Lemma 63.6.5 (0F6U)—The Stacks project

Loading web-font TeX/Math/Italic

Table of contents
Part 3: Topics in Scheme Theory
Chapter 63: More Étale Cohomology
Section 63.6: Upper shriek for locally quasi-finite morphisms
Lemma 63.6.5 (cite)

previous lemma
next remark

numberstags

history
statistics

Lemma 63.6.5. Consider a cartesian square

$\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y }$

of schemes with $f$ locally quasi-finite. For any abelian sheaf $\mathcal{F}$ on $Y'_{\acute{e}tale}$ we have $(g')_*(f')^!\mathcal{F} = f^!g_*\mathcal{F}$ .

Proof. By uniqueness of adjoint functors, this follows from the corresponding (dual) statement for the functors $f_!$ . See Lemma 63.4.10. $\square$

Comments (0)

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).

Comments (0)

Post a comment