$T$-equivariant resolution of coherent sheaf

Question

Let $X := \{x_1x_2= 0\} \subset \mathbb{C}^2$ and let $\mathcal{F} := \mathcal{O}_0$ where $0\in X$ is the origin. Let $T = \left\{\begin{pmatrix} t_1 & 0 \\ 0 & t_2\end{pmatrix}\right\}\subset \operatorname{GL}_2$ act on $X$ diagonally.

What is the minimal $T$-equivariant resolution of $\mathcal{F}$?

Answer 1

The resolution has the form $$ \dots \to \mathcal{O} \oplus \mathcal{O} \stackrel{\left(\begin{smallmatrix} x_1 & 0 \\ 0 & x_2\end{smallmatrix}\right)}\longrightarrow \mathcal{O} \oplus \mathcal{O} \stackrel{\left(\begin{smallmatrix} x_2 & 0 \\ 0 & x_1\end{smallmatrix}\right)}\longrightarrow \mathcal{O} \oplus \mathcal{O} \stackrel{\left(\begin{smallmatrix} x_1 & x_2\end{smallmatrix}\right)}\longrightarrow \mathcal{O} \longrightarrow \mathcal{F} \longrightarrow 0. $$ It is obvious that it has a $T$-equivariant structure.