Let $X$ be a smooth, projective variety over an algebraically closed field $k$ (of characteristic zero), $B$ a connected, noetherian scheme (possibly non-reduced) and $U$ an open subscheme of $X \times_k B$ such that for every closed point $b \in B$, the complement of $U_b:=U \cap (X \times \{b\})$ is of codimension at least $2$ in $X_b:=X \times \{b\}$. Choose a closed point $o \in B$ and denote by $i:X_o \to X \times B,$ $j:U \to X \times B$ and $j':U_o \to X_o$ the natural immersions. Is it true that $i^*(j_*\mathcal{O}_U) \cong j'_*\mathcal{O}_{U_o}$?
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$\begingroup$ In fact, the natural homomorphisms $\mathcal{O}_{X_o}\to (j')_*\mathcal{O}_{U_0}$ and $\mathcal{O}_{X\times B} \to j_*\mathcal{O}_U$ are both isomorphisms. This is the type of result discussed in EGA IV, Section 5.9 and 5.10. If Grothendieck does not convince you, you can also consult Proposition 3.5 of Hassett-Kov'acs, "Reflexive Pull-Backs and Base Extension". $\endgroup$– Jason StarrCommented Oct 9, 2016 at 16:32
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$\begingroup$ @JasonStarr Thank you, this answers my question. $\endgroup$– user45397Commented Oct 9, 2016 at 17:02
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I am just posting my comment as an answer.
In fact, the natural homomorphisms $\mathcal{O}_{X_o}\to (j')_*\mathcal{O}_{U_o}$ and $\mathcal{O}_{X\times B} \to j_*\mathcal{O}_U$ are both isomorphisms. This is the type of result discussed in EGA IV, Sections 5.9 and 5.10. If Grothendieck does not convince you, you can also consult Proposition 3.5 of Hassett-Kovács, "Reflexive Pull-Backs and Base Extension".