Lemma 34.3.18 (0212)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 34: Topologies on Schemes
Section 34.3: The Zariski topology
Lemma 34.3.18 (cite)

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Lemma 34.3.18. Given schemes $X$ , $Y$ , $Z$ in $(\mathit{Sch}/S)_{Zar}$ and morphisms $f : X \to Y$ , $g : Y \to Z$ we have $g_{big} \circ f_{big} = (g \circ f)_{big}$ and $g_{small} \circ f_{small} = (g \circ f)_{small}$ .

Proof. This follows from the simple description of pushforward and pullback for the functors on the big sites from Lemma 34.3.16. For the functors on the small sites this is Sheaves, Lemma 6.21.2 via the identification of Lemma 34.3.12. $\square$

Comments (2)

Comment #7402 by Floris Ruijter on May 29, 2022 at 05:27

"Given schemes X, Y, Y" should be "[..] X, Y, Z"

Comment #7584 by Stacks Project on August 12, 2022 at 15:23

Thanks, this was fixed a while back here.

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