The homogenized ideal of a marked ideal of maximal order

Let $(\mathcal{I},E,\mu )$ be a marked ideal of maximal order with $\mu \ge 1$ and put $T(\mathcal{I})={\mathcal{D}}^{\mu -1}(\mathcal{I})$ (Marked ideals of maximal order, tangent directions and transversality to the exceptional divisors). The homogenized ideal is the ideal sheaf $H(\mathcal{I}):=\mathcal{I}+\mathcal{D}(\mathcal{I})\cdot T(\mathcal{I})+\cdots +{\mathcal{D}}^i(\mathcal{I})\cdot T(\mathcal{I}{)}^i+\cdots +{\mathcal{D}}^{\mu -1}(\mathcal{I})\cdot T(\mathcal{I}{)}^{\mu -1},$ the products and sums being products and sums of ideal sheaves; the homogenized marked ideal is $H(\mathcal{I},\mu ):=(H(\mathcal{I}),E,\mu )$. It satisfies $H(\mathcal{I})\supseteq \mathcal{I}$ and $T(H(\mathcal{I}))=T(\mathcal{I})$ for $\mu \ge 1$. If $\mu >1$ and $K$ has characteristic zero or perfect characteristic $p>\mu$, then $\mathcal{D}(H(\mathcal{I}))\subseteq H(\mathcal{D}(\mathcal{I}),\mu -1),$ where $(\mathcal{D}(\mathcal{I}),\mu -1)$ is again of maximal order, so the right-hand homogenization is defined. It is designed so that it looks the same from every tangent direction and is equivalent to $(\mathcal{I},\mu )$ (proved below).