process

Torsion classes of finite type and spectra 4052. Assume that B is a set of homogeneous finitely generated ideals. The set B 0 offinite products of ideals belonging to B is a basis for a t-filter of finite type.Proof. 1. For any homogeneous element a 2 I we have .IJ W a/ J ,soIJ 2 F byT3and the fact that every homogeneous ideal containing an ideal from F must belongto F.2. We follow [17, VI.6.10]. We must check that the set F of homogeneous idealscontaining ideals in B 0 is a t-filter of finite type. T1 is plainly satisfied. Let a be ahomogeneous element in A and I 2 F. There is an ideal I 0 2 B 0 contained in I . Then.I W a/ I 0 and therefore .I W a/ 2 F, hence T2is satisfied as well.Next we verify that F satisfies T3. Suppose that I is a homogeneous ideal and thereexists J 2 F such that .I W a/ 2 F for every homogeneous element a 2 J . We mayassume that J 2 B 0 . Let a 1 ;:::;a n be generators of J . Then .I W a i / 2 F, i n,and .I W a i / J i for some J i 2 B 0 . It follows that a i J i I for each i, and henceJJ 1 :::J n J.J 1 \\J n / I ,soI 2 F.6 Torsion modules and the category QGr ALet A be a commutative graded ring. In this section we introduce the category QGr A,which is analogous to the category of quasi-coherent sheaves on a projective variety.The non-commutative analog of the category QGr A plays a prominent role in “noncommutativeprojective geometry" (see, e.g., [1], [16], [19]).Recall that the projective scheme Proj A is a topological space whose points are thehomogeneous prime ideals not containing A C . The topology of Proj A is defined bytaking the closed sets to be the sets of the form V.I/ DfP 2 Proj A j P I g for I ahomogeneous ideal of A. We set D.I / WD Proj A n V.I/.In the remainder of this section the homogeneous ideal A C A is assumed tobe finitely generated. This is equivalent to assuming that A is a finitely generatedA 0 -algebra. The space Proj A is spectral and the quasi-compact open sets are thoseof the form D.I / with I finitely generated (see, e.g., [9, 5.1]). Let Tors A denote thetensor torsion class of finite type corresponding to the family of homogeneous finitelygenerated ideals fA n C g n1 (see Proposition 5.4). We refer to the objects of Tors A astorsion graded modules.Let QGr A D Gr A= Tors A. Let Q denote the quotient functor Gr A ! QGr A.We shall identify QGr A with the full subcategory of Tors-closed modules. The shiftfunctor M 7! M.n/ defines a shift functor on QGr A for which we shall use the samenotation. Observe that Q commutes with the shift functor. Finally we shall writeO D Q.A/. Note that QGr A is a locally finitely generated Grothendieck categorywith the family, fQ.M /g M 2gr A , of finitely generated generators (see [7, 5.8]).The tensor product in Gr A induces a tensor product in QGr A, denoted by . Moreprecisely, one setsX Y WD Q.X ˝ Y/for any X; Y 2 QGr A.