Timeline for Multiplicative Structures On Free Resolutions
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Jul 22, 2011 at 8:58 | vote | accept | Hanno | ||
| Dec 4, 2010 at 13:43 | comment | added | Tyler Lawson | For some blasted reason I have written the word "nonassociative" absolutely everywhere, when I instead mean "associative (noncommutative)" - and obliviously have not figured out why until now. Sorry. | |
| Nov 30, 2010 at 15:06 | history | edited | Hanno | CC BY-SA 2.5 |
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| Nov 30, 2010 at 0:17 | comment | added | Hanno | Ok, I will ask more specifically, hopefully it's not too basic: First, what do you mean by the "free nonassociative resolution"? The dg-algebra I thought of is the free associative algebra with generators and relations given above, but this doesn't seem to be what you are talking about. Secondly, what's the precise relation to the general two-sided bar construction you mentioned, und what are good references for the latter? | |
| Nov 29, 2010 at 19:14 | comment | added | Tyler Lawson | Unfortuantely I don't remember references immediately - esp for the equivalence between the Koszul algebra and the nonassociative one, I only know the argument - but am sure someone else probably does. So far as functorial resolutions, a standard technique called the "two-sided bar construction" using the forgetful-free adjoint pair between A-modules and sets (or A-modules and some base algebra over which S and S/f are flat) works. And model categories/Quillen equivalences have quite a number of references now. If there are specifics please let me know and I'll try to track something down. | |
| Nov 29, 2010 at 18:27 | comment | added | Hanno | That sounds very intersting, Tyler, thank you! Could you give references? | |
| Nov 29, 2010 at 16:46 | answer | added | Jesse Burke | timeline score: 8 | |
| Nov 29, 2010 at 16:32 | comment | added | Tyler Lawson | Whenever f is not a zero divisor, your "free nonassociative" resolution is indeed weakly equivalent to S/f, and freeness allows it to automatically lift to the endomorphism algebra of a free resolution of any S/f-module. So far as functoriality, whether you use the Koszul algebra or the free nonassociative algebra there are functorial resolutions of M that you want - the obstructions only come if you're trying to work with a specific resolution such as a minimal resolution. In fact, S/f, the Koszul algebra, and the nonassoc. alg. all have equivalent derived categories (Quillen equivalence!). | |
| Nov 29, 2010 at 9:43 | history | asked | Hanno | CC BY-SA 2.5 |