# depth and extension of sections

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 $$X$$?

• 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. – Sándor Kovács Apr 22 at 23:13
• 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. – Sándor Kovács Apr 22 at 23:20