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This is a question that came up in the comments section of here. 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 $n$)

This is a question that came up in the comments section of here. 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 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?

This is a question that came up in the comments section of here. 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 $n$)

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user127776
user127776
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On locally 3-syzygy sheaves

This is a question that came up in the comments section of here. 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 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?