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.



Pull-back on a blow-up
The easiest way to compute is the following. Assume that $A$ is the zero locus of a regular section $s$ of a vector bundle $E$ on $X$ (this is always the case locally). Then there is an embedding $i:Y \to P\_X(E)$ over $X$. Then $i\_\*O\_Y$ has a nice resolution. 


Indeed, consider the relative Grothendieck line bundle $O(-1)$ on $P\_X(E)$. We have a natural embedding $O(-1) \to p^\*E$, where $p:P\_X(E) \to X$ is the projection. Consider the cokernel of this morphism, $Q$ (in fact $Q$ is isomorphic to the relative tangent bundle up to a twist). On the other hand, $p^\*$ is a section of $p^\*E$, so it gives a section $s'$ of $Q$. Then $i(Y)$ is the zero locus of $s'$ on $P\_X(E)$. Moreover, this section is regular, so we have the following Koszul resolution
$$
0 \to \Lambda^{r-1}Q^\* \to \dots \to \Lambda^2Q^\* \to Q^\* \to O\_{P\_X(E)} \to i\_\*O\_Y \to 0.
$$
This resolution can be used to compute $i\_\*Lf^\*M$ for any quasicoherent sheaf $M$ on $X$. Indeed, we have
$$
i\_\*Lf^\*M = i\_\*Li^\*Lp^\*M = Lp^\*M\otimes^L i\_\*O\_Y.
$$
Since $i$ is a closed embedding this gives a nice control over $Lf^\*M$.


EDIT. In case $X = A^n$, $A = \{0\}$ we can take $E = O^n$, $s = (x\_1,\dots,x\_n)$, where $x\_i$ are the coordinates. Then $P\_X(E) = A^n\times P^{n-1}$ and $Q = T\_{P^{n-1}}(-1)$. So, $i\_\*Lf^\*M$ is quasiisomorphic to the following complex
$$
M\boxtimes\Omega^{n-1}(n-1) \to \dots \to M\boxtimes\Omega^2(2) \to M\boxtimes\Omega^1(1) \to M\boxtimes O.
$$
For example, if $M$ is a skyscraper in $0$ then the above complex is supported on the exceptional divisor $P^{n-1} \subset Y$ and all differential vanish, so one concludes that $L\_pf^\*M \cong j\_\*\Omega^p(p)$, where $j:P^{n-1} \to Y$ is the embedding.


