TITLE: Computing intersection multiplicity using Tor - explicit example
QUESTION [12 upvotes]: When trying to compute the (Serre-generalized) intersection number of two varieties at a closed point, I came to a need to compute the following $\operatorname{Tor}$:

Let $k$ be an algebrically closed field, $A=k[x_1,x_2,x_3,x_4]$ and $\mathfrak m=(x_1,x_2,x_3,x_4)$. Let $M = k[x_1,x_2,x_3,x_4]/(x_1x_3,x_1x_4,x_2x_3,x_2x_4)$, $N=k[x_1,x_2,x_3,x_4]/(x_1-x_3,x_2-x_4)$. I want to compute $\operatorname{Tor}^i_{A_{\mathfrak m}}(M_{\mathfrak m},N_{\mathfrak m})$.

Any ideas how to do this?
I first noted that $N$ is gotten from $A$ by quotient by a a regular sequence, so that the Koszul complex of $(x_1-x_3,x_2-x_4)$ is a free resolution of $N$ over $A$. However, tensoring with $M$ computations became too hard, and I was not able to find the cohomology of the resulting complex.
Any ideas?
Thanks!

REPLY [2 votes]: You can do this computation without thinking much, really.
The two generators of the ideal which defines $N$ form a regular sequence, so you can use a Koszul complex to projectively resolve $N$. Tensor it with $M$ and just compute the homology of the resulting complex.