# Reference request: excess normal bundle and derived pullback

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.

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.

• Perfect, thanks. Aug 3, 2020 at 17:39

Perhaps, Proposition 1.28 in https://arxiv.org/abs/1411.7994 may help.

• Thanks Sasha, this is very nice too! Aug 3, 2020 at 17:44
• I have an idle dream of writing a sequel to Huybrechts' Fourier-Mukai book that includes all these little things that I need all the time. Or maybe you can do it, you're much energetic than I am. Aug 3, 2020 at 17:50
• I have no experience in writing books, so I would be afraid to start. Aug 3, 2020 at 18:00