Stacks project -- Comments

#7458 on tag 0FKF by ?

Mon, 27 Jun 2022 01:48:43 GMT

?The most difficult part is omitted?

#7457 on tag 0GWI by nhw

nhw — Sun, 26 Jun 2022 12:25:54 GMT

In the first line should "categori" be category?

#7456 on tag 031T by mi

mi — Sun, 26 Jun 2022 04:36:01 GMT

In the statement of (1), shall we also say a is not a unit?

#7455 on tag 0A38 by WhatJiaranEatsTonight

WhatJiaranEatsTonight — Sun, 26 Jun 2022 01:03:50 GMT

I think here we shall add $\cap_{i\in I}U_i$ and $gcd(n_i)$ but not $\cup_{i\in I}U_i$ .

#7454 on tag 026B by Hao Peng

Hao Peng — Sat, 25 Jun 2022 03:52:35 GMT

I am confused by the cocycle condition. My worry is that the composition doesn't make sence: While $pr^\star_{01}\phi_{ij}$ is a morphism from $(U_{ijk}\to U_{ij})^\star(U_{ij}\to U_i)^\star X_i$ to $(U_{ijk}\to U_{ij})^\star(U_{ij}\to U_j)^\star X_j$ , $pr^\star_{12}\phi_{jk}$ is a morphism from $(U_{ijk}\to U_{jk})^\star(U_{jk}\to U_j)^\star X_j$ to $(U_{ijk}\to U_{jk})^\star(U_{jk}\to U_k)^\star X_k$ . Thus it is preassumed that $(U_{ijk}\to U_{ij})^\star (U_{ij}\to U_j)^\star X_j=(U_{ijk}\to U_{jk})^\star (U_{jk}\to U_j)^\star X_j$ , which is not in general true. Is this a mistake or we insect a natural isomorphism between them, or we can choose the cleavage such that this is true for any fiber products?

#7453 on tag 0539 by Hao Peng

Hao Peng — Fri, 24 Jun 2022 04:24:47 GMT

Sorry, in the argument above, change " $R$ is a domain" to " $M$ is torsion-free".

#7452 on tag 0539 by Hao Peng

Hao Peng — Fri, 24 Jun 2022 04:14:36 GMT

I noticed that after minor change this lemma holds for a Bezout domain too:

Let $\sum a_ix_i=0$ where $a_i\in R^*, x_i\in M$ , then consider $(a_1, \ldots, a_n)=(a)$ , thus $a_i=ab_i$ for some $b_i\in R$ , and $\sum c_ib_i=1$ for some $c_i\in R$ . Because $R$ is a domain, $\sum_i b_ix_i=0$ , so we can take $\overrightarrow{y}=A\overrightarrow{x}$ , where $A=1-\overrightarrow{c}\overrightarrow{b}^t$ . Then $\overrightarrow{b}^tA=\overrightarrow{b}^t-\overrightarrow{b}^t\overrightarrow{c}\overrightarrow{b}^t=0$ .

#7451 on tag 0AMA by Logan Hyslop

Logan Hyslop — Thu, 23 Jun 2022 04:46:34 GMT

I believe there may be a rather minor typo in Lemma 14.19.2, specifically "Taking $n=k$ " should say "Taking $n=k=k^\prime$ ."

#7450 on tag 0E7M by Christophe Marciot

Christophe Marciot — Thu, 23 Jun 2022 01:25:03 GMT

In the intro at \emph{we conclude these points are nodes and smooth points on both} $C$ and $X'$ , would it a bit better for comprehension to add a \emph{respectively} after the $X'$ ?

#7449 on tag 032M by Laurent Moret-Bailly

Laurent Moret-Bailly — Thu, 23 Jun 2022 09:30:42 GMT

@ #7448: True, but if $R$ is a noetherian domain with perfect fraction field $K$ of char. $p>0$ , then $R=K$ . Otherwise, by Krull-Akizuki, there is a discrete valuation ring $V$ between $R$ and $K$ , and a uniformizer of $V$ cannot be a $p$ -th power in $K$ . (There may be simpler arguments).