Stacks project -- Comments https://stacks.math.columbia.edu/recent-comments.xml Stacks project, see https://stacks.math.columbia.edu en stacks.project@gmail.com (The Stacks project) pieterbelmans@gmail.com (Pieter Belmans) https://stacks.math.columbia.edu/static/stacks.png Stacks project -- Comments https://stacks.math.columbia.edu/recent-comments.rss #7457 on tag 0GWI by nhw https://stacks.math.columbia.edu/tag/0GWI#comment-7457 A new comment by nhw on tag 0GWI. In the first line should "categori" be category?

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nhw Sun, 26 Jun 2022 12:25:54 GMT
#7456 on tag 031T by mi https://stacks.math.columbia.edu/tag/031T#comment-7456 A new comment by mi on tag 031T. In the statement of (1), shall we also say a is not a unit?

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mi Sun, 26 Jun 2022 04:36:01 GMT
#7455 on tag 0A38 by WhatJiaranEatsTonight https://stacks.math.columbia.edu/tag/0A38#comment-7455 A new comment by WhatJiaranEatsTonight on tag 0A38. I think here we shall add $\cap_{i\in I}U_i$ and $gcd(n_i)$ but not $\cup_{i\in I}U_i$.

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WhatJiaranEatsTonight Sun, 26 Jun 2022 01:03:50 GMT
#7454 on tag 026B by Hao Peng https://stacks.math.columbia.edu/tag/026B#comment-7454 A new comment by Hao Peng on tag 026B. 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?

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Hao Peng Sat, 25 Jun 2022 03:52:35 GMT
#7453 on tag 0539 by Hao Peng https://stacks.math.columbia.edu/tag/0539#comment-7453 A new comment by Hao Peng on tag 0539. Sorry, in the argument above, change "$R$ is a domain" to "$M$ is torsion-free".

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Hao Peng Fri, 24 Jun 2022 04:24:47 GMT
#7452 on tag 0539 by Hao Peng https://stacks.math.columbia.edu/tag/0539#comment-7452 A new comment by Hao Peng on tag 0539. 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 $\overrightarrow{b}^tA=\overrightarrow{b}^t-\overrightarrow{b}^t\overrightarrow{c}\overrightarrow{b}^t=0$.

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Hao Peng Fri, 24 Jun 2022 04:14:36 GMT
#7451 on tag 0AMA by Logan Hyslop https://stacks.math.columbia.edu/tag/0AMA#comment-7451 A new comment by Logan Hyslop on tag 0AMA. I believe there may be a rather minor typo in Lemma 14.19.2, specifically "Taking $n=k$" should say "Taking $n=k=k^\prime$."

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Logan Hyslop Thu, 23 Jun 2022 04:46:34 GMT
#7450 on tag 0E7M by Christophe Marciot https://stacks.math.columbia.edu/tag/0E7M#comment-7450 A new comment by Christophe Marciot on tag 0E7M. 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 $X'$ ?

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Christophe Marciot Thu, 23 Jun 2022 01:25:03 GMT
#7449 on tag 032M by Laurent Moret-Bailly https://stacks.math.columbia.edu/tag/032M#comment-7449 A new comment by Laurent Moret-Bailly on tag 032M. @ #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).

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Laurent Moret-Bailly Thu, 23 Jun 2022 09:30:42 GMT
#7448 on tag 032M by Haohao Liu https://stacks.math.columbia.edu/tag/032M#comment-7448 A new comment by Haohao Liu on tag 032M. From the proof we see that the condition "fraction field has characteristic zero " can be relaxed to "fraction field is perfect".

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Haohao Liu Thu, 23 Jun 2022 03:33:54 GMT