Is it possible to do intersection theory in the derived category of a scheme? Let's say we are given a smooth scheme $X$ over $\mathbb{C}$, is it possible to do intersection theory in the (bounded) derived category of coherent sheaves? 
I want to know if there is a way to extend the naive thought that for two subschemes $V$ and $Z$ of $X$, $\mathcal{O}_V \otimes^L \mathcal{O}_Z$ is the "intersection" of $V$ and $Z$. Reference guides are also welcome.
 A: This is certainly possible--in fact, it's essentially Grothendieck's original approach to defining the intersection product on the (rational) Chow groups of regular schemes.  Grothendieck shows (see these nice notes of Gillet) that there is a graded isomorphism $$\bigoplus_k CH^k(X)_\mathbb{Q}\to Gr^*_{\gamma} K_0(X)$$ where $\gamma$ is a certain filtration on $K_0(X)$; this map is obtained via a sort of Chern character.  (As Leo Alonso notes, this may be found in SGA 6.)
How does the derived category come in?  Well first, one may recover $K_0(X)$ from the derived category of (bounded) complexes of coherent sheaves on $X$, by taking the free abelian group on objects modulo the relation that $[X]+[Z]=[Y]$ if $$X\to Y\to Z\to \Sigma X$$ is a distinguished triangle.  It's not too hard to see that this is the usual $K_0(X)$ (see e.g. 3.1.4 in these notes of Schlichting).  The product in $K_0(X)$ is precisely given by $[F]\cdot [G]=[F\otimes^L G]$.
Combining these two results, we exactly realize your dream of doing intersection theory (in the usual sense) via the derived category.  (Of course for actual applications, one should check that this product on $CH^k(X)_\mathbb{Q}$ agrees with a more geometric definition, e.g. via deformation to the normal cone as in Fulton.)
