Canonical resolutions commute with embeddings of ambient smooth schemes

The invariant of the embedded ideal. Running Steps 2a and 1bb of the algorithm for $({\mathcal{I}}^{\mathrm{\prime }},\mathrm{\varnothing },1)$ at a point $x\in X$: in Step 2a the ideal is of maximal order with ${\mathrm{ord}}_N=1$, so the companion ideal is ${\mathcal{I}}^{\mathrm{\prime }}$ itself and the invariant receives the first coordinate $1$; in Step 1bb (non-boundary case $s(x)=0$) the resolution passes to a hypersurface of maximal contact. Passing successively to the tangent directions $u_1,\dots ,u_k$, each passage adjoins the source pair $(1,0)$, encoded as the four rational coordinates $(0,1,0,0)$ under [F3], to the invariant by [F2], and after $k$ passages one arrives at the restriction to $X$, where the invariant of $(\mathcal{I},1)$ appears. Hence ${\mathrm{inv}}_{{\mathcal{I}}^{\mathrm{\prime }}}(x)=(0,1,0,0;\dots ;0,1,0,0;{\mathrm{inv}}_{\mathcal{I}}(x))$ with $(0,1,0,0)$ repeated $k$ times, and analogously ${\nu }_{{\mathcal{I}}^{\mathrm{\prime }}}(x)={\nu }_{\mathcal{I}}(x)$ and ${\rho }_{{\mathcal{I}}^{\mathrm{\prime }}}(x)={\rho }_{\mathcal{I}}(x)$.

The resolutions correspond. By [F3] the centers of the resolution of $({\mathcal{I}}^{\mathrm{\prime }},1)$ are the maximal loci of the invariants computed in step 1.1; because the first $4k$ encoded coordinates of ${\mathrm{inv}}_{{\mathcal{I}}^{\mathrm{\prime }}}$ are constant on the image of the resolution of $(\mathcal{I},1)$, the maximal locus over $X$ is the image of the maximal locus of ${\mathrm{inv}}_{\mathcal{I}}$, and blowing it up in $X^{\mathrm{\prime }}$ restricts to blowing it up in $X$. Iterating, the canonical resolution of $({\mathcal{I}}^{\mathrm{\prime }},1)$ restricts over $X$ to the canonical resolution of $(\mathcal{I},1)$, with the invariant comparison of step 1.1 (the prefixed constant pairs are removed on restriction).