Let $X$ be a noetherian scheme. Consider the smallest class $\mathcal{S}$ of coherent sheaves on $X$ which has the following closure properties:

• Every locally free sheaf is in $\mathcal{S}$.
• If $A,B \in \mathcal{S}$, then $A \otimes B \in \mathcal{S}$.
• If $C$ is a quotient of $B \in \mathcal{S}$, then $C \in \mathcal{S}$.
• If $0 \to A \to B \to C \to 0$ is an exact sequence with $A,C \in \mathcal{S}$, then $B \in \mathcal{S}$.

Do we have $\mathcal{S}=\mathsf{Coh}(X)$? If not, is there any more explicit description of the sheaves in $\mathcal{S}$? Of course we have $\mathcal{S}=\mathsf{Coh}(X)$ when $X$ has the resolution property, but I am interested in the general case.

Let $X$ be a noetherian scheme. Consider the smallest class $\mathcal{S}$ of coherent sheaves on $X$ which has the following closure properties:

• Every locally free sheaf is in $\mathcal{S}$.
• If $A,B \in \mathcal{S}$, then $A \otimes B \in \mathcal{S}$.
• If $C$ is a quotient of $B \in \mathcal{S}$, then $C \in \mathcal{S}$.
• If $0 \to A \to B \to C \to 0$ is an exact sequence with $A,C \in \mathcal{S}$, then $B \in \mathcal{S}$.

Do we have $\mathcal{S}=\mathsf{Coh}(X)$? If not, is there any more explicit description of the sheaves in $\mathcal{S}$? Of course we have $\mathcal{S}=\mathsf{Coh}(X)$ when $X$ has the resolution property, but I am interested in the general case.

For example let $X$ be the affine plane with a doubled origin. Then every locally free sheaf is trivial. But what is $\mathcal{S}$?

Let $X$ be a noetherian scheme. Consider the smallest class $\mathcal{S}$ of coherent sheaves on $X$ which has the following closure properties:

• Every locally free sheaf is in $\mathcal{S}$.
• If $A,B \in \mathcal{S}$, then $A \otimes B \in \mathcal{S}$.
• If $C$ is a quotient of $B \in \mathcal{S}$, then $C \in \mathcal{S}$.
• If $0 \to A \to B \to C \to 0$ is an exact sequence with $A,C \in \mathcal{S}$, then $B \in \mathcal{S}$.

Do we have $\mathcal{S}=\mathsf{Coh}(X)$? If not, is there any more explicit description of the sheaves in $\mathcal{S}$? Of course we have $\mathcal{S}=\mathsf{Coh}(X)$ when $X$ has the resolution property, but I am interested in the general case. If necessary, you can also assume that $X$ is smooth over a field or any convenient assumption.

Let $X$ be a noetherian scheme. Consider the smallest class $\mathcal{S}$ of coherent sheaves on $X$ which has the following closure properties:

• Every locally free sheaf is in $\mathcal{S}$.
• If $A,B \in \mathcal{S}$, then $A \otimes B \in \mathcal{S}$.
• If $C$ is a quotient of $B \in \mathcal{S}$, then $C \in \mathcal{S}$.
• If $0 \to A \to B \to C \to 0$ is an exact sequence with $A,C \in \mathcal{S}$, then $B \in \mathcal{S}$.

Do we have $\mathcal{S}=\mathsf{Coh}(X)$? If not, is there any more explicit description of the sheaves in $\mathcal{S}$? Of course we have $\mathcal{S}=\mathsf{Coh}(X)$ when $X$ has the resolution property, but I am interested in the general case.