## Bloch-Kato conjecture and motivic cohomology with . . . (1995)

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@MISC{Suslin95bloch-katoconjecture,

author = {A. Suslin and V. Voevodsky},

title = {Bloch-Kato conjecture and motivic cohomology with . . . },

year = {1995}

}

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485 | Étale Cohomology
- Milne
- 1980
(Show Context)
Citation Context ...lies in Z \Theta S. Denote by T 0 the inverse image of T under the morphism X 0 \Theta S f \Theta1 S \Gamma\Gamma\Gamma! X \Theta S. The scheme T 0 is etale over the henselian scheme T and hence (see =-=[M]-=-) splits into a disjoint sum T 0 = T 0 ` T 0 0 ` ::: ` T 0 n , where the image of T 0 in T does not contain t 0 and the schemes T 0 i are finite and etale over T and henselian. Since t 0 2 Z \Theta S ... |

450 | Théorie des topos et cohomologie étale des schémas, Séminaire de Géometrie Algébrique du Bois-Marie 1963–1964 - Artin, Grothendieck, et al. |

189 |
Lectures on Algebraic Topology
- Dold
- 1972
(Show Context)
Citation Context ...homology presheaves of Cs(F) implies easily that the natural embedding i : Cs(F) ,! T ot C ; (F) is a quasiisomorphism. The quasiisomorphism i has a canonical left inverse called the shuffle map (cf. =-=[D]-=- ch. 6, x 12), whose construction we are going to remind. Every strictly increasing map (OE; /) : [p + q] ! [p] \Theta [q] defines a linear isomorphism of schemes \Delta p+q ! \Delta p \Theta \Delta q... |

114 |
Critères de platitude et de projectivité
- Raynaud, Gruson
- 1971
(Show Context)
Citation Context ... (1 U \Theta p) \Gamma1 (S " (U \Theta (X n Z)) in U \Theta BZ (X). The scheme T is clearly proper and surjective over U , but need not be finite over U . According to the Platification Theorem -=-= see [R-G]-=- there exists a blow up (not necessarily with smooth center) U 0 ! U such that the proper inverse image T 0 of T in U 0 \Theta BZ (X) 30 ANDREI SUSLIN AND VLADIMIR VOEVODSKY is flat over U 0 . Since T... |

109 | Triangulated categories of motives over a field, in “Cycles, Transfers, and Motivic Cohomology Theories” Princeton Univ - Voevodsky - 2000 |

78 | Singular homology of abstract algebraic varieties
- SUSLIN, VOEVODSKY
- 1996
(Show Context)
Citation Context ...eld F and any integral separated scheme of finite type X over F there exists a proper surjective morphism Y ! X with Y smooth over F . It should be noted that this theorem implies that all results of =-=[S-V]-=- hold over an algebraically closed field of arbitrary characteristic. Hypercohomology. Motivic cohomology are defined as Zariski (or Nisnevich) hypercohomology with coefficients in a complex of sheave... |

70 | Algebraic cycles and higher K-theory - Bloch - 1986 |

46 |
The Milnor ring of a global field
- Bass, Tate
- 1973
(Show Context)
Citation Context .... Theorem 3.4. The natural homomorphism T (F ) ! \Phi 1 n=0 H n;n M (SpecF ) defines an isomorphism K Ms(F ) \Gamma! \Phi 1 n=0 H n;n M (SpecF ) (here K Ms(F ) is the Milnor ring of the field F - see =-=[B-T]-=-). Proof. We'll use the notation H ; M (F ) for H ; M (SpecF ). Note first of all that motivic cohomology H M (F; Z(n)) coincides up to a shift of degrees with homology of the complex Cs(Z tr (G n m )... |

40 | Cohomological theory of presheaves with transfers - Voevodsky - 2000 |

39 |
Etude Locale des Schemas et des Morphismes de Schemas (EGA 4
- Grothendieck, Dieudonne
(Show Context)
Citation Context ...S1 ) = lim \Gamma! ji Hom S j (X \Theta S i S j ; Y \Theta S i S j ) and furthermore for each X1 2 Sch=S1 there exists an index i 2 I and a scheme X i 2 Sch=S i such that X1 = X i \Theta S i S1 - see =-=[EGA-4]-=-. Moreover Corollary 12.2.2 shows that given a finite family of morphisms fX k i ! X i g K k=1 over S i , the corresponding family fX k i \Theta S i S1 ! X i \Theta S i S1 g K k=1 is a cdh-covering if... |

37 |
Gersten conjectures and the homology of schemes, Annals
- Bloch, Ogus
- 1974
(Show Context)
Citation Context ...we conclude that H 2 M (P 1 F ; Z=l(1)) = H 1 Zar (P 1 F ; H 1 (1)). Furthermore Zariski cohomology of any smooth scheme with coefficients in H 1 (1) may be computed using the Gersten complexes - see =-=[B-O]-=-. In particular H Zar (P 1 F ; H 1 (1)) may be computed using the Gersten complex H 1 et (F (P 1 );sl ) \Gamma! M x2(P 1 F ) 1 Z=l Here and in the sequell we denote by U i the set of points of codimen... |

37 |
The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory
- Nisnevich
- 1989
(Show Context)
Citation Context ...e reasoning to an arbitrary scheme etale over S we see that for q ! d the Nisnevich sheaf R q ffs(F) is supported in dimensionsd \Gamma q and hence H p Nis (S; R q ffs(F)) = 0 for p ? d \Gamma q -see =-=[Nis]-=-. Furthermore for qsd the Nisnevich sheaf R q ffs(F) is supported in dimension 0 and hence H p Nis (S; R q ffs(F)) = 0 for p ? 0. The Leray spectral sequence shows now that for n ? d we have an isomor... |

34 | Homology of schemes - Voevodsky - 1996 |

31 | A spectral sequence for motivic cohomology - Bloch, Lichtenbaum - 1994 |

30 | Height pairing between algebraic cycles - Beilinson - 1987 |

28 |
Homology of the general linear group over a local ring, and Milnor’s K-theory. English translation
- Nesterenko, Suslin
- 1990
(Show Context)
Citation Context ...any closed point. The residue field F (v) is a finite extension of F and we set `(v) = N F (v)=F (fX 1 (v); :::; X n (v)g) 2 K M n (F ). The usual argument involving the Weil reciprocity formula (see =-=[N-S]-=-) shows that the homomorphism ` : Z tr (G \Thetan m )(F ) ! K M n (F ) killes the image of @ 0 \Gamma @ 1 and thus defines a homomorphisms` : H n;n M (F ) ! K M n (F ) inverse to F . x 4. Fundamental ... |

25 |
V.: Bivariant cycle cohomology. Cycles, Transfers, and Motivic Homology Theories
- Friedlander, Voevodsky
- 2000
(Show Context)
Citation Context ...morphism. We define a map OE : (Z tr (X)=Z tr (X n Z))(S) ! (Z tr (X 0 )=Z tr (X 0 n Z 0 ))(S) setting OE([T ]) = [T 0 0 ]. The above discussion shows that OEfs= Id; fsOE = Id. We refer the reader to =-=[F-V]-=- for the proof of the following important Theorem. Theorem 4.12 (Quasiinvertibility of the Tate object). Assume that resolution of singularities holds over F . Then for any complexes A ffl ; B ffl 2 D... |

25 | Algebraic K-theory and the norm residue homomorphism - Suslin - 1985 |

14 | Higher Chow groups and etale cohomology, Cycles, Transfers and Motivic Homology Theories - Suslin - 1999 |

14 | Relative cycles and Chow sheaves. In Cycles, transfers, and motivic homology theories, volume 143 - Suslin, Voevodsky - 2000 |

13 | The norm residue homomorphism of degree three - Merkurjev, Suslin - 1991 |

10 | Values of zeta-functions at nonnegative integers - LICHTENBAUM - 1984 |

8 |
Affine analog of the proper base change theorem
- Gabber
(Show Context)
Citation Context ...l scheme T and denote by I the defining ideal of S. Let T h I be the henselization of T along I. The scheme S embeds canonically as a closed subscheme of T h I and according to a Theorem of O. Gabber =-=[G]-=- the restriction homomorphism in etale cohomology H n et (T h I ; \Omega n l ) ! H n et (S; \Omega n l ) is an isomorphism. On the other hand the scheme T h I is a filtered 54 ANDREI SUSLIN AND VLADIM... |

8 | Hilbert 90 for K3 for degree-two extensions - Rost - 1986 |

7 | Hilbert 90 for K 3 for degree-two extensions - Rost - 1986 |

3 | The Milnor ring of a global - Bass, Tate - 1973 |

3 | K-cohomology of Severy-Brauer varieties and the norm residue homomorphism - Merkurjev, Suslin - 1983 |

1 |
On the Norm Residue Homomorphism for
- Merkurjev
- 1996
(Show Context)
Citation Context ...s implies the validity of the Bloch-Kato Conjecture (and hence of the Beilinson-Lichtenbaum Conjecture as well). In the special case l = 2 such kind of statement was proved previously by A. Merkurjev =-=[Me]-=- using very different technique. We start with the following elementary Lemma. Lemma 11.1. The following conditions are equivalent (1) All Bockstein homomorphisms fi 1;k : H n et (F; \Omega n l ) \Gam... |

1 | Nesterenko and A.Suslin. Homology of the general linear group over a local ring and Milnor K-theory. Izv - unknown authors |

1 | Triangulated categories of motives over a Cycles, transfers and motivic homology theories - Voevodsky - 1995 |