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.