## Lecture notes on motivic cohomology (2006)

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Venue: | of Clay Mathematics Monographs. American Mathematical Society |

Citations: | 22 - 2 self |

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@INPROCEEDINGS{Mazza06lecturenotes,

author = {Carlo Mazza and Vladimir Voevodsky and Charles Weibel},

title = {Lecture notes on motivic cohomology},

booktitle = {of Clay Mathematics Monographs. American Mathematical Society},

year = {2006}

}

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### Abstract

From the point of view taken in these lectures, motivic cohomology with coefficients in an abelian group A is a family of contravariant functors H p,q (−, A) : Sm/k → Ab from smooth schemes over a given field k to abelian groups, indexed by

### Citations

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Algebraic geometry
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Citation Context ...an irreducible closed subset of T which is finite over S. Then f (W) is closed and irreducible in T ′ and finite over S. If W is finite and surjective over S, then so is f (W). PROOF. By Ex.II.4.4 of =-=[Har77]-=-, f (W) is closed in T ′ and proper over S. Since f (W) has finite fibers over S, it is finite over S by [EGA3, 4.4.2]. If W → S is surjective, so is f (W) → S. □ Given elementary correspondences V ∈ ... |

945 | Categories for the working mathematician - Lane - 1971 |

506 | Étale Cohomology
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Citation Context ...ts sequence: 0 ✲ H p,q (X,Z)/l ✲ H p,q (X,Z/l) ✲ l H p+1,q (X,Z) ✲ 0. COROLLARY 4.8. There is a quasi-isomorphism of complexes of étale sheaves Z/l(1) ét ≃ µ l . PROOF. Since sheafification is exact (=-=[Mil80]-=- p. Z(1) ét ≃ O∗ ét [−1], and hence Z/l(1) ét ≃ O ∗ ét[−1] ⊗ L Z/l ≃ µ l . 63), theorem 4.1 gives □ COROLLARY 4.9. If 1/l ∈ k and X is smooth, then H p,1 (X,Z/l) = 0 for p = 0,1,2 while: H 0,1 (X,Z/l... |

464 |
in: Théorie des Topos et Cohomologie Etale des Schémas, Springer Lect. Notes 269
- Grothendieck, Verdier, et al.
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Citation Context ...ption that R is a commutative ring and that cdR(k) < ∞, i.e., k is a field having finite étale cohomological dimension for coefficients in R. This assumption allows us to invoke a classic result from =-=[SGA4]-=-. Lemma 9.25. ([SGA4], [Mil80]) Let X be a scheme of finite type over k. If k has finite R-cohomological dimension d then cdR(X) ≤ d + 2 dimk X. Corollary 9.26. Ext n (Rtr(X), F ) = 0 when n ≫ 0. Proo... |

278 |
An introduction to homological algebra
- Weibel
- 1994
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Citation Context ... correspondence from X to Y (both connected), the transfer A → A is multiplication by the degree of W over X. The following theorem is a special case of a well known result on functor categories, see =-=[Wei94]-=- 1.6.4 and Exercises 2.3.7 and 2.3.8. Theorem 2.3. The category PST(k) is abelian and has enough injectives and projectives. Example 2.4. The sheaf O ∗ of global units and the sheaf O of global functi... |

210 | Sur quelques points d’algèbre homologique - Grothendieck - 1957 |

163 |
Des catégories dérivées des catégories abéliennes, thèse de doctorat d’état
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- 1967
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Citation Context ... PROPOSITION 8A.4. Let A be an additive symmetric monoidal category. Then the chain homotopy category K − (A ) of bounded above cochain complexes is a tensor triangulated category. EXAMPLE 8A.5. (See =-=[Ver96]-=-.) Let A be the category of modules over a commutative ring, or more generally over a scheme. Then not only is K − (A ) a tensor triangulated category, but the total tensor product ⊗ L makes the deriv... |

150 | Elements of homotopy theory - Whitehead - 1978 |

116 | Critères de platitude et de projectivite - Gruson, Raynaud - 1971 |

111 | Triangulated categories of motives over a field, Cycles, Transfers and Motivic Homology Theories
- Voevodsky
- 2000
(Show Context)
Citation Context ... As a presheaf, the map Ztr(X)(U) → Ztr(X)(V ) associated to V → U is just the pullback of cycles along V × X → U × X. The notation Ztr(X) was introduced in [SV00] while the notation L(X) was used in =-=[TriCa]-=- and cequi(X/ Spec k, 0) in [RelCy] and [CohTh]. We will write Z for the presheaf with transfers Ztr(Spec k); it is just the constant presheaf Z on Sm/k, equipped with the transfer maps of 2.2. Thus t... |

100 |
Algebraic K-theory and quadratic forms
- Milnor
(Show Context)
Citation Context ...p = 0 (2) one has H p,1 ⎧ ⎨ (X,Z) = ⎩ O∗ (X) for p = 1 Pic(X) for p = 2 0 for p = 1,2 (3) for a field k, one has H p,p (Spec(k),A) = KM p (k) ⊗ A where KM p (k) is the p-th Milnor K-group of k (see =-=[Mil70]-=-). (4) for a strictly Hensel local scheme S over k and an integer n prime to char(k), one has H p,q { µ ⊗q (S,Z/n) = n (S) for p = 0 0 for p = 0 where µn(S) is the groups of n-th roots of unity in S.... |

92 | conjecture and motivic cohomology with finite coefficients. In The arithmetic and geometry of algebraic cycles
- Suslin, Voevodsky
- 1998
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Citation Context ...at Ztr(X) is a projective object in PST(k). As a presheaf, the map Ztr(X)(U) → Ztr(X)(V ) associated to V → U is just the pullback of cycles along V × X → U × X. The notation Ztr(X) was introduced in =-=[SV00]-=- while the notation L(X) was used in [TriCa] and cequi(X/ Spec k, 0) in [RelCy] and [CohTh]. We will write Z for the presheaf with transfers Ztr(Spec k); it is just the constant presheaf Z on Sm/k, eq... |

82 | Singular homology of abstract algebraic varieties
- Suslin, Voevodsky
- 1996
(Show Context)
Citation Context ..., q) ̸= (0, 0), (1, 1), (2, 1) ⎪⎨ Z(X) (p, q) = (0, 0) (X, Z) = ⎪⎩ O∗ (X) (p, q) = (1, 1) Pic(X) (p, q) = (2, 1) This theorem will follow from lemmas 4.3 − 4.6 below. An alternative proof is given in =-=[SV96]-=-. Let us define the functor M ∗ (P 1 ; 0, ∞) : Sm/k → Ab which sends a scheme X to the group of rational functions on X × P 1 which equal 1 on X × {0, ∞}. Clearly M ∗ (P 1 ; 0, ∞) is a sheaf for the Z... |

81 |
Derived categories and stable equivalence
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- 1989
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Citation Context ...r to [GM03] or [Wei94] for basic facts about derived categories. We will also need the notion of a thick subcategory, which was introduced by Verdier in [Ver96]. We will use Rickard’s definition (see =-=[Ric89]-=-); this is slightly different from, but equivalent to, Verdier’s definition. DEFINITION 9.1. A full additive subcategory E of D − is thick if: (1) Let A → B → C → A[1] be a distinguished triangle. The... |

80 |
Vector bundles and projective modules
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(Show Context)
Citation Context ...nsfers F would have F (Φ)f ∗ (x) = 2x for all x ∈ F (X). Let L be a line bundle on a smooth variety X satisfying L 2 ∼ = OX but [L ⊕ L] ̸= [OX ⊕ OX] in K0(X). It is well-known that such L exists; see =-=[Swa62]-=-. It is also well-known that there is an étale cover f : Y → X of degree 2 with Y = Spec(OX ⊕ L); see [Har77, IV Ex.2.7]. Since f ∗ L ∼ = OY , the element x = [L] − [OX] of K0(X) satisfies f ∗ (x) = 0... |

64 |
Correspondences, motifs and monoidal transformations
- Manin
- 1968
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Citation Context ... it is a finite correspondence from X to Z by lemma 1.4. We can easily check that id X is the identity of Cor(X,X), and that the composition of finite correspondences is associative and bilinear (see =-=[Man68]-=- and [Ful84, 16.1]). DEFINITION 1.5. Let Cor k be the category whose objects are the smooth separated schemes of finite type over k and whose morphisms from X to Y are elements1. THE CATEGORY OF FINI... |

54 |
Friedlander Cycles, Transfers, and Motivic Homology Theories
- Voevodsky, Suslin, et al.
- 1999
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Citation Context ...nvariants. The idea of these lectures was to define motivic cohomology and to give careful proofs for the elementary results in the second family. In this sense they are complimentary to the study of =-=[VSF00]-=-, where the emphasis is on the properties of motivic cohomology itself. In the process, the structure of the proofs force us to deal with the main properties of motivic cohomology as well (such as hom... |

48 |
Categories derivees, etat 0
- Verdier
- 1977
(Show Context)
Citation Context .... Proposition 8A.4. Let A be an additive symmetric monoidal category. Then the chain homotopy category K − (A) of bounded above cochain complexes is a tensor triangulated category. Example 8A.5. (See =-=[Ver77]-=-.) Let A be the category of modules over a commutative ring, or more generally over a scheme. Then not only is K − (A) a tensor triangulated category, but the total tensor product ⊗ L makes the derive... |

40 | Cohomological theory of presheaves with transfers
- Voevodsky
- 2000
(Show Context)
Citation Context ...), is that the sheaf of finite cycles Ztr(X) is the free object generated by X. This idea led to a group of results which is represented here by lemma 6.23. The second one which is the main result of =-=[CohTh]-=- is represented here by theorem 13.7. Taken together they allow one to efficiently do homotopy theory in the category of sheaves with transfers. A considerable part of the first half of the lectures i... |

38 |
Algébre locale multiplicités
- Serre
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Citation Context ...he Wi are the irreducible components of the support of Z which are surjective over a component of X and ni is the geometric multiplicity of Wi in Z, i.e., the length of the local ring of Z at Wi (See =-=[Ser65]-=- or [Ful84]). □ We will now define an associative and bilinear composition for finite correspondences between smooth schemes. For this, it suffices to define the composition W ◦V of elementary corresp... |

38 |
The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory. In: Algebraic K-theory: Connections with geometry and topology
- Nisnevich
- 1989
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Citation Context ...i be the decomposition of U into its irreducible components and let Ti = T × U Ui . Refining T → U as in 12.25, we may assume that each Ti → Ui is a proper birational cdh cover. By platification (see =-=[RG71]-=- or 1A.1) applied to T → X, there is a blow-up X ′ → X along a Z ⊆ X such that the proper transform T ′ i of each Ti is flat over X ′. We set U ′ i = Ui × X X ′. The situation is described by the foll... |

34 | The additivity of traces in triangulated categories
- May
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Citation Context ...rposes. However, it is possible to add extra axioms in order to work 6566 APPENDIX 8A - TENSOR TRIANGULATED CATEGORIES with a richer structure. For example, many more axioms are postulated by May in =-=[May01]-=-. EXERCISE 8A.2. Show that the canonical isomorphisms l i r j ,r j l i : C[i] ⊗ D[ j] ∼ = (C ⊗ D)[i + j] differ by (−1) i j , and are interchanged by the twist isomorphism τ on C ⊗ D and C[i] ⊗ D[ j].... |

29 | Suslin, Homology of the general linear group over a local ring - Nesterenko, A - 1989 |

23 | Homotopy algebraic K-theory, in ‘Algebraic K-Theory and Algebraic Number Theory - Weibel - 1989 |

23 | Une suite exacte de Mayer-Vietoris en K-théorie algébrique (French), Algebraic K-theory - Jouanolou - 1972 |

19 |
Global class field theory of arithmetic schemes. Applications of Algebraic K-theory to Algebraic Geometry and Number Theory
- Kato, Saito
- 1986
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Citation Context ... 6.1]. ¿From this description it follows that Speck has Nisnevich cohomological dimension zero. This implies that the Nisnevich cohomological dimension of any Noetherian scheme X is at most dimX; see =-=[KS86]-=-. LEMMA 12.3. If {U i → X} is a Nisnevich covering then there is a nonempty open V ⊂ X and an index i such that U i | V → V has a section. PROOF. For each generic point x of X, there is a generic poin... |

17 |
Values of zeta-functions at non-negative integers, Number Theory
- Lichtenbaum
- 1983
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Citation Context ...by integers p and q. The idea of motivic cohomology goes back to P. Deligne, A. Beilinson and S. Lichtenbaum. Most of the known and expected properties of motivic cohomology (predicted in [ABS87] and =-=[Lic84]-=-) can be divided into two families. The first family concerns properties of motivic cohomology itself – there are theorems concerning homotopy invariance, Mayer-Vietoris and Gysin long exact sequences... |

14 |
Bloch’s higher Chow groups revisited, K-theory
- Levine
- 1992
(Show Context)
Citation Context ...lectures. See [Blo94], [BL94], [FS02], [Lev98] and [SV00]. Deeper comparison results include the theorem of M. Levine comparing CH i (X, j;Q) with the graded pieces of the gamma filtration in K∗(X)⊗Q =-=[Lev94]-=-, and the construction of the spectral sequence relating motivic cohomology and algebraic K-theory for arbitrary coefficients in [BL94] and [FS02]. The lectures in this book may be divided into two pa... |

11 |
Mennicke symbols and their applications in the K-theory of fields
- Suslin
- 1980
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Citation Context ...NE/F : E ∗ → F ∗ .If F ⊂ E is a finite field extension, there is a “norm map” NE/F : KM n (E) → KM n (F ) satisfying the analogue of lemma 5.3. In addition, it satisfies the following condition (see =-=[Sus82]-=-). Theorem 5.4 (Weil Reciprocity). Suppose that L is an algebraic function field over k. For each discrete valuation w on L there is a map and for all x ∈ K M n+1(L): ∂w : K M n+1(L) → K M n (k(w)), ∑... |

8 | Cancellation theorem
- Voevodsky
(Show Context)
Citation Context ... approach, which needs resolution of singularities, was developed in the book “Cycles, Transfers and Motivic Homology Theories” [VSF00]. A second recent approach is to use the Cancellation Theorem of =-=[Voe02]-=- and the Gersten resolution 23.11 for motivic cohomology sheaves. A third approach, which is the one we shall develop here, uses Bloch’s higher Chow groups CH i (X, m) to establish the more general is... |

3 | Some relations between higher K-functors - Swan - 1972 |

3 |
Categories for the working mathematician
- Soc, Providence
- 1994
(Show Context)
Citation Context ...from Y to Y ′ , then the cycle associated to the subscheme V ×W by 1.3 gives a finite correspondence from X ⊗Y to X ′ ⊗Y ′ . It is easy to verify that ⊗ makes Cor k a symmetric monoidal category (see =-=[Mac71]-=-). EXERCISE 1.10. If S = Speck then Cor k (S,X) is the group of zero-cycles in X. If W is a finite correspondence from A 1 to X, and s,t : Speck → A 1 are k-points, show that the zero-cycles W ◦ Γs an... |

3 |
conjecture and motivic cohomology with finite coefficients, The Arithmetic and Geometry of Algebraic Cycles
- Bloch-Kato
(Show Context)
Citation Context ...nd higher Chow groups leads to connections between motivic cohomology and algebraic K-theory, but we do not discuss these connections in the present lectures. See [Blo94], [BL94], [FS02], [Lev98] and =-=[SV00]-=-. Deeper comparison results include the theorem of M. Levine comparing CH i (X, j;Q) with the graded pieces of the gamma filtration in K∗(X)⊗Q [Lev94], and the construction of the spectral sequence re... |

2 |
motives
- Mixed
- 1998
(Show Context)
Citation Context ...cohomology and higher Chow groups leads to connections between motivic cohomology and algebraic K-theory, but we do not discuss these connections in the present lectures. See [Blo94], [BL94], [FS02], =-=[Lev98]-=- and [SV00]. Deeper comparison results include the theorem of M. Levine comparing CH i (X, j;Q) with the graded pieces of the gamma filtration in K∗(X)⊗Q [Lev94], and the construction of the spectral ... |

2 |
Higher Chow Groups and Étale
- Suslin
(Show Context)
Citation Context ...mensional cycles The next step in the proof of theorem 19.1 (that motivic cohomology and higher Chow groups agree) is the reduction to equidimensional cycles. The main references for this lecture are =-=[HighCh]-=- and [FS02]. DEFINITION 18.1. For an equidimensional X, and i ≤ dimX, we write zi equi (X,m) for zequi (X,dimX −i)(∆m ), the free abelian group generated by all codimension i subvarieties on X × ∆m wh... |

1 |
Higher Chow Groups and
- Suslin
(Show Context)
Citation Context ...mensional cycles The next step in the proof of theorem 19.1 (that motivic cohomology and higher Chow groups agree) is the reduction to equidimensional cycles. The main references for this lecture are =-=[HigCh]-=- and [FS00]. Definition 18.1. For an equidimensional X, and i ≤ dim X, we write z i equi(X, m) for zequi(X, dim X − i)(∆ m ), the free abelian group generated by all codimension i subvarieties on X × ... |

1 | K-theory of a space with coefficients in a (discrete - Rector - 1971 |

1 | SGA1] Revêtements étales et groupe fondamental - Multiplicités, Berlin - 1965 |

1 | Vector bundles and projective modules, Trans - Gordon, Müller-Stach, et al. |

1 | Cancellation theorem, Preprint. http://www.math.uiuc.edu/K-theory/541, 2002. [Wei89] [Wei94] C. Weibel, Homotopy algebraic K-theory, Algebraic K-theory and algebraic number theory - Voevodsky - 1987 |