Timeline for Determinantal identities for perfect complexes
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 7, 2020 at 17:53 | answer | added | Marc Hoyois | timeline score: 11 | |
| Mar 5, 2020 at 22:52 | comment | added | Damian Rössler | @François Brunault. Thank you for the reference. The Knudsen-Mumford does just that though. My question concern the KM determinant. | |
| Mar 5, 2020 at 22:50 | comment | added | Damian Rössler | @Evgeny Shinder. Thank you for your comment. Yes I agree. These mild conditions (eg quasi-projective) imply that perfect complexes are isomorphic in the derived category to finite complexes of vector bundles and then one can reduce to vector bundles (as in the example). But even then, I need a canonical iso., invariant under base change. | |
| Mar 5, 2020 at 22:35 | comment | added | François Brunault | In the article "Tamagawa numbers for motives with (non-commutative) coefficients" by Burns and Flach emis.de/journals/DMJDMV/vol-06/21.html the authors extend some determinant functor from the category of projective $R$-modules to perfect complexes. I don't know whether this gives something in the direction you want, but maybe the approach could be useful. | |
| Mar 5, 2020 at 21:55 | comment | added | Evgeny Shinder | ...and under not very restrictive conditions on $X$, e.g. quasiprojective we have $K_0(Perf(X)) = K_0(VB(X))$ ($VB$ stands for vector bundles). Hence if an additive identity is checked on vector bundles, it seems to follow for perfect complexes. | |
| Mar 5, 2020 at 21:47 | comment | added | Evgeny Shinder | One simple-minded comment is that $det$ is a homomorphism from $K_0(Perf(X))$ to $Pic(X)$, hence | |
| Mar 5, 2020 at 9:58 | history | asked | Damian Rössler | CC BY-SA 4.0 |