Voevodsky s Construction Important Concepts (Mazza, Voevodsky, Weibel)

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1 Motivic Cohomology 1. Triangulated Category of Motives (Voevodsky) 2. Motivic Cohomology (Suslin-Voevodsky) 3. Higher Chow complexes a. Arithmetic (Conjectures of Soulé and Fontaine, Perrin-Riou) b. Mixed Tate Motives (Relations with motives associated to π 1 (P 1 S); Deligne, Goncharov...) c. Limiting mixed motives. 1

2 Voevodsky s Construction Important Concepts (Mazza, Voevodsky, Weibel) 1. Additive category of correspondences Cor k. Objects smooth varieties over k. Morphisms: f i X Z i Y Here f i finite surjective, and Z = n i Z i Hom Cor (X, Y ). Example: Z X Y graph of f : X Y, Sm k Cor k. 2. Presheaves with transfers PST: F : Cor op k Ab 2

3 Examples of presheaves with transfer 1. Representable Z tr (X), X Ob(Sm k ). Z tr (X)(U) := Hom Cor (U, X). 2. Z tr ((X 1, x 1 ) (X n, x n )) Coker( Ztr (X 1 X i X n ) Z tr ( ) X i ) 3. Z tr ( n G m ). Take X i = A 1 {0}, x i = 1. 3

4 Chains, Homotopy Invariance n = Spec k[x 0,..., x n ]/( x i 1) For F Ob(PST) C (F )(U) := i : n 1 n... F (U 2 ) F (U 1 ) F (U). F is homotopy invariant if i 0 = i 1 : F (U A 1 ) F (U). Lemma 1 In general, i 0 = i 1 : C F (U A 1 ) C F (U) are chain homotopic. Corollary 2 The homology presheaves H n (C F ) are homotopy invariant. 4

5 A 1 -Homotopy f, g : X Y in Cor k. H : X A 1 Y. H X 0 = f; H X 1 = g Equivalence relation in Cor. f g. Proposition 3 f g Then f, g chain homotopy maps C Z tr (X) C Z tr (Y ). 5

6 Motivic Cohomology Z(q) := C Z tr ( q Gm )[ q]; q 0. Complex of Zariski sheaves on X for any X. Definition 4 The motivic cohomology groups H p,q (X) := H p Zar (X, Z(q)). Products: Z(q) Z(s) Z(q + s), H p,q (X) H r,s (X) H p+r,q+s (X). Low degrees: H 0,0 (X) = Z(π 0 (X)), H p,0 = 0; p 1. Z(1) = O [ 1]. Relation to Milnor K-theory: Theorem 5 H n,n (k) = Kn Milnor (k). 6

7 Triangulated Category of Motives Definition 6 A Nisnevich cover U X is an étale cover such that for any field K, the induced map on K-points is surjective. Tensor Products in PST: Z tr (X) Z tr (Y ) := Z tr (X Y ). Z tr (X) projective in PST; α Z tr(x α ) F R F F, R G G resolutions by representable sheaves. F G := T ot(r F R G ) = F L G. A 1 -weak equivalence: D := D (Sh Nis (Cor k )) Quotient D by smallest thick subcategory W containing cones of Z(X A 1 ) Z(X). Definition 7 DM eff Nis(k) := D Sh Nis (Cor k )[W 1 ]. 7

8 Motives (Continued) Definition 8 M(X) := Z tr (X) DM eff Nis(k). DM eff geo(k) DM eff Nis(k) is the thick subcategory generated by the M(X). Why the Nisnevich topology? Proposition 9 U X Nisnevich cover. Then Z tr (U X U) Z tr (U) Z(X) 0 exact sequence of Nisnevich sheaves. Suffices to check for X = Spec A, A henselian local. Now use the fact that a Nisnevich cover of Spec A is split. 8

9 Properties of DM eff Nis(k) Mayer-Vietoris: M(U V ) M(U) M(V ) M(X) M(U V )[1]. Kunneth: M(X Y ) = M(X) M(Y ). Vector Bundle: M(X) = M(V ) when V X vector bundle. Cancellation: Define M(q) = M Z(q). Assume varieties over k admit resolution of singularities. Then Hom(M, N) = Hom(M(q), N(q)). Projective Bundle: n i=0 M(X)(i)[2i] = M(P(V )). Chow Motives: X, Y smooth projective. Then Hom(M(X), M(Y )) = CH dim X (X Y ). 9

10 Relation with Motivic Cohomology Theorem 10 X/k smooth. Then H p,q (X) = Hom DM eff Nis (k)(z tr (X), Z tr (q)[p]). 10

11 Algebraic Cycles cosimplicial variety. Z q (X, n) = codim. q algebraic cycles on X n in good position for faces. Z q (X, ) : Z q (X, 2) Z q (X, 1) Z q (X, 0). CH q (X, n) := H n (Z q (X, )). Theorem 11 (Voevodsky) X smooth, k perfect. Then H p,q (X) = CH q (X, 2q p). Better to work with Z q (X, )[ 2q]. H p,q (X) = H p (Z q (X, )[ 2q]). Beilinson, Soulé conjecture that this shifted complex has cohomological support in degrees [0, 2q]. 11

12 Relation with K-Theory CH q (X, n) Q = gr q γk n(x) Q. Relation with Étale Cohomology Theorem 12 (Suslin) Let k = k and let X/k be smooth. Let m 1 be relatively prime to char. k. Then Het(X, p Z/mZ(q)) = H p( ) Z q (X, )[ 2q] Z/mZ. 12

13 Arithmetic Conjectures Conjecture 13 (Soulé) Let X be an arithmetic variety of dimension n. Define the scheme-theoretic zeta function ζ X (s) := x X xclsd. 1 1 N(x) s. Then the Euler-Poincaré characteristic of the complex χ(x, q) of Z q (X, )[ 2q] is finite, and we have χ(x, q) = ord s=n q ζ X (s). Example 14 X = O k, ring of integers in a number field, n = 1. By Borel 1 p = q = 0 rkh p,q (O k ) = r 1 + r 2 1 q = p = 1 r 1 + r 2 p = 1; q = 2m + 1 r 2 p = 1; q = 2m 0 else. 13

14 Arithmetic Conjectures (Conjectures of Beilinson; Bloch, Kato; Fontaine, Perrin-Riou) Regulator map: k C, X smooth projective. r : H p,q (X) H p,q D (X). Assume p < 2q (negative weight). Then 0 H p 1 (X, C)/(F q + H p 1 (X, Z(q)) H p,q D (X) Hp (X, Z(q)) tors 0. Must also take invariants under real conjugation F : H p,q D (X) R Volume form given (upto Q ) by rational structure on H p DR /F q. Take p < 2q 1 for simplicity. Let L(H p 1 (X, Q l ), s) be the Hasse-Weil L-function. Let L (s = p q) be the first non-vanishing term in the Taylor series at s = p q. 14

15 Arithmetic Conjectures (cont.) Conjecture 15 (i) Image of r is cocompact in H p,q D (X) R. (ii) Rank of H p,q (X) Q is the order of vanishing of L(H p 1 (X, Q l ), s) at s = p q. (iii) Volume of H p,q D (X) R/Image(r) L (H p 1 (X, Q l (q)), s = p q) Q. Fontaine, Perrin-Riou: Reformulate in terms of (roughly) a metric on det H p,q (X). (The metric depends on the trivialization of a certain Betti-DR line.) Problem: Construct this metric directly on Z q (X, )[ 2q]. 15

16 Mixed Tate Motives k field. M motive over k. (Conjectural) t-structure on DM(k), M core. Again conjecturally, M will have a weight filtration W M. Definition 16 M is mixed Tate if gr W M = i Z(n i). Problem: Give a synthetic construction of mixed Tate motives over k. 1. (Deligne, Goncharov,...) Look at subquotients of the groupring Z[π 1 (P 1 k S, p 0)]. 2. (Bloch, Kriz) (a) Construct a DG version q 0 N (q) of q Zq (Spec k, )[ 2q] Q. (b) Consider corepresentations of the the commutative Hopf algebra H = H 0 (Bar( q N (q))). I H augmentation ideal. Conjecturally, L := (I/I 2 ) is the pro-lie algebra associated to the Tannaka group of mixed Tate motives/k. 16

17 Limiting Mixed Motives (Important work on this subject by Ayoub, Bondarko,Vologodsky,... What follows is a modest attempt to interpret LMM in terms of cycle complexes. ) X/k smooth, Y = r i=1 Y i X normal crossings divisor, X = X Y. Write N(X) = N (X, ) (i.e. ignore gradings) 0 N(Y ) N(X) N(X ) C 0 Here C 0. Weight filtration: Y I := i I Y i, Y I := j I Y j. N(X, Y I ) := N(X)/N(Y I ) N(X) Y I N(X) cycles meeting Y I properly (including faces). N(X, Y I ) Y I = N(X) Y I /N(X) Y I N(Y I ) W p N(X, Y ) := I =p N(X, Y I) Y I N(X, Y ). W 0 = N(X) Y N(X). W r = N(X, Y ) N(X ). 17

18 Limiting Mixed Motives (cont.) Theorem 17 Y (I) := i I Y i, Y ( ) := X. Then gr W p N(X, Y ; q) I =p N(Y (I); q p)[ p]. Analogy: W N(X, Y ) W Ω X (log Y ). Now assume Y : t = 0 principal. Steenbrink double complex: A pq := Ω p+q+1 X (log Y )/W q. d = d : A p,q A p+1,q ; d = dt/t : A p,q A p,q+1 Motivic analog: N = N(X, Y ), M = r 1 i=0 N/W i. W a M := r 1 b=0 W 2b+a+1N/W b N. gra W M r 1 b=0 I =2b+a+1 H, (Y (I))[ 2b a 1]. 18

19 Motivic analog (cont.) A M. Construct B : M M, B 2 = 0, B(W a ) W a. Then B d + d in Steenbrink. Definition 18 The limiting mixed motivic cohomology is H, (M, B). Corollary 19 Steenbrink spectral sequence : E 1 = r 1 b=0 I =2b+a+1 H, (Y (I))[ 2b a 1] H, (M, B) Warning: B depends on a choice of homotopy. I have not yet checked the dependence of the complex (M, B) on this choice. 19