This is a question that came up in the comments section of [here](https://mathoverflow.net/questions/379562/kernels-of-surjections-from-a-vector-bundle-to-a-reflexive-sheaf). A reflexive sheaf $E$ is called "locally $3$-syzygy" if it fits into an exact sequence $0\rightarrow E \rightarrow F\_1\rightarrow F\_2 \rightarrow F\_3$ where $F\_i$s are vector bundles. These reflexive sheaves are vector bundle on the complement of a codimension $4$ subvariety, the question is whether the converse is true i.e. when a reflexive sheaf that is vector bundle on the complement of a closed set of codimension $\geq 4$ is locally 3-syzygy?


I am able to prove through some long method that if the converse is true you get $K\_1(k)=0$ (if I'm not making any mistakes in my arguments!). First algebraic $K$-group of fields are trivial but obviously it is wrong. This specially implies that there are counter-examples of the form $\mathbb{A}^n$ with the closed subvariety of the form $\mathbb{A}^k$ where $n-k\geq 4$. So the counter-examples seem to be ubiquitous. My question is, is this almost always wrong or there is some chance of it being true?


Edit: Another relevant question that I cannot answer. Does having a really high codimension wrt the ambient variety guarantee that any reflexive sheaf that is a vector bundle on the complement of the closed variety is locally 3-syzygy? (For example let $Z$ be 3 dimensional variety embedded in $Z\times \mathbb{P}^n$ for a very large $n$)



On locally 3-syzygy sheaves
Locally, the depth increases along syzygies, so there are always counter examples as long as there is a closed point whose local ring has depth at least 3.


For instance, let $R=k[x\_1,...,x\_n]$ for $n\geq 3$. Let $E$ be the second syzygy of $R/(x\_1,...,x\_n)$. Then $E$ is reflexive and locally free on $Spec(R)-{m}$ where $m=(x\_1,...,x\_n)$. But it will not be $3$rd syzygy because the depth of $E\_m$ is only 2.


Now some good news: when depth is no more than 2, being Gorenstein would be enough to guarantee. That is because any module of maximal depth on a Gorenstein local ring would be an $n$-syzygy for any $n$.


