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Torsion classes of finite type and spectra 409Proposition 7.2. Let R (respectively A) be a commutative ring (respectively gradedcommutative ring which is finitely generated as an A 0 -algebra). Then L tor .Mod R/and L tor .QGr A/ are ideal lattices.Proof. The spaces Spec R and Proj A are spectral. Thus Spec R and Proj A arespectral, also L open .Spec R/ and L open .Proj A/ are ideal lattices by Proposition 7.1.By Theorems 2.2 and 6.4 we have isomorphisms L open .Spec R/ Š L tor .Mod R/ andL open .Proj A/ Š L tor .QGr A/. Therefore L tor .Mod R/ and L tor .QGr A/ are ideallattices.Corollary 7.3. Let R (respectively A) be a commutative ring (respectively the gradedcommutative ring which is finitely generated as an A 0 -algebra). The points ofSpec L tor .Mod R/ (respectively Spec L tor .QGr A/) are the \-irreducible torsionclasses of finite type in Mod R (respectively tensor torsion classes of finite type inQGr A) and the mapsf W Spec R ! Spec L tor .Mod R/; P 7! S P DfM 2 Mod R j M P D 0g;f W Proj A ! Spec L tor .QGr A/; P 7! S P DfM 2 QGr A j M P D 0gare homeomorphisms of spaces.Proof. This is a consequence of Theorems 2.2, 6.4 and Propositions 7.1, 7.2.8 Reconstructing affine and projective schemesLet R (respectively A) be a commutative ring (respectively graded commutative ringwhich is finitely generated as an A 0 -algebra). We shall write Spec.Mod R/ WDSpec L tor .Mod R/ (resp. Spec.QGr A/ WD Spec L tor .QGr A/) and supp.M / WDfP 2 Spec.Mod R/ j M 62 P g (resp. supp.M / WD fP 2 Spec.QGr A/ j M 62 P g)for M 2 Mod R (resp. M 2 QGr A). It follows from Corollary 7.3 thatsupp R .M / D f 1 .supp.M // (resp. supp A .M / D f 1 .supp.M //):Following [4], [5], we define a structure sheaf on Spec.Mod R/ (Spec.QGr A/)asfollows. For an open subset U Spec.Mod R/ (U Spec.QGr A/), letS U DfM 2 Mod R.QGr A/ j supp.M / \ U D;gand observe that S U DfM j M P D 0 for all P 2 f1 .U /g is a (tensor) torsion class.We obtain a presheaf of rings on Spec.Mod R/ (Spec.QGr A/) byU 7! End Mod R=SU .R/.End QGr A=SU .O//:If V U are open subsets, then the restriction mapEnd Mod R=SU .R/ ! End Mod R=SV .R/.End QGr A=SU .O/ ! End QGr A=SV .O//
410 G. Garkusha and M. Prestis induced by the quotient functorMod R=S U ! Mod R=S V .QGr A=S U ! QGr A=S V /:The sheafification is called the structure sheaf of Mod R (QGr A) and is denoted byO Mod R (O QGr A ). This is a sheaf of commutative rings by [13, XI.2.4]. Next letP 2 Spec.Mod R/ and P WD f1 .P /. WehaveO Mod R;P D lim !P 2UEnd Mod R=SU .R/ D lim !f …PEnd Mod R=SD.f /.R/ Š lim !f …PR f D O R;P :Similarly, for P 2 Spec.QGr A/ and P WD fO QGr A;P Š O Proj A;P :1 .P / we haveThe next theorem says that the abelian category Mod R (QGr A) contains all thenecessary information to reconstruct the affine (projective) scheme .Spec R;O R / (respectively.Proj A; O Proj A /).Theorem 8.1 (Reconstruction). Let R (respectively A) be a ring (respectively gradedring which is finitely generated as an A 0 -algebra). The maps of Corollary 7.3 induceisomorphisms of ringed spacesandf W .Spec R;O R /f W .Proj A; O Proj A /! .Spec.Mod R/; OMod R /! .Spec.QGr A/; OQGr A /:Proof. The proof is similar to that of [5, 9.4]. Fix an open subset U Spec.Mod R/and consider the composition of the functorsF W Mod R . /! Qcoh Spec R. /j f 1 .U /! Qcoh f 1 .U /:Here, for any R-module M , we denote by M its associated sheaf. By definition,the stalk of M at a prime P equals the localized module M P . We claim that Fannihilates S U . In fact, M 2 S U implies f 1 .supp.M //\f 1 .U / D;and thereforesupp R .M /\f 1 .U / D;. Thus M P D 0 for all P 2 f 1 .U / and therefore F.M/ D0. It follows that F factors through Mod R=S U and induces a map End Mod R=SU .R/ !O R .f 1 .U // which extends to a map O Mod R .U / ! O R .f 1 .U //. This yields themorphism of sheaves f ] W O Mod R ! f O R .By the above f ] induces an isomorphism f ]P W O Mod R;f.P/ ! O R;P at each pointP 2 Spec R. We conclude that f ]Pis an isomorphism. It follows that f is an isomorphismof ringed spaces if the map f W Spec R ! Spec.Mod R/ is a homeomorphism.This last condition is a consequence of Propositions 7.1 and 7.2. The same argumentsapply to show thatf W .Proj A; O Proj A /! .Spec.QGr A/; OQGr A /is an isomorphism of ringed spaces.
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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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On K 1 of a Waldhausen category 95(
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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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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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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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Page 452 and 453: List of contributorsArthur Bartels,
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