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. 18,
¶
Line 2 of proof of Prop 17.33
|
It should say "Let $\pi:A\to B$ *be* the canonical projection". | Yuhao Cheng | |
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p. 21,
¶
Line -3
|
"where $K$ is field" should be "where $K$ is a field". | Yuhao Cheng | |
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p. 22,
¶
Line 4
|
It should read "morphism of S-schemes". | Jan Willing | |
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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 | |
|
p. 23,
¶
Line 4, line 23
|
Add periods at the end of sentences. | Erik Nikolov | |
|
p. 24,
¶
Line 12
|
Remove the second "graded" in "graded commutative graded algebra". | Jan Willing | |
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p. 28,
¶
Line 16
|
Replace $\mathscr E$ by $\mathscr E''$ in the top entry of the commutative triangle. | L. Potter | |
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p. 28,
¶
Line -13
|
Remove "a" at the beginning of this line. | Yuhao Cheng | |
|
p. 29,
¶
Exercise 17.8
|
Replace $\Omega_{A/R}$ by $\Omega^1_{A/R}$. | Javier de la Bodega | |
|
p. 29,
¶
Ex. 17.6., line -8
|
Replace "is of the from" by "is of the from". | Jan Willing | |
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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 | |
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p. 32,
¶
Remark 18.2(5)
|
"(17.3)" should be "(see Section (17.3))". | Yuhao Cheng | |
|
p. 36,
¶
Line -15
|
"(17.2)" should be "(see Section(17.2))". | Yuhao Cheng | |
|
p. 36,
¶
Proof of Proposition 18.15, (i) ⇒ (ii)
|
In “$\operatorname{id}_A$ can be lifted to an $A$-algebra homomorphism,” it should be $R$-algebra homomorphism. | Elías Guisado | |
|
p. 42,
¶
Line 7
|
Replace $h\colon T\to Y$ by $h\colon T\to X$. | Yunhao Sun | |
|
p. 43,
¶
Lemma 18.37
|
In the statement, replace $A_0$ by $A/I$. | Elías Guisado | |
|
p. 50,
¶
Theorem 18.56(ii)
|
"$g$ is étale in $x$" should be "$g$ is étale at $x$". | ||
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p. 60,
¶
Line -14
|
It should be "is" rather than "ist". | L. Potter | |
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p. 62,
¶
Proposition 18.79
|
Replace "morphisms" by "morphism". | Gabriel Dill | |
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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 | |
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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 | |
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p. 66,
¶
Exercise 18.24
|
Replace `Björn Poonen' by `Bjorn Poonen'. | U. Görtz | |
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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 | |
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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 | |
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p. 75,
¶
Line 5
|
In the phrase "Then $i$ is called a quasi-regular at $z$", the word "a" should be removed. | Benjamin Diamond | |
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p. 83,
¶
Line 23
|
Replace $-\otimes {\rm id}_{\mathbb A^m}$ with $-\otimes {\rm id}_{\mathbb A^n}$. | ||
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p. 85,
¶
Line 14
|
Replace 'local intersection ring' by 'complete intersection ring'. | U. Görtz | |
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p. 89,
¶
Formulation of Exercise 19.8, second line
|
It should be $\mathscr{L}$ instead of $\mathscr{F}$ | Branislav Sobot | |
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p. 106,
¶
Definition 20.37
|
Replace $U\to S$ by $g\colon U\to S$. | woo | |
|
p. 109,
¶
Line -2
|
The word "minimal" is missing: "Hence $x$ is contained in a single irreducible component if and only if $\mathscr O_{X,x}$ contains a unique minimal prime ideal." | Branislav Sobot | |
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p. 112,
¶
First line after Definition 20.54
|
Replace ''approximation theory" with "approximation property". | Niklas Arppe | |
|
p. 148,
¶
Exercise 20.30, Line 3
|
Replace $x_ix_i$ by $x_ix_j$. | Erik Nikolov | |
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p. 149,
¶
Exercise 20.39(3)
|
It should say "$q_Y \circ f = \pi_0(f) \circ q_X$". | Javier de la Bodega | |
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p. 152,
¶
Line -9
|
Replace "...for an $A$-module or a complex of $A$-module..." by "...for an $A$-module or a complex of $A$-modules...". | Isaac K. Martin | |
|
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. 160,
¶
Line 7
|
Replace "Remark F.205" by "Corollary F.205". | LY | |
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p. 161,
¶
Third line of Remark 21.19.
|
"if" missing from the end of "if and only if" | Yiliu Liu | |
|
p. 163,
¶
Line 7
|
Replace let by Let | LY | |
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p. 165,
¶
line -1
|
The last term in the exact sequence should not have a subscript $X$. | L. Potter | |
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p. 166,
¶
After proof of Proposition 21.41
|
One needs to flip the domain and codomain of the morphism φ and of the functor φ_*. | Elías Guisado | |
|
p. 166,
¶
Proof of Proposition 21.40
|
Replace “and $H^1_{\rm Tors}(X, \mathscr F^0)=0$ by Lemma 21.33” by "and $H^1_{\rm Tors}(X, \mathscr A^0)=0$ ...". In the final line of the proof, replace $\mathscr A$ by $\mathscr F$, and in that chain of equations the Coker term should be written in the middle. | Elías Guisado / U.G. | |
|
p. 169,
¶
Proof of Proposition 21.48
|
At the end of the first sentence, change “open subscheme” to “open subset.” | Elías Guisado | |
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p. 169,
¶
Line 2
|
Replace "Theorem 21.7" by "Proposition 21.7". | Jan Willing | |
|
p. 172,
¶
Proof of Lemma 21.52
|
In the proof of (i), third sentence, we need to restrict the equality $\mathscr F_{\kappa_\alpha}(s)=\mathscr F_{\kappa'_\alpha}(s')$ to $U_\alpha$. | Elías Guisado | |
|
p. 174,
¶
Coro. 21.56 (2), line 1
|
Replace $f\colon Y \to Y$ by $f\colon X \to Y$. | LY | |
|
p. 177,
¶
Equation (21.13.3)
|
in the right hand side, change $\Gamma_Z(\mathscr F)$ (this notation hasn't been defined) to $\Gamma_Z(X, \mathscr F)$. | Elías Guisado | |
|
p. 177,
¶
Remark 21.60
|
Right after (21.13.2), change “For $U\subseteq X$ open” to “For $U\subseteq X$ open such that $U\cap Z$ is closed in $U$” (the inclusion $\Gamma_{U\cap Z}(U, \mathscr F_{|U})\subseteq\Gamma(U, \mathscr F)$ only makes sense in the latter case). Add "since" before "the restriction". Extra remark: Another way of showing that $i^{-1}(\mathscr F) \supseteq i^!(\mathscr F)$ is by noting that $\mathscr F \supseteq \mathscr F^Z$ implies $i^{-1}(\mathscr F) \supseteq i^{-1}(\mathscr F^Z) = i^!(\mathscr F)$, where notation ℱ^Z and last equality come from the proof of Proposition 21.61. |
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. 183,
¶
Proof of Lemma 21.76
|
In the second-to-last sentence of the proof, for the definition of $h$ we should rather write \begin{equation*} h(s)_{i_0\dots i_{n+2}}=\begin{cases} 0,&\text{if }i_0\neq i^0\\ s_{i_1\dots i_{n+2}},&\text{if }i_0=i^0. \end{cases} \end{equation*} In other words, $h$ is the sum of the maps \begin{align*} \mathscr{O}_{U_\mathbf{i}}^p(W)\to\mathscr{O}_{U_{(i^0,\mathbf{i})}}^p(W),\quad\mathbf{i}\in I^{n+1}, \end{align*} which is the zero map $0\to 0$ if $W\not\subset U_\mathbf{i}$ and the identity $\mathscr{O}(W)\to\mathscr{O}(W)$ if $W\subset U_{(i^0,\mathbf{i})}$ (equivalently, if $W\subset U_{\mathbf{i}}$). | Elías Guisado | |
|
p. 185,
¶
Final line of the proof of The 21.78
|
Replace (21.17.4) by (21.17.5) | LY | |
|
p. 185,
¶
Remark 21.79
|
In “The proof of Theorem 21.78 shows that the morphism (21.17.6) is the edge morphism...” it should be “the morphism (21.17.7).” | Elías Guisado | |
|
p. 185,
¶
Proof of Theorem 21.78
|
In the first paragraph, instead of $R^p \check{H}^0(\mathscr U, -) = \check{C}^p(\mathscr U, -)$ it should be $R^p \check{H}^0(\mathscr U, -) = \check{H}^p(\mathscr U, -)$ | Elías Guisado | |
|
p. 186,
¶
Proposition 21.83
|
In the statement, in the hypothesis “suppose that $\check{H}(U, \mathscr F)=0$ for all $p > 0$ and $U \in \mathscr B$” it should be $\check{H}(U, \mathscr F_{|U})=0$ instead (the notation $\check{H}(U, \mathscr F)$ hasn't been defined). In the proof, last sentence, in “the set of all open coverings” it should be “the class of all open coverings.” | Elías Guisado | |
|
p. 187,
¶
Fifth paragraph
|
Replace “corresponding modules over a ringed spaces” by “corresponding to modules over a ringed space”. | Elías Guisado | |
|
p. 190,
¶
Remark 21.90 (2)
|
Replace $\mathscr{F}$ by $\mathscr{G}$ (three times). | Niklas Arppe | |
|
p. 190,
¶
line 19
|
After line (21.19.2): replace "isomorphim" by "isomorphisms". | Jan Willing | |
|
p. 191,
¶
line 4
|
Replace "...where he..." by "...where the...". | Jan Willing | |
|
p. 192,
¶
Lemma 21.96
|
In the proof of (2), after “choose a quasi-isomorphism $\mathscr{Q}\to\mathscr{F}$” we should say “with $\mathscr{Q}$ K-flat.” | Elías Guisado | |
|
p. 194,
¶
Right before Remark 21.102
|
Instead of “we conclude by Proposition B.15” it should be “we conclude by Proposition B.16.” | Elías Guisado | |
|
p. 194,
¶
Third line in the proof of Prop. 21.100
|
Replace the first $\le$ by $<$. | LY | |
|
p. 194,
¶
Second line in Proof of Prop. 21.100
|
Add $\mathscr G$ after “complex of $\mathscr O_X$-modules”. | Jan Willing | |
|
p. 194,
¶
line 7
|
Include \mathcal G after "every complex" UG: Duplicate | Jan Willing | |
|
p. 196,
¶
Lemma 21.106
|
In the statement, in “let $\mathscr{I}$ be a K-injective complexes” the last word should be “complex.” | Elías Guisado / Yiliu Liu | |
|
p. 196,
¶
Line 13
|
Replace ${\rm Hom}$ by $\mathscr Hom$. | U. Görtz | |
|
p. 196,
¶
Remark 21.109, (1)
|
Change “if $G$ is an exact complex of $\mathscr{O}_{X,x}$-modules” to “if $F$ is an exact complex of $\mathscr{O}_{X,x}$-modules.” | Elías Guisado | |
|
p. 197,
¶
Lines -3, -2
|
Replace $L(f^*)\circ L(g_*)$ by $L(f^*)\circ L(g^*)$ (twice). | Elías Guisado | |
|
p. 198,
¶
Line 3
|
Replace "$G$-acyclicity" by "$g_*$-acyclicity". | LY | |
|
p. 199,
¶
Last paragraph of Coro 21.116
|
Replace $R^i f_* \mathscr F$ by $R^i g_* \mathscr F$. | LY | |
|
p. 201,
¶
Line 2
|
Replace “complexes $\mathscr{O}_Y$-modules” by “complexes of $\mathscr{O}_Y$-modules.” | Elías Guisado | |
|
p. 204,
¶
Remark 21.123
|
In “choose also a quasi-isomorphism $\mathcal{K} \to \mathscr{H}om(\mathscr{E}, \mathscr{I})$ with $\mathcal{K}$ a K-flat complex” it should be $\mathscr{K}$ rather than $\mathcal{K}$. | Elías Guisado | |
|
p. 206,
¶
Proof of Theorem 21.128
|
In the commutative square involving $\alpha$, $\beta$, $\gamma$, in the right lower object there is $RHom$ (italics) but it should be $R{\rm Hom}$. | Elías Guisado | |
|
p. 211,
¶
Proof of Lemma 21.134
|
In the beginning of the third paragraph replace “in general there is a split exact sequence of complexes,” by “... termwise split exact sequence.” | Elías Guisado | |
|
p. 211,
¶
Proof of Lemma 21.134
|
In the proof of (2): In the displayed distinguished triangle, $\mathscr{F}_{|V}$ is placed where $\mathscr{G}_{|V}$ should stand at and vice versa. In “it suffices to show that locally on $X$ the composition $\mathscr{E}\to\mathscr{F}\to C_u$,” replace $\mathscr{F}$ by $\mathscr{G}$. | Elías Guisado | |
|
p. 213,
¶
Corollary 21.141
|
In the statement, in “suppose $\mathscr{F}(\mathscr{O}_X)\to\mathscr{G}(\mathscr{O}_X)$ is an isomorphism,” the correct morphism is $\Phi(\mathscr{O}_X)\to\Psi(\mathscr{O}_X)$. In the proof, last sentence, replace “Lemma 21.139” with “Proposition 21.139.” | Elías Guisado | |
|
p. 214,
¶
Proof of Lemma 21.144, lines -3, -2
|
Replace $\mathscr Hom$ by ${\rm Hom}$ twice. | LY | |
|
p. 217,
¶
Proof of Proposition 21.154
|
In the proof, first paragraph, last line, change $\sigma^{m+1}K$ to $\sigma^{m+1}\mathscr{K}$. | Elías Guisado / LY | |
|
p. 222,
¶
Equation (21.35.2)
|
Remove the subscript "$g$" from the first "tor-amp". | LY | |
|
p. 225,
¶
Proof of Lemma 21.177
|
In "we proceed by induction an b−a" it should be "on b-a." | Elías Guisado | |
|
p. 225,
¶
Line -12
|
Replace "analogues" by "analogous". | U. Görtz | |
|
p. 225,
¶
Line -10
|
Citation for complexes of finite tor-dimension should be Proposition 21.172, not Proposition 21.163. | LY / L. Potter | |
|
p. 226,
¶
Line 6
|
Replace U by V. | Jan Willing | |
|
p. 227,
¶
Exercise 21.5
|
In the third line, replace “objective” by “object.” | Elías Guisado | |
|
p. 228,
¶
Exercise 22.14
|
Replace "spaces" by "space". | Jan Willing | |
|
p. 229,
¶
Exercise 21.21
|
In the Exercise replace “{\rm Tor}_A^{-p}” by “{\rm Tor}_{-p}^A”. | Jan Willing | |
|
p. 229,
¶
Exercise 21.18
|
Replace “$R$-modules” by “$A$-modules” | Jan Willing | |
|
p. 236,
¶
Line 5
|
Replace $H^i(\mathcal U, \mathscr F)$ by $\check{H}^i(\mathcal U, \mathscr F)$. | LY | |
|
p. 242,
¶
Theorem 22.22
|
In (3) it must say the $R$-submodule instead of $A$-submodule. | T. Wedhorn | |
|
p. 250,
¶
Line 5
|
should be "and hence is equal" instead of "and is hence is equal" | Yiliu Liu | |
|
p. 252,
¶
Line 21
|
Should be $D\textrm{QCoh}(X)$ instead of $D\textrm{QCoh}(()X)$. | Bianca Fürstenau | |
|
p. 254,
¶
Second line in section 22.12
|
should be "plump", not "plumb" | Saskia Kern | |
|
p. 273,
¶
Line 7
|
In the map target, replace $\mathscr{F}\otimes^L Rf_*\mathscr{G}$ by $\mathscr{F}\otimes^L Lf^*\mathscr{G}$. | Elías Guisado / Immanuel Klevesath and Matthis Scholz | |
|
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. 280,
¶
Equation (*)
|
In the middle terms, replace $G_s$ by $G_{s'}$ twice. | LY | |
|
p. 282,
¶
Line 12
|
Replace if by is. | LY | |
|
p. 282,
¶
Equation (22.24.1)
|
The last term should be $Lu'^*E$, instead of $Lu^*E$. | LY | |
|
p. 284,
¶
Equation (22.25.1)
|
Should be $X'_{ijl}$ instead of $X_{ijl}$. | LY | |
|
p. 285,
¶
Lines -13, -6, -4
|
Replace $A \otimes_R R'$ by $A'\otimes_A B$. | LY | |
|
p. 287,
¶
Line 11
|
Add missing "$E$" in the third term. | LY | |
|
p. 294,
¶
Exercise 22.26
|
The hint (resp. remark) refer to (b) (resp. (a)). This should be (2) (resp. (1)). | T. Wedhorn | |
|
p. 298,
¶
Line 5
|
Should be "exactness of the direct image functor" instead of "exactness if the direct image functor" | Yiliu Liu | |
|
p. 305,
¶
Line -7
|
Replace $S' := A[T_1, \dots, T_r]$ by $S':=A[T_0,...,T_r]$. | LY | |
|
p. 307,
¶
Line -3 (statement of Cor. 23.18)
|
Replace "finite generated" by "finitely generated". | Matthieu Romagny | |
|
p. 310,
¶
Line -1
|
In the diagram, replace the lower right entry $X$ by $Y$. | LY | |
|
p. 315,
¶
Line 13, 16
|
Replace $D^?_{\mathcal A}(\mathcal A')$ by $D^?(\mathcal A)$ three times. | LY | |
|
p. 320,
¶
Line -5,-6
|
Replace "acyclic" by "with differentials all zero" (twice). | LY | |
|
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. 335,
¶
Line 1
|
Replace (c) by (d). | LY | |
|
p. 336,
¶
Line -14
|
The reference should be to Definition 23.57 instead of Definition 23.100. | LY | |
|
p. 338,
¶
Line -4
|
Replace "\subseteq \cdots" by $\supseteq \cdots$. | LY | |
|
p. 338,
¶
Line 8
|
Replace $c_1^{K_0}(L)$ by $c_1^{K_0}(\mathscr L)$. Replace $Ls_0^*$ by $s_0^*$. | LY | |
|
p. 346,
¶
Line -6
|
Replace $T(M)$ by $\beta(M)$. | LY | |
|
p. 349,
¶
Lines 2, 3
|
Replace $\kappa$ by $\kappa(s)$ (twice). | U. Görtz | |
|
p. 358,
¶
Theorem 23.140
|
In the beginning, a point $s \in S$ is fixed. Later on, twice in the expression “$s \mapsto \dim_{κ(s)} H^i(X_s, \mathcal{F}_s)$” and once in “for all $s \in S$”, the variable is reused as if the point had not been fixed. | Bianca Fürstenau | |
|
p. 369,
¶
Exercise 23.16, final line
|
Replace $F_R$ by $\mathcal F_R$. | LY | |
|
p. 375,
¶
Exercise 23.45 (1)
|
Replace $D\otimes_SS'$ by $D \times_S S^\prime$. | Haoyang Yuan | |
|
p. 383,
¶
Proof of Prop. 24.10, line -4
|
Replace Theorem 22.1 by Lemma 22.1. | Matthieu Romagny | |
|
p. 408,
¶
Proposition 24.69, line 4
|
Replace "$\mathrm{Pic}(\mathbb{P}(E))$" to "$\mathrm{Pic}(\mathbb{P}(\mathscr{E}))$". | Xiaolong Liu | |
|
p. 416,
¶
Line 1/2, Definition 24.82
|
Replace "an inductive system of $S_n$-morphism" by "a system of $S_n$-morphisms". | Matthieu Romagny / Alice Bouillet | |
|
p. 419,
¶
Proposition 24.88 (2)
|
Replace "$M$ it is of rank $r$ if and only if..." by "$M$ is of rank $r$ if and only if..." | Matthieu Romagny | |
|
p. 466,
¶
Proposition 25.62
|
Add assumption that $f$ is "separated morphism of finite type of noetherian schemes" for $f^!$ to be defined. | Yiliu Liu | |
|
p. 466,
¶
Second Paragraph of Example 25.63
|
Replace $p\colon \mathbb P^1_S \to S$ by $p\colon \mathbb P^n_S\to S$. | Matthis Scholz | |
|
p. 467,
¶
Line before Equation (25.12.2)
|
Add assumption that $g$ is "separated morphism of finite type between noetherian schemes". | Yiliu Liu | |
|
p. 469,
¶
Line -13
|
Should be either "(Proposition 25.87)" instead of "(Proposition 25.17)". | Yiliu 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. 496,
¶
Corollary 25.131, line 5 of Proof
|
Remove the extra comma between $K$ and the derived tensor product on the right hand side of the "=". | Yingying Wang | |
|
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. 542,
¶
Throughout the page
|
Replace $\Omega$ by $\Omega^1$ several times. | Javier de la Bodega | |
|
p. 543,
¶
Throughout the page
|
Replace $\Omega$ by $\Omega^1$ several times. | Javier de la Bodega | |
|
p. 545,
¶
Line -14
|
Change $X\to X^{(p)}$ to $X^{(p)}\to X$ (twice). | U. Görtz | |
|
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. 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. 557,
¶
Line 21 (Case 2)
|
Add condition $y_1=y_2\neq 0$. | Yunhao Sun | |
|
p. 559,
¶
Line -9
|
Replace $g^*\mathscr O(1)$ by $f^*\mathscr O(1)$. | Yingying Wang | |
|
p. 574,
¶
in the proof of Theorem 26.133
|
There are two small typos, where both times "e_i" should actually be "d_i". They are located in the line showing that the Ext group equals 0 – once in the direct sum over the Ext groups and once in the direct sum over the cohomology groups. | Matthias Schempp | |
|
p. 577,
¶
Last line of Example 26.144
|
in the equality for deg(\mathcal{E}), there is a plus missing in the sum (right before d_r) | Saskia Kern | |
|
p. 578,
¶
Line 3
|
The grammar is off: two "we"' where there should only be one. | Saskia Kern | |
|
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. 640,
¶
Proof of Proposition 27.105
|
It should be $Y_s$ instead of $G_s$. | Branislav Sobot | |
|
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. 670,
¶
Line 5
|
Replace "of $S$-group scheme" by "of $S$-group schemes". | Matthis Scholz | |
|
p. 687,
¶
The first displaymath of section 27.40
|
Replace $f^t\colon X^t \to Y^t$ with $f^t\colon Y^t \to X^t$. | Matthis Scholz | |
|
p. 693,
¶
first line of the proof of lemma 27.226
|
Replace $p^{t,-1}(y)$ by $(p')^{-1}(y)$. | Matthis Scholz | |
|
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. 765,
¶
Definition/Remark F.101
|
In Line -3, it is clearer to rephrase as "for every pair $p, q\in \mathbb Z$ of integers, there exists an integer $r(p, q)\ge 0$". In Line -2, in the final term the subscript $r$ is missing. |
Elías Guisado / UG | |
|
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. 771,
¶
Line 7, i.e. Eqn. (F.25.1)
|
Add superscript $k$ for the second ${\rm Tot}(X)$. | LY | |
|
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. 778,
¶
Line −8
|
The label of the morphism $-w[-1]$ has insufficient vertical space above; it clips into line −9. | Bianca Fürstenau | |
|
p. 779,
¶
Remark F.135 (and probably elsewhere)
|
The superscripts $-^{\rm op}$ and $-^{\rm opp}$ are mixed. Should make the same choice everywhere. | U. Görtz | |
|
p. 779,
¶
Line 3
|
Delete opening parenthesis in front of Hom. | Jhan-Cyuan Syu | |
|
p. 783,
¶
Line 5
|
Replace $S$ by $S^Y$ in the subscript of the colim. | Jan Willing | |
|
p. 784,
¶
Line -13, =second line in (F.36)
|
Replace "systems" by "system". | Jan Willing | |
|
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 | |
|
p. 805,
¶
Line 24
|
"$R^0(A)$" should be "$R^0F(A)$" | LY |
Remarks
| Page | Description | Submitted by | Ed. |
|---|---|---|---|
|
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. 40,
¶
Proof of Proposition 18.27
|
In the proof of the equivalence of (iv) with the other conditions one must use that $\mathcal{O}_{X_s,x}\cong\mathcal{O}_{X,x}\otimes_{\mathcal{O}_{S,s}}\kappa(s)\cong\mathcal{O}_{X,x}/\mathfrak{m}_s\mathcal{O}_{X,x}$; here $s=f(x)$ (I think this formula for the stalk of the fiber isn't obvious if one is not told about it in advance). | Elías Guisado | |
|
p. 58,
¶
Line -7
|
Should refer to Prop. 18.67 instead of Thm. 18.56. | ||
|
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. 174,
¶
After proof of Corollary 21.55
|
Instead of “using that quasi-flasque $\mathscr O_X$-modules are also right $f_*$-acyclic by Proposition 21.27, ...” it would be more precise to say “using the fact that an $\mathscr O_X$-module $\mathscr G$ such that $\mathscr G_{f^{-1}(V)}$ is quasi-flasque for all $V\in \mathscr B$ is right $f_*$-acyclic by Proposition 21.27, ...” | Elías Guisado | |
|
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. 689,
¶
Remark 27.218 (2)
|
It should be explained that to show that $m^*\mathscr P$ is isomorphic to $p_1^*\mathscr P \otimes p_2^*\mathscr P$, one needs to use that $\mathscr P$ is rigidified. Indeed, from $\mathscr P \in {\rm Pic}_{X/S}(X^t)$ being an $X^t$-valued point of $X^t$, it follows only that these two line bundles are isomorphic up to tensoring by the pull-back of a line bundle on $X^t$. Restricting to $0\times 0\times X^t$ and using the rigidifcation, one sees that this line bundle is trivial. | Gabriel Dill | |
|
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. 783,
¶
Line -3
|
To be consistent with the above, one should define $T_X$ as the category with objects $X' -> X$ etc. and replace $T$ by $T_X$ in the subscript of the colim. | Jan Willing | |
|
p. 787,
¶
(F.37.2) and construction below
|
It might be helpful to make it clearer that (F.37.2) is supposed to be a short exact sequence in $C(\mathcal A)$ (not in the derived category where this notion does not make sense), and likewise, below, the isomorphism $f_0\colon X \rightarrow {\rm Im}(f)$ is understood as an isomorphism in the category of complexes. | Jan Willing | |
|
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 |