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Deformations of gerbes on smooth manifolds 383we denote by DR. xC .J X /Œ1/ ! the sheaf of DGLA with the underlying graded Liealgebra X ˝ xC .J X /Œ1 and the differential ı Cr can C ! . Letg DR .J X / ! D .XI DR. xC .J X /Œ1/ ! /;be the corresponding DGLA of global sections.Suppose now that U isacoverofX; let W N U ! X denote the canonical map.For W Œn ! letG DR .J/ ! D .N .n/UI DR. xC .J X /Œ1/ ! /:For W Œm ! and a morphism W Œm ! Œn in such that D ı the map.m/ ! .n/ induces the map W G DR .J/ ! ! G DR.J/ ! :For n D 0;1;:::let G DR .J/ n ! D Q Œn !G DR .J/ ! . The assignment Œn 7! G DR.J/ n !extends to a cosimplicial DGLA G DR .J/ ! . If we need to explicitly indicate the coverwe will also denote this DGLA by G DR .J/ ! .U/.Lemma 7.8. The cosimplicial DGLA G DR .J/ ! is acyclic, i.e. satisfies the condition(3.8).Proof. Consider the cosimplicial vector space V withV n D .N n UI DR. xC .J X /Œ1/ ! /and the cosimplicial structure induced by the simplicial structure of N U. The cohomologyof the complex .V ;@/is the Čech cohomology of U with the coefficients inthe soft sheaf of vector spaces ˝ xC .J X /Œ1 and, therefore, vanishes in the positivedegrees. G DR .J/ ! as a cosimplicial vector space can be identified with yV in thenotations of Lemma 2.1. Hence the result follows from Lemma 2.1.We leave the proof of the following lemma to the reader.Lemma 7.9. The map W g DR .J X / ! ! G DR .J/ 0 ! induces an isomorphism of DGLAg DR .J X / ! Š ker.G DR .J/ 0 ! G DR.J/ 1 ! /where the two maps on the right are .@ 0 0 / and .@ 0 1 / .Two previous lemmas together with Corollary 3.13 imply the following:Proposition 7.10. Let ! 2 .XI . 2 X ˝ J X=O X / cl / and let m be a commutativenilpotent ring. Then the map induces equivalence of groupoids:MC 2 .g DR .J X / ! ˝ m/ Š Stack.G DR .J/ ! ˝ m/:
384 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganFor ˇ 2 .XI 1 ˝ J X =O X / there is a canonical isomorphism of cosimplicialDGLA exp.ˇ /W G DR .J/ !Cr canˇ ! G DR .J/ ! . Therefore, G DR .J/ ! depends only onthe class of ! in H 2 DR .J X=O X /.The rest of this section is devoted to the proof of the following theorem.Theorem 7.11. Suppose that .U; A/ is a descent datum representing a twisted formS of O X . There exists a quasi-isomorphism of cosimplicial DGLA G DR .J/ ŒS !G DR .J.A//.7.4 Quasiisomorphism. Suppose that .U; A/ is a descent datum for a twisted formof O X . Thus, A is identified with O N0 U and A 01 is a line bundle on N 1 U.7.4.1 Multiplicative connections. Let C .A 01 / denote the set of connections r onA 01 which satisfy1. Ad A 012 ..pr 2 02 / r/ D .pr 2 01 / r˝Id C Id ˝ .pr 2 12 / r,2. .pr 0 00 / r is the canonical flat connection on O N0 U.Let Isom 0 .A 01˝J N1 U; J.A 01 // denote the subset of Isom 0 .A 01˝J N1 U; J.A 01 //which consists of which satisfy1. Ad A 012 ..pr 2 02 / / D .pr 2 01 / ˝ .pr 2 12 / ,2. .pr 0 00 / D IdNote that the vector space xZ 1 .UI 1 / of cocycles in the normalized Čech complexof the cover U with coefficients in the sheaf of 1-forms 1 acts on the set C .A 01 /,with the action given by˛ r DrC˛: (7.3)Here r2C .A 01 /, ˛ 2 xZ 1 .UI 1 / 1 .N 1 U/.Similarly, the vector space xZ 1 .UI J 0 / acts on the set Isom 0 .A 01˝J N1 U; J.A 01 //,with the action given as in Corollary 7.4.Note that since the sheaves involved are soft, cocycles coincide with coboundaries:xZ 1 .UI 1 / D xB 1 .UI 1 /, xZ 1 .UI J 0 / D xB 1 .UI J 0 /.Proposition 7.12. The set C .A 01 / (respectively, Isom 0 .A 01 ˝ J N1 U; J.A 01 //)is an affine space with the underlying vector space being xZ 1 .UI 1 / (respectively,xZ 1 .UI J 0 /).Proof. Proofs of the both statements are completely analogous. Therefore we explainthe proof of the statement concerning Isom 0 .A 01 ˝ J N1 U; J.A 01 // only.We show first that Isom 0 .A 01 ˝ J N1 U; J.A 01 // is nonempty. Choose an arbitrary 2 Isom 0 .A 01 ˝ J N1 U; J.A ij // such that .pr 0 00 / D Id. Then, by Corollary7.4, there exists c 2 .N 2 UI J 0 / such that c.Ad A 012 ..d 1 / 02 // D .d 0 / 01˝.d 2 / 12 . It is easy to see that c 2 xZ 2 .UI exp J 0 /. Since the sheaf exp J 0 is soft,
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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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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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On a class of Hilbert C*-manifolds
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228 U. Bunke, T. Schick, M. Spitzwe
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434 C. WeibelWe need to see that th
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List of contributorsArthur Bartels,
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List of participantsNatalia Abad Sa
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