Abstract
Given a positively graded commutative coherent ring A = ⊕ j ⩾ 0 A j , finitely generated as an A 0 -algebra, a bijection between the tensor Serre subcategories of qgr A and the set of all subsets Y ⊆ Proj A of the form Y = ⋃ i ∈ Ω Y i with quasi-compact open complement Proj A ∖ Y i for all i ∈ 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 ( Proj A , O Proj A ) → ∼ ( Spec ( qgr A ) , O qgr A ) , where ( Spec ( qgr A ) , O qgr A ) is a ringed space associated to the lattice L Serre ( qgr A ) of tensor Serre subcategories of qgr A.
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