Timeline for $M \oplus N$ is of finite type if $M,N$ are of finite type?
Current License: CC BY-SA 2.5
11 events
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| Nov 16, 2010 at 5:39 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
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| Nov 16, 2010 at 2:56 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
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| Nov 15, 2010 at 23:55 | comment | added | Sándor Kovács | @Torsten: I think I fixed it. | |
| Nov 15, 2010 at 23:53 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
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| Nov 15, 2010 at 23:38 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
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| Nov 15, 2010 at 21:23 | comment | added | Torsten Ekedahl | 1) It is true that direct limits are always right exact. 2) I don't see why $M'\to Q_i$ is surjective which seems to be needed to get the surjectivity of $\mathrm{lim} K'_i \to M'$. 3) Note that without Ab5 finite generation should be formulated as $\mathrm{lim}M_i\to M$ surjective implies $M_j=M$ for some $j$. | |
| Nov 15, 2010 at 20:57 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
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| Nov 15, 2010 at 20:53 | comment | added | Sándor Kovács | @Torsten: Thanks! Yeah, I thought this would be the issue, but I did not want to clutter it with proposed "sticky points". However, in some sense, this is an issue already where I indicated, as I wrote $\Leftrightarrow$ even though there only $\Rightarrow$ is needed, but if it were true there it would have worked for the $K_i$. | |
| Nov 15, 2010 at 20:50 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
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| Nov 15, 2010 at 20:45 | comment | added | Torsten Ekedahl | Your proposed sticky point is not one I think. However, without Ab5 directed colimits are not left exact so I don't think your conclusion that $\mathrm{lim} Q_i = 0$ implies $\mathrm{lim}K_i=M$ is justified (there seems to be problems with both injectivity and surjectivity of $\mathrm{lim}K_i\to M$). | |
| Nov 15, 2010 at 20:28 | history | answered | Sándor Kovács | CC BY-SA 2.5 |