Stacks project -- Comments

#9489 on tag 012X by Elías Guisado

Elías Guisado — Tue, 02 Jul 2024 05:39:31 GMT

After Lemma 12.25.1, in “these spectral sequences define two filtrations on $H^n(\operatorname{Tot}(K^{\bullet,\bullet}))$ . We will denote these $F_I$ and $F_{II}$ ,” perhaps one could spell out $F_J^pH^n(\operatorname{Tot}(K^{\bullet,\bullet}))=\operatorname{Im}\left(H^n(F_J^p\operatorname{Tot}(K^{\bullet,\bullet}))\to H^n(\operatorname{Tot}(K^{\bullet,\bullet}))\right)$ , for $J=I,II$ . That is, we are plugging $(\operatorname{Tot}(K^{\bullet,\bullet}),F_J)$ into Definition 12.24.5.

#9488 on tag 012Q by Elías Guisado

Elías Guisado — Tue, 02 Jul 2024 03:17:35 GMT

Analogously as in Comment #9487:

In the statement, instead of “the limit $E_{\infty}$ exists” shouldn't we specify “the limit $E_{\infty}$ of the spectral sequence of Lemma 12.24.2 exists”?
In the proof, second sentence, instead of $Z_r=\bigoplus Z_r^{p,q}$ , $B_r=\bigoplus B_r^{p,q}$ , shouldn't one write $Z_r=\left(Z_r^{p,q}\right)$ , $B_r=\left(Z_r^{p,q}\right)$ ? (We are working over an abelian category $\mathcal{A}$ that might not satisfy AB3.)

#9487 on tag 012F by Elías Guisado

Elías Guisado — Mon, 01 Jul 2024 12:42:03 GMT

In the statement, instead of “the limit $E_\infty$ exists” shouldn't we specify “the limit $E_\infty$ of the spectral sequence of Lemma 12.23.2 exists”?
In the proof, second sentence, instead of $Z_r=\bigoplus Z_r^p$ , $B_r=\bigoplus B_r^p$ , shouldn't one write $Z_r=(Z_r^p)$ , $B_r=(Z_r^p)$ ? (We are working over an abelian category $\mathcal{A}$ that might not satisfy AB3.)

#9486 on tag 012C by Elías Guisado

Elías Guisado — Mon, 01 Jul 2024 09:20:02 GMT

Typo: In the statement, I think the very last equality should be $E_1^p=H(\operatorname{gr}^pK,\operatorname{gr}^pd)$ .

#9485 on tag 01JA by Ziqiang Cui

Ziqiang Cui — Sat, 29 Jun 2024 01:11:56 GMT

Can you please explain why in the proof of lemma 26.14.1 the commutative diagram imply the isomorphism of sheaves $\mathscr O_{U_i}|_{U_i\cap U_j}\rightarrow \mathscr O_{U_j}|_{U_i\cap U_j}$ . What I think is all three morphisms in the commutative diagram are isomorphisms and hence induce a sequence of isomorphisms $\mathscr O_{U_i}|_{U_i\cap U_j}\cong \varphi _i|{U_{ij}}^*\mathscr O_i|_{U_{ij}}\cong \varphi _j{U_{ji}}^*\mathscr O_{j}|_{U_{ji}}\cong \mathscr O_{U_{j}}|_{U_i\cap U_j}$ . But I could be wrong.

#9484 on tag 00AK by Ziyu Lü

Ziyu Lü — Sat, 29 Jun 2024 05:34:25 GMT

Maybe a easy question but why $\varphi_i$ in Lemma 6.33.2 is isomorphism. Let $W\subset U_i$ be open for fixed $i$ . Let $(s_j)_{j\in I}\in \mathscr F|_{U_i}$ then for any other index $j\in I$ , $s_j$ is uniquely determined by the image of $s_i$ . In detail, we construct $\alpha: \mathscr F|_{U_i}\rightarrow \mathscr F_i: (s_j)_{j\in I}\rightarrow s_i$ and $\beta: \mathscr F_i\rightarrow \mathscr F_{U_i}: s\rightarrow (s_j)_{j\in I}$ , where in the second map $s_j=s$ if $j=i$ , otherwise set $s_j=\varphi(s|{W\cap U_j})$ . And they are mutually inverses.

#9483 on tag 01CM by Huang

Huang — Sat, 29 Jun 2024 01:27:16 GMT

Should Lemma 17.22.2.(1) be correct only for $F$ being coherent(or finite presentation), then it follows from the stalk isomorphism of intern Hom, while the general case the proof would not go through since surjective of sheaves only implies surjective on the stalk level?

#9482 on tag 0AMJ by Elías Guisado

Elías Guisado — Fri, 28 Jun 2024 05:28:34 GMT

The diagram should compile like this. I don't know why Mathjax doesn't compile it well here in the Stacks Project (same happened to me in #6829).

#9481 on tag 0AMJ by Elías Guisado

Elías Guisado — Fri, 28 Jun 2024 05:21:58 GMT

How one can describe exactly how $g'$ is induced? What I thought is to define $g'$ as the connecting homomorphism arisen from the snake lemma applied to the commutative diagram with exact rows $\require{AMScd} \begin{CD} @.TE@>{Tf}>> TA@>{T\alpha}>>A'@>>> 0\\ @.@V{Td}VV@V{Tg}VV@VV{0}V@.\\ 0@>>>\operatorname{Ker}TS d@>>> TSE@>{\smash{ TSd}}>> TS^2E \end{CD}$ Note that this also gives a description of $g'$ in Lemma 12.21.2 if $T=S=\operatorname{id}_\mathcal{A}$ . I don't know if there is another way, though. The footnote seems to be using something else, instead of the snake lemma.

#9480 on tag 011U by Elías Guisado

Elías Guisado — Thu, 27 Jun 2024 12:02:06 GMT

Maybe in the list of properties (1) – (3) to be found before Definition 12.22.5 one could mention Lemma 12.21.4? It seems it is what one needs.