K p q k(x) K n(x) x X p

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1 oc 5. Lecture Quien s ocaization theorem and Boch s formua. Our next topic is a sketch of Quien s proof of Boch s formua, which is aso a a brief discussion of aspects of Quien s remarkabe paper [4]. For K 2, this formua was proved by Boch in [2]. We remind the reader that in this paper Quien introduces the Quien Q- construction QE of an exact category E and defines the K-groups of E to be the homotopy groups of the cassifying space of QE. Of most importance to us are the abeian category M X of coherent sheaves on a Noetherian scheme O X and the exact subcategory P X M X, yieding K (X) = π (BQP X ), K (X) = π (BQM X ). A key ingredient in Quien s proof of Boch s formua is the ocaization sequence for K, extending known ocaization sequences for ow K-groups. Quien formuates his resuts in an abstract, categorica setting. Theorem 5.1. (Locaization Theorem of Quien, cf. [4]) Let A be an abeian category and B A a Serre subcategory with quotient category A/B. Then there is a ong exact sequence of Quien K-groups K 1 (A) K 1 (A/B) K 0 (B) K 0 (A) K 0 (A/B) 0. In conjunction with Quien s devissage theorem, this ocaization theorem impies the foowing: Theorem 5.2. Consider X (Sch/k) and et M r (X) denote the Serre subcategory of the category M X consisting of coherent sheaves whose support has codimension r. Then there is a natura ong exact sequence K i+1 k(x) K i (M r+1 (X)) K i (M r (X)) K i k(x) x X r x X r Here, X r denotes the set of points of X of codimension r. Consequenty, there is a spectra sequence 1 (X) = K p q k(x) K n(x) x X p reating the K-theory of the residue fieds of points of X to the K theory of X. Proof. Quien s devissage theorem tes us that K i (M r (X)/M r+1 (X)) is naturay isomorphic to X r K i k(x). The asserted exact sequences patch together to give an exact coupe, with the indexing of the spectra sequence determined by this exact coupe. Definition 5.3. The Gersten compex for K n is the compex 0 K nx K n k(x) K n 1 k(x) K 0 k(x) 0 x X 0 x X 1 x X n determined by the exact sequences of Theorem 5.2. Essentiay by inspection, we have the foowing thereom concerning the reationship of the spectra sequence of Theorem 5.2 and the exactness of the Gersten compex. 1

2 2 Proposition 5.4. Let X (Sch/k). Then the foowing conditions are equivaent: 1.) For every r 0, the incusion M r+1 (X) M r (X) induces the zero map on K-groups. 2.) In the spectra sequence of Theorem 5.2, for a q, 2 = 0 for p > 0 and the edge homomorphism K qx E 0,q 2 X is an isomorphism. 3.) The Gersten compex for X is exact. Here is Quien s theorem estabishing the vaidity of Boch s formua. Theorem 5.5. (Boch s formua by Quien [4]) Let X Sch(k) be reguar. Then there is a cannoica isomorphism H q (X, K q ) CH q (X) where K q is the sheaf on X (for the Zariski topoogy) associated to the presheaf U K q (U). Proof. Granted the above anaysis of the Quien spectra sequence, there are two additiona ingredients in the proof. The first is Quien s theorem that the Gersten resoution is exact for Spec O X,x whenever X Sch(k) and x X is a reguar point. This tes us that the Gersten compex for K n(x) becomes a resoution of resoution of K n (X) by fasque sheaves 0 K n X i x K n k(x) i x K n 1 k(x) x X 0 x X 1 Consequenty, the E 2 -term of the Quien spectra sequence has the form 2 (X) = Hp (X, K q ) K p q (X). The second is Quien s determination of the ast differentia in the Gersten compex K 1 k(x) d1 K 0 k(x) = Z q (X). x X q 1 x X q Quien concudes that the image of this map is precisey the codimension q cyces rationay equivaent to Derived categories. In order to formuate motivic cohomoogy, we need to introduce the anguage of derived categories. Let A be an abeian category (e.g., the category of modues over a fixed ring) and consider the category of chain compexes CH (A). We sha index our chain compexes so that the differentia has degree +1. We assume that A has enough injectives and projectives, so that we can construct the usua derived functors of eft exact and right exact functors from A to another abeian category B. For exampe, if F : A B is right exact, then we define L i F (A) to be the i-th homoogy of the chain compex F (P ) obtained by appying F to a projective resoution P A of A; simiary, if G : A B is eft exact, then R j G(A) = H j (I ) where A I is an injective resoution of A. The usua verification that these derived functors are we defined up to canonica isomorphism actuay proves a bit more. Namey, rather take the homoogy of the compexes F (P ), G(I ), we consider these compexes themseves and observe that they are independent up to quasi-isomorphism of the choice of resoutions. Reca, that a map C D is a quasi-isomorphism if it induces an isomorphism on homoogy; ony in specia cases is a compex C quasi-isomorphic to its homoogy H (C ) viewed as a compex with trivia differentia.

3 3 We define the derived category D(A) of A as the category obtained from the category of CH (A) of chain compexes of A by inverting quasi-isomorphisms. Of course, some care must be taken to insure that such a ocaization of CH (A) is we defined. Let Hot(CH (A)) denote the homotopy category of chain compexes of A: maps from the chain compex C to the chain compex D in H(CH (A)) are chain homotopy equivaence casses of chain maps. Since chain homotopic maps induce the same map on homoogy, we see that D(A) can aso be defined as the category obtained from Hot(CH (A)) by inverting quasi-isomorphisms. The derived category D(A) of the abeian category CH (A) is a trianguated category. Namey, we have a shift operator ( )[n] defined by (A [n]) j A n+j. This indexing is very confusing (as woud be any other); we can view A [n] as A shifted down or to the eft. We aso have distinguished trianges A B C A [1] defined to be those trianges quasi-isomorphic to short exact sequences 0 A B C 0 of chain compexes. This notation enabes us to express Ext-groups quite neaty as Ext i A(A, B) = H i (Hom A (P, B)) = Hom D(A) (A[ i], B) = Hom D(A) (A, B[i]) = H i (Hom A (A, P )) Boch s Higher Chow Groups. From our point of view, motivic cohomoogy shoud be a cohomoogy theory which bears a reationship to K (X) anaogous to the roe Chow groups CH (X) bear to K 0 (X) (and anaogous to the reationship of H sing (T ) to K top(t )). In particuar, motivic cohomoogy wi be douby indexed. We now discuss a reativey naive construction by Spencer Boch of higher Chow groups which satisfies this criterion. We sha then consider a more sophisticated version of motivic cohomoogy due to Susin and Voevodsky. We work over a fied k and define n to be Spec k[x 0,..., x n ]/( i x i 1), the agebraic n-simpex. As in topoogy, we have face maps i : n 1 n (sending the coordinate function x i k[ n ] to 0) and degeneracy maps σ j : n+1 n (sending the coordinate function x j k[ n ] to x j + x j+1 k[ n+1 ]). More generay, a composition of face maps determines a face F i n. Of course, n A n. Boch s idea is to construct a chain compex for each q which in degree n woud be the codimension q-cyces on X n. In particuar, the 0-th homoogy of this chain compex shoud be the usua Chow group CH q (X) of codimension q cyces on X moduo rationa equivaence. This can not be done in a competey straightforward manner, since one has no good way in genera to restrict a genera cyce on X [n] via a face map i to X n 1. Thus, Boch ony considers codimension q cyces on X n which restrict propery to a faces (i.e., to codimension q cyces on X F ). Definition 5.6. Let X be a variety over a fied k. For each p 0, we define a compex z p (X, ) which in degree n is the free abeian group on the integra cosed subvarieties Z X n with the property that for every face F n dim k (Z (X F )) dim k (F ) + p.

4 4 The differentia of z p (X, ) is the aternating sum of the maps induced by restricting cyces to codimension 1 faces. Define the higher Chow homoogy groups by CH p (X, n) = H n (z p (X, )), n, p 0. If X is ocay equi-dimensiona over k (e.g., X is smooth), et z q (X, n) be the free abeian group on the integra cosed subvarieties Z X n with the property that for every face F n codim X F (Z (X F )) q. Define the higher Chow cohomoogy groups by CH q (X, n) = H n (z q (X, )), n, q 0, where the differentia of z q (X, ) is defined exacty as for z p (X, ). Boch, with the aid of Marc Levine, has proved many remarkabe properties of these higher Chow groups. Theorem 5.7. Let X be a quasi-projective variety over a fied. Boch s higher Chow groups satisfy the foowing properties: CH p (, ) is covarianty functoria with respect to proper maps. CH q (, ) is contravarianty functoria on Sm k, the category of smooth quasi-projective varieties over k. CH p (X, 0) = CH p (X), the Chow group of p-cyces moduo rationa equivaence. (Homotopy invariance) π : CH p (X, ) CH p+1 (X A 1 ). (Locaization) Let i : Y X be a cosed subvariety with j : U = X Y X the compement of Y. Then there is a distinguished triange z p (Y, ) i z p (X, ) j z p (U, ) z p (Y, )[1] (Projective bunde formua) Let E be a rank n vector bunde over X. Then CH (P(E) ) is a free CH (X, )-modue on generators 1, ζ,..., ζ n 1 CH 1 (P(E), 0). For X smooth, K i (X) Q q CH q (X, i) Q for any i 0. Moreover, for any q 0, (K i (X) Q) (q) CH q (X, i) Q. If F is a fied, the K M n (F ) CH n (Spec F, n). The most difficut of these properties, and perhaps the most important, is ocaization. The proof requires a very subte technique of moving cyces. Observe that z p (X, ) z p (U, ) is not surjective because the conditions of proper intersection on an eement of z p (U, n) (i.e, a cyce on U n ) might not continue to hod for the cosure of that cyce in X n Beiinson s Conjectures. We give beow a ist of conjectures due to Beiinson which reate motivic cohomoogy and K-theory. Boch s higher Chow groups go some way toward providing a theory which satisfies these conjectures. Namey, Beiinson conjectures the existence of compexes of sheaves Γ Zar (r) whose cohomoogy (in the Zariski topoogy) H p (X, Γ Zar (r)) one coud ca motivic cohomoogy. If we set H p (X, Γ Zar (r)) = CH r (X, 2r p),

5 5 then many of the cohomoogica conjectures Beiinson makes for his conjectured compexes are satisfied by Boch s higher Chow groups CH (X, ). Conjecture 5.8. (Beiinson [1]) Let X be a smooth variety over a fied k. Then there shoud exist compexes of sheaves Γ Zar (r) of abeian groups on X with the Zariski topoogy, we defined in D(AbSh(X Zar )), functoria in X, and equipped with a graded product, which satisfy the foowing properties: (1) Γ Zar (1) = Z; Γ Zar (1) G m [ 1]. (2) H 2n (X, Γ zar (n)) = CH n (X). (3) H i (Spec k, Γ Zar (i)) = K M i k, Minor K-theory. (4) (Motivic spectra sequence) There is a spectra sequence of the form 2 = H p q (X, Γ Zar (q)) K p q (X) which degenerates after tensoring with Q. Moreover, for each prime, there is a mod- version of this spectra sequence 2 = H p q (X, Γ Zar (q) L Z/) K p q (X, Z/) (5) grγ(k r j (X) Q H 2r j (X Zar, Γ Zar (r)) Q. (6) (Beiinson-Lichtenbaum Conjecture) Γ Zar L Z/ τ r Rπ (µ r ) in the derived category D(AbSh(X Zar )) provided that is invertibe in O X, where π : X et X Zar is the change of topoogy morphism. (7) (Vanishing Conjecture) Γ Zar (r) is acycic outside [1, r] for r 1. These conjectures require considerabe expanation, of course. Essentiay, Beiinson conjectures that agebraic K-theory can be computed using a spectra sequence of Atiyah-Hirzebruch type (4) using motivic compexes Γ Zar (r) whose cohomoogy pays the roe of singuar cohomoogy in the Atiyah-Hirzebruch spectra sequence for topoogica K-theory. I have indexed the spectra sequence as Beiinson suggests, but we coud equay index it in the Atiyah-Hirzebruch way and write (by simpy re-indexing) 2 = H p (X, Γ Zar ( q/2)) K p q (X). where Γ Zar ( q/2) = 0 if q is not an even non-positive integer and Γ Zar ( q/2) = Γ Zar (i) is q = 2i 0. (1) and (2) just normaize our compexes, assuring us that they extend usua Chow groups and what is known in codimensions 0 and 1. Note that (1) and (2) are compatibe in the sense that H 2 (X, Γ Zar (1)) = H 2 (X, O X[ 1]) = H 1 (X, O X) = P ic(x). (3) asserts that for a fied k, the n-th cohomoogy of Γ Zar (n) the part of highest weight with respect to the action of Adams operations shoud be Minor K-theory. This has been verified for Boch s higher Chow groups by Susin-Nesterenko and Totaro. The (integra) spectra sequence of (4) has been estabished thanks to the work of many authors. This spectra sequence coapses at the E 2 -eve when tensored with Q, so that E 2 Q = E Q. (5) asserts that this coapsing can be verified by using Adams operations, interpreted using the γ-fitration. The vanishing conjecture of (7) is the most probematic, and there is no consensus on whether it is ikey to be vaid. However, (6) incorporates the mod- version of the vanishing conjecture and has apparenty been proved by Rost and Voevodsky.

6 6 (6) asserts that if we consider the compexes Γ Zar (r) moduo (in the sense of the derived category), then the resut has cohomoogy cosey reated to etae cohomoogy with µ r coefficients, where µ is the etae sheaf of -th roots of unity (isomorphic to Z/ if a -th roots of unity are in k. If the terms in the mod- spectra sequence were simpy etae cohomoogy, then we woud get etae K- theory which woud vioate the vanishing conjectured in (7) (and which woud impy periodicity in ow degrees which we know to be fase). So Beiinson conjectures that the terms moduo shoud be the cohomoogy of compexes which invove a truncation. More precisey, Rπ F is a compex of sheaves for the Zariski topoogy (given by appying π to an injective resoution F I of etae sheaves) with the property that HZar (X, Rπ F ) = Het(X, F ). Now, the n-th truncation of Rπ F, τ n Rπ F, is the truncation of this compex of sheaves in such a way that its cohomoogy sheaves are the same as those of Rπ F in degrees n and are 0 in degrees greater than n. (We do this by retaining coboundaries in degree n+1 and setting a higher degrees equa to 0.) If X = Speck, then H p (Speck, τ n Rπ µ n ) equas Het(Speck, p µ n ) for p n and is 0 otherwise. For a positive dimensiona variety, this truncation has a somewhat mystifying effect on cohomoogy. It is worth emphasizing that one of the most important aspects of Beiinson s conjectures is its expicit nature: Beiinson conjectures precise vaues for agebraic K-groups, rather than the conjectures which preceded Beiinson which required the degree to be arge or certain torsion to be ignored. Such a precise conjecture shoud be much more amenabe to proof. References Bei [1] A. Beiinson, Height Paring between agebraic cyces. K-theory, arithmetic, and geometry. Lecture Notes in Mathematics #1289, springer (1986). B2 [2] S. Boch, K 2 and agebraic cyces. Ann. of Math. 99 (1974), B [3] S. Boch, Agebraic cyces and higher K-theory. Advances in Math 61 (1986), Q [4] D. Quien, Higher agebraic K-theory, I. L.N.M # 341, pp Department of Mathematics, Northwestern University, Evanston, IL E-mai address: eric@math.northwestern.edu