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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.
400 G. Garkusha and M. Prest(A thick subcategory is a triangulated subcategory closed under direct summands).Given a perfect complex X 2 D per .R/ denote by supp.X/ DfP 2 Spec R jX ˝LR R P ¤ 0g. It is easy to see thatsupp.X/ D [ n2Zsupp R .H n .X//;where H n .X/ is the nth homology group of X.Theorem 4.1 (Thomason [18]). Let R be a commutative ring. The assignmentsT 2 L thick .D per .R// 7! [supp.X/andX2TV 2 L.Spec R/ 7! fX 2 D per .R/ j supp.X/ V gare mutually inverse lattice isomorphisms.Given a subcategory X in Mod R, we may consider the smallest torsion class offinite type in Mod R containing X. This torsion class we denote byp \X D fS Mod R j S X is a torsion class of finite typeg:Theorem 4.2. (cf. Garkusha–Prest [8, Theorem C]) Let R be a commutative ring.There are bijections between• the set of all open subsets Y .Inj fg R/ ,• the set of all torsion classes of finite type in Mod R,• the set of all thick subcategories of D per .R/.These bijections are defined as follows:´S D¹M j .M / Y º;Y 7!T D¹X 2 D per .R/ j .H n .X// Y for all n 2 ZºI8ˆ< Y D [ /;S 7! M 2S.Mˆ:T D¹X 2 D per .R/ j H n .X/ 2 S for all n 2 ZºI8ˆ< Y D [.H n .X//;T 7! X2T ;n2Zˆ:S D p ¹H n .X/ j X 2 T ;n2 Zº:
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EMS Series of Congress ReportsEMS S
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Editors:Guillermo CortiñasDepartam
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ContentsPreface....................
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IntroductionSince its inception 50
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Introductionxiwith D. Blecher, he c
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Program list of speakers and topics
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Categorical aspects of bivariant K-
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72 H. Emerson and R. MeyerIn this s
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74 H. Emerson and R. MeyerK-theoret
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76 H. Emerson and R. MeyerExample 4
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78 H. Emerson and R. Meyerand exact
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80 H. Emerson and R. Meyercondition
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82 H. Emerson and R. MeyerAs a resu
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84 H. Emerson and R. MeyerLet h and
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86 H. Emerson and R. MeyerCorollary
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88 H. Emerson and R. Meyer5 Dual-Di
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On K 1 of a Waldhausen categoryFern
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118 M. Karoubi[35], M. F. Atiyah an
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120 M. Karoubihere by GrK n .A/, ar
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122 M. Karoubi(resp. 1) in A. In th
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124 M. Karoubithe class ˛ of A in
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126 M. Karoubiested in the complex
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128 M. KaroubiThe same method may b
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130 M. KaroubiBy the usual excision
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132 M. KaroubiLet A be a graded twi
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134 M. Karoubithe same ideas as in
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136 M. KaroubiTheorem 6.2. 25 The (
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138 M. KaroubiIn order to show this
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140 M. KaroubiThe following theorem
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142 M. KaroubiWe would like to poin
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144 M. KaroubiZ=n identified with t
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146 M. Karoubiwhere n is the ring
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148 M. Karoubi[6] M. F. Atiyah, R.
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Equivariant cyclic homology for qua
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174 C. Voigtand using ˇ.x; y/ D .S
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176 C. Voigtalgebra A. The space Cn
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178 C. VoigtKK-theory [16]. Hence,
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A Schwartz type algebra for the tan
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202 J. CuntzWe denote by N the set
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204 J. CuntzConsider the action of
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206 J. Cuntzprime numbers p and q,
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208 J. Cuntz6 Representations as cr
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210 J. CuntzProof. The spectral pro
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212 J. CuntzThe operator s 1 plays
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214 J. CuntzProposition 7.4. The K-
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On a class of Hilbert C*-manifoldsW
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On a class of Hilbert C*-manifolds
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Page 364 and 365: Deformations of gerbes on smooth ma
Page 408 and 409: Torsion classes of finite type and
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Page 452 and 453: List of contributorsArthur Bartels,
Page 454 and 455: List of participantsNatalia Abad Sa
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