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 following 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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0F8T. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 38.43.2 as pdf