Free Resolution of this determinantal variety.  I am looking for a free resolution of the ideal generated by $2\times 2$-minors of a $3\times 3$ -matrix. More precisely let $M$ be a matrix (sorry but I cannot write a matrix for some TeX technical reason)
$$
M=\begin{bmatrix}
x_{1}& x_{2}& x_{3} \\\
x_{4}& x_{5}& x_{6} \\\ 
x_{7}& x_{8}& x_{9}
\end{bmatrix}
$$
whose entries are indeterminates. I would like to find a free resolution of the ideal generated by $2\times 2$-minors of $M$ in the ring $\mathbb{C}[x_{1},\dots,x_{9}]$. 
Does anyone know the reference for this resolution? Or make one for me. 
 A: The standard way to construct a resolution is the following. Let $X = A^9$ and let $E$ be a 3-dim (trivial) vector bundle on $X$ such that $M \in End(E)$. Consider the relative Grassmannian $p:Gr(2,E) \to X$ and let $U$ be its tautological rank 2 bundle. Let $Z$ be the zero locus of the morphism $U \to p^*E \stackrel{\ M\ }\to p^*E$ on $Gr(2,E)$. Then it is clear that you ask for a resolution of $p(Z)$. What is good is that it is easy to write a resolution for $Z$ on $Gr(2,E)$, and then you can ush it forward to $X$. 
Indeed, $Z$ is the zero locus of a regular section of the vector bundle $U^*\otimes p^*E$. Consequently, $O_Z$ has a resolution by the Koszul complex
$$
\dots \to \Lambda^2(U\otimes p^*E^*) \to U\otimes p^*E^* \to O_{Gr(2,E)} \to O_Z\to 0.
$$
It remains to note that $\Lambda^k(U\otimes p^*E) = \oplus \Sigma^\alpha U\otimes p^*\Sigma^{\alpha^T}E^*$, the sum is over all Young diagrams in a $2\times 3$ rectangle, $\alpha^T$ is the transposed diagram, and $\Sigma^\alpha$ is the Schur functor. To push forward to $X$ one can use the projection formula and Borel--Bott--Weil to compute the cohomology on $Gr(2,E)$. As a result you will get
$$
0 \to O_X \to
E\otimes E^* \to 
sl(E) \oplus sl(E) \to 
E^*\otimes E \to
O_X \to O_{p(Z)} \to 0.
$$
