Given a morphism of schemes $f: U \rightarrow X$, can one determine when $f$ is an isomorphism of $U$ onto an open subscheme of $X$ in terms of some induced functors between the categories of quasicoherent modules $Qcoh(U)$ and $Qcoh(X)\ ?$

To begin, it might be helpul to simply assume that $U \subset X$ is an open subscheme, and consider some properties of the resulting functors. I can think of one interesting functor: there is an exact functor $Qcoh(X) \rightarrow Qcoh(U)$ given by restriction of sheaves. I assume this is a special case of some more general construction (direct or inverse image functor?) that probably has an adjoint on some side.

Can we continue to list enough functors and properties of these functors to the point where we have determined precisely when the above map f is an isomorphism onto an open subscheme?

Given a morphism of schemes f: U → X, can one determine when f is an isomorphism of U onto an open subscheme of X in terms of some induced functors between the categories of quasicoherent modules Qcoh(U) and Qcoh(X)?

To begin, it might be helpul to simply assume that U ⊆ X is an open subscheme, and consider some properties of the resulting functors. I can think of one interesting functor: there is an exact functor Qcoh(X) → Qcoh(U) given by restriction of sheaves. I assume this is a special case of some more general construction (direct or inverse image functor?) that probably has an adjoint on some side.

Can we continue to list enough functors and properties of these functors to the point where we have determined precisely when the above map f is an isomorphism onto an open subscheme?

Given a morphism of schemes f: U → X, can one determine when f is an isomorphism of U onto an open subscheme of X in terms of some induced functors between the categories of quasicoherent modules Qcoh(U) and Qcoh(X)?

To begin, it might be helpul to simply assume that U ⊆ X is an open subscheme, and consider some properties of the resulting functors. I can think of one interesting functor: there is an exact functor Qcoh(X) → Qcoh(U) given by restriction of sheaves. I assume this is a special case of some more general construction (direct or inverse image functor?) that probably has an adjoint on some side.

Can we continue to list enough functors and properties of these functors to the point where we have determined precisely when the above map f is an isomorphism onto an open subscheme?

Given a morphism of schemes $f: U \rightarrow X$, can one determine when $f$ is an isomorphism of $U$ onto an open subscheme of $X$ in terms of some induced functors between the categories of quasicoherent modules $Qcoh(U)$ and $Qcoh(X)\ ?$

To begin, it might be helpul to simply assume that $U \subset X$ is an open subscheme, and consider some properties of the resulting functors. I can think of one interesting functor: there is an exact functor $Qcoh(X) \rightarrow Qcoh(U)$ given by restriction of sheaves. I assume this is a special case of some more general construction (direct or inverse image functor?) that probably has an adjoint on some side.

Can we continue to list enough functors and properties of these functors to the point where we have determined precisely when the above map f is an isomorphism onto an open subscheme?