Structure sheaves in different point are not isomorphic?

Question

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 $X$.

Answer 1

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.