Reconstructing projective schemes from Serre subcategories

Given a positively graded commutative coherent ring A=⊕j⩾0AjA=⊕j⩾0Aj, finitely generated as an A0A0-algebra, a bijection between the tensor Serre subcategories of qgrA   and the set of all subsets Y⊆ProjAY⊆ProjA of the form Y=⋃i∈ΩYiY=⋃i∈ΩYi with quasi-compact open complement ProjA∖YiProjA∖Yi for all i∈Ωi∈Ω is established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of isomorphism classes of indecomposable injective graded modules are used in an essential way. Also, there is constructed an isomorphism of ringed spaces(ProjA,OProjA)→∼(Spec(qgrA),OqgrA), where (Spec(qgrA),OqgrA)(Spec(qgrA),OqgrA) is a ringed space associated to the lattice LSerre(qgrA)LSerre(qgrA) of tensor Serre subcategories of qgrA.