process

Torsion classes of finite type and spectra 399D.I / is quasi-compact in Spec R by [2, Chapter 1, Ex. 17 (vii)]. We see that D.I / DSi2 0D.I i / for some finite subset 0 .Assume E 2 D fg .I / n S i2 0D fg .I i /. It follows from Theorem 3.5 that E P.E/ 2D fg .I /n S i2 0D fg .I i /. But E P.E/ 2 D fg .I /\Spec R D D.I / D S i2 0D.I i /, andhence it is in D.I i0 / D D fg .I i0 /\Spec R for some i 0 2 0 , a contradiction. So D fg .I /is quasi-compact. It also follows that the intersection D fg .I / \ D fg .J / D D fg .IJ / oftwo quasi-compact open subsets is quasi-compact. Furthermore, every quasi-compactopen subset in Inj fg R must therefore have the form D fg .I / with I finitely generated.Finally, it follows from Remark 3.4 and Theorem 3.5 that a subset V of Inj fg R isZariski-closed and irreducible if and only if there is a prime ideal Q of R such thatV DfE j P.E/ Qg. This obviously implies that the point E Q 2 V is generic.Notice that Inj fg R is not a spectral space in general, for it is not necessarily T 0 .Lemma 3.7. Let the ring R be commutative. Then the mapsandSpec R V7! Q V DfE 2 Inj R j P.E/ 2 V g.Inj fg R/ Q 7! V Q DfP.E/ 2 Spec R j E 2 Qg DQ \ Spec Rinduce a 1-1 correspondence between the lattices of open sets of Spec R and those of.Inj fg R/ .Proof. First note that E P 2 Q V for any P 2 V (see [14, 9.2]). Let us check thatQ V is an open set in .Inj fg 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, Exercise 17 (vii)]), so there are finitely generated ideals I R such thatV D S V.I /. Since the points E and E P.E/ are, by Theorem 3.5, indistinguishablein .Inj fg R/ we see that Q V D S V fg .I /, hence this set is open in .Inj fg R/ .The same arguments imply that V Q is open in Spec R. It is now easy to see thatV QV D V and Q VQ D Q.4 Torsion classes and thick subcategoriesWe shall write L.Spec R/, L..Inj fg R/ /, L thick .D per .R//, L tor .Mod R/ to denote:• the lattice of all open subsets of Spec R,• the lattice of all open subsets of .Inj fg R/ ,• the lattice of all thick subcategories of D per .R/,• the lattice of all torsion classes of finite type in Mod R, ordered by inclusion.