Lemma 38.43.2. In Situation 38.43.1 let $h : Y \to X$ be a morphism of schemes such that the pullback $E = h^{-1}D$ of $D$ is defined (Divisors, Definition 31.13.12). Let $(U, A, f, M^\bullet )$ is an affine chart for $(X, D, M)$. Let $V = \mathop{\mathrm{Spec}}(B) \subset Y$ is an affine open with $h(V) \subset U$. Denote $g \in B$ the image of $f \in A$. Then

$(V, B, g, M^\bullet \otimes _ A B)$ is an affine chart for $(Y, E, Lh^*M)$,

$I_ i(M^\bullet , f)B = I_ i(M^\bullet \otimes _ A B, g)$ in $B$, and

if $(X, D, M)$ is a good triple, then $(Y, E, Lh^*M)$ is a good triple.

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