Let $X$ be a regular scheme (all local rings are regular). Let $Y,Z$ be two closed subschemes defined by ideals sheaves $\mathcal I,\mathcal J$. Serre gave a beautiful formula to count the intersection multiplicity of $Y,Z$ at a generic point $x$ of $Y\cap Z$ as:
$$\sum_{i\geq 0} (-1)^i\text{length}_{\mathcal O_{X,x}} \text{Tor}_i^{\mathcal O_{X,x}}(\mathcal O_{X,x}/\mathcal I_x, \mathcal O_{X,x}/\mathcal J_x)$$
It takes quite a bit of work to show that this is the right definition (even that the sum terminates is a non-trivial theorem of homological algebra): it is non-negative, vanishes if the dimensions don't add up correctly, positivity etc. In fact, some cases are still open as far as I know. See here for some reference.
I have heard one of the great things about Lurie's thesis is setting a framework for derived algebraic geometry. In fact, in the introduction he used Serre formula as a motivation (it is pretty clear from the formula that a "derived" setting seems natural). However, I could not find much about it aside from the intro, and Serre formula was an old flame of mine in grad school. So my (somewhat vague):
Question: Does any of the desired properties of Serre formula follow naturally from Lurie's work? If so (since things are rarely totally free in math), where did we actually pay the price (in terms of technical work to establish the foundations)? EDIT: Clark's answer below greatly clarifies and gives more historical context to my question, highly recommend!)
Best Answer
There are a number of comments to make about Serre's intersection formula and its relation to derived algebraic geometry.
First, we should be a little more cautious about attribution. The idea of using "derived rings" to give an intrinsic version of the Serre intersection formula is not recent. The idea goes back at least to thoughts of Deligne, Kontsevich, Drinfeld, and Beilinson in the 1980s (and possibly earlier). These ideas have been made precise in a number of ways, in particular in work of Kapranov & Ciocan-Fontaine, and Toën & Vezzosi. EDIT: As Ben-Zvi reminded me below, one should also mention Behrend and Behrend-Fantechi on DG schemes and virtual fundamental classes. Of course Lurie's work has been the most comprehensive and powerful in its treatment of the foundations of DAG, but it's important to understand that his work arose in the context of these fascinating ideas.
Now, just to provide a little context, let me try to recall how Serre's formula arises from DAG considerations. Let's start by using the notation above, but let's assume for simplicity that $X$, $Y$, and $Z$ are all local schemes. (Some of the technicalities of DAG arise in making sheaf theory work with some sort of "derived rings," so our discussion will be easier if we ignore that for now.) So we write $X=\mathrm{Spec}(A)$, $Y=\mathrm{Spec}(B)$, and $Z=\mathrm{Spec}(C)$ for local rings $A$, $B$, and $C$.
Now if our aim is to intersect $Y$ and $Z$ in $X$, we know how to do that algebro-geometrically. We form the fiber product $Y\times_XZ=\mathrm{Spec}(B\otimes_AC)$. The tensor product that appears here is really the thing we're going to alter. To do that, we're going to regard $B$ and $C$ as (discrete) simplicial (commutative) $A$-algebras, and we're going to form the derived tensor product. This produces a new simplicial commutative ring $B\otimes^{\mathbf{L}}_AC$ whose homotopy groups are exactly the groups $\mathrm{Tor}^A_i(B,C)$. The intersection multiplicity is simply the length of $B\otimes^{\mathbf{L}}_AC$ as a simplicial $A$-module.
As Ben Webster says, the real joy of DAG is in thinking of the geometry of our new derived ring $B\otimes^{\mathbf{L}}_AC$ as a single unit instead of thinking only of its disembodied homotopy groups. The question you're asking seems to be: does thinking geometrically about this gadget help us to prove Serre's multiplicity conjectures in a more conceptual manner?
The short answer is: I don't know. I do not think a new proof of any of these has been announced using DAG (and it's definitely not in any of Lurie's papers), and in any case I do not think DAG has the potential to make the conjectures "easy." But let me see if I can make a case for the following idea: revisiting Serre's original method of reduction to the diagonal in the context of DAG.
Recall that, if $k$ is a field, if $A$ is a $k$-algebra, and if $M$ and $N$ are $A$-modules, then $$M\otimes_AN=A\otimes_{A\otimes_kA}(M\otimes_kN).$$ Hence to understand $\mathrm{Tor}^A_{\ast}(M,N)$, it suffices to understand $\mathrm{Tor}^{A\otimes_kA}_{\ast}(A,-)$. This allowed Serre to reduce to the case of the diagonal in $\mathrm{Spec}(A\otimes_kA)$. The key point here is that everything is flat over $k$, so Serre could only use this to prove the multiplicity conjectures for $A$ essentially of finite type over a field. Observe that the same equality holds if we work in the derived setting: if $M$ and $N$ are simplicial $A$-modules, and $A$ is an $R$-algebra, then the derived tensor product of $M$ and $N$ over $A$ can be computed as $$A\otimes^{\mathbf{L}}_{A\otimes^{\mathbf{L}}_RA}(M\otimes^{\mathbf{L}}_RN).$$ The gadget on the right (or, strictly speaking, its homotopy) has a name familiar to toplogists; it's the Hochschild homology $\mathrm{HH}^R(A,M\otimes^{\mathbf{L}}_RN)$.
The hope is that we've chosen $R$ cleverly enough that $B\otimes^{\mathbf{L}}_RC$ is "less complicated" than $B\otimes^{\mathbf{L}}_AC$. (More precisely, we want the $\mathrm{Tor}$-amplitude of $M$ and $N$ to decrease when we think of them as $R$-modules. There's a particular way of building $R$, but let me skip over this point.)
Has our situation improved? Perhaps only a little: we've turned our problem of looking at the derived intersection $Y\times^h_XZ$ into the study of the derived intersection of the diagonal inside $X\times^h_RX$ with some simpler derived subscheme $Y\times^h_RZ$ thereof. But now we can try to iterate this, working inductively.
I don't know whether this can be made to work, of course.