Timeline for Derived Nakayama for complete modules
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 8, 2020 at 21:23 | comment | added | Leonid Positselski | Yes, the augmented Čech complex $\check C(A;\mathbf s)$ is the direct limit of the Koszul complexes $Kos(A;s_1^n,\dotsc,s_m^n)$ over $n\to\infty$. | |
| Jun 8, 2020 at 9:29 | comment | added | user20948 | Thanks a lot to you along with @Anonymous. If I am not mistaken, the Čech complex in question is closely related to the Koszul complex, as the second proof shows. | |
| Jun 7, 2020 at 18:56 | comment | added | Leonid Positselski | @Anonymous Yes, this simplified version of the argument also works. | |
| Jun 7, 2020 at 15:43 | comment | added | Anonymous | It is not necessary to use MGM duality. Say $C \otimes_A^L A/I = 0$ with $C$ derived $I$-complete. Choose generators $x_1,...,x_r \in I$. Then $C \otimes_A^L Kos(A; x_1,...,x_r) = 0$ as the Koszul complex has finitely many homology groups and all of those are $A/I$-modules. Now note that $C \otimes_A^L Kos(A;f) = 0$ iff $f$ acts invertibly on $C$; thus, if $C$ is also derived $f$-complete (i.e., $Rlim_f C = 0$), this forces $C = 0$. Using this observation, proceed by descending induction to show that $C \otimes_A Kos(A;x_1,...,x_i) = 0$ for $i \geq 0$. The $i=0$ case implies the lemma. | |
| Jun 7, 2020 at 13:54 | history | edited | Leonid Positselski | CC BY-SA 4.0 |
A reference made more precise.
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| Jun 7, 2020 at 13:49 | comment | added | Leonid Positselski | @Yai0Phah I've edited the answer above, adding an answer to your question. | |
| Jun 7, 2020 at 13:48 | history | edited | Leonid Positselski | CC BY-SA 4.0 |
A (long) answer to the question about the unbounded case asked in the comments is added.
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| Jun 7, 2020 at 9:11 | comment | added | user20948 | What about unbounded case? In topology, if I am not mistaken, given a $p$-complete spectrum without boundedness conditions, if $X/p\simeq0$, then $X\simeq0$. | |
| May 14, 2019 at 13:11 | vote | accept | slinshady | ||
| May 14, 2019 at 13:05 | history | answered | Leonid Positselski | CC BY-SA 4.0 |