process

Torsion classes of finite type and spectra 397Clearly, F P is of finite type. The F P -torsion modules are characterized by the propertythat M P D 0 (see [17, p. 151]).More generally, let P be a subset of Spec R. ToP we associate a Gabriel filterF P D \F P DfI j P \ V.I/ D;g:P 2PThe corresponding torsion class consists of all modules M with M P D 0 for all P 2 P .Given a family of injective R-modules E, denote by F E the Gabriel filter determinedby E. By definition, this corresponds to the localizing subcategory S E DfM 2 Mod R j Hom R .M; E/ D 0 for all E 2 Eg.Proposition 3.1. A Gabriel filter F is of finite type if and only if it is of the form F Pwith P a closed set in Spec R. Moreover, F P is determined by E P DfE P j P 2 P gvia F P DfI j Hom R .R=I; E P / D 0g.Proof. This is a consequence of Theorem 2.2.Proposition 3.2. Let P be the closure of P in Spec R. Then P DfQ 2 Spec R jQ P g. Also F P D F P .Proof. This is direct from the definition of the topology.Recall that for any ideal I of a ring, R, and r 2 R we have an isomorphismR=.I W r/ Š .rR C I /=I , where .I W r/ Dfs 2 R j rs 2 I g, induced by sending1 C .I W r/ to r C I .Proposition 3.3. Let E be an indecomposable injective module and let P.E/ be theprime ideal defined before. Let I be a finitely generated ideal of R. Then E 2 V fg .I /if and only if E P.E/ 2 V fg .I /.Proof. Let I be such that E D E.R=I/. For each r 2 R n I we have, by the remarkjust above, that the annihilator of r CI 2 E is .I W r/and so, by definition of P.E/;wehave .I W r/ P.E/. The natural projection .rR C I /=I Š R=.I W r/ ! R=P.E/extends to a morphism from E to E P.E/ which is non-zero on r C I . Forming theproduct of these morphisms as r varies over R n I , we obtain a morphism from E toa product of copies of E P.E/ which is monic on R=I and hence is monic. ThereforeE is a direct summand of a product of copies of E P.E/ and so E 2 V fg .J / impliesE P.E/ 2 V fg .J /, where J is a finitely generated ideal.Now, E P.E/ 2 V fg .I /, where I is a finitely generated ideal, means that there isa non-zero morphism f W R=I n ! E P.E/ for some n. Since R=P.E/ is essential inE P.E/ the image of f has non-zero intersection with R=P.E/ so there is an ideal J ,without loss of generality finitely generated, with I n