Let $A \rightarrow B$ and $C \rightarrow B$ be two maps of schemes. How can I compute the derived fiber product $A \otimes^L_B C$? I'm guessing this is a dg-scheme.

For instance - let $B=\mathbb{A}^1$ and $A = 0 \in \mathbb{A}^1$, and $C \in \mathbb{A}^1$ some point (either $0$ or $1$).

**Question**: The example I'm really interested in is $\tilde{\mathfrak{g}} \otimes^L_{\mathfrak{g}} 0$, where $\mathfrak{g}=\mathfrak{sl}_2$ and $\tilde{\mathfrak{g}}=\{ X,(0 \subset V \subset \mathbb{C}^2) | X \in \mathfrak{sl}_2, X V \subset V \}$ is the Grothendieck-Springer resolution.