Elementary properties of the homogenized ideal

Clauses (1) and (2). If $\mu =1$ then $T(\mathcal{I})={\mathcal{D}}^0\mathcal{I}=\mathcal{I}$ and the defining sum has the single term $\mathcal{I}$, so $H(\mathcal{I})=\mathcal{I}$, which is (1). For (2), extend the defining sum to all $i\ge 0$. For each $i\ge \mu$, the $i$-th term ${\mathcal{D}}^i(\mathcal{I})T(\mathcal{I}{)}^i$ is contained in $T(\mathcal{I}{)}^i$ because ${\mathcal{D}}^i(\mathcal{I})\subseteq {\mathcal{O}}_X$, and hence is contained in $T(\mathcal{I}{)}^{\mu }$. Since $T(\mathcal{I})={\mathcal{D}}^{\mu -1}(\mathcal{I})$, the $i=\mu -1$ term is exactly $T(\mathcal{I}{)}^{\mu }$. Thus every later term is contained in the last retained term, and the full sum equals its truncation.

Clause (5). For a local product generator of a summand ${\mathcal{D}}^i(\mathcal{I})T(\mathcal{I}{)}^i$, applying a coordinate derivative of total order at most $\mu -1$ gives sums of products with $a_0$ derivatives on the first factor and $a_1,\dots ,a_i$ derivatives on the $i$ tangent factors, where $a_0+{\sum }_{\mathrm{\ell }}{a}_{\mathrm{\ell }}\le \mu -1$. The first factor lies in ${\mathcal{D}}^{i+a_0}(\mathcal{I})$. If $i+a_0\le \mu -1$, this is contained in $T(\mathcal{I})={\mathcal{D}}^{\mu -1}(\mathcal{I})$ because derivative ideals increase with their index. If $i+a_0>\mu -1$, then ${\sum }_{\mathrm{\ell }}{a}_{\mathrm{\ell }}\le \mu -1-a_0<i$, so at least one tangent factor is undifferentiated and the product contains a factor of $T(\mathcal{I})$. In both cases the resulting product lies in $T(\mathcal{I})$, proving ${\mathcal{D}}^{\mu -1}(H(\mathcal{I}))\subseteq T(\mathcal{I})$. The reverse inclusion follows from $\mathcal{I}\subseteq H(\mathcal{I})$ and monotonicity of derivative ideals. Thus $T(H(\mathcal{I}))=T(\mathcal{I})$, proving (5).

Clause (3). Each product ${\mathcal{D}}^i(\mathcal{I},\mu )(T(\mathcal{I}),1{)}^i$ has underlying ideal $J_i={\mathcal{D}}^i(\mathcal{I})T(\mathcal{I}{)}^i$ and mark $\mu$. Their literal ideal sum is $H(\mathcal{I})$ with mark $\mu$. Its support is the intersection of the supports of $(J_i,\mu )$, since the order of an ideal sum is the minimum of the summand orders. At a common admissible center the controlled transform of the literal sum distributes termwise, because all marks are $\mu$. Induction therefore identifies its test sequences and induced supports with the simultaneous ones for the summands. Under AC, [F4] gives exactly those supports and test sequences for the iterated marked sum. Thus the displayed operation represents $H(\mathcal{I},\mu )$ up to marked equivalence, proving (3); no literal equality between a sum of ideal powers and a power of an ideal sum is used.

Clause (4). Assume $\mu >1$ and $K$ has characteristic zero or perfect characteristic $p>\mu$. The maximal-order criterion in [F3] gives ${\mathcal{D}}^{\mu }(\mathcal{I})={\mathcal{O}}_X$, so $(\mathcal{D}(\mathcal{I}),\mu -1)$ is maximal order and $T(\mathcal{D}(\mathcal{I}))={\mathcal{D}}^{\mu -2}(\mathcal{D}(\mathcal{I}))=T(\mathcal{I})$. Leibniz gives $\mathcal{D}(H(\mathcal{I}))\subseteq {\sum }_{i=0}^{\mu -1}{\mathcal{D}}^{i+1}(\mathcal{I})T(\mathcal{I}{)}^i+{\sum }_{i=1}^{\mu -1}{\mathcal{D}}^i(\mathcal{I})T(\mathcal{I}{)}^{i-1}\mathcal{D}(T(\mathcal{I}))\mathrm{.}$ Since $\mathcal{D}(T(\mathcal{I}))={\mathcal{D}}^{\mu }(\mathcal{I})={\mathcal{O}}_X$, the second sum is the sum of the terms ${\mathcal{D}}^i(\mathcal{I})T(\mathcal{I}{)}^{i-1}$ for $1\le i\le \mu -1$. In the first sum, the terms with $1\le i+1\le \mu -1$ are precisely the terms ${\mathcal{D}}^j(\mathcal{D}(\mathcal{I}))T(\mathcal{I}{)}^j$ of $H(\mathcal{D}(\mathcal{I}),\mu -1)$ after setting $j=i$; its remaining top term is ${\mathcal{D}}^{\mu }(\mathcal{I})T(\mathcal{I}{)}^{\mu -1}=T(\mathcal{I}{)}^{\mu -1}$, already the $i=\mu -1$ term of the second sum. The second sum itself consists of the terms ${\mathcal{D}}^{i-1}(\mathcal{D}(\mathcal{I}))T(\mathcal{I}{)}^{i-1}$ of $H(\mathcal{D}(\mathcal{I}),\mu -1)$. Hence the derivative ideal is contained in that homogenized ideal, proving (4).