By the Gabriel-Rosenberg reconstruction theorem a scheme can be reconstructed from the category of quasicoherent sheaves on . If is a subscheme of then we can identify the category quasicoherent sheaves on with certain abelian subcategory of quasicoherent sheaves on . One attempts to abstractly deefine the properties of such a subcategory which guarantee that it corresponds to a subcategory of that kind.
(Alexander Rosenberg in his 1996 book and MPI preprints with Lunts from 1996)
If a subscheme is also reflective then we call it Zariski closed.
If an abelian category has the Gabriel’s property (sup) then every
(a) The intersection of any set of subschemes of is a subscheme.
(b) The intersection of any set of Zariski closcd suhschemes of is a Zariski closed subscheme.