Lemma 38.43.10. Let $X, D, \mathcal{I}, M$ be as in Situation 38.43.1. If $(X, D, M)$ is a good triple, then there exists a closed immersion

\[ i : T \longrightarrow D \]

of finite presentation with the following properties

$T$ scheme theoretically contains $D \cap W$ where $W \subset X$ is the maximal open over which the cohomology sheaves of $M$ are locally free,

the cohomology sheaves of $Li^*L\eta _\mathcal {I}M$ are locally free, and

for any point $t \in T$ with image $x = i(t) \in W$ the rank of $H^ i(M)_ x$ over $\mathcal{O}_{X, x}$ and the rank of $H^ i(Li^*L\eta _\mathcal {I}M)_ t$ over $\mathcal{O}_{T, t}$ agree.

**Proof.**
Let $(U, A, f, M^\bullet )$ be an affine chart for $(X, D, M)$ such that $I_ i(M^\bullet , f)$ is a principal ideal for all $i \in \mathbf{Z}$. Then we define $T \cap U \subset D \cap U$ as the closed subscheme defined by the ideal

\[ J(M^\bullet , f) = \sum J_ i(M^\bullet , f) \subset A/fA \]

studied in More on Algebra, Lemmas 15.96.8 and 15.96.9; in terms of the second lemma we see that $T \cap U \to D \cap U$ is given by the ring map $A/fA \to C$ studied there. Since $(X, D, M)$ is a good triple we can cover $X$ by affine charts of this form and by the first of the two lemmas, this construction glues. Hence we obtain a closed subscheme $T \subset D$ which on good affine charts as above is given by the ideal $J(M^\bullet , f)$. Then properties (1) and (2) follow from the second lemma. Details omitted. Small observation to help the reader: since $\eta _ fM^\bullet $ is a complex of locally free modules by More on Algebra, Lemma 15.96.5 we see that $Li^*L\eta _\mathcal {I}M|_{T \cap U}$ is represented by the complex $\eta _ fM^\bullet \otimes _ A C$ of $C$-modules. The statement (3) on ranks follows from Cohomology, Lemma 20.55.10.
$\square$

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