Suppose $X$ is a smooth projective variety over $\mathbb{C}$. How can one understand that in $D(Coh(X))$, the structure sheaves corresponding to different points of $X$ are all non-isomorphic? Here by structure sheaf I mean that points get the usual structure of closed subschemes of $X$.
Asked
Active
Viewed 84 times
3
-
Is $D$ the derived category and you're interpreting a sheaf and a complex centered in degree $0$? – Jim Mar 23 '14 at 22:46
-
Yes indeed, that's what I mean. – Karsten Mar 23 '14 at 23:05
1 Answers
2
First note that an isomorphism between elements of the derived category induces an isomorphism between the homologies of those elements, so if two sheaves are isomorphic in the derived category then they're isomorphic as sheaves.
Then to see that they're not isomorphic as sheaves just note that their supports are all different.
I don't believe the fact that $X$ is smooth projective over $\mathbb C$ has anything to do with this. I think it's true in general.
Jim
- 30,682