process

Torsion classes of finite type and spectra 401Proof. That S Y DfM j .M / Y g is a torsion class follows because it is defined asthe class of modules having no non-zero morphism to a family of injective modules,E WD Inj R n Y . By Lemma 3.7, E \ Spec R D U is a closed set in Spec R, that isP.E/ 2 U for all E 2 E. S Y is also determined by the family of injective modulesfE P g P 2U . Indeed, any E 2 E is a direct summand of some power of E P.E/ by the proofof Proposition 3.3. Therefore Hom R .M; E P.E/ / D 0 implies Hom R .M; E/ D 0. ByProposition 3.1 S Y is of finite type. Conversely, given a torsion class of finite type S, theset Y S D S M 2S .M / is plainly open in .Inj fg R/ . Moreover, S YS D S and Y D Y SY .Consider the following diagram:L.Spec R/ L..Inj fg R/ / ıL thick .D per .R//L tor .Mod R/;where ; are as in Lemma 3.7, ; are as in Theorem 4.1 and the remaining maps arethe corresponding maps indicated in the formulation of the theorem. We have D 1by Theorem 4.1, D 1 by Lemma 3.7, and D ı 1 by the above.By construction,.V/ D ˚X j S n2Z supp R.H n .X// V DfX j supp.X/ V gfor all V 2 L.Spec R/. Thus D . Since , , are bijections so is .On the other hand,ı.T / D [supp R .H n .X// D [supp.X/X2T ;n2ZX2Tfor any T 2 L thick .D per .R//. We have used here the relation[supp R .M / D [supp R .H n .X//:M 2.T /X2T ;n2ZOne sees that ı D . Since ı; ; are bijections so is . Obviously, D 1and the diagram above yields the desired bijective correspondences. The theorem isproved.To conclude this section, we should mention the relation between torsion classesof finite type in Mod R and the Ziegler subspace topology on Inj R (we denote thisspace by Inj zg R). The latter topology arises from Ziegler’s work on the model theoryof modules [20]. The points of the Ziegler spectrum of R are the isomorphism classesof indecomposable pure-injective R-modules and the closed subsets correspond tocomplete theories of modules. It is well known (see [14, 9.12]) that for every coherentring R there is a 1-1 correspondence between the open (equivalently closed) subsets