Calculating intersection multiplicities explicitly with Serre's formula

Question

Let's say I want to explicitly calculate the intersection multiplicity of two subvarieties of $\Bbb A^n_k$ using Serre's Tor-formula (involving as little homological algebra as possible).
A typical example [Hartshorne, p.428] consists of two planes in $\Bbb A^4_k$ (we can assume $k$ algebraically closed, if needed) meeting at a point, $X=V(I)$, with $$I=(x,y)\cap(z,w)=(x,y)(z,w)=(xz,xw,yz,yw)\subset R=k[x,y,z,w],$$ intersecting with a third plane $Y=V(J)$, with $J=(x-z,y-w)$.

The "naive", scheme-theoretic intersection is given by $$\frac{R}{I+J}=\frac{k[x,y,z,w]}{(xz,xw,yz,yw,x-z,y-w)}\cong \frac{k[x,y]}{(x^2,xy,y^2)},$$ which has length 3 (while, clearly, we expect the intersection multiplicity to be 2).
Serre's formula gives us the correct multiplicity by defining $$i(X,Y)=\sum_i (-1)^i \operatorname{length} \operatorname{Tor}^R_i(R/I,R/J).$$

1. We know that for $i=1$, $\operatorname{Tor}^R_1(R/I,R/J)=(I\cap J)/{IJ}$. How do we compute this quotient conveniently?
   (Embarassingly enough, I'm having some trouble calculating generators for $I\cap J$.)
2. What do we know about $\operatorname{Tor}_i$ for $i\geq 2$?

Answer 1

I couldn't think of any way to compute $I\cap J$ in general other than by using Groebner basis (probably since it is my favorite). Probably you already know this but the idea is to compute the ideal $tI+(1-t)J$ and eliminate the variable $t$ by lex-order.

This way, we can compute $I\cap J =(yz-xw,y^2w-yw^2,xyw-xw^2,x^2w-xzw,x^2z-xz^2)$ and since all of its generators except $yz-xw$ are in $IJ$, $I\cap J /IJ$ is generated by $yz-xw$.

For computing higher Tor, we can consider the short exact sequence $0\to J\to R\to R/J\to 0$, which induces the long exact sequence on Tor, and note that $\operatorname{Tor}^R_i(R/I,R)=0$ for $i\ge 1$, we have that $$\operatorname{Tor}^R_{i+1}(R/I,R/J)=\operatorname{Tor}^R_i(R/I,J) , \quad \forall i\ge 1.$$

The free resolution for $J$ is $\ldots \to 0 \to 0 \to R^1 \to R^2 \to J \to 0$, so $\operatorname{Tor}^R_i(R/I,J)=0$ for all $i\ge 2$.

Lastly, to compute $\operatorname{Tor}^R_{2}(R/I,R/J)=\operatorname{Tor}^R_1(R/I,J)$, consider the short exact sequence $0\to I\to R\to R/I\to 0$, we have that $\operatorname{Tor}^R_1(R/I,J) = \operatorname{Ker}(I\otimes J \to R\otimes J) =0 $.