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 #3395 on tag 00I7 by Matthieu Romagny https://stacks.math.columbia.edu/tag/00I7#comment-3395 A new comment by Matthieu Romagny on tag 00I7. Remove "algebraically" in the statement of the Lemma.

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Matthieu Romagny Sun, 17 Jun 2018 04:40:03 GMT
#3394 on tag 0DTJ by Matthieu Romagny https://stacks.math.columbia.edu/tag/0DTJ#comment-3394 A new comment by Matthieu Romagny on tag 0DTJ. Remove "for" in statement of condition (1).

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Matthieu Romagny Sun, 17 Jun 2018 03:05:47 GMT
#3390 on tag 0BNH by Vignesh https://stacks.math.columbia.edu/tag/0BNH#comment-3390 A new comment by Vignesh on tag 0BNH. Sorry about the typo in my comment, in the second paragraph, I wanted to say:

By Lemma 10.86.1, we get that $0\to \lim K/(I^nR^t\cap K)\to (R^t)^\wedge \to M^\wedge \to 0$ is exact. By Artin-Rees lemma, $\lim K/(I^nR^t\cap K)=K^\wedge$ . Therefore we get $0\to K^\wedge\to (R^t)^\wedge \to M^\wedge \to 0$ is exact as required.

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Vignesh Fri, 08 Jun 2018 12:59:34 GMT
#3389 on tag 0BNH by Vignesh https://stacks.math.columbia.edu/tag/0BNH#comment-3389 A new comment by Vignesh on tag 0BNH. A different presentation of the proof of Lemma 10.96.1:

Consider a presentation of module M (which is f.g. by assumption) $0\to K\to R^t\to M\to 0$ We get that $0\to K/(I^nR^t\cap K)\to R^t/(I^nR^t)\to M/(I^nM)\to 0$ is exact. (Here $K$ is viewed as a submodule of $R^t$.) One sees this by observing that $I^nR^t$ maps surjectively onto $I^nM$.

By Lemma 10.86.1, we get that $0\to \lim K/(I^nR^t\cap K)\to R^\wedge \to M^\wedge \to 0$ is exact. By Artin-Rees lemma, $\lim K/(I^nR^t\cap K)=K^\wedge$ Therefore we get $0\to K^\wedge\to R^\wedge \to M^\wedge \to 0$ is exact as required.

A correction in the proof of Lemma 10.96.2:

Here I think you mean an arbitrary ideal $J$ in R (and not specifically the ideal $I$ w.r.t. which $R$ is completed): $J \otimes _ R R^\wedge \to R \otimes _ R R^\wedge = R^\wedge$.

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Vignesh Fri, 08 Jun 2018 12:43:52 GMT
#3388 on tag 03QR by Dario https://stacks.math.columbia.edu/tag/03QR#comment-3388 A new comment by Dario on tag 03QR. Typo: Let $K^{sep}$ a separable closure...missing be

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Dario Fri, 01 Jun 2018 06:09:35 GMT
#3387 on tag 09YS by Dario https://stacks.math.columbia.edu/tag/09YS#comment-3387 A new comment by Dario on tag 09YS. Typo in (1): the map on stalks should also go from F to G

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Dario Fri, 01 Jun 2018 05:33:58 GMT
#3386 on tag 09XY by Dario https://stacks.math.columbia.edu/tag/09XY#comment-3386 A new comment by Dario on tag 09XY. Typo: Moreo generally... Just above 0BDS

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Dario Fri, 01 Jun 2018 04:54:37 GMT
#3385 on tag 07Z6 by shanbei https://stacks.math.columbia.edu/tag/07Z6#comment-3385 A new comment by shanbei on tag 07Z6. In the third line of proof of (7) in Lemma 07ZA, perhaps you meant the rank of image is less than n instead of \leq?

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shanbei Sun, 27 May 2018 01:01:13 GMT
#3384 on tag 06LB by Daniel Litt https://stacks.math.columbia.edu/tag/06LB#comment-3384 A new comment by Daniel Litt on tag 06LB. Lines (1), (2), (7), and (11) seem not to be rendering correctly on my computer.

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Daniel Litt Sat, 26 May 2018 05:58:58 GMT
#3382 on tag 03LE by Reimundo Heluani https://stacks.math.columbia.edu/tag/03LE#comment-3382 A new comment by Reimundo Heluani on tag 03LE. In Lemma 38.12.2, $f$ should be $f:Y\rightarrow X$.

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Reimundo Heluani Wed, 23 May 2018 12:16:47 GMT