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410 G. Garkusha and M. Prestis induced by the quotient functorMod R=S U ! Mod R=S V .QGr A=S U ! QGr A=S V /:The sheafification is called the structure sheaf of Mod R (QGr A) and is denoted byO Mod R (O QGr A ). This is a sheaf of commutative rings by [13, XI.2.4]. Next letP 2 Spec.Mod R/ and P WD f1 .P /. WehaveO Mod R;P D lim !P 2UEnd Mod R=SU .R/ D lim !f …PEnd Mod R=SD.f /.R/ Š lim !f …PR f D O R;P :Similarly, for P 2 Spec.QGr A/ and P WD fO QGr A;P Š O Proj A;P :1 .P / we haveThe next theorem says that the abelian category Mod R (QGr A) contains all thenecessary information to reconstruct the affine (projective) scheme .Spec R;O R / (respectively.Proj A; O Proj A /).Theorem 8.1 (Reconstruction). Let R (respectively A) be a ring (respectively gradedring which is finitely generated as an A 0 -algebra). The maps of Corollary 7.3 induceisomorphisms of ringed spacesandf W .Spec R;O R /f W .Proj A; O Proj A /! .Spec.Mod R/; OMod R /! .Spec.QGr A/; OQGr A /:Proof. The proof is similar to that of [5, 9.4]. Fix an open subset U Spec.Mod R/and consider the composition of the functorsF W Mod R . /! Qcoh Spec R. /j f 1 .U /! Qcoh f 1 .U /:Here, for any R-module M , we denote by M its associated sheaf. By definition,the stalk of M at a prime P equals the localized module M P . We claim that Fannihilates S U . In fact, M 2 S U implies f 1 .supp.M //\f 1 .U / D;and thereforesupp R .M /\f 1 .U / D;. Thus M P D 0 for all P 2 f 1 .U / and therefore F.M/ D0. It follows that F factors through Mod R=S U and induces a map End Mod R=SU .R/ !O R .f 1 .U // which extends to a map O Mod R .U / ! O R .f 1 .U //. This yields themorphism of sheaves f ] W O Mod R ! f O R .By the above f ] induces an isomorphism f ]P W O Mod R;f.P/ ! O R;P at each pointP 2 Spec R. We conclude that f ]Pis an isomorphism. It follows that f is an isomorphismof ringed spaces if the map f W Spec R ! Spec.Mod R/ is a homeomorphism.This last condition is a consequence of Propositions 7.1 and 7.2. The same argumentsapply to show thatf W .Proj A; O Proj A /! .Spec.QGr A/; OQGr A /is an isomorphism of ringed spaces.