Lemma 79.12.9 (04QF)—The Stacks project

Processing math: 100%

Table of contents
Part 4: Algebraic Spaces
Chapter 79: More on Groupoids in Spaces
Section 79.12: The finite part of a morphism
Lemma 79.12.9 (cite)

previous lemma
next lemma

numberstags

history
statistics
1 tag refers to this tag

Lemma 79.12.9. Let $S$ be a scheme. Let

$\xymatrix{ X' \ar[d] \ar[r] & X \ar[d] \\ Y' \ar[r] & Y }$

be a fibre product square of algebraic spaces over $S$ . Then

$\xymatrix{ (X'/Y')_{fin} \ar[d] \ar[r] & (X/Y)_{fin} \ar[d] \\ Y' \ar[r] & Y }$

is a fibre product square of sheaves on $(\mathit{Sch}/S)_{fppf}$ .

Proof. It follows immediately from the definitions that the sheaf $(X'/Y')_{fin}$ is equal to the sheaf $Y' \times _ Y (X/Y)_{fin}$ . $\square$

Comments (0)

There are also:

2 comment(s) on Section 79.12: The finite part of a morphism

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