Consider a fiber square
$\require{AMScd}$
\begin{CD}
 X' @>i'>> Y'\\
 @V g V V @VV f V\\
 X @>>i> Y,
\end{CD}
where $i$ and $i'$ are regular immersions, and consider the *excess normal bundle* defined by the exact sequence
$$ 0 \to N\_{X'/Y'} \to N\_{X/Y} \to E \to 0, $$
which measures the failure of $f$ to be transverse to $i(X)$ in the sense of differential topology.


Does anyone know a reference for the fact that $L\_j f^\*(i\_\* \mathcal O\_X) = \Lambda^j E^\*$?


If $f$ is also a regular immersion then this is SGA 6, VII, Proposition 2.5, although that's not the friendliest reference. If need be I can derive the fact I want from that special case, but I'd rather just have it off the shelf.



Reference request: excess normal bundle and derived pullback
See Lemma 3.2 in the following paper: R. W. Thomason, *Les K-groupes d'un schéma éclaté et une formule d'intersection excédentaire*, Invent. Math. **112**, 195--215 (1993), [DOI](https://doi.org/10.1007/BF01232430).


