After learning the trditional intersection theory (W. Fulton's Intersection Theory and D. Eisenbud & J. Harris's 3264 and all that), I have some biased thinking about what I have learned in this period of time：

The first is the excess intersection formula:

Theorem 1. (Excess intersection formula) For regularly embedded subschemes $X_i\subset Y$ , a closed subscheme $V\subset Y$ and a connected component $Z\subset V\cap\bigcap_iX_i$, we have $$(X_1\cdot\ldots\cdot X_r\cdot V)^Z=\left\{s(Z,V)\cap\prod_{i=1}^rc(N_{X_i}Y|_Z)\right\}_{\dim V-\sum\mathrm{codim}(X_i,Y)}$$

Using this formula we can compute the contribution of degree support on the wrong dimension connected component. But is there some explanations using derived-philosophy? I guess may be the intersection information of bad dimensions (like self-intersection and excess intersection formula as above) can be reflected in some higher ranks of the derived intersection product.

Second, we all know the Serre's formula about intersection multiplicity:

Theorem 2. (Serre) If $A,B\subset X$ are two closed subschemes of smooth variety $X$ with expected dimensions and $Z$ be an irreducible component of $A\cap B$, then $$i(Z;A,B;X)=\sum_{i=0}^{\dim X}(-1)^i\mathrm{length}_{\mathscr{O}_{A\cap B,Z}}\mathrm{Tor}_i^{\mathscr{O}_{X,Z}}(\mathscr{O}_{A,Z},\mathscr{O}_{B,Z}).$$

Note that the zero-degree term is just $\mathrm{length}_{\mathscr{O}_{A\cap B,Z}}\mathscr{O}_{X,Z}/(\mathscr{I}_{A},\mathscr{I}_{B})$. Now is there relationship between this formula and some derived tensor product (maybe $\mathscr{O}_{A,Z}\otimes^{\mathbf{L}}_{\mathscr{O}_{X,Z}}\mathscr{O}_{B,Z}$)? What is the meaning of this tensor product? (or some derived intersection product?)

Maybe these things are related to the derived algebraic geometry (or the derived intersection theory I guess), is there some references (papers/notes/books) about derived-aspect intersection theory (char$=0$ is OK)?

Any help will be grateful!