Exact sequence of normal cones

Question

Suppose that we have a sequence $i: X \hookrightarrow Y$, $j: Y \hookrightarrow Z$ of closed embeddings of varieties such that $i$ is regular. In this case, do we have an exact sequence of cones of the form

$N_{X}Y \rightarrow C_{X}Z \rightarrow i^{*} C_{Y}Z$,

where $i^{*} C_{Y}Z := C_{Y}Z \times _{Y} X$? What is the reference for this fact?

The above seems to be used implicitly in the proof of the analogous statement for the intrinsic normal cone of Behrend-Fantechi. On the other hand, the corresponding fact for the Segre classes seems to be wrong (as in 4.2.8 in Fulton), but perhaps what goes wrong is that the Segre class of the pullback of the cone along a regular embedding is not the same as the Gysin homomorphism of the cone itself.

Answer 1

This is false, a sequence like that need not necessarily be exact.

The counterexample is the same as in Fulton, with $Z$ a cone over a quadric, $Y$ a line through the vertex and $X$ the vertex itself; see Virtual Classes for the Working Mathematician, 3.5.1.b).