# Extending locally free sheaves and compatibility with fibers

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 $i^*(j_*\mathcal{O}_U) \cong j'_*\mathcal{O}_{U_o}$?

• 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". – Jason Starr Oct 9 '16 at 16:32
• @JasonStarr Thank you, this answers my question. – user45397 Oct 9 '16 at 17:02

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".