The Stacks project

Lemma 20.19.3 (Proper base change for sheaves of sets). Consider a cartesian square of topological spaces

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

Assume that $f$ is proper and separated. Then $g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}$ for any sheaf of sets $\mathcal{F}$ on $X$.

Proof. We argue exactly as in the proof of Theorem 20.19.2 and we find it suffices to show $(f_*\mathcal{F})_ y = \Gamma (X_ y, \mathcal{F}|_{X_ y})$. Then we argue as in Lemma 20.19.1 to reduce this to the $p = 0$ case of Lemma 20.17.3 for sheaves of sets. The first part of the proof of Lemma 20.17.3 works for sheaves of sets and this finishes the proof. Some details omitted. $\square$


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