Homogenization commutes with smooth pullback

Derivative ideals commute with smooth pullback. A projection $\pi \text{\hspace{0.17em}⁣}:X\times {\mathbb{A}}^r\to X$ satisfies ${\pi }^{\ast }{\mathcal{D}}^i(\mathcal{A})={\mathcal{D}}^i({\pi }^{\ast }\mathcal{A})$ because differentiating a pulled-back function in the $X$-directions gives the pulled-back derivatives and the new ${\mathbb{A}}^r$-coordinate derivatives annihilate the pulled-back generators of $\mathcal{A}$. For an étale morphism this is [F1]. A general smooth germ factors locally as an étale morphism after a projection by [F2], so for smooth $\phi$ one has ${\phi }^{\ast }{\mathcal{D}}^i(\mathcal{A})={\mathcal{D}}^i({\phi }^{\ast }\mathcal{A})$ for every coherent ideal $\mathcal{A}$ and every $i\ge 0$.

Maximal order is preserved. By [F2, F3], at every $x^{\mathrm{\prime }}\in X^{\mathrm{\prime }}$ one has ${\mathrm{ord}}_{x^{\mathrm{\prime }}}({\phi }^{\ast }\mathcal{I})={\mathrm{ord}}_{\phi (x^{\mathrm{\prime }})}(\mathcal{I})\le \mu$, so the pullback is of maximal order in every characteristic. Step 1.1 also gives $T({\phi }^{\ast }\mathcal{I})={\mathcal{D}}^{\mu -1}({\phi }^{\ast }\mathcal{I})={\phi }^{\ast }T(\mathcal{I})$.

Homogenization commutes. Using step 1.1 termwise, ${\phi }^{\ast }H(\mathcal{I})={\sum }_{i=0}^{\mu -1}{\phi }^{\ast }({\mathcal{D}}^i(\mathcal{I})T(\mathcal{I}{)}^i)={\sum }_{i=0}^{\mu -1}{\mathcal{D}}^i({\phi }^{\ast }\mathcal{I})({\phi }^{\ast }T(\mathcal{I}){)}^i={\sum }_{i=0}^{\mu -1}{\mathcal{D}}^i({\phi }^{\ast }\mathcal{I})T({\phi }^{\ast }\mathcal{I}{)}^i=H({\phi }^{\ast }\mathcal{I})$, where step 2.1 identified $T({\phi }^{\ast }\mathcal{I})={\phi }^{\ast }T(\mathcal{I})$.