Let $U$ be a dense open subscheme of an integral noetherian scheme $X$ and let $E$ be a vector bundle on $U$. Suppose that the complement $Y$ of $U$ has codimension $\textrm{codim}(Y,X) \geq 2$. Let $F$ be a vector bundle on $X$ extending $E$, i.e., $F|\_{U} = E$. 


Is any extension of $E$ to $X$ isomorphic to $F$?



Extending vector bundles on a given open subscheme
This is true if $X$ satisfies Serre's condition $S\_2$, i.e. $\mathcal O\_X$ is $S\_2$. Then a vector bundle is $S\_2$ since locally it is isomorphic to $\mathcal O\_X^n$.


More generally, a coherent sheaf $F$ on a Japanese scheme (for example: $X$ is of finite type over a field) which is $S\_2$ has a unique extension from an open subset $U$ with $\operatorname{codim} (X\setminus U)\ge 2$. This follows at once from the cohomological characterization of $S\_2$.


Thus, another name for the $S\_2$-sheaves: they are sheaves which are *saturated in codimension 2*, and another name for the $S\_2$-fication: *saturation in codimension 2*.


P.S. Of course, by Serre's criterion, normal = $S\_2+R\_1$. So the above statement is true for any normal (e.g. smooth) variety.


P.P.S. And of course, Gorenstein implies Cohen-Macaulay implies $S\_2$. So the statement is also true for hypersurfaces and complete intersections, which could be very singular and non-reduced.




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Edit to define some terms:


1. A Japanese (or Nagata) ring is a ring obtained from a ring finitely generated over a field or $\mathbb Z$ by optionally applying localizations and completions. The property used here is that for a Japanese ring $R$, its integral closure (normalization) $\tilde R$ is a finitely generated $R$-module. This is important because the $S\_2$-fication $S\_2(R)$ lies between $R$ and $\tilde R$.
2. A coherent sheaf $F$ satisfies $S\_n$ if for any point $x\in Supp(F)$, one has
$$ depth\_x (F) \ge \min(\dim\_x Supp(F),n) $$
If $F$ locally corresponds to an $R$-module $M$, and $x$ to a prime ideal $p$, then the depth is the length of a maximal regular sequence $(f\_1,\dots, f\_k)$ of elements of $R\_p$ for $M\_p$ (so, $f\_1$ is a nonzerodivisor in $M\_p$, etc.).


