Exterior tensor product of structure sheaves

I am reading the book "Fourier-Mukai transforms in algebraic geometry" by Daniel Huybrechts and to solve one of its questions, I came up to show that $$\mathcal{O}_{X_1}\boxtimes\mathcal{O}_{X_2}=\mathcal{O}_{X_1\times X_2}$$ i.e. $${\pi_{X_1}}^*\mathcal{O}_{X_1}\otimes{\pi_{X_2}}^*\mathcal{O}_{X_2}=\mathcal{O}_{X_1\times X_2}$$ where here $\pi_{X_1}$ (resp. $\pi_{X_2}$) is the projection from $X_1\times X_2\to X_1$ (resp. $X_1\times X_2\to X_2$) and I am looking at $\mathcal{O}_{X_1}$, $\mathcal{O}_{X_2}$ and $\mathcal{O}_{X_1\times X_2}$ as objects in $D^b(X_1)$, $D^b(X_2)$ and $D^b(X_1\times X_2)$, respectively. My question is that is this equality true and if yes, why?

Solution 1:

Yes. The way you compute a derived pullback is to take a flat resolution and then pull that back, but the structure sheaf is trivially flat and pulls back to the structure sheaf.