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394 G. Garkusha and M. Prest2. the set of all torsion classes of finite type in Mod R (respectively tensor torsionclasses of finite type in QGr A).This theorem says that Spec R and Proj A contain all the information about finitelocalizations in Mod R and QGr A respectively. The next result says that there isa 1-1 correspondence between the finite localizations in Mod R and the triangulatedlocalizations in D per .R/ (cf. [11], [8]).Theorem. Let R be a commutative ring. The mapinduces a bijection betweenS 7! T DfX 2 D per .R/ j H n .X/ 2 S for all n 2 Zg1. the set of all torsion classes of finite type in Mod R,2. the set of all thick subcategories of D per .R/.Following Buan–Krause–Solberg [5] we consider the lattices L tor .Mod R/ andL tor .QGr A/ of (tensor) torsion classes of finite type in Mod R and QGr A, as well astheir prime ideal spectra Spec.Mod R/ and Spec.QGr A/. These spaces come naturallyequipped with sheaves of rings O Mod R and O QGr A . The following result says that theschemes .Spec R;O R / and .Proj A; O Proj A / are isomorphic to .Spec.Mod R/; O Mod R /and .Spec.QGr A/; O QGr A / respectively.Theorem (Reconstruction). Let R (respectively A) be a commutative ring (respectivelycommutative graded ring which is finitely generated as an A 0 -algebra). Then there arenatural isomorphisms of ringed spacesand.Spec R;O R /.Proj A; O Proj A /! .Spec.Mod R/; OMod R /! .Spec.QGr A/; OQGr A /:2 Torsion classes of finite typeWe refer the reader to the appendix for necessary facts about localization and torsionclasses in Grothendieck categories.Proposition 2.1. Assume that B is a set of finitely generated ideals of a commutativering R. The set of those ideals which contain a finite products of ideals belonging toB is a Gabriel filter of finite type.Proof. See [17, VI.6.10].Given a module M , we denote by supp R .M / DfP 2 Spec R j M P ¤ 0g. HereM P denotes the localization of M at P , that is, the module of fractions M Œ.R n P/ 1 .