process

Torsion classes of finite type and spectra 395Note that V.I/ DfP 2 Spec R j I P g is equal to supp R .R=I / for every ideal Iandsupp R .M / D [V.ann R .x//; M 2 Mod R:x2MRecall from [10] that a topological space is spectral if it is T 0 , quasi-compact, ifthe quasi-compact open subsets are closed under finite intersections and form an openbasis, and if every non-empty irreducible closed subset has a generic point. Given aspectral topological space, X, Hochster [10] endows the underlying set with a new,“dual", topology, denoted X , by taking as open sets those of the form Y D S i2 Y iwhere Y i has quasi-compact open complement X nY i for all i 2 . Then X is spectraland .X / D X (see [10, Proposition 8]). The spaces, X, which we shall consider arenot in general spectral; nevertheless we make the same definition and denote the spaceso obtained by X .Given a commutative ring R, every closed subset of Spec R with quasi-compactcomplement has the form V.I/ for some finitely generated ideal, I ,ofR (see [2,Chapter 1, Ex. 17 (vii)]). Therefore a subset of Spec R is open if and only if it is of theform S V.I / with each I finitely generated. Notice that V.I/with I a non-finitelygenerated ideal is not open in Spec R in general. For instance (see [18, 3.16.2]), letR D CŒx 1 ;x 2 ;:::and m D .x 1 ;x 2 ;:::/. It is clear that V .m/ Dfmg is not open inSpec CŒx 1 ;x 2 ;:::.For definitions of terms used in the next result see the Appendix to this paper.Theorem 2.2 (Classification). Let R be a commutative ring.betweenThere are bijections1. the set of all open subsets V Spec R,2. the set of all Gabriel filters F of finite type,3. the set of all torsion classes S of finite type in Mod R.These bijections are defined as follows:´FV D¹I R j V.I/ V º;V 7!S V D¹M 2 Mod R j supp R .M / V ºI8ˆ< V F D [ V.I/;F 7!I 2Fˆ:S F D¹M 2 Mod R j ann R .x/ 2 F for every x 2 M ºI8ˆ< F S D¹I R j R=I 2 Sº;S 7! V ˆ: S D [ supp R .M /:M 2S