Timeline for Does projective imply flat?
Current License: CC BY-SA 3.0
17 events
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| Jul 1, 2015 at 0:23 | vote | accept | Theo Johnson-Freyd | ||
| Jun 30, 2015 at 22:15 | comment | added | Theo Johnson-Freyd | @EricWofsey Ah, good point. | |
| Jun 30, 2015 at 22:14 | comment | added | Eric Wofsey | @TheoJohnson-Freyd: A projective non-flat object can never have a flat resolution, since by projectivity the first map in the resolution will split and a direct summand of a flat object is flat. | |
| Jun 30, 2015 at 22:06 | comment | added | Theo Johnson-Freyd | Well, grumble. And this gives an example where I don't even have flat resolutions, since by the same logic $(N,1)$ fails to be flat for every $N$. (Tensoring with $(M,0)$ and then projecting onto the $\mathcal B$ factor still gives the functor $F$.) | |
| Jun 30, 2015 at 21:10 | comment | added | thel | yeah, book-keeping problem on my side, sorry. Both ways work. | |
| Jun 30, 2015 at 21:06 | history | edited | Eric Wofsey | CC BY-SA 3.0 |
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| Jun 30, 2015 at 21:01 | comment | added | Eric Wofsey | @ThelSeraphim: I don't see how the $V\otimes W$ term causes any trouble; it is natural to include it in the special case described in the penultimate paragraph. I think you're right that the "square-zero" version without it also works, though. | |
| Jun 30, 2015 at 20:46 | comment | added | thel | I think your $V \otimes W$ factor may be spurious? Doesn't it spoil associativity? I'm pretty sure the square-zero extension version of what you're saying works. Beautiful example. | |
| Jun 30, 2015 at 20:34 | history | edited | Eric Wofsey | CC BY-SA 3.0 |
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| Jun 30, 2015 at 20:29 | history | edited | Eric Wofsey | CC BY-SA 3.0 |
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| Jun 30, 2015 at 20:21 | history | edited | Eric Wofsey | CC BY-SA 3.0 |
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| Jun 30, 2015 at 20:16 | history | edited | Eric Wofsey | CC BY-SA 3.0 |
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| Jun 30, 2015 at 19:38 | history | edited | Eric Wofsey | CC BY-SA 3.0 |
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| Jun 30, 2015 at 19:25 | history | edited | Eric Wofsey | CC BY-SA 3.0 |
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| Jun 30, 2015 at 19:20 | history | edited | Eric Wofsey | CC BY-SA 3.0 |
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| Jun 30, 2015 at 19:12 | history | edited | Eric Wofsey | CC BY-SA 3.0 |
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| Jun 30, 2015 at 19:07 | history | answered | Eric Wofsey | CC BY-SA 3.0 |