The Stacks project

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

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

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

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

Proof. The first statement follows from the folowing observations: $g$ is a nonzerodivisor in $B$ which defines $E \cap V \subset V$ and $M^\bullet \otimes _ A B$ represents $M^\bullet \otimes _ A^\mathbf {L} B$ and hence represents the pullback of $M$ to $V$ by Derived Categories of Schemes, Lemma 36.3.8. Part (2) follows from part (1) and More on Algebra, Lemma 15.96.3. Combined with More on Algebra, Lemma 15.96.3 we conclude that the second statement of the lemma holds. $\square$

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