To study the geometry of the moduli space $\mathcal M$ of semi-stable sheaves on a variety $X$ with fixed Hilbert polynomial, it is useful to have a locally free resolution of the structure sheaf $\mathcal O_\Delta$ of the diagonal $\Delta \subset \mathcal M \times \mathcal M$. (I am interesting in $X=\mathbb{CP^2}$, then for a general Hilbert polynomial the moduli space $\mathcal M$ is a smooth projective variety).
King, Walter On Chow Rings of Fine Moduli Spaces of Modules (statement of thm. 1 and section 1) give the construction of the class $[\Delta]$ of the diagonal in Chow ring $A^*(\mathcal M \times \mathcal M)$ given the resolution of universal bundle $\mathcal U$ on $\mathcal M \times X$. Is it possible to generalize it to obtain the class not in the Chow ring but in the derived category $D^b(\mathcal M)$?