process

More documents

Recommendations

Info

Axioms for the norm residue isomorphism 433Corollary 3.3. The structural map H 1; 1 .M / ! H 1; 1 .k/ D k is injective.Proof. By (0.4) and Lemma 3.2, it suffices to show that Hom.Z;D.bC 1/Œ2b C 1/ D 0.By (0.5) this results from the vanishing of both Hom.Z;M.b C 1/Œ2b C 1/ andHom.Z; X ˝ L bCdC1 / which follows from Lemma 3.1.Lemma 3.4. H 0;1 .X/ D H 2;1 .X/ D 0 and H 1;1 .XI Z/ Š H 1;1 .Spec kI Z/ Š k .Proof. The spectral sequence E p;q1D H q .X pC1 I Z.1// ) H pCq;1 .XI Z/ degenerates,all rows vanishing except for q D 1 and q D 2, because Z.1/ Š O Œ 1; see[2, 4.2]. We compare this with the spectral sequence converging to H pCq .XI G m /;Hzar q .Y; O / ! H q ét . ; G m/ is an isomorphism for q D 0; 1 (and an injection forq D 2). Hence for q 2 we have H q;1 .X/ D H q;1ét.X/ D H q;1ét.k/ D H q 1ét.k; G m /.Remark. The proof of 3.4 also shows that H 3;1 .X/ injects into H 2 ét .k; G m/ D Br.k/.Lemma 3.5. H 2dC1;dC1 .DI Z/ is the kernel of H 1; 1 .M /y! H 1; 1 .k/ D k .Proof. From (0.5) we get an exact sequence with coefficients Z:H 2d;dC1 .X˝L d / ! H 2dC1;dC1 .D/ ! H 2dC1;dC1 .M / Dy ! H 2dC1;dC1 .X˝L d /:The first group is H 0;1 .X/, which is zero by 3.4, so it suffices to show that the map Dyidentifies with the structural map y. This follows from Axiom 0.3 (b), because for anyu in H 1; 1 .M / D Hom.Z; M.1/Œ1/, X tensored with the compositeX ˝ L d Dy ! M ˝ L d u ! Z.1/Œ1 ˝ L d D Z.d C 1/Œ2d C 1is the X-dual of X. 1/Œ 1u! M y ! X.Corollary 3.6. H 2dC1;dC1 .D/ D H 2dC1;dC1 .DI Z/ ˝ Z .`/ D 0.4 Between motivic and étale cohomologyDefinition 4.1 ([10, p. 90]). Let L.n/ denote the truncation nC1 Z ét .n/ of the complexin DM eff representing étale motivic cohomology; i. e., H p . ; L.n// Š H p;nét.`/. /for p n C 1. Let K.n/ denote the mapping cone of the canonical map Z .`/ .n/ !L.n/, and consider the triangle Z .`/ .n/ ! L.n/ ! K.n/ ! Z .`/ .n/Œ1.Lemma 4.2. The map H nC1 .X; K.n//y! H nC1 .M; K.n// is an injection.Proof. By triangle (0.4) we have an exact sequence:H n .D ˝ L b ; K.n// ! H nC1 .X; K.n//y! H nC1 .M; K.n//:
434 C. WeibelWe need to see that the left term vanishes. By (0.5), it is sufficient to show thatH n .M ˝ L b ; K.n// and H n 1 .X ˝ L b`; K.n// vanish. This follows from [10, 6.12],which says that H .Y.1/; K.n// D 0 for every smooth Y , the assumption that Mis a summand of X, and the consequent collapsing of the spectral sequence E p;q1DH q .X p ; K.n// ) H pCq .X; K.n//.Corollary 4.3. The map H nC1 .X; K.n// ! H nC1 .k.X/; K.n// is an injection.Proof. The map H nC1 .X; K.n// ! H nC1 .k.X/; K.n// is an injection by [2, 11.1,13.8, 13.10], or by [10, 7.4]. The corollary follows from Lemma 4.2, since H .X; / !H .M; / is a summand of H .X; / ! H .X; /.Proposition 4.4. There is an exact sequenceH nC1;n .X/ ! H nC1;nét.k/ ! H nC1;n .k.X//:Proof. By 1.5, X ! Spec.k/ is an isomorphism on étale motivic cohomology, so wehave H nC1;nét.k/ Š H nC1 .X; L.n//.The map Spec k.X/ ! X induces a commutative diagram with exact rows:étH nC1;n .X/H nC1 .X; L.n//H nC1 .X; K.n//4:3 into0=H nC1;n .k.X// H nC1 .k.X/; L.n// H nC1 .k.X/; K.n//:By 4.3, the right vertical map is an injection. The proposition now follows by a diagramchase.Acknowledgements. This paper is an attempt to clarify Voevodsky’s preprint [8]. It isdirectly inspired by the work of Markus Rost, especially [4], who explicitly suggestedthe line of attack, based upon Axiom 0.3 (c), which is presented in this paper. I amgreatly indebted to both of them.References[1] C. Haesemeyer and C. Weibel, Norm Varieties and the Chain Lemma (after Markus Rost),Preprint 2008, available at http://www.math.uiuc.edu/0900.[2] C. Mazza,V.Voevodsky and C. Weibel, Lecture notes on motivic cohomology, Clay Monogr.Math. 2, Amer. Math. Soc. , Providence, R.I., 2006.[3] M. Rost, Chain lemma for splitting fields of symbols, Preprint 1998, available athttp://www.math.uni-bielefeld.de/~rost/chain-lemma.html[4] M. Rost, On the Basic Correspondence of a Splitting Variety, Preprint 2006, available athttp://www.math.uni-bielefeld.de/~rost[5] N. Steenrod and D. Epstein, Cohomology Operations, Ann. of Math. Stud. 50, PrincetonUniversity Press, Princeton, N.J., 1962.
Page 3 and 4:
EMS Series of Congress ReportsEMS S
Page 5 and 6:
Editors:Guillermo CortiñasDepartam
Page 8 and 9:
ContentsPreface....................
Page 10 and 11:
IntroductionSince its inception 50
Page 12 and 13:
Introductionxiwith D. Blecher, he c
Page 14 and 15:
Program list of speakers and topics
Page 16 and 17:
Categorical aspects of bivariant K-
Page 18 and 19:
Page 20 and 21:
Page 22 and 23:
Page 24 and 25:
Page 26 and 27:
Page 28 and 29:
Page 30 and 31:
Page 32 and 33:
Page 34 and 35:
Page 36 and 37:
Page 38 and 39:
Page 40 and 41:
Page 42 and 43:
Page 44 and 45:
Page 46 and 47:
Page 48 and 49:
Page 50 and 51:
Page 52 and 53:
Page 54:
Page 57 and 58:
42 A. Bartels, S. Echterhoff, and W
Page 59 and 60:
Page 61 and 62:
Page 63 and 64:
Page 65 and 66:
Page 67 and 68:
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
72 H. Emerson and R. MeyerIn this s
Page 89 and 90:
74 H. Emerson and R. MeyerK-theoret
Page 91 and 92:
76 H. Emerson and R. MeyerExample 4
Page 93 and 94:
78 H. Emerson and R. Meyerand exact
Page 95 and 96:
80 H. Emerson and R. Meyercondition
Page 97 and 98:
82 H. Emerson and R. MeyerAs a resu
Page 99 and 100:
84 H. Emerson and R. MeyerLet h and
Page 101 and 102:
86 H. Emerson and R. MeyerCorollary
Page 103 and 104:
88 H. Emerson and R. Meyer5 Dual-Di
Page 106 and 107:
On K 1 of a Waldhausen categoryFern
Page 108 and 109:
On K 1 of a Waldhausen category 93t
Page 110 and 111:
On K 1 of a Waldhausen category 95(
Page 112 and 113:
On K 1 of a Waldhausen category 97s
Page 114 and 115:
On K 1 of a Waldhausen category 992
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
Page 130:
Page 133 and 134:
118 M. Karoubi[35], M. F. Atiyah an
Page 135 and 136:
120 M. Karoubihere by GrK n .A/, ar
Page 137 and 138:
122 M. Karoubi(resp. 1) in A. In th
Page 139 and 140:
124 M. Karoubithe class ˛ of A in
Page 141 and 142:
126 M. Karoubiested in the complex
Page 143 and 144:
128 M. KaroubiThe same method may b
Page 145 and 146:
130 M. KaroubiBy the usual excision
Page 147 and 148:
132 M. KaroubiLet A be a graded twi
Page 149 and 150:
134 M. Karoubithe same ideas as in
Page 151 and 152:
136 M. KaroubiTheorem 6.2. 25 The (
Page 153 and 154:
138 M. KaroubiIn order to show this
Page 155 and 156:
140 M. KaroubiThe following theorem
Page 157 and 158:
142 M. KaroubiWe would like to poin
Page 159 and 160:
144 M. KaroubiZ=n identified with t
Page 161 and 162:
146 M. Karoubiwhere n is the ring
Page 163 and 164:
148 M. Karoubi[6] M. F. Atiyah, R.
Page 166 and 167:
Equivariant cyclic homology for qua
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186:
Page 189 and 190:
174 C. Voigtand using ˇ.x; y/ D .S
Page 191 and 192:
176 C. Voigtalgebra A. The space Cn
Page 193 and 194:
178 C. VoigtKK-theory [16]. Hence,
Page 196 and 197:
A Schwartz type algebra for the tan
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
Page 204 and 205:
Page 206 and 207:
Page 208 and 209:
Page 210 and 211:
Page 212 and 213:
Page 214:
Page 217 and 218:
202 J. CuntzWe denote by N the set
Page 219 and 220:
204 J. CuntzConsider the action of
Page 221 and 222:
206 J. Cuntzprime numbers p and q,
Page 223 and 224:
208 J. Cuntz6 Representations as cr
Page 225 and 226:
210 J. CuntzProof. The spectral pro
Page 227 and 228:
212 J. CuntzThe operator s 1 plays
Page 229 and 230:
214 J. CuntzProposition 7.4. The K-
Page 232 and 233:
On a class of Hilbert C*-manifoldsW
Page 234 and 235:
On a class of Hilbert C*-manifolds
Page 236 and 237:
Page 238 and 239:
Page 240:
Page 243 and 244:
228 U. Bunke, T. Schick, M. Spitzwe
Page 245 and 246:
Page 247 and 248:
Page 249 and 250:
Page 251 and 252:
Page 253 and 254:
Page 255 and 256:
Page 257 and 258:
Page 259 and 260:
Page 261 and 262:
Page 263 and 264:
Page 265 and 266:
Page 267 and 268:
Page 269 and 270:
Page 271 and 272:
Page 273 and 274:
Page 275 and 276:
Page 277 and 278:
Page 279 and 280:
Page 281 and 282:
Page 283 and 284:
Page 285 and 286:
Page 287 and 288:
Page 289 and 290:
Page 291 and 292:
Page 293 and 294:
Page 295 and 296:
Page 297 and 298:
Page 299 and 300:
Page 301 and 302:
Page 303 and 304:
Page 305 and 306:
Page 307 and 308:
Page 309 and 310:
Page 311 and 312:
Page 313 and 314:
Page 315 and 316:
Page 317 and 318:
Page 319 and 320:
Page 321 and 322:
Page 323 and 324:
Page 325 and 326:
Page 327 and 328:
Page 329 and 330:
Page 331 and 332:
Page 333 and 334:
Page 335 and 336:
Page 337 and 338:
Page 339 and 340:
Page 341 and 342:
Page 343 and 344:
Page 345 and 346:
Page 347 and 348:
Page 349 and 350:
Page 351 and 352:
Page 353 and 354:
Page 355 and 356:
Page 357 and 358:
Page 359 and 360:
Page 361 and 362:
Page 364 and 365:
Deformations of gerbes on smooth ma
Page 366 and 367:
Page 368 and 369:
Page 370 and 371:
Page 372 and 373:
Page 374 and 375:
Page 376 and 377:
Page 378 and 379:
Page 380 and 381:
Page 382 and 383:
Page 384 and 385:
Page 386 and 387:
Page 388 and 389:
Page 390 and 391:
Page 392 and 393:
Page 394 and 395:
Page 396 and 397:
Page 398 and 399: Deformations of gerbes on smooth ma
Page 408 and 409: Torsion classes of finite type and
Page 428 and 429: Parshin’s conjecture revisitedTho
Page 430 and 431: Parshin’s conjecture revisited 41
Page 440: Parshin’s conjecture revisited 42
Page 443 and 444: 428 C. WeibelGiven (0.4) and axiom
Page 445 and 446: 430 C. WeibelAs pointed out in the
Page 447: 432 C. WeibelConsider the cohomolog
Page 452 and 453: List of contributorsArthur Bartels,
Page 454 and 455: List of participantsNatalia Abad Sa
show all

process

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?