process

396 G. Garkusha and M. PrestProof. The bijection between Gabriel filters of finite type and torsion classes of finitetype is a consequence of a theorem of Gabriel (see, e.g., [7, 5.8]).Let F be a Gabriel filter of finite type. Then the set ƒ F of finitely generated idealsI belonging to F is a filter basis for F. Therefore V F D S I 2ƒ FV.I/ is open inSpec R.Now let V be an open subset of Spec R. Let ƒ denote the set of finitely generatedideals I such that V.I/ V . By definition of the topology V D S I 2ƒV.I/ andI 1 I n 2 ƒ for any I 1 ;:::;I n 2 ƒ. We denote by F 0 Vthe set of ideals I Rsuch that I J for some J 2 ƒ. By Proposition 2.1 F 0 Vis a Gabriel filter of finitetype. Clearly, F 0 V F V DfI R j V.I/ V g. Suppose I 2 F V n F 0 V; by [17,VI.6.13–15] (cf. the proof of Theorem 6.4) there exists a prime ideal P 2 V.I/ suchthat P 62 F 0 V . But V.I/ V and therefore P J for some J 2 ƒ, soP 2 F0 V ,acontradiction. Thus F 0 V D F V .Clearly, V D V FV for every open subset V Spec R. Let F be a Gabriel filterof finite type and I 2 F. Clearly F F VF and, as above, there is no ideal belongingto F VF n F. We have shown the bijection between the sets of all Gabriel filters of finitetype and all open subsets in Spec R. The description of the bijection between the setof torsion classes of finite type and the set of open subsets in Spec R is now easilychecked.3 The fg-topologyLet Inj R denote the set of isomorphism classes of indecomposable injective modules.Given a finitely generated ideal I of R, we denote by S I the torsion class of finitetype corresponding to the Gabriel filter of finite type having fI n g n1 as a basis (seeProposition 2.1 and Theorem 2.2). Note that a module M has S I -torsion if and only ifevery element x 2 M is annihilated by some power I n.x/ of the ideal I . Let us setD fg .I / WD fE 2 Inj R j E is S I -torsion freeg; V fg .I / WD Inj R n D fg .I /(“fg” referring to this topology being defined using only finitely generated ideals).Let E be any indecomposable injective R-module. Set P D P.E/to be the sum ofannihilator ideals of non-zero elements, equivalently non-zero submodules, of E. SinceE is uniform the set of annihilator ideals of non-zero elements of E is closed underfinite sum. It is easy to check ([14, 9.2]) that P.E/ is a prime ideal and P.E P / D P .Here E P stands for the injective hull of R=P. There is an embedding˛ W Spec R ! Inj R; P 7! E P ;which need not be surjective. We shall identify Spec R with its image in Inj R.If P is a prime ideal of a commutative ring R its complement in R is a multiplicativelyclosed set S. Given a module M we denote the module of fractions MŒS 1 by M P . There is a corresponding Gabriel filterF P DfI j P … V.I/g: