Timeline for Koszul duality and coherent sheaves on projective space
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 1, 2018 at 20:18 | comment | added | user105178 | @dhy I see. Thank you for clarifying! | |
| Jul 1, 2018 at 19:43 | comment | added | dhy | What version of inhomogeneous Koszul duality are you using? The only version I know of requires $A$ to be a filtered deformation of a quadratic algebra, in which case what is your filtration on $A$? The only one that comes to mind for me makes $A$ a homogeneous quadratic algebra. Also I was under the impression that in this case you still necessarily have the identity that the power series of dimensions of grade components of $A$ and $B$ multiplied to $1$, which is evidently false here. | |
| Jul 1, 2018 at 19:22 | comment | added | user105178 | @dhy Also, I seem to remember that the exterior algebra B can be obtained from an exceptional collection, which also comes from the resolution of the diagonal. | |
| Jul 1, 2018 at 19:06 | comment | added | user105178 | @dhy The Koszul dual of B is indeed a polynomial algebra. However, since A an inhomogeneous quadratic algebra, there is still a chance that the Koszul dual of A is B, just like the relationship between the enveloping algebra of a Lie algebra and the Chevalley complex. | |
| Jul 1, 2018 at 18:46 | comment | added | dhy | I don't think there is any relationship between $A$ and $B$. The Koszul dual of $B$ is the symmetric algebra, which looks completely different from $A$ (in particular, it is not finite dimensional.) $B$ is also very canonical in a sense, while $A$ is not (it depends highly on an exceptional collection.) | |
| Jul 1, 2018 at 17:43 | comment | added | meh | I'm not sure what the Beilinson quiver module might be, but you should look at Eisenbud's book on syzygies in which there is a duality established, one side of which is what you call B. | |
| Jul 1, 2018 at 3:43 | history | asked | user105178 | CC BY-SA 4.0 |