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406 G. Garkusha and M. PrestLemma 6.1. Given X; Y 2 Gr A there is a natural isomorphism in QGr A: Q.X/ Q.Y / Š Q.X ˝ Y/. Moreover, the functor Y W QGr A ! QGr A is right exactand preserves direct limits.Proof. See [9, 4.2].As a consequence of this lemma we get an isomorphism X.d/ Š O.d/ X forany X 2 QGr A and d 2 Z.The notion of a tensor torsion class of QGr A (with respect to the tensor product) is defined analogously to that in Gr A. The proof of the next lemma is like that ofLemma 5.2 (also use Lemma 6.1).Lemma 6.2. A torsion class S is a tensor torsion class of QGr A if and only if it isclosed under shifts of objects, i.e. X 2 S implies X.n/ 2 S for any n 2 Z.Given a prime ideal P 2 Proj A and a graded module M , denote by M P thehomogeneous localization of M at P .Iff is a homogeneous element of A,byM f wedenote the localization of M at the multiplicative set S f Dff n g n0 .Lemma 6.3. If T is a torsion module then T P D 0 and T f D 0 for any P 2 Proj A andf 2 A C . As a consequence, M P Š Q.M / P and M f Š Q.M / f for any M 2 Gr A.Proof. See [9, 5.5].Denote by L tor .Gr A; Tors A/ (respectively L tor .QGr A/) the lattice of the tensortorsion classes of finite type in Gr A with torsion classes containing Tors A (respectivelythe tensor torsion classes of finite type in QGr A) ordered by inclusion. The map`W L tor .Gr A; Tors A/ ! L tor .QGr A/;S 7! S= Tors Ais a lattice isomorphism, where S= Tors A DfQ.M / j M 2 Sg (see, e.g., [7, 1.7]).We shall consider the map ` as an identification.Theorem 6.4 (Classification). Let A be a commutative graded ring which is finitelygenerated as an A 0 -algebra. Then the mapsV 7! S DfM 2 QGr A j supp A .M / V g and S 7! V D [ supp A .M /M 2Sinduce bijections between1. the set of all open subsets V Proj A,2. the set of all tensor torsion classes of finite type in QGr A.Proof. By Lemma 5.3 it is enough to show that the mapsV 7! F V DfI 2 A j V.I/ V g and F 7! V F D [ V.I/I 2F