Let $X$ be a smooth algebraic variety, $A \subset X$ a smooth subvariety, $f:Y \to X$ the blow-up of $X$ along $A$ and $M$ a quasi-coherent $O_X$-module (in the case I'm interested in, $M$ is actually a $D_X$-module but it doesn't change a thing).

Question: Is there a natural resolution of $O_Y$ as a $f^{-1}O_X$-module to compute the inverse image $Lf^*M = O_X \otimes^L_{f^{-1}O_Y} f^{-1}M$?

Any reference where I could find such a computation? The elementary case of $A = 0$ inside $X = \mathbb{A}^n$ would already be a great help.

Let $X$ be a smooth algebraic variety, $A \subset X$ a smooth subvariety, $f:Y \to X$ the blow-up of $X$ along $A$ and $M$ a quasi-coherent $O_X$-module (in the case I'm interested in, $M$ is actually a $D_X$-module but it doesn't change a thing).

Question: Is there a natural resolution of $O_Y$ as a $f^{-1}O_X$-module to compute the inverse image $Lf^*M = O_Y \otimes^L_{f^{-1}O_X} f^{-1}M$?

Any reference where I could find such a computation? The elementary case of $A = 0$ inside $X = \mathbb{A}^n$ would already be a great help.