TITLE: Extending vector bundles on a given open subscheme, reprise
QUESTION [20 upvotes]: In this question, Ariyan asks about the question of uniqueness of extensions of vector bundles when they exist.
Sasha's answer suggests that extensions of vector bundles don't always exist.
More precisely, if $F$ is a vector bundle on an open subscheme $U$, there does not always exist a vector bundle $F'$ on the ambient space $X$ such that $F'|_U \cong F$.
Can anyone give me a simple example of such an $F$?
I am mainly interested in the case when $X$ is a variety (over $\mathbb{C}$), and $U$ is an open subvariety. Probably I want $X$ to be smooth.

REPLY [3 votes]: Generalizing @Sasha's example.

Let $X$ be a regular projective scheme over a noetherian ring $A$, and let $Z$ be a closed subscheme of $X$ of codimension $\ge 3$.
  The ideal sheaf $I_Z$ is generated by sections after tensoring by $O_X(l)$ for some $l \gg 0$, it follows that we have an exact sequence
  $$
0 \to E \to O_X(l)^m \to O_X \to O_Z \to 0
$$
   for some $m$ large.
  If $U \subset X - Z$ is any open subset whose complement is of codimension $\ge 2$, then $E|U$ is locally free but does not extend to the whole of $X$, or to any open set containing an associated point of $Z$.


For any $x\in X$, the ring $O_{X,x}$ is regular and thus the module $E_x$ is of finite projective dimension. 
For module with finite projective dimension, being a second syzygy is equivalent to being reflexive. 
It follows that $E_x$ is reflexive for all $x\in X$ and thus $E$ is reflexive. 
Clearly $E$ is locally free on $X-Z$.
Let $z$ be an associated point of $Z$, then we claim $E_z$ is not free at $z$.
The module $O_{Z,z}$ has finite projective dimension pdim $O_{Z,z}$  = depth $O_{X,z}$ - depth $O_{Z,z}$ by the Auslander-Buchsbaum formula. Since $z$ is an associated point of $Z$, we have depth $O_{Z,z} = 0$. Since Z has codimension $\ge 3$, and $O_{X,z}$ is Cohen-Macaulay, we conclude that pdim $O_{Z,z}$ =  depth $O_{X,z}$ = dim $O_{X,z} \ge 3$. It follows that $E_z$ cannot be free, otherwise 
$$
0 \to  E_z \to O(l)^m_z \to O_{X,z} \to O_{Z,z} \to 0
$$ would be a free resolution of $O_{Z,z}$ of length 2.
On a scheme satisfying (G1) (Gorenstein in codimension one) and Serre's condition (S2), e.g. normal, Cohen-Macaulay, regular, etc., a reflexive sheaf admits a unique extension from an open set whose complement has codimension $\ge 2$ to the whole space.