Let $S$ be an affine scheme, $X$ smooth affine over $S$ and $U$ an open subset of $X$, fiberwise of codimension at least two.

Suppose that we have a function on $U$, can we extend it to $X$?

  • $\begingroup$ You probably mean that the complement of $U$ has fiberwise codimension at least two, right? In that case, it also has codimension at least two in $X$ and you can see that $S$ is a red herring and this is the usual $S_2$ property of smooth schemes. $\endgroup$ – Sándor Kovács Apr 22 at 23:13
  • $\begingroup$ It is a more interesting question if you don't assume that $X$ is smooth, only that it is flat over $S$ and that the fibers are $S_2$. In that case the same statement is still true. See Prop 3.5 in "Reflexive pull-backs and base extension", Journal of Algebraic Geometry 13 (2), 233-248. $\endgroup$ – Sándor Kovács Apr 22 at 23:20

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