MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES

JIANQIANG ZHAO

1. Introduction

Let F be a �eld and n> 2 be a positive integer. A simplex in the projective spacePnF is an ordered set of hyperplanes L ¼ ðL0; . . . ; LnÞ. A face of L is any non-empty intersection of the hyperplanes. A pair of simplices is admissible if they donot have common faces of the same dimension. It is a generic pair if all the facesof the two simplices are in general position.

In their seminal paper [2, 4] Beilinson et al. initiated the study of themotivic cohomology of the admissible pairs of simplices ðL;M Þ on P2

F which isde�ned as a graded comodule

L2j¼0H

�motðL;M Þj over a suitable graded algebra

AðF Þ�;Q :¼ AðF Þ� �Q. The cohomology is the arithmetico-algebraic analog of therelative Betti cohomology and is de�ned via mysterious complexes formed by thegeometric combinatorial data of the pair. The reason for this is that the gradedpieces of the cohomology groups can actually be obtained from a spectral sequenceconverging to H �ðPnC n L;M n L;QÞ with degenerate E2-term. In this paper wepropose a generalization of the above from P2

F to PnF .Brie:y speaking, the group AnðF Þ is generated by admissible pairs of simplices

in PnF , subject to a set of relations (see De�nition 2.1 for the detail). The de�ning(double scissors congruence) relations re:ect (conjecturally all of) the functionalequations of Aomoto polylogarithms �rst studied in [1]. These groups are expectedto form a Hopf algebra and are closely related to algebraic K-theory by thefollowing conjecture [4, p. 550] (after slight reformulation).

Conjecture 1.1. The restricted coproduct induces a complex

A>0 �!A>0 � A>0 �!A>0 � A>0 � A>0 �! . . .

whose graded n-piece

An �!Mn�1k¼1

Ak �An�k �! . . .

provides the isomorphism

HiðnÞðA�;QÞ ffi gr�nK2n�iðF ÞQ;

where � stands for the �-�ltration of K-groups.

Beilinson et al. proved this conjecture up to K3ðF Þ in [2, 4]. We now have a lotof evidence up to gr�4K7ðF Þ (see [10, 14]).

Received 8 October 2001; revised 21 October 2002.

2000 Mathematics Subject Classi�cation 14F42, 16W30 (primary), 13D03, 13D25 (secondary).

This research was partially supported by NSF grant DMS0139813

PLMS 1470---10/2/2004---SRUMBAL---89152

Proc. London Math. Soc. (3) 88 (2004) 313--354 q 2004 London Mathematical SocietyDOI: 10.1112/S0024611503014709

According to the Tannakian formalism the category MTMðF Þ of mixed Tatemotives over a �eld F is supposed to be equivalent to the category of gradedmodules over a certain graded commutative Hopf algebra A� (see [3] and [9,Chapter 3]). Therefore the Ext groups in the category MTMðF Þ are isomorphicto the cohomology of the Hopf algebra A�. Beilinson et al. conjecture thatA� :¼

Ln> 0An is isomorphic to A� and therefore the graded object A� �Q

should have a Hopf algebra structure over Q. This is the primary motivation tostudy the groups An in this paper.

In [13] by using a geometric combinatorial method we analyzed the generic partA0� of A� (see page 316 for the precise de�nition of A0

�). In particular, we provedthat the coproduct is well de�ned on the generic part. However, the proofs arecomplicated due to their computational nature. In this paper we will explore adiIerent path by applying the theory developed in [2, 3] to study the motiviccohomology of all the admissible pairs of simplices, not only the generic ones. Itfollows from the general theory of framed mixed Hodge structures that thecoproduct map on An should exist. This idea will then be used to �nd the explicitde�nition of the coproduct map on An for n6 4.

We now describe the content of this paper in some detail. Section 2 is a briefreview of the double scissors congruence groups and the framed Hodge--Tatestructures. In x 3, for a non-degenerate admissible pair ðL;M Þ in PnC we explicitlyde�ne a linearly constructible motivic perverse sheaf SðL;M Þ without monodromyin the sense of [3] and prove that SðL;M Þ provides the Hodge--Tate structure ofthe relative Betti cohomology H �ðPnC n L;M n L;QÞ when L and M are in generalposition. Essentially SðL;M Þ is the E0-term of a spectral sequence whichconverges to H �ðPnC n L;M n L;QÞ. It is shown in [3] that SðL;M Þ is uniquelydetermined by a 2-dimensional diagram which satis�es some compatibilityconditions. We denote this diagram also by SðL;M Þ. Let K�;�2j be the bicomplexinside SðL;M Þ corresponding to the 2jth weight graded piece and let K�2j be itstotal complex. Then we have the following result.

THEOREM 3.10. Let ðL;M Þ be a pair of simplices in PnC in general position.Then the relative Betti cohomology

Hnþ�ðPn n L;M n L;QÞ ffiMnj¼0

H �ðK�2jðL;M ÞÞ:

The bene�t of the diagram SðL;M Þ is that it can be described in purelygeometric combinatorial terms. For a general �eld F we can therefore recoverthe motivic cohomology H �motðL;M Þ using a similarly constructed 2-dimensionaldiagram to de�ne

HqmotðL;M Þ :¼

Mnj¼0

HqðK�2jðL;M ÞÞ

graded by the weight. When F ¼ C this is consistent with Theorem 3.10. Themain result about the structure of the motivic cohomology H �motðL;M Þ is given bythe following.

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO314

MAIN THEOREM 3.25. The complex K�2j is concentrated on degrees ½j� n; j�and is exact everywhere except at K0

2j. Moreover, H0ðK�0ðL;M ÞÞ ffiH0ðK�2nðL;M ÞÞ ffi Q and H0ðK�2jðL;M ÞÞ has dimension at most ðnjÞ

2 for0 < j < n. The equality holds when ðL;M Þ is a generic pair and there exists acanonical choice of basis in this case.

In x 4, from the above explicit construction of the motivic cohomology we areable to recover the de�nition of the coproduct on the generic part A0

� which is inagreement with [3]. One of the advantages of our approach in this generic case isthat it follows directly from our de�nition that the coproduct is well de�nedalthough it is not obviously so at a �rst glance.

For A2 our general theory also enables us to handle all the non-generic cases inx 5 which greatly simpli�es the computation for pairs of triangles in [2].

One of the most important applications of our motivic cohomology theory isgiven in x 6. There, we �rst reduce the problem of de�ning the coproduct on A3 tothe generic case and �nitely many non-generic cases. Then for each of them we areable to de�ne the coproduct explicitly. Having the clear picture of A3 in our mindwe are able to generalize to A3 a few deep results about A2 in another paper [14].

Using an argument similar to that used in dealing with A3 we can further de�nethe coproduct explicitly on all of A4. Nevertheless, in the last section, we contentourselves with only two crucial examples of A4 to show how our general theorycan be applied in these cases. One example is a part of the classicaltetralogarithmic pair of simplices. The other is essentially the only obstacle tothe construction of the Aomoto tetralogarithm by using classical tetralogarithmsand products of polylogarithms of lower weights.

This paper grew out of the attempt to give a conceptually clearer solution tothe problem of de�ning the coproduct on An than the one presented in my thesis.My approach to the relative motivic cohomology in x 3.2 is inspired by severalconversations with C.-L. Chai to whom I want to express my hearty gratitude.The anonymous referee provided some very helpful comments which greatlyimprove the exposition of the paper.

2. Preliminaries and notation

We shall begin with some notation. Let F be a �eld. A simplex in the projectivespace PnF is an ordered set of hyperplanes L ¼ ðL0; . . . ; LnÞ. It is non-degenerate ifthe intersection of all the hyperplanes Li is empty. A face of L is a non-emptyintersection of some ordered hyperplanes. For example, the face of L1 \ L2 can berepresented as ðL1; L2Þ ¼ �ðL2; L1Þ. For any ordered index set

I ¼ ði1; . . . ; ikÞ � ½½n�� :¼ f0; 1; . . . ; ng

we set LI ¼Ti2I Li or LI ¼ ðLi1 ; . . . ; LikÞ (ordered set of faces of L) depending on

the context. We also adopt the convention that Lij ¼ Li \ Lj and Li;j ¼ ðLi; LjÞand so on. Given a non-degenerate simplex L we may choose the coordinatesystem ½t0; . . . ; tn� in PnF such that Li ¼ fti ¼ 0g for 06 i6n. Such a simplex iscalled the standard simplex.

A pair of simplices is admissible if they do not have common faces of the samedimension. It is a generic pair if all the faces of the two simplices are ingeneral position.

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 315

2.1. Double scissors congruence groups AnðF ÞThese groups are �rst introduced in [3] and then modi�ed in [2].

DEFINITION 2.1. De�ne A0ðF Þ ¼ Z. If n > 0 then AnðF Þ is the abelian groupgenerated by admissible pairs of n-simplices ðL;M Þ subject to the following relations.

(R1) Non-degeneracy. We have ðL;M Þ ¼ 0 if and only if L or M is degenerate.(R2) Skew symmetry. For every permutation � of ½½n��,

ð�L;M Þ ¼ ðL;�M Þ ¼ sgnð�ÞðL;M Þ;

where �L ¼ ðL�ð0Þ; . . . ; L�ðnÞÞ:(R3) Additivity in L and M. For any nþ 2 hyperplanes L0; . . . ; Lnþ1 and

n-simplex M in Pn, Xnþ1j¼0ð�1ÞjððL0; . . . ;cLjLj; . . . ; Lnþ1Þ;M Þ ¼ 0

if every pair ððL0; . . . ;cLjLj; . . . ; Lnþ1Þ;M Þ is admissible. A similar relation is satis�edfor M.

(R4) Projective invariance. For every g 2 PGLnþ1ðF Þ,

ðgL; gM Þ ¼ ðL;M Þ:

Denote by ½L;M� the class of ðL;M Þ in AnðF Þ.

CONVENTION. Sometimes we need to produce simplices by intersectinghyperplanes (for example, see the next proposition). Suppose L0; . . . ; Ln andM0; . . . ;Mn are hyperplanes in Pnþ1. If N ¼ Li or N ¼Mi for some i thenwe always throw away ðNjL;M Þ :¼ ððL0 \N; . . . ; Ln \NÞ; ðM0 \N; . . . ;Mn \NÞÞbecause they are not pairs of simplices.

The interested readers may �nd that we have replaced the relations in theTrivial Intersection axiom of [13, x 2] by the above convention to make thede�nition logically better. They can also �nd a detailed analysis of this de�nitionand the relations between our formulation and those of other authors. There, weproved the following.

PROPOSITION 2.2 (Intersection additivity). For n hyperplanes M1; . . . ;Mn

and any nþ 1 hyperplanes L0; . . . ; Ln in Pn,Xni¼0ð�1ÞiðLijðL0; . . . ;cLiLi; . . . ; LnÞ;M Þ ¼ 0

if every pair ðLijðL0; . . . ;cLiLi; . . . ; LnÞ;M Þ is admissible on Li ffi Pn�1. A similarrelation holds for M.

One often considers another family of groups A0nðF Þ (called the generic part of

AnðF Þ) de�ned similarly to AnðF Þ except that(i) the generators ðL;M Þ are required to be generic pairs,

(ii) in (R3), all ðL0; . . . ;cLiLi; . . . ; Lnþ1;M Þ are generic pairs.

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO316

2.2. Framed Hodge--Tate structures: a brief review

We will use [2] as our primary reference in this subsection.Recall that a non-trivial Hodge--Tate structure is a mixed Hodge structure whose

quotient of weight �2k is isomorphic to some non-zero copies of ZðkÞ. We say thatH is an n-framed Hodge--Tate structure if it is equipped with a non-zero vectorv 2 grW2nH and a non-zero covector evv 2 ðgrW0 HÞ�, that is, a pair of morphisms

v : Zð�nÞ �! grW2nH1 and evv : ðgrW0 HÞ� �! Zð0Þ:

Consider the �nest equivalence relation on the set of all n-framed Hodge--Tatestructures for which H1 � H2 if there is a map H1 ! H2 respecting frames. Forexample, any n-framed Hodge--Tate structure is equivalent to some H withW�2H ¼ 0 and W2nH ¼ H. The equivalence classes form an abelian group Hn asfollows: if two classes of Hn are represented by H and H 0 and their correspondingframes are denoted by ðv; evv Þ and ðv 0; evv 0Þ, then ½H� þ ½H 0 � is represented by theHodge--Tate structure H �H 0 with frame ðvþ v 0; evvþ evv 0Þ and �½H� is the class ofH with frame ð�v; evv Þ (or ðv;�evv Þ). The tensor product of mixed Hodge structuresinduces the commutative multiplication � : Hi �Hj ! Hiþj.

One can de�ne the coproduct

� ¼Miþj¼n

�i;j : Hn �!Miþj¼n

Hi �Hj

as follows. Let ðH; v; evv Þ be as above and R � grW2iH be a lattice and R� � ðgrW2iHÞ�the dual lattice. One can de�ne the homomorphisms

’ : R�!Hi; : R� �!Hj:

Namely, for x 2 R, ’ðxÞ is the class of the sub-Hodge structure H, consisting ofvectors in W2i whose projection in grW2iH is proportional to x, with framing fx; evvg.For y 2 R�, ðyÞ is the class of the quotient-structure ðH=W2i�1Þ= ker y withframing fv; yg.

Let fejg and fejg be dual bases in R and R� respectively. Then one de�nesthe coproduct

�i;n�ið½H�Þ :¼Xj

’ðejÞ � ðejÞ:

It is easy to see that � and � are compatible, that is,

�ð�ða� bÞÞ ¼ ðð�� �Þ � $Þð�ðaÞ � �ðbÞÞ

where $ða� b� c� dÞ ¼ a� c� b� d. Set H0 :¼ Z. Then H� :¼L

n> 0Hn is agraded Hopf algebra with multiplication � and coproduct �.

The following result is the main reason for considering the Hopf algebra H�.More details can be found in [3].

THEOREM 2.3. The category of mixed Q-Hodge --Tate structures is canonicallyequivalent to the category of �nite-dimensional graded H�;Q-comodules.

This theorem implies that the equivalence assigns to a Hodge structure H thegraded comodules MH and MH

n ¼ grW2nðHÞ with H�-action MH � ðMHÞ� ! H�

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 317

given by the formula

xi � yj 7�! fclass of mixed Hodge structures framed by xi; yjg:

Furthermore, the dual Hopf algebra H��;Q is isomorphic to the universal enveloping

algebra of the graded Lie algebra �LðF Þ where Ln ¼ ðHn=P

16 k<nHk � Hn�kÞ�Q.Conjecturally one should be able to replace Hn by the double scissors congruencegroups AnðF Þ everywhere in this subsection. The crucial piece of knowledgecurrently missing is how to de�ne the coproduct explicitly in general.

3. Motivic cohomology of an admissible pair of simplices

This is the main theoretical part of this paper. We will apply its main results toa series of constructions in the next few sections.

3.1. The linearly constructible motivic perverse sheaf

Let F be an arbitrary �eld. Let ðL;M Þ be a non-degenerate admissible pair ofsimplices in PnF . Denote the set of generic pairs of sub-simplices of ðL;M Þ by

GPðL;M Þ ¼ fðLJ ;MKÞ : I; J � ½½n��; codimðLJ \MKÞ ¼ jJ j þ jKjg:For 06 j; k6n write

LjMk ¼ LJ \MK : jJj ¼ j; jKj ¼ k; ðLJ ;MKÞ 2 GPðL;M Þf g:

Remarks 3.1. (i) We emphasize that one should understand LJ \MK as an‘ordered’ intersection, for example, L0;1 \M1 ¼ �L1;0 \M1.

(ii) By convention, LjM0 ¼ Lj, L0Mk ¼Mk and L0 ¼M0 ¼ fPng.

Following the idea of [3, x 2.2] we �rst construct the linearly constructiblemotivic perverse sheaves without monodromy SðL;M Þ as follows. Brie:yspeaking, SðL;M Þ is the 2-dimensional diagram shown in Figure 1 consisting of(A) Q-vector spaces Si;d, with i; d 2 Z; and(B) linear maps (diIerentials) u and v.

Figure 1. Linearly constructible motivic perverse sheaf SðL;MÞ.

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO318

(A) The space Si;d is a direct sumLSiðNÞ where N ranges over d-dimensional

projective subspaces of PnF which are in Ls�dMn�s, with d6 s6n. For all otherprojective subspaces N of PnF we set SiðNÞ ¼ 0. Let LJ \MK 2 LjMk, withjþ k6n. Then we set

SiðLJ \MKÞ ¼0 if iþ k� j 6¼ n,hðLJ \MKÞi if iþ k� j ¼ n,

�where hðLJ \MKÞi is the Q-vector space generated by the symbol ðLJ \MKÞ.This de�nition forces us to identify

sgnððJ;KÞ ! ðJ inc; KincÞÞðLJ \MKÞ ¼ sgnððeJJ; eKKÞ ! ðeJJ inc; eKK incÞÞðLeJ \MeKÞ ð1Þwhenever jeJJ j ¼ jJ j, j eKKj ¼ jKj and geometrically LeJ \MeK ¼ LJ \MK after

forgetting the order. Here J inc is the rearrangement of J in the increasingorder. It is straightforward to see that

Si;d ¼M 0

L 02Lði�dÞ=2;M 02Mn�ðiþdÞ=2

SiðL 0 \M 0Þ ð2Þ

is a �nite-dimensional Q-vector space, whereL 0 means that SiðL 0 \M 0Þ is

identi�ed with Sið eLL 0 \ eMM 0Þ by (1).

PROPOSITION 3.2. The space Si;d ¼ 0 unless it lies in the above trianglebounded by d ¼ 0, i ¼ d and iþ d ¼ 2n and satis�es i # d ðmod 2Þ.

Proof. This is clear. �

Remarks 3.3. To understand the diagram in Figure 1 geometrically weobserve the following.

(i) The dth row of the diagram is constructed only from d-dimensional linearsubspaces of Pn.

(ii) In general, the further left or right we move, the more contributions comefrom M-faces or L-faces respectively. For example, S0;0 ¼ hðM-verticesÞi andS2n;0 ¼ hðL-verticesÞi.

(B) For each projective subspace N 0 of Pn, with N a codimension 1 subspace ofN 0, and each integer i we now de�ne linear maps

uðN;N 0Þ : SiðNÞ �! Si�1ðN 0Þ

and

vðN 0; NÞ : Siþ1ðN 0Þ �! SiðNÞ:

For jþ k6n and j; k> 0 let

J ¼ ð*1; . . . ; *jÞ; K ¼ ð+1; . . . ; +kÞ and LJ \MK 2 LjMk:

Let J 0 ¼ J n f*jg and K 0 ¼ K n f+kg. By the structure of Si we only need to

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 319

de�ne the following. Put i ¼ nþ j� k:uðLJ \MK;LJ \MK 0 Þ ¼ 0; vðLJ 0 \MK;LJ \MKÞ ¼ 0;

uðLJ \MK;LJ 0 \MKÞ : SiðLJ \MKÞ �! Si�1ðLJ 0 \MKÞ;ðLJ \MKÞ 7�! ð�1ÞkðLJ 0 \MKÞ;

vðLJ \MK 0 ; LJ \MKÞ : Siþ1ðLJ \MK 0 Þ �! SiðLJ \MKÞ;ðLJ \MK 0 Þ 7�! ðLJ \MKÞ:

Remarks 3.4. (i) The linear maps u and v are well de�ned. We only needto observe that if LJ \MK 0 ¼ LJ 0 \MK then LJ \MK 0 �M+k and thereforecodimðLJ \MKÞ ¼ codimðLJ \MK 0 Þ < jþ k which is contradictory to our choiceof LJ and MK .

(ii) If the element in J n J 0 is not the last entry in J then we can reorder J andmake it happen. The same applies for K 0.

PROPOSITION 3.5. The linear maps u and v satisfy the following conditions,whenever the compositions named make sense:(1a) uðN;N 0Þ � vðN 0; NÞ ¼ 0,(1b) vðN 0; NÞ � uðN;N 0Þ ¼ 0,(2a) for �xed N and N 00,

PN�N 0�N 00 vðN 0; NÞ � vðN 00; N 0Þ ¼ 0,

(2b) for �xed N and N 00,P

N�N 0�N 00 uðN 0; N 00Þ � uðN;N 0Þ ¼ 0,(2c) for �xed N and N 00, vðN 00; N 02Þ � uðN 01; N 00Þ þ uðN;N 02Þ � vðN 01; NÞ ¼ 0.

Proof. The conditions (1a) and (1b) are easily checked by de�nition andRemark 3.4. To verify the rest we let LJ , LJ 0 , MK and MK 0 be as above and letJ 00 ¼ J 0 n f*j�1g and K 00 ¼ K 0 n f+k�1g.

(2a) Let ðLJ ;MKÞ 2 GPðL;M Þ and N ¼ LJ \MK . Cases where N00 ¼ LJ 00 \MK

or N 00 ¼ LJ 0 \MK 0 are trivial by de�nition. Assume N 00 ¼ LJ \MK 00 . Then wehave only two possible choices of N 0:

(i) N 01 ¼ LJ \MK 0 or(ii) N 02 ¼ LJ \MeK 0 where eKK 0 ¼ ð+1; . . . ; +k�2; +kÞ.

In Case (i) we have

vðN 01; NÞ � vðN 00; N 01ÞðLJ \MK 00 Þ ¼ vðN 01; NÞðLJ \MK 0 Þ ¼ ðLJ \MKÞ;whereas in Case (ii),

vðN 02; NÞ � vðN 00; N 02ÞðLJ \MK 00 Þ ¼ vðN 02; NÞðLJ \MeK 0 Þ ¼ �ðLJ \MKÞ;

because MK ¼ �M+1;...; +k�2; +k;+k�1 : So (2a) is proved.

(2b) This is similar to (2a).(2c) Let ðLJ ;MKÞ 2 GPðL;M Þ andN ¼ LJ \MK . Cases whereN

00 ¼ LJ 00 \MK

or N 00 ¼ LJ \MK 00 are trivial by de�nition. Assume N 00 ¼ LJ 0 \MK 0 . Then wehave only two possible choices of N 0:

(i) N 01 ¼ LJ 0 \MK and N 02 ¼ LJ \MK 0 ; or(ii) N 01 ¼ LJ \MK 0 and N 02 ¼ LJ 0 \MK .

Case (i) is trivial to verify. In Case (ii) we have

vðN 00; N 02Þ � uðN 01; N 00ÞðN 01Þ ¼ vðN 00; N 02Þðð�1Þk�1N 00Þ ¼ ð�1Þk�1N 02;

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO320

and

uðN;N 02Þ � vðN 01; NÞðN 01Þ ¼ uðN;N 02ÞðNÞ ¼ ð�1ÞkN 02:

Thus (2c) is veri�ed in this case too. �

Remark 3.6. There are some misprints on page 705 of [3] in condition (2c).Pictorially, condition (2c) says that the following diagram anti-commutes:

N 00

v u

N 02 N 01

u vN

DEFINITION 3.7. For Si;d in the triangular region of Figure 1 let j ¼ 12 ði� dÞ

and k ¼ n� 12 ðiþ dÞ with i # d ðmod 2Þ. We de�ne

u : Si;d �! Si�1;dþ1;N 7�! ð�1Þk

X 0

N 02Lj�1Mk;N�N 0N 0;

v : Si;d �! Si�1;d�1;N 7�!

X 0

N 02LjMkþ1;N0�N

N 0:

HereP 0 means that N 0 ranges over distinct linear subspaces of Pn.

Geometrically, the map u sends a linear subspace produced by faces of L and Mto all the subspaces of one dimension higher by removing one (highestdimensional) L-face each time; while the map v sends such a linear subspace toall the subspaces of one dimension lower by intersecting it with one (highestdimensional) M-face each time.

By de�nition we see that the 2-dimensional diagram SðL;M Þ satis�es thecondition of a double complex except at terms on the bottom line. This follows fromProposition 3.5 and the fact that u is a �nite sum of uðN;N 0Þ’s, and v is a �nite sum ofvðN 0; NÞ’s. Moreover, we can put a weight �ltration W� on the diagram by settingW2k of the diagram to be that part of it lying on or to the lower left of those terms Si;dwith iþ d ¼ 2k for 06 k6n (compare [3, x 2.2.1]). Then we de�ne W2kþ1 ¼W2k for06 k < n,Wi ¼ 0 for i < 0, andWi ¼ W2n for i > 2n. Thus the weight graded piecegrWi is trivial unless i ¼ 0; 2; . . . ; 2n� 2; 2n. Now according to [3, x 2.2.3] we make thefollowing de�nition.

DEFINITION 3.8. For any integer 06 j6n we construct a cochain complexK�2jðL;M Þ as follows. In the 2-dimensional diagram of Figure 1, we extract theparallelogram-shaped region with lower vertex at S2j;0 so that it becomes a doublecomplex K�;�2j ðL;M Þ such that

Kp;q2j ¼ S2j�p�q;q�p:

������!

������! ������!

������!

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 321

Label the associated single complex K�2jðL;M Þ so that the groups S2j;k in thecolumn over S2j;0 lie in degree zero.

Obviously, Kq2jðL;M Þ is non-trivial if and only if j� n6 q6 j because K p;q

2j ¼ 0unless ðp; qÞ lies in the rectangular area of Figure 2.

We are now ready to de�ne the motivic cohomology of a pair ofadmissible simplices.

DEFINITION 3.9. Let ðL;M Þ be a non-degenerate admissible pair of simplicesin PnF . Then we de�ne the qth rational motivic cohomology of ðL;M Þ by

HqmotðL;M Þ :¼

Mnj¼0

HqmotðL;M Þj; Hq

motðL;M Þj :¼ HqðK�2jðL;M ÞÞ; ð3Þ

graded by the weights 2j.

From the general theory of framed Hodge--Tate structures of x 2.2 we expectthat the weight �ltration W� makes the motivic cohomology H �motðL;M Þ into agraded comodule

Lnj¼0H

�motðL;M Þj over the graded (commutative) Hopf algebra

A�;Q (see equation (23)).

3.2. The motivation

This subsection is logically dependent on Main Theorem 3.25 concerning thestructure of H �motðL;M Þ. Readers are recommended to skip this section in the �rstreading. The goal of the present subsection is to motivate our construction in theprevious subsection by proving the following theorem.

THEOREM 3.10 (compare [3, Theorem 2.2.3]). Let F ¼ C and Pn ¼ PnC. LetðL;M Þ be an admissible pair of simplices in Pn in general position. Then therelative Betti cohomology

HnþqðPn n L;M n L;QÞ ffiMnj¼0

HqðK�2jðL;M ÞÞ

graded by the weight.

Proof. For any complex projective variety X and any closed subset N of X

let jU : U ¼ X nN ,!X and iN;X : N ,!X be the inclusions. Recall that if F is a

Figure 2. The non-trivial terms of K�;�2j , for 06 j6n.

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO322

constant sheaf on an irreducible X then we have the exact sequence

0�!H0NðFÞ �!F �! j�ðF jUÞ �! 0; ð4Þ

where H0NðX;FÞ is the subsheaf of F with supports in N (see [12, Exercise II.1.20]).

We use the same Z to denote the constant sheaf of Z on Pn. As usual, we havea long exact sequence for Betti cohomology

. . .�!Hq�1ðM n L;ZÞ �!HqðPn n L;M n L;ZÞ�!�!HqðPn n L;ZÞ�!res HqðM n L;ZÞ �! . . .

where res is the restriction map. Hence the cohomology groups H �ðM n L;ZÞ andH �ðPn n L;ZÞ are linked together and yield a two-column (p ¼ 0; 1) �rst quadrantspectral sequence converging to the relative cohomology

Ep;q1 ¼ H

qðMðpÞ n L;ZÞ ¼)HpþqðPn n L;M n L;ZÞ; ð5Þwhere Mð0Þ ¼ Pn and Mð1Þ ¼M.

To decodeHpðM n L;ZÞwe let Ti ¼Mi n L and for any subset

I ¼ fi1; . . . ; isg � ½½n��we put TI ¼

Tj Tij ¼MI n L. Set MILJ ¼ H0

MILJðZjMI

Þ for any I; J � ½½n��, where if

I ¼ ; thenMI ¼ LI ¼ Pn.Then sinceMI is irreducible in theZariski topology, it canbeproved easily by using (4) that we have the projective resolutions

0�!MjJ j¼n

LJ �!@

. . .�!@MjJ j¼1

LJ �!@

Z�!@ jPnnL� ðZjPnnLÞ �! 0

and

0�!M

jJ j¼n�jIjMILJ �!

@. . .�!@

MjJ j¼1

MILJ �!@

MI �!@jTI� ðZjTI Þ �! 0:

Here the diIerentials @ are induced by the maps u from the previous subsection.Hence we can compute the following cohomology groups by these resolutions:

HpðPn n L;ZÞ ffi RpLðPn; jPnnL� ðZjPnnLÞÞ; ð6Þ

HpðMI n L;ZÞ ffi RpLðMI; jMInL� ðZjMInLÞÞ: ð7Þ

We now connect the cohomology H �ðM n L;ZÞ to H �ðMI n L;ZÞ for I runningthrough the non-trivial subsets of ½½n��. For any complex variety X covered by twoopen subsets U1 and U2 we have the Mayer--Vietoris long exact sequence

. . .�!Hq�1ðU1 \ U2;ZÞ �!HqðX;ZÞ�!�!HqðU1;ZÞ �HqðU2;ZÞ �!HqðU1 \ U2;ZÞ �! . . .

which yields another two-column �rst quadrant spectral sequence (p ¼ 0; 1)

Ep;q1 ¼

MI�f1;2g;jIj¼pþ1

HqðUI;ZÞ ¼)HpþqðX;ZÞ;

where Ufig ¼ Ui and Uf1;2g ¼ U1 \ U2. It is straightforward to extend this to thecase where X is covered by �nitely many open subsets and get

Ep;q1 ¼

MI�½½n��;jIj¼pþ1

HqðTI;ZÞ ¼)HpþqðM n L;ZÞ: ð8Þ

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 323

Here all the horizontal diIerentials dp;q1 are induced by the maps v from theprevious subsection.

Splicing (5) and (8) together we get the �rst quadrant spectral sequence with

Ep;q1 ðL;M Þ ¼

MI�½½n��;jIj¼p

HqðTI;ZÞ ¼)HpþqðPn n L;M n L;ZÞ:

Here if jIj ¼ 0 then TI ¼ Pn n L. Notice that the horizontal maps from the �rstcolumn to the second column are induced from the restriction maps Pn !Mi.Applying (6) and (7) we can de�ne the E0-term of this spectral sequence as

Ep;q0 ðL;M Þ ¼

MjIj¼p;jJ j¼q

LðMI;MILJÞ:

Let Bp;q be the corresponding bicomplex whose single total complex is �ltered bythe columns

ðfilkðB�ÞÞp;q ¼Bp;q if p>n� k,0 if p < n� k.

�We now put the increasing weight �ltration W on B by letting W2k ¼ filk andW2kþ1 ¼W2k for all k. De�ne the bicomplex A�;� by Ap;q ¼ Ln�qMp withdiIerentials provided by the maps u and v from the previous subsection. It isclear that

Ep;q2 ðL;M Þ �Q ¼ Hp

hHn�qv ðB�;�Þ �Q ffi Hp

hHqvðA�;�Þ;

where Hh and Hv denote the horizontal and vertical cohomology, respectively. Onthe other hand,

Ep;q2 ðL;M Þ �Q ffi grW2n�2pH

nþp�qðPn n L;M n L;QÞ

because the spectral sequence EðL;M Þ degenerates at the E2-term since

HphH

qvðA�;�Þ ¼ H0ðK�2n�2pÞ if p ¼ q,

0 otherwise.

�To see this consider the descriptive picture of A�;� given by Figure 3.

Clearly, we may assume 06 p6n and p6 q6n. If p < q then Ap;q lies inside thebicomplex K�;�2n�2p. Hence Hq

vðAp;�Þ ¼ Hq�pv ðK0;�

2n�2pÞ and HphH

qvðA�;�Þ is a sub-

quotient of

H0hH

q�pv ðK�;�2n�2pÞ ffi grW2n�2pH

q�pv ðK�2n�2pÞ;

where the weight �ltration on K is inherited from that of B. But HaðK�2n�2pÞ ¼ 0

Figure 3. The bicomplex A�;�.

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO324

if a > 0 by Main Theorem 3.25. By the same theorem we know that H0ðK�2n�2pÞ isof pure weight 2n� 2p and therefore if p ¼ q then

HphH

pv ðA�;�Þ ¼ H0

hH0v ðK�;�2n�2pÞ ¼ H

0ðK�2n�2pÞ:

This proves the required result and concludes the proof of our theorem. �

The above theorem is the motivation of De�nition 3.9 in the previoussubsection. From the proof we see that the space HnðPn n L;M n L;ZÞ has acanonical mixed Hodge structure [7, (8.3.10)] which we denote by HðL;M Þ. Theweights of HðL;M Þ that are of interest are 2k for 06 k6n and the correspondinggraded quotients are pure Hodge--Tate structures isomorphic to Zð�kÞck which isck copies of the Tate structure of weight 2k for some non-negative integer ck. Nowconsider the complex ðEp;0

1 �Q; dp;01 Þ. It is nothing but the injective resolution ofjPnnM! ðQjPnnMÞ where Q is the constant sheaf of Q on Pn. Therefore we see that

grW0 HðL;M ÞQ ffi En;02 �Q ¼ Hn

cptðPn nM;QÞ

which is the compactly supported cohomology of Pn nM and therefore is dual tothe Borel --Moore homology HB:M:

n ðPn nM;QÞ. On the other hand,

grW2nHðL;M Þ ffi E0;n2 ffi E

0;n1 ¼ H

nðPn n L;ZÞ:

Therefore the mixed Hodge structure HðL;M Þ has the following frame: adistinguished covector

NM : grW0 HðL;M Þ!� Zð0Þ;

where NM is the class of the oriented chain representing a generator of theBorel --Moore homology HB:M:

n ðPn nM;ZÞ; and a distinguished vector

½!L� : Zð�nÞ!� grW2nHðL;M Þ;

where ½!L� is the class of

ð25iÞ�nd logðt1=t0Þ ^ . . . ^ d logðtn=t0Þ; Li ¼ fti ¼ 0g; for 06 i6n:

Observe that ð25iÞn½!L� lies in HnðPn n L;ZÞ under the isomorphism (cf. [6, (3.2.2)])

HnðPn;O�PnðlogLÞÞ ffi HnðPn n L;CÞ:

Finally, we notice that the Aomoto polylogarithmÐNM

!L satis�es similar axiomsto those in the de�nition of An (De�nition 2.1). Thus the map ðL;M Þ 7!HðL;M Þproviding the homomorphism An ! Hn is a formal analog of the Aomotopolylogarithm.

3.3. A pairing

As observed in [2] we can de�ne the pairings on the Q-vector spaces

Kq2n�2iðL;M Þ ( K�q2i ðM;LÞ���!

hh�; �iiQ

for �i6 q6n� i as follows: if x and y can be represented by the sameðLJ \MKÞ 2 LjMk then

hhx; yii ¼ ð�1Þjk:

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 325

For all other x and y we de�ne hhx; yii ¼ 0: Then we linearly extend hh�; �ii toKq

2n�2iðL;M Þ ( K�q2i ðM;LÞ.

PROPOSITION 3.11. The pairing hh�; �ii provides an isomorphism between thetwo complexes of Q-vector spaces

K2n�2iðL;M Þ ffi K2iðM;LÞ�:Here the dual is obtained by applying the contravariant functor HomQð�;QÞ.

Proof. We only need to verify that the pairings are compatible withthe diIerentials. Let u� and v� be the diIerentials for K2iðM;LÞ�: SupposeðLJ \MKÞ 2 LjMk with J ¼ f*1; . . . ; *jg and K ¼ f+1; . . . ; +kg. Let J 0 ¼ J n f*jgand K 0 ¼ K n f+kg. Then we have

hhuðN;N 0ÞðLJ \MKÞ; ðLJ 0 \MKÞii ¼ 0;

hhðLJ \MKÞ; v�ðN 0; NÞðLJ 0 \MKÞii ¼ 0;

unless N ¼ ðLJ \MKÞ and N 0 ¼ ðLJ 0 \MKÞ in which case we have

hhuðN;N 0ÞðLJ \MKÞ; ðLJ 0 \MKÞii ¼ ð�1Þjk;hhðLJ \MKÞ; v�ðN 0; NÞðLJ 0 \MKÞii ¼ ð�1Þjk:

Similarly, one can prove that

hhðLJ \MKÞ; vðN 0; NÞðLJ \MK 0 Þii ¼ hhu�ðN;N 0ÞðLJ \MKÞ; ðLJ \MK 0 Þiiwhich is either ð�1Þjk if N ¼ ðLJ \MKÞ and N 0 ¼ ðLJ \MK 0 Þ or 0 otherwise.

The proposition is now proved. �

Applying Proposition 3.11 we readily get the following corollary.

COROLLARY 3.12. Let ðL;M Þ be an admissible pair of simplices. For 06 j6nwe set GjðL;M Þ ¼ H0ðK�2jðL;M ÞÞ. Then there is an isomorphism

GjðL;M Þ ffi Gn�jðM;LÞ�:

REMARKS 3.13. (i) One can imagine the isomorphism pictorially as a :ippingof the 2-dimensional diagram in Figure 1 about the center column (that is, the nthcolumn) followed by reversing all the arrows and changing u to v� and v to u�.

(ii) Our de�nition of the complex K2jðL;M Þ and the above pairing does notagree with [2] when n ¼ 2. But it is obvious that up to signs they are the same.We make these changes for the sake of easy generalization.

3.4. Construction of the complex CðL;M ÞIn this subsection we will recast the objects Si;d as abelian groups

Dði�dÞ=2;n�ðiþdÞ=2 de�ned by generators and relations and de�ne diIerentials d 0

and d 00 corresponding to u and v respectively. They provide an integral version ofSðL;M Þ and, more importantly, are very convenient for computation.

Recall that for a subset f+1; . . . ; +kg of ½½n�� we set M+1;...; +k ¼ ðM+1 ; . . . ;M+kÞwhich is an M-face with codimension k (geometrically, it is

Tks¼1M+s). We call a

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO326

chain of inclusions of the faces

M+1;...; +n �M+1;...; +n�1 � . . . �M+1; +2 �M+1

an M-,ag, denoted by M½+1; . . . ; +k� or ðM+1;...; +k ; . . . ;M+1Þ. There are ðnþ 1Þ!diIerent M-:ags. Partial :ags are in one-to-one correspondence with the faces ofM by

M½+1; . . . ; +k� ¼ ðM+1;...; +k ; . . . ;M+1Þ !M+1;...; +k :

We de�ne L-:ags similarly.

DEFINITION 3.14. Set Dj;k ¼ 0 if jþ k>nþ 1 or j < 0 or k < 0. SetD0;0 ¼ hðPnÞi. For k 6¼ 0 or j 6¼ 0, Dj;k ¼ Genðj; kÞ=Relðj; kÞ where Genðj; kÞ isthe group generated by �nite Z-linear combinations of pairs of :ags:X

Lj�...�L1; Ls2LsMk�...�M1; Mt2Mt

aðLj;...;L1;Mk;...;M1ÞðLj; . . . ; L1;Mk; . . . ;M1Þ: ð9Þ

Here the coeQcients aðLj;...;L1;Mk;...;M1Þ 2 Z should satisfy two kinds of conditions.

(i) For any �xed l < j, �xed Ls 2 Ls (with s 6¼ l), �xed Mt 2Mt (with16 t6 k), and variable Ll 2 Ll,X

Llþ1�Ll�Ll�1aðLj;...;L1;Mk;...;M1Þ ¼ 0: ð10Þ

(ii) If Lj \ eMMt ¼ Lj \Mt for all 16 t6 k then

aðLj;...;L1;Mk;...;M1Þ ¼ aðLj;...;L1;eMMk;...;eMM1Þ: ð11Þ

The group Relðj; kÞ is generated by two kinds of relations.(i) For every �xed l < k, every �xed Mt 2Mt (with t 6¼ l), every �xed Ls 2 Ls

(with 16 s6 j), and variable Ml 2Ml,XMlþ1�Ml�Ml�1

ðLj; . . . ; L1;Mk; . . . ;M1Þ ¼ 0: ð12Þ

(ii) If eLLs \Mk ¼ Ls \Mk for all 16 s6 j then

ðLj; . . . ; L1;Mk; . . . ;M1Þ ¼ ð eLLj; . . . ; eLL1;Mk; . . . ;M1Þ: ð13Þ

The diIerentials d 0 and d 00 are de�ned as follows:

d ¼ dj;k ¼ d 00 þ d 0 :Dj;k �!Dj;kþ1 �Dj�1;k;

ðLj; . . . ; L1;Mk; . . . ;M1Þ7�!

XM 02Mkþ1;M

0�Mk

M 0\ðLj;...;L1Þ distinct

ðLj; . . . ; L1;M 0;Mk; . . . ;M1Þ

þ ð�1Þk

]ðRÞX

ðeMk;...;eM1Þ\Lj¼ðMk;...;M1Þ\LjðLj�1; . . . ; L1; eMMk; . . . ; eMM1Þ;

where ]ðRÞ is the number of terms in the sum. Then d is linearly extended to Dj;k.

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 327

Here if j ¼ 0 then d 0 ¼ 0 and if jþ k ¼ n then d 00 ¼ 0. By convention we set

d : D1;0 �!D0;0 �D1;1;

ðL1Þ 7�! ðPnÞ þX

M12M1

ðL1;M1Þ;

d : D0;0 �!D0;1;

ðPnÞ 7�!X

M12M1

ðM1Þ:

REMARKS 3.15. (i) It is not hard to see that the fraction appearing in theabove de�nition is super�cial. Indeed, if the :ag ðLj; . . . ; L1;Mk; . . . ;M1Þ appearsin x 2 Dðj; kÞ with coeQcient a, then all the :ags ðLj; . . . ; L1; eMMk; . . . ; eMM1Þ, withLj \ eMMt ¼ Lj \Mt for all 16 t6 k, must appear in x with the same coeQcient aby (11). Hence every term of dðxÞ has integer coeQcient.

(ii) In the de�nition of d 00 we only allow distinct M 0 \ ðLj; . . . ; L1Þ to appearbecause ultimately we will be concerned only with the geometric con�gurationof ðL;M Þ.

(iii) The key idea is that we should keep skew-symmetry on both L and M by(10) and (12) (see Lemma 3.18). The relations (11) and (13) take the non-genericconditions into account.

In the next de�nition we construct the complex CðL;M Þ using groups Dj;k asbuilding blocks.

DEFINITION 3.16. Let 16 j6n. We de�ne the complex C�2jðL;M Þ as follows:for j� n6 k6 j,

. . .��! C 2j�n�k2j ��������!

d2j�n�k2jC2j�n�kþ1

2j �����! . . . . . .���!

Mr2ðkÞr¼r1ðkÞ

Dr;r�k ���!d Mr2ðk�1Þ

r¼r1ðk�1ÞDr;r�kþ1 ���! . . .

where if 2j� n> 0 then

ðr1ðkÞ; r2ðkÞÞ ¼ð0; nþ k� jÞ if j� n6 k6 0,

ðk; nþ k� jÞ if 06 k6 2j� n,ðk; jÞ if 2j� n6 k6 j,

8><>:and if 2j� n6 0 then

ðr1ðkÞ; r2ðkÞÞ ¼ð0; nþ k� jÞ if j� n6 k6 2j� n,ð0; jÞ if 2j� n6 k6 0,

ðk; jÞ if 06 k6 j.

8><>:Here we use the fact that if r>nþ k� j and j� n6 k then 2r� kþ 1>nþ 1and therefore Dr;r�kþ1 ¼ 0:

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO328

LEMMA 3.17. Let ðL;M Þ be an admissible pair of simplices. Then for all j andk we can de�ne Dj;k by using only pairs of sub-simplices in GPðL;M Þ.

Proof. We want to show that Dj;k can actually be generated by sums as in (9)where equations (10) and (11) hold and, moreover, ðLj;MkÞ 2 GPðL;M Þ wheneveraðLj;...;L1;Mk;...;M1Þ 6¼ 0. Suppose there appears a pair ðLj;MkÞ =2GPðL;M Þ. Then

codimðLj \ MkÞ < jþ k and j; k> 1. Write the :ag

ðLj; . . . ; L1;Mk; . . . ;M1Þ ¼ ðL½*1; . . . ; *j�;M½+1; . . . ; +k�Þ:

We now have two cases.

Case (i): Lj 6�M1 ¼M+1 . Then we must have some l < k such that

M+1...+l \ Lj ¼M+1...+l�1+lþ1 \ L

j: ð14Þ

Indeed, if M+1...+l \ Lj 6�M+lþ1 for all 16 l < k then we see that

codimðM+1...+lþ1 \ LjÞ ¼ codimðM+1...+l \ L

jÞ þ 1; for 16 l < k:

Thus

codimðM+1...+k \ LjÞ ¼ codimðM1 \ LjÞ þ k� 1 ¼ jþ k

which is a contradiction. By equation (12) we get

ðLj; . . . ; L1;M½+1; . . . ; +k�Þ þ ðLj; . . . ; L1;M½+1; . . . ; +l�1; +lþ1; +l; +k�Þ ¼ 0;

while by equations (14) and (11) the coeQcients of these two terms are equal in theexpression (9). Thus the :ag ðLj; . . . ; L1;M½+1; . . . ; +k�Þ cannot essentially appearin (9).

Case (ii): Lj �M1. Then

L*2*3...*j \M1 ¼ L*1*3...*j \M

1:

By (13) we have

ðL½*1; . . . ; *j�;M½+1; . . . ; +k�Þ ¼ ðL½*2; *1; *3; . . . ; *j�;M½+1; . . . ; +k�Þ:

But by (11),

aðL½*1;...;*j�;M½+1;...; +k�Þ þ aðL½*2;*1;*3;...;*j�;M½+1;...; +k�Þ ¼ 0:

Hence again the :ag ðLj; . . . ; L1;M½+1; . . . ; +k�Þ cannot essentially appear in (9). �

LEMMA 3.18. Every element in Dj;k satis�es skew-symmetry on both its L-partand its M-part. Namely, for any permutations � of f1; . . . ; jg and $ of f1; . . . ; kg,

ðL½*�ð1Þ; . . . ; *�ðjÞ�;M½+$ð1Þ; . . . ; +$ðkÞ�Þ¼ sgnð�Þsgnð$ÞðL½*1; . . . ; *j�;M½+1; . . . ; +k�Þ:

Proof. First we deal with the M-part. By de�nition, for any +1; . . . ; +k 2 ½½n��,any �xed Ls 2 Ls (with 16 s6 j), any �xed Mt ¼M+1...+t 2Mt (with 16 t6 k,

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 329

t 6¼ l), and variable Ml 2Ml;

0 ¼X

Mlþ1�Ml�Ml�1

ðLj; . . . ; L1;Mk; . . . ;M1Þ

¼ ðLi; . . . ; L1; M½+1; . . . ; +k�Þ þ ðLi; . . . ; L1; M½+1; . . . ; +lþ1; +l; +lþ2; . . . ; +k�Þ;

that is, Dðj; kÞ satis�es skew-symmetry on the part of M. In fact it is clear thatskew-symmetry on the M-part is equivalent to (12). The assertion on the L-partcan be veri�ed using duality between L and M (see Proposition 3.11). Thus skew-symmetry on the L-part is equivalent to (10). �

For example,

ðM012;M½01�Þ ¼ �ðM012;M½10�Þ ¼ ðM012;M½12�Þ¼ �ðM012;M½21�Þ ¼ ðM012;M½20�Þ ¼ �ðM012;M½02�Þ:

Now suppose ðL012; L01; L0Þ has coeQcient a, then ðL012; L01; L1Þ has coeQcient�a, then ðL012; L12; L1Þ has coeQcient a, and so on.

In order to compare Q-vector spaces Si;d with D-groups we need to look at theirbases. We �rst look for an optimal way to write down the basis of each of Si;d. ForðLJ ;MKÞ 2 GPðL;M Þ, let

EðLJ ;MKÞ ¼ fðLeJ ;MeKÞ 2 GPðL;M Þ : eJJ ¼ eJJ inc; eKK ¼ eKK inc; jeJJ j ¼ jJ j;j eKKj ¼ jKj; LeJ \MeK ¼ LJ \MKg:

DEFINITION 3.19. For �xed j; k > 0 let us put the usual lexicographic order+ on fðI1; I2Þ : jI1j ¼ j; jI2j ¼ k; I1; I2 � ½½n��g. We say that ðJ;KÞ is smaller thanðJ 0; K 0Þ if ðJ;KÞ + ðJ 0; K 0Þ: Then in the set EðLJ ;MKÞ there is a unique pairðeJJ; eKKÞ which is smallest. We call the pair ðLeJ ;MeKÞ of subsimplices of ðL;M Þ anoptimal pair. Denote the set of all optimal pairs by OPðL;M Þ.

By formula (2) we get the following lemma.

LEMMA 3.20. The �nite-dimensional Q-vector space Si;d has a basis

fðLJ \MKÞ : ðLJ ;MKÞ 2 OPðL;M Þ; 2jJ j ¼ 2n� d� i; 2jKj ¼ i� dg:

Proof. By de�nition, the increasing order of indices takes care of the skew-symmetry on both L and M; ‘optimal’ restriction is aimed at the non-generic conditions. �

Remark 3.21. The optimality is a non-trivial condition. For example, take anon-degenerate admissible pair of tetrahedra ðL0; . . . ; L3;M0; . . . ;M3Þ in P3 withthe only non-generic condition

L1;2 \M1;2 6¼ ;:

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO330

Then, in S2;0 we have

ðL1 \M1;2Þ ¼ ðL2 \M1;2Þ ¼ ðL1 \M2;1Þ ¼ ðL2 \M2;1Þ:

Among them only ðL1 \M1;2Þ 2 OPðL;M Þ:

LEMMA 3.22. If ðLJ ;MKÞ is optimal then ðLeJ ;MeKÞ is optimal for any subseteJJ � J and eKK � K in the increasing order.

Proof. We �rst show that the lemma is true if eKK ¼ K is �xed. The lemmafollows from this because, by the same argument, we may �x eJJ and let eKK vary.

Let J ¼ ð*1; . . . ; *jÞ. Suppose on the contrary that there is an index subseteJJ � J such that ðLeJ ;MKÞ is not optimal. Then there exists an optimal ðLJ1 ;MK1Þ

with J1 ¼ ðr1; . . . ; rpÞ + eJJ and

LJ1 \MK1¼ LeJ \MK: ð15Þ

Step (i). We �rst show that eJJ � J1. Suppose on the contrary that eJJ 6� J1 andtherefore there exists an *s 2 eJJ n J1. Then

LJ1 \MK1¼ LeJ \MK � L*s

and thus

LJ1nfrpg;*s \MK1¼ LJ1 \MK1

:

Hence *s > rp by optimality of ðLJ1 ;MK1Þ. We claim that this in turn implies that

J1 � J . Suppose on the contrary that ri =2J for some 16 i6 p. Then

LJ \MK � LeJ \MeK � Lri :Thus LJ \MK ¼ LJ2 \MK with

J2 ¼ ð*1; . . . ; *s�1; ri; *sþ1; . . . ; *jÞ + J

because ri6 rp < *s. This is contradictory to the assumption that ðLJ ;MKÞ isoptimal. Hence J1 � J .

To reach a contradiction under the assumption eJJ 6� J1 we now observe that

because J1 + eJJ we can reorder the index set ðJ n eJJÞ [ J1 to get J2 satisfyingJ2 + J . But this is impossible because ðLJ ;MKÞ is optimal while

LJ \MK ¼ LJneJ \ ðLeJ \MKÞ ¼ LJneJ \ ðLJ1 \MK1Þ ¼ LJ2 \MK1

:

Step (ii). Step (i) implies that eJJ ¼ J1 because J1 + eJJ � J1. We can now assumeon the contrary that ðLeJ ;MKÞ is not optimal and therefore LeJ \MK ¼ LeJ \MK1

with K1 + K by (15). Then

LJ \MK ¼ LJneJ \ ðLeJ \MKÞ ¼ LJneJ \ ðLeJ \MK1Þ ¼ LJ \MK1

;

which contradicts the assumption that ðLJ ;MKÞ is optimal.This completes the proof of the lemma. �

Let us turn to the D-groups. In order to �nd bases of them we set Lð*Þ ¼ L*and ALð*Þ ¼ Pn for 06*6n. For s> 1 and 06*0 < . . . < *s6n we de�ne

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 331

recursively

ALð*0; . . . ; *sÞ ¼Xsi¼0ð�1ÞiLð*0; . . . ; b*i*i; . . . ; *sÞ; for s> 1;

and

Lð*0; . . . ; *sÞ ¼ ðL*0...*s ; ALð*0; . . . ; *sÞÞ; for s> 1:

LEMMA 3.23. If ðL;M Þ is a generic pair of simplices then Dj;k is has thefollowing basis:�

ðLð*1; . . . ; *jÞ;M½+1; . . . ; +k�Þ : 06*1 < . . . < *j6n;

06 +1 < . . . < +k6n�: ð16Þ

Proof. By Lemma 3.18, each element in the set (16) is clearly the uniquerepresentation corresponding to the face L*1;...;*j \M+1;...; +k . The linear independencealso follows easily. �

To �nd a basis of Dj;k in general, let us put

I ¼ fð+1; . . . ; +kÞ : 06 +1 < . . . < +k6ng:

For a �nite set B ¼ fbi : i 2 Ig indexed by I and a partition P ¼ fI1; . . . ; I sg ofI we de�ne the partition sum-set of B corresponding to P as

BP ¼�er ¼

Xi2I r

bi : 16 r6 s

�:

Then by de�nition it is not diQcult to see that there is always some partitionsum-set BP of (16), say,

BP ¼�er ¼

XK2I rðLðJÞ;M½K�Þ

�;

that forms the basis of Dj;k. Here if both ðLðJÞ;M½K1�Þ and ðLðJÞ;M½K2�Þappear in er then LJ \MK1

¼ LJ \MK2. Let ðLðJÞ;M½K�Þ be any one of the

terms appearing in er. Then there is a unique element

ðJOP; KOPÞ 2 OPðL;M Þ \ EðLðJÞ;M½K�Þ

so that we can rewrite

er ¼XK2IrðLðJOPÞ;M½K�Þ ¼ eOP

r :

Therefore Dj;k can be generated by all such eOP and they are linearly independent.

PROPOSITION 3.24. For all j and k we have

Dj;k �Q ffi Snþj�k;n�j�k: ð17Þ

Moreover, for each i ¼ 0; 1; . . . ; n, we have the isomorphism of complexes

C2iðL;M Þ �Q ffi K2iðL;M Þ:

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO332

Proof. De�ne a Q-linear map

S 0 : Dj;k �Q�! Snþj�k;n�j�k;ðLð*1; . . . ; *jÞ;M½+1; . . . ; +k�Þ 7�! ðL*1;...;*j \M+1;...; +kÞ:

Then we extend it linearly to Dj;k. It is easy to see that

S 0ðeOPr Þ ¼ jI rj

�LJOP ;MKOPÞ:

Setting SðeOPr Þ ¼ S 0ðeOP

r Þ=jI rj proves (17) immediately thanks to Lemma 3.20.We now show that the diIerentials are compatible with S. Using the above

notation we put

x ¼ er 2 Dj;k; xi ¼ ðLðJ n f*igÞ;M½K�Þ 2 Dj�1;k; for 16 i6 j:

Then

S � d 0ðxÞ ¼ð�1ÞkXji¼1ð�1Þi�1SðxiÞ

¼XK2I r

Xji¼1ð�1Þkþi�1ðL*1;...;b*i*i;...;*j \M+1;...; +kÞ

¼ð�1Þj�1Xji¼1

uðSðxÞ;SðxiÞÞðSðxÞÞ:

Thus, whenever the compositions on Dj;k make sense,

S � d 0 ¼ ð�1Þj�1u � S: ð18ÞSimilarly, letting exx ¼ ðLðJÞ; eMM;M½K�Þ 2 Dj;kþ1 we have

S � d 00ðxÞ ¼X 0

eM2Mkþ1; eM�MK

SðexxÞ¼

X 0

+kþ12½½n��nKðLJ \MK;+kþ1Þ

¼X 0

eM2Mkþ1; eM�MK

vðSðxÞ;SðexxÞÞ � SðxÞ:Here

P 0 means that LJ \MK;+kþ1 are all distinct. Thus, whenever thecompositions make sense,

S � d 00 ¼ v � S: ð19ÞEquations (18) and (19) show that the diIerentials of the complexes arecompatible with S. In particular, using Proposition (3.5) we see that d2 ¼ 0. Theproof of the proposition is now complete. �

3.5. The Main Theorem

Recall that the qth rational motivic cohomology of an admissible pair ofsimplices ðL;M Þ is de�ned as

HqmotðL;M Þ ¼

Mnj¼0

HqðK�2jðL;M ÞÞ;

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 333

where the complex K�2j is constructed according to De�nition 3.9. In thissubsection we prove the following.

MAIN THEOREM 3.25. (I) The complex K�2jðL;M Þ is exact everywhere exceptat K0

2j. Moreover, H0ðK�0ðL;M ÞÞ ffi H0ðK�2nðL;M ÞÞ ffi Q and H0ðK�2jðL;M ÞÞ has

dimension at most ðnjÞ2 for 0 < j < n.

(II) Let Sn denote the set of index subsets of f1; . . . ; ng in the increasing order.If ðL;M Þ is a generic pair then the set

eðI; JÞ ¼ ðALð0; IÞ;M½J �Þ : jIj ¼ j; jJ j ¼ n� j; I; J 2 Snf g ð20Þis a basis of H0ðC �2jðL;M ÞÞ. In particular, H0ðK�2jðL;M ÞÞ has dimension ðnjÞ

2. HereðALðKÞ;PnÞ ¼ ALðKÞ and ðPn;MKÞ ¼ ðMKÞ for any index set K.

Remark 3.26. In the generic case, we will call L0 and M0 the special L-faceand M-face respectively, when the basis is given as in the theorem. If ðL;M Þ isnot a generic pair then it might be necessary to choose other faces than L0 or M0

as the special faces. Strictly speaking, the non-generic relations between L and M(for example, an M-vertex lies in some L-face) bring in extra relations among theotherwise linearly independent generators of the cohomology group. This is thereason why H0ðK�2jðL;M ÞÞ may have smaller dimension when ðL;M Þ is a non-generic pair. See the computations in the next three sections.

In what follows we will prove several lemmas before giving the proof of theMain Theorem at the end of this subsection. Thanks to Proposition 3.24 thetheorem can be proved using either CðL;M Þ or KðL;M Þ because we can alwaysthrow away torsions if necessary. We �rst handle the special cases j ¼ 0 and nbecause they are in fact generic cases.

LEMMA 3.27. One has HnðC �2nðL;M ÞÞ ¼ 0. If 06 k6n� 1 then

ker d k2n ¼ hALð0; *1; . . . ; *n�kÞ : 16*1 < . . . < *n�k6ni:Hence HqðC �2nðL;M ÞÞ ¼ 0 unless q ¼ 0 for which we have

H0ðC�2nðL;M ÞÞ ¼ hALð0; . . . ; nÞi:

Proof. It is clear that

C n�12n ¼ hðLiÞ : 06 i6ni �! C n

2n ¼ hðPnÞiis surjective. Therefore HnðC�2nðL;M ÞÞ ¼ 0.

Suppose 06 k6n� 1 and d k2nðxÞ ¼ 0 for some x 2 Ck2n ¼ Dn�k;0. By Proposition

3.24 we may write x ¼ x 0 þP

16 i1<...<ik 6n ai1;...;ikLð1; . . . ; bi1i1; . . . ; bikik . . . ; nÞ where L0

appears in every term of x 0. Since L0 appears in every term of

x�X

16 i1<...<ik 6n

ai1;...;ikALð0; 1; . . . ; bi1i1; . . . ; bikik . . . ; nÞand by de�nition ALð. . .Þ ¼ d 0ðLð. . .ÞÞ, we may assume that x ¼ x 0, without loss ofgenerality. Thus

x ¼X

16 i0<...<ik 6n

ci0;...;ikLð0; . . . ; bi0i0; . . . ; bikik . . . ; nÞ:

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO334

Now we can write d k2nðxÞ ¼ 0 as

yþX

16 i0<...<ik 6n

ci0;...;ikLð1; . . . ; bi0i0; . . . ; bikik . . . ; nÞ ¼ 0

where L0 appears in every term of y. Then we must have y ¼ 0 by Proposition3.24 and X

16 i0<...<ik 6n

ci0;...;ikLð1; . . . ; bi0i0; . . . ; bikik; . . . ; nÞ ¼ 0

which implies that ci0;...;ik ¼ 0 for all i0 < . . . < ik because, again by Proposition 3.24,the LðJÞ are linearly independent for all increasing index sets J 2 ½½n��. Thus x ¼ 0.The lemma then follows from the fact that, for 16 k6n� 1,

ALð0; *1; . . . ; *n�kÞ ¼ dk�12n ðLð0; *1; . . . ; *n�kÞÞ: ��

The lemma implies that the complex C�2n is exact at C q2n whenever there is a

non-trivial d 0 mapping to it. So the only exception is C 02n and ALð0; . . . ; nÞ

generates H0ðC �2nðL;M ÞÞ ¼ ker d02n. We will see that in the general situation we

may apply the above argument to u on KðL;M Þ.

LEMMA 3.28. For all 06 k6n� 1 we have

ker dk�n0 ¼� X

Mk2Mk

ðMk;Mk�1; . . . ;M1Þ : Mt 2Mt; 16 t6 k� 1

�where we set ker d�n0 ¼ 0. Hence HqðC�0 ðL;M ÞÞ ¼ 0 unless q ¼ 0 for which wehave H0ðC�0 ðL;M ÞÞ ¼ hM½1; . . . ; n�i:

Proof. This follows from Lemma 3.27 by taking the dual and usingProposition 3.11. �

The lemma implies that the complex C �0 is exact at Cq0 whenever it has a non-

trivial d 00. So the only exception is at C 00 and H0ðC�0 ðL;M ÞÞ is generated by

M½1; . . . ; n�. A similar result holds for v on KðL;M Þ in general.

Proof of the Main Theorem 3.25. By Lemma 3.17 all the D-groups inC�ðL;M Þ can be generated by generic pairs ðLj;MkÞ. We now assume this.

Thanks to Lemmas 3.27 and 3.28 we may assume 0 < j < n. We �rst look atthe boundary cohomologies of C �2jðL;M Þ. We deal with them as special casesbecause there is only one D-component at degree j� n and j.

(i) When k ¼ j� n we have

C�2jðL;M Þ : . . .�!D0;n�j�1 �D1;n�j���!dj�12j

D0;n�jð¼ Cj2jÞ �! 0:

Clearly D1;n�j is not empty since j 6¼ 0. Therefore

dj�12j ðL0;Mn�j; . . . ;M1Þ ¼ ðMn�j; . . . ;M1Þ

implies that dj�12j is surjective and HjðC�2jðL;M ÞÞ ¼ 0.

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 335

(ii) When k ¼ j we have

C �2jðL;M Þ : 0�! ðCj�n2j ¼ÞDj;0���!

dj�n2jDj�1;0 �Dj;1 �! . . . :

Note that j < n so the component Dj;1 6¼ 0. Thus, for any J ¼ ð*1; . . . ; *jÞ,

dj�n2j ðLðJÞÞ ¼X

M12M1;M1\LJ distinct

ðLðJÞ;M1Þ 6¼ 0:

Hence ker dj�n2j ¼ 0 and dj�n2j is injective. Therefore Hj�nðC �2jðL;M ÞÞ ¼ 0.

In what follows we provide a proof, using KðL;M Þ, which has a stronggeometric :avor.

We only need to look at the terms Si;d in the interior of each complexK�2iðL;M Þ (for 16 i6n� 1) extracted from Figure 1.

Suppose ðuþ vÞðPxrÞ ¼ 0 where xr 2 Si;r and ði; rÞ runs down the ith column

of the 2-dimensional diagram in Figure 1, say r1 > r> r2. Then xr1 is on the upperboundary of the parallelogram and we can apply Lemmas 3.28 and 3.27 and �ndthat xr1 is an image of an element sitting at the boundary of the ðiþ 1Þth column.Thus we may assume xr1 ¼ 0. By induction on r we now may assume xr ¼ 0 forall r1 > r > r2. Let us prove that xr2 is an image too. Suppose that

xr2 ¼X

ðLJ ;MKÞ2OPðL;M ÞaðJ ;KÞðLJ \MKÞ

with rational coeQcients aðJ ;KÞ. It is clear that either u or v is non-trivial becausexr2 is sitting on one of the two lower sides of the parallelogram de�ningK�2iðL;M Þ, but not on the vertices. Now let us consider the two cases separately.

(i) u is non-trivial. Set J ¼ ð*1; . . . ; *jÞ. Then we have

uðxr2Þ ¼X

ðLJ ;MKÞ2OPðL;M ÞaðJ ;KÞð�1Þkþj�i

Xji¼1ðLJnf*ig \MKÞ ¼ 0: ð21Þ

We observe that if 0 =2J then ðL0;J \MKÞ 2 GPðL;M Þ. Indeed, if ðL0;J ;MKÞ isdegenerate then L0;J \MK � L0 and we can replace the �rst entry in J by 0which contradicts the optimality of ðLJ \MKÞ. Moreover, ðL0;J \MKÞ 2 OPðL;M Þbecause otherwise there is some eJJ + J with ðL0;J \MKÞ ¼ ðL0;eJ \MKÞ whichimplies that ðLJ \MKÞ ¼ ðLeJ \MKÞ by de�nition, another contradiction. Now set

x 0r2 ¼ xr2 � ð�1ÞjX0 =2J

aðJ ;KÞuððL0;J \MKÞÞ

where J ranges over those indices appearing in (21) (the sign ð�1Þj is to cancel allthe terms ðLJ \MKÞ satisfying 0 =2 J in x 0r2 ). Then L0 appears in every term of x 0r2 .Hence by forgetting the images we may assume that L0 actually appears in everyterm of xr2 . Rewriting J as f0; Jg with J ¼ ð*2; . . . ; *jÞ in (21) we get, by theassumption uðxr2Þ ¼ 0,

0 ¼ uðxr2Þ ¼X

ðL0;J ;MKÞ2OPðL;M Það0;J;KÞð�1Þkþj�1ðLJ \MKÞ

þX

ðL0;J ;MKÞ2OPðL;M Það0;J ;KÞð�1Þkþj�i

Xji¼2ðL0;Jnf*ig \MKÞ:

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO336

Notice that every term inside the second sum is optimal by Lemma 3.22 and thereforeis linearly independent from the terms in the �rst sum which are also optimal. Toprove this, notice that all the indices are in increasing order so that if somecancellation occurs between the �rst and second sum we must have ðLJ \MKÞ 2 L0

for some J . This contradicts the optimality of ðLJ \MKÞ. ThusXðL0;J ;MKÞ2OPðL;M Þ

að0;J;KÞð�1Þkþj�1ðLJ \MKÞ ¼ 0:

This implies that all að0;J;KÞ ¼ 0 because all the terms are optimal and therefore arelinearly independent. Consequently xr2 ¼ 0.

(ii) v is non-trivial. This case is dual to (i) by Proposition 3.11.The exactness claim in Part (I) of the theorem now follows immediately. Part

(II) and the dimension estimate in (I) are straightforward and we leave the detailsto interested readers. �

4. The coproduct on A� �Q

By Main Theorem 3.25 the cohomology HqmotðL;M Þ is trivial for a 6¼ 0 and

GjðL;M Þ ¼ H0ðC �2jðL;M ÞÞ is free of rank rj6 ðnjÞ2. Rewriting (3) for q ¼ 0 using

Corollary 3.24, we get

H0motðL;M Þ ¼

Mnj¼0

GjðL;M ÞQ:

Let fei;j : 16 i6 rjg be a basis of GjðL;M Þ. We expect that there exist well-de�ned maps

’ij : GiðL;M Þ� �GjðL;M Þ �!Aj�i; for 06 i6 j6n;

such that the coproduct map on An can be de�ned as

�k;n�k : An �!An�k � Ak;

½L;M� 7�!Xrki¼1

’kn½e�i;k� � ’0k½ei;k�;ð22Þ

so that A�;Q becomes a graded commutative Hopf algebra. Here and in the rest ofthe paper,

’0k½ei;k� :¼’0k½ðM½1; . . . ; n�Þ� � ei;k�;

’kn½e�i;k� :¼’kn½e�i;k � ðALð0; . . . ; nÞÞ�:

The maps ’ij will also induce the actions

HqmotðL;M Þj � A�j�i;Q �!Hq

motðL;M Þi ð23Þ

which makes H �motðL;M Þ into a graded comodule over the graded Hopf algebraA�;Q. Here if q 6¼ 0 and i 6¼ j then it is the zero map. One can also de�ne thePoincarTee duality on the motivic cohomology (see [2, x 2.18]).

4.1. The coproduct on the generic part

We carry out the above program in the generic case as follows.

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 337

DEFINITION 4.1. Let ðL;M Þ be a generic pair. For 06 i6 j6n we can de�ne

’ij : GiðL;M Þ� �GjðL;M Þ �!Aj�i;

eðI; J 0Þ� � eðI 0; JÞ 7�! sgnðIÞsgnðJÞ½LI \MJ jL0;I 0nI ;M0;J 0nJ �;

where eðI; JÞ are de�ned in formula (20). Here for any I � f1; . . . ; ng,sgnðIÞ ¼ sgnðI; f1; . . . ; ng n IÞ:

LEMMA 4.2. The maps ’ij are well de�ned.

Proof. The group GiðL;M Þ is de�ned modulo images of d�12i which provideadditivity and modulo the relations of the component groups of C 0

2i whichprovides skew symmetry by Lemma 3.18. These two properties are also satis�edby Aj�i. �

When ði; jÞ ¼ ð0; kÞ or ðn� k; nÞ one gets

’0k : G0ðL;M Þ� �GkðL;M Þ �!Ak;

ðM½1; . . . ; n�Þ� � eðI; JÞ 7�! sgnðJÞ½MJ jL0;I ;M0;J �;’kn : GkðL;M Þ� �GnðL;M Þ �!An�k;

eðI; JÞ� � ðALð0; . . . ; nÞÞ 7�! sgnðIÞ½LI jL0;I ;M0;J �;

where J ¼ f1; . . . ; ng n J and I ¼ f1; . . . ; ng n I are in increasing order. Thisde�nition is essentially the same as the one given on page 708 of [3] (it waspointed out in [2] that jk and jn�k should be switched there). We would like tomention that everywhere the tensor product is over Q such that in the tensorproduct Am � An the signs multiply. In particular, under the isomorphism A1 ffi F (the negative sign is mapped to the inverse map in F ( .

DEFINITION 4.3. The coproduct maps are de�ned by

�n�k;k : A0n �! A0

n�k �A0k;

½L;M� 7�!XI;J

’kn½eðI; JÞ�� � ’0k½eðI; JÞ�:

In the above de�nition we have �xed a basis for each GjðL;M Þ. However, wehave the following result.

PROPOSITION 4.4. The coproduct map does not depend on the choice of thebasis of GjðL;M Þ (for 06 j6n).

Proof. This follows directly from Lemma 4.2. A combinatorial proof of this isprovided as [13, Proposition 3.3]. �

4.2. The coproduct on the non-generic part

Let ðL;M Þ be a non-generic pair of non-degenerate admissible simplices. Byadditivity on both L and M we may assume that non-generic conditions occur inonly one L-:ag and only one M-:ag. We call such a pair reduced.

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO338

For any given 0 < k < n it is generally quite diQcult to �nd a basis of GkðL;M Þð¼ H0ðC �2kðL;M ÞÞÞ because it depends crucially on the con�guration of ðL;M Þ. Ifa basis fei;k : 16 i6 rkGkðL;M Þg has been found then ’0k and ’kn should bealways de�nable in a way similar to the generic case. Then the coproduct would begiven by (22). In fact, we believe that if ðL;M Þ is reduced then one can relabel L andM such that one of the following two cases should appear:

(i) for each I � ½½n�� and jIj ¼ kþ 1, there is a special M-face MjI such thateach element in the basis of GkðL;M Þ has the form

ei;k ¼ eðI; JÞ ¼ ðALð0; IÞ;M½J �Þ; where I � ½½n�� n f0g; J � ½½n�� n fjIg;

(ii) for each J � ½½n�� and jJ j ¼ n� k, there is a special L-face LiJ such that eachelement in the basis has the form

ei;k ¼ eðI; JÞ ¼ ðALðiJ ; IÞ;M½J �Þ; where I � ½½n�� n fiJg; J � ½½n�� n f0g:

Then in Case (i) setting I ¼ f1; . . . ; ng n I and J ¼ f0; . . . ; ng n ðfjIg [ JÞwe de�ne

’0k½eðI; JÞ� ¼ sgnðjI; J; JÞ½MJ jL0;I ;MjI;J�;

’kn½eðI; JÞ�� ¼ sgnðIÞ½LI jL0;I ;MjI;J �:

In Case (ii) setting I ¼ f0; . . . ; ng n ðfiJg [ IÞ and J ¼ f1; . . . ; ng n J we de�ne

’0k½eðI; JÞ� ¼ sgnðJÞ½MJ jLiJ ;I ;M0;J �;’kn½eðI; JÞ�� ¼ sgnðiJ ; I; IÞ½LI jLiJ ;I ;M0;J �:

Note that in the generic case we have iJ ¼ jI ¼ 0.Applying the above idea in the next three sections we are going to deal with all

possible con�gurations when n ¼ 2; 3 and some non-generic con�gurations whenn ¼ 4, which settles these three cases explicitly.

5. Admissible pairs of triangles in P2F

In this section, we use the theory developed in the previous sections to handlearbitrary con�gurations of admissible pairs of triangles in P2

F . This greatlysimpli�es the calculation given in [2, 4] for pairs of triangles. In particular, noauxiliary functions to de�ne the coproduct map �1;1 on A2 will be needed becauseof the way we de�ne ’01 and ’12 on the generators of G1ðL;M Þ.

5.1. De�nition of coproduct on A2

Let ½L;M� 2 A2 be a non-degenerate admissible pair. If ðL;M Þ is not a genericpair then we further assume that the non-generic conditions occur in only one :agof L and only one :ag of M. Hence we have the following three cases to consider:

(I) ðL;M Þ is a generic pair;(II) M01 2 L1;(III) L01 2M1.

Here, the condition in Case (II) is the only non-generic condition for the pair inthat case. The same is true for Case (III).

In the following we only need to �nd the maps ’01 and ’12 by equation (22).

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 339

(I) ðL;M Þ is a generic pair. Then one has

G1ðL;M Þ ¼ hðALð0iÞ;MjÞ : i; j ¼ 1; 2i:

The coproduct is de�ned in De�nition 4.3.(II) M01 2 L1. Then

ðL1;MiÞ =2C02 for i ¼ 0; 1 but ðL1;M0Þ þ ðL1;M1Þ 2 C0

2 : ð24ÞHence

G1 ¼ hðALð02Þ;M1Þ; ðALð0iÞ;M2Þ; ði ¼ 1; 2Þi:

Thus rkG1 ¼ 3 and we can de�ne

’01 : ðALð02Þ;M1Þ 7�! ½M1jL0;2;M0;2�;ðALð01Þ;M2Þ 7�!�½M2jL0;1;M0;1�;ðALð02Þ;M2Þ 7�!�½M2jL0;2;M0;1�;

’12 : ðALð02Þ;M1Þ 7�!�½L2jL0;1;M0;1�;ðALð01Þ;M2Þ 7�! ½L1jL0;2;M0;2�;ðALð02Þ;M2Þ 7�!�½L2jL0;1;M0;2�:

Here we abused the notation a little because what we really mean in the aboveis to de�ne ’01ððM½12�Þ�;�Þ and ’01ð�; ðALð012ÞÞÞ where ðM½12�Þ and ðALð012ÞÞare the generators of G0ðL;M Þ and G2ðL;M Þ respectively. We will use the samenotation in what follows.

By the general theory, ðALð01Þ;M1Þ is another possible element in G1. But inour case d0

2 : ðL1;M1Þ 7! � ðM0Þ � ðM1Þ: Also notice that the image of thiselement under ’12 obtained in the generic situation, ½L1jL0;2;M0;1�, degeneratesbecause L0 \M0 ¼ L0 \M1. From the maps ’01 and ’12 we see that thecoproduct �1;1ð½L;M�Þ has only three terms. This is a general phenomenon,namely, the coproduct �k;n�kðL;M Þ always has rkGkðL;M Þ terms for admissiblepairs ðL;M Þ in Pn.

(III) L01 2M1. This can be treated by taking the dual (that is, exchange theletter L with M everywhere) in Case (II) due to Proposition 3.11.

5.2. Examples

As the �rst example we look at the pair of simplices U2ðtÞ ¼ ðL;M Þcorresponding to the dilogarithms Li2ðtÞ de�ned by Figure 4.

Example 5.1 (Li2). In Figure 4 one has M01 2 L1, L01 2M2 and L02 2M1.Then using equation (24) and ðALð02Þ;M1Þ ¼ 0 we can see that

G1 ¼ hðALð02Þ;M2Þi:

Thus rkG1 ¼ 1 and we can de�ne

’01 : ðALð02Þ;M2Þ 7�!�½M2jL0;2;M0;1� ¼ �½1; 0; 1� t; 1�;’12 : ðALð02Þ;M2Þ 7�!�½L2jL0;1;M0;2� ¼ �½1; 0; 1; t�:

Notice that ½L1jL0;2;M0;1�, ½M1jL0;2;M0;2� and ½M2jdL0;1;M0;1� are degenerate.

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO340

From equation (22) we get

�1;1ðU2ðtÞÞ ¼ �ð½1; 0; 1; t� � ½1; 0; 1; 1� t�Þ ¼ �½t� ð1� tÞ�

under the isomorphism A1 ffi F ( by the cross ratio. This agrees with [4, 2.14].When F ¼ C the con�guration in this example corresponds to the classical

dilogarithm function

Li2ðzÞ ¼ðz0

dt

1� t �dt

t

which is an iterated integral from 0 to z in the sense of K.-T. Chen [5]. Whenz ¼ t is real this integral is equal to the 2-dimensional integralð

0<x<y<t

dx

1� xdy

y¼ �

ð0<1�x<y<t

dx

x

dy

y:

As the second example we look at the con�guration corresponding to the doublelogarithms Li1;1. See Figure 5.

Example 5.2 (Li1;1). In Figure 5 one has L01 2M1 and L02 2M2. It is easyto see that

G1 ¼ hðALð01Þ;M2Þ; ðALð02Þ;M1Þi:By equation (22),

�1;1ð½L;M�Þ ¼½L1jL0; L2;M0;M2� � ð�½M2jL0; L1;M0;M1�Þþ ð�½L2jL0; L1;M0;M1�Þ � ½M1jL0; L2;M0;M2�:

To see the relation between this con�guration and the double logarithm wetake F ¼ C, x ¼ a2=a1 and y ¼ 1=a2 where a1 6¼ 0 and a2 6¼ 0; 1. Then thedouble logarithm

Li1;1ðx; yÞ ¼X

0<m<n

xmyn

mn

can be analytically continued to C2 as a meromorphic function by the iteratedintegral in the sense of K.-T. Chen (see [16]). When x and y are real numbers this

Figure 4. Dilogarithmic pair of simplices U2ðtÞ.

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 341

integral becomes the 2-dimensional integralð0<t1<t2<1

dt1t1 � a1

dt2t2 � a2

¼ð0<t1þa1<t2þa2<1

dt1t1

dt2t2

corresponding to Figure 5. Notice that if a2 ¼ 0 then it becomes the classicaldilogarithm treated in Example 5.1.

Let us �rst assume a1 6¼ a2, that is, x 6¼ 1. By de�nition a1 6¼ 0 and a2 6¼ 1 andthus ½L;M� is admissible. So we have

�1;1ð½L;M�Þ ¼½1; 0; a2 � a1;�a1� � ð�½1; 0;�a2; 1� a2�Þþ ð�½1; 0; a1 � a2; 1� a2�Þ � ½1; 0; 1� a1;�a1�

which corresponds to the formal equation (setting Li1ðtÞ ¼ � logð1� tÞ)ða20

dt

t� a1�

ð10

dt

t� a2þ

ð1a1

dt

t� a2�

ð10

dt

t� a1

¼ Li1ðxÞ � Li1ðyÞ þ Li1�1� xy1� x

�� Li1ðxyÞ:

This is consistent with the double logarithm variation of mixed Hodge structuresgiven by the matrix

M1;1ðx; yÞ ¼

1 0 0 0Li1ðyÞ 1 0 0Li1ðxyÞ 0 1 0

Li1;1ðx; yÞ Li1ðxÞ Li1

�1� xy1� x

�1

0BBBB@1CCCCAdiag½1; 25i; 25i; ð25iÞ2�:

This is essentially the same as de�ned in [8, x 2]. Notice that there is an error inthat the term 25i log x in the matrix M1;1ðx; yÞ should be replaced by25i logð1� xÞ. The explicit variations of mixed Hodge structures related tomultiple logarithms Li1;...;1 are given in [15].

It is interesting to see what happens when a1 ¼ a2 ¼ t, that is, x ¼ 1. Thecon�guration is the dual to Example 5.3. Setting

a ¼ rðM1jL1;0;M0;2Þ ¼ rð0;1; 1� t;�tÞ

Figure 5. Double-logarithmic pair of simplices.

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO342

one gets (recalling that y ¼ 1=t), by the computation in Example 5.3,

�1;1ð½L;M�Þ ¼ ��1;1ð½M;L�Þ ¼ �½a� ð�aÞ� ¼ �½logð1� yÞ � ð� logð1� yÞÞ�:On the other hand, when x ¼ 1 we get the one-variable function

Li1;1ð1; yÞ ¼X

0<m<n

yn

mn

which converges when jyj < 1. Then by an easy computation

dLi1;1ð1; yÞ ¼X

0<m<n

yn�1

mdy ¼ � logð1� yÞd logð1� yÞ:

We can compare this with the variation of mixed Hodge structures given by thematrix M1;1ð1; yÞ (see [15, p. 186]):

M1;1ð1; yÞ ¼1 0 0

Li1ðyÞ 1 0Li1;1ð1; yÞ Li1ðyÞ 1

0@ 1Adiag½1; 25i; ð25iÞ2�:

As the last example in this subsection we treat the con�guration denoted byKðaÞ in [4, 2.13]. See Figure 6.

Example 5.3. In Figure 6 one has M01 2 L2, M02 2 L0, and M12 2 L1. It isnot hard to see that

G1 ¼ hðL2;M2Þ þ ðL1;M0Þ þ ðL0;M1Þ � ðP2Þi:

To calculate �1;1ð½L;M�Þ we can cut M into three parts as follows. Let li be theL-vertex facing Li and mi the M-vertex facing Mi. Let M3 ¼ l1;m0, M4 ¼ l2;m2,P ¼M3 \M4, and M5 ¼ P;m1. By additivity we can express M as the sum of theoriented triangles

41 ¼ ðM2;M5;M3Þ; 42 ¼ ðM0;M4;M5Þ; 43 ¼ ðM1;M3;M4Þ:

It is easy to see that �1;1ðL;43Þ ¼ 0 and

�1;1ðL;41Þ ¼ � ½L1jL2;0;M2;5� � ½M5jL2;1;M2;3�;

�1;1ðL;42Þ ¼ � ½L2jL1;0;M0;5� � ½M5jL1;2;M0;4�:

Figure 6. Con�guration KðaÞ: M01 2 L2, M02 2 L0, and M12 2 L1.

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 343

Set A ¼ L1 \M5, B ¼ L2 \M5 and b ¼ ½M5jL1;2;M0;4� ¼ ½A;B;m1; P �. Thenclearly ½M5jL2;1;M2;3� ¼ ½B;A;m1; P � ¼ b�1. By projecting from l1 we have

½L1jL2;0;M2;5� ¼ ½B;m1;P;A� ¼�b1� b ;

and by projecting from l2,

½L2jL1;0;M0;5� ¼ ½A;m1;P;B� ¼1

1� b :

Thus

�1;1ð½L;M�Þ ¼ ��b1� b� b

�1� �

� 1

1� b� b� �

¼ ð�bÞ � b:

To express b intrinsically by quantities only from ðL;M Þ we follow [4, 2.13] to seta ¼ rðL1jL0;2;M1;0Þ. By projective invariance, we can change L to the standardsimplices and get the right-hand con�guration of Figure 6. It is easy to see thatM0 \ L1 has coordinates ½a; 0; 1�.

Let P ¼ ½1; 1; 1�. Then from the parallelogram ðP;m2; a; BÞ we get jl0; Aj ¼ aþ 1.By projecting to the x-axis one has

b ¼ ½A;B;m1; P � ¼ ½aþ 1; 0;1; 1� ¼ �a�1:Therefore

�1;1ð½L;M�Þ ¼ a�1 � ð�a�1Þ ¼ ða� aÞ � ða� ð�1ÞÞ ¼ a� ð�aÞ

which agrees with [4, 2.13].

6. Admissible pairs of tetrahedra in P3F

In this section, we calculate the motivic cohomology of all admissible pairs ofsimplices in A3 from which we can easily de�ne the coproduct

� ¼M3

k¼0�k;3�k : A3 �!

M3

k¼0Ak � A3�k where �k;3�k : A3 �!Ak �A3�k

using equation (22). At the end of this section we will �nd the coproduct map onthe multiple polylogarithmic pairs corresponding to Li1;1;1, Li2;1, Li1;2 and Li3as corollaries.

6.1. De�nition of coproduct on A3

Let ½L;M� 2 A3 be a non-degenerate admissible pair. As for A2, by additivityon both L and M we have the following twelve cases to consider:

ðIÞ ðL;M Þ is a generic pair, ðIIÞ M012 2 L1;

ðIIIÞ M012 2 L12; ðIVÞ M01 � L1 and M012 2 L12;

ðVÞ M012 2 L12 �M1; ðVIÞ L012 2M1;

ðVIIÞ L012 2M12; ðVIIIÞ L01 �M1 and L012 2M12;

ðIXÞ L012 2M12 � L1; ðXÞ M01 \ L01 6¼ ;;ðXIÞ M01 2 L1; ðXIIÞ L01 2M1;

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO344

where the condition(s) in each of the above cases are the only non-genericconditions for the pair in that case.

In view of the duality of L and M provided by Proposition 3.11, we can reduce(VI) to (II), (VII) to (III), (VIII) to (IV), and (IX) to (V). Further, Case (X) canbe reduced to Case (IV): Let E ¼M01 \ L01. By additivity M is a sum (ordiIerence) of M 0 and M 00 where we get M 0 and M 00 by replacing M012 and M123,respectively, with E. Then ðL;M Þ is a sum (or diIerence) of two pairs in Case(IV). Similarly, (XI) can be reduced to Case (V) and (XII) can be reduced to (VI).

We deal with Case (I) to (V) separately. We only need to show that either Case (i)or Case (ii) in x 4.2 appears, so that we may de�ne the maps ’01, ’13, ’02 and ’23 bythe procedure presented in x 4.2. Then the coproduct is de�ned by equation (22).

(I) ðL;M Þ is a generic pair. Then one has

G1ðL;M Þ ¼ hðALð0kÞ;M½ij�Þ : 16 i < j6 3; k ¼ 1; 2; 3i;and

G2ðL;M Þ ¼ hðALð0ijÞ;MkÞ : 16 i < j6 3; k ¼ 1; 2; 3i:

The coproduct is de�ned in De�nition 4.3.(II) M012 2 L1. Then

ðL1;M½ij�Þ =2C02 ; for 06 i; j6 2; ðL1;M½01�Þ þ ðL1;M½02�Þ 2 C 0

2 ;

ðL1;M½10�Þ þ ðL1;M½12�Þ 2 C02 ; ðL1;M½20�Þ þ ðL1;M½21�Þ 2 C 0

2 :

One has

M012 2 L1 ¼) ðALð01Þ;M½ij�Þ =2 ker d02 ; for 06 i; j6 2: ð25Þ

Hence rkG1 ¼ 8 and

G1 ¼ hðALð0kÞ;M½i3�Þ; i ¼ 1; 2; k ¼ 1; 2; 3; ðALð0kÞ;M½12�Þ; k ¼ 2; 3i:

Therefore one can de�ne

’01 : ðALð0kÞ;M½ij�Þ 7�! ð�1ÞcðijÞ�1½MijjL0;k;M0;cðijÞ�;’13 : ðALð0kÞ;M½ij�Þ 7�! ð�1Þk�1½LkjL0;cðkÞ;M0;i;j�:

Here for any index subset S of f1; 2; 3g in increasing order, cðSÞ is thecomplementary subset in increasing order.

By the general theory ðALð01Þ;M½12�Þ is another possible element in G1. Butin our case L1 \M01 ¼ L1 \M02 ¼ L1 \M12 implies that ½L1jL0;2;3;M0;1;2� isdegenerate. Therefore there are only eight terms in �1;2ð½L;M�Þ.

Since the condition clearly has no eIect on G2ðL;M Þ by de�nition, we see thatrkG2 ¼ 9 and �2;1ð½L;M�Þ is de�ned exactly the same as in the generic case.

(III) M012 2 L12. Then M012 2 L1 and M012 2 L2, so one can use Case (I) todeduce that

G1ðL;M Þ ¼ hðL0;M½i3�Þ � ðLk;M½i3�Þ; i ¼ 1; 2; k ¼ 1; 2; 3;

ðL0;M½12�Þ � ðL3;M½12�Þi:

Thus rkG1 ¼ 7, so there are exactly seven terms in �1;2ð½L;M�Þ. Note that½L1jL0;2;3;M0;1;2� and ½L2jL0;1;3;M0;1;2� are degenerate.

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 345

To �nd the basis of G2ðL;M Þ we notice that, for i ¼ 1; 2,

ðL12; Li;MkÞ =2C 04 ; for k ¼ 0; 1; 2; but

X2k¼0ðL12; Li;MkÞ ¼ 0:

We see that

M012 2 L12 ¼) ðALð012Þ;MkÞ =2 ker d04 for k ¼ 0; 1; 2: ð26Þ

Therefore

G2ðL;M Þ ¼ hðALð0i3Þ;MkÞ; i ¼ 1; 2; k ¼ 1; 2; 3; ðALð012Þ;M3Þi:Therefore rkG2 ¼ 7. Thus there are exactly seven terms in �2;1ð½L;M�Þ also. Notethat for k ¼ 1; 2, both ½L12jL0;3;M0;k� are degenerate.

(IV) M01 � L1 and M012 2 L12. Then

M01 � L1 ¼) ðL1;M½i2�Þ =2C 02 ; i ¼ 0; 1;

P1i¼0ðL1;M½i2�Þ 2 C 0

2 ;

ðL1;M½i3�Þ =2C 02 ; i ¼ 0; 1;

P1i¼0ðL1;M½03�Þ 2 C0

2 ;

(ð27Þ

and from Case (III) one can deduce that

G1ðL;M Þ ¼ hðALð0kÞ;M½ij�Þ; 16 i; j; k6 3;

ði; j; kÞ 6¼ ð1; 2; 1Þ; ð1; 2; 2Þ; ð1; 3; 1Þi:

Hence rkG1 ¼ 6 and there are exactly six terms in �1;2ð½L;M�Þ. Note that the followingare all degenerate: ½L1jL0;2;3;M0;1;2�, ½L1jL0;2;3;M0;1;3� and ½L2jL0;2;3;M0;1;2�.

To �nd the basis of G2ðL;M Þ one can use

M01 � L1 ¼) ðALð01iÞ;M1Þ =2 ker d04 for i ¼ 2; 3;

ðALð012Þ;M1Þ þ ðALð013Þ;M1Þ 2 ker d04 ;

(ð28Þ

and equation (26) to get

G2ðL;M Þ ¼ hðALð012Þ;M3Þ; ðALð013Þ;MkÞ; k ¼ 2; 3;

ðALð023Þ;MkÞ; k ¼ 1; 2; 3i:

Therefore rkG2 ¼ 6. Thus there are exactly six terms in �2;1ð½L;M�Þ also. Notethat for k ¼ 2; 3, ½L1kjL0;cð1;kÞ;M0;1� are degenerate, and so is ½L12jL0;3;M0;2�.

(V) M012 2 L12 �M1. Then one can use Case (III) to get

G1ðL;M Þ ¼ hðALð03Þ;M½12�Þ; ðALð1kÞ;M½13�Þ; k ¼ 0; 3;

ðALð0kÞ;M½23�Þ; k ¼ 1; 2; 3i:

Thus rkG1 ¼ 6, so there are exactly six terms in �1;2ð½L;M�Þ. Clearly there is noway one can use the formula in the generic case. We have to choose L1 as thespecial plane for M13 which belongs to Case (ii) in x 4.2. Hence we get

’01 : ðALð10Þ;M½13�Þ 7�!�½M13jL1;0;M0;2�;ðALð13Þ;M½13�Þ 7�!�½M13jL1;3;M0;2�;

’13 : ðALð10Þ;M½13�Þ 7�!�½L0jL1;2;3;M0;1;3�;ðALð13Þ;M½13�Þ 7�!�½L3jL1;0;2;M0;1;3�:

The missing terms ½Lkj . . . ;M0;1;2� for k ¼ 1; 2 and ½M13jL1;2;M0;2� are degenerate.

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO346

We now have

L12 �M1 ¼) ðALð12iÞ;M1Þ ¼ 0 for i ¼ 2; 3: ð29Þ

From equation (26) we know that ðALð012Þ;M2Þ =2 ker d04 . Thus

G2ðL;M Þ ¼ hðALð103Þ;M1Þ; ðALð0i3Þ;M2Þ; i ¼ 1; 2;

ðALð0ijÞ;M3Þ; 16 i; j6 3i:

Therefore rkG2 ¼ 6 also. Hence there are exactly six terms in �2;1ð½L;M�Þ also.We de�ne:

’02 : ðALð103Þ;M1Þ 7�! ½M1jL1;0;3;M0;2;3�;ðALð0i3Þ;M2Þ 7�! �½M2jL0;i;3;M0;1;3�; i ¼ 1; 2;

ðALð0ijÞ;M3Þ 7�! ½M3jL0;i;j;M0;1;2�;’23 : ðALð103Þ;M1Þ 7�! ½L03jL1;2;M0;1�;

ðALð0i3Þ;M2Þ 7�! ð�1Þi½Li;3jL0;cði3Þ;M0;2�; i ¼ 1; 2;

ðALð0ijÞ;M3Þ 7�! ð�1ÞcðijÞ�1½LijjL0;cðijÞ;M0;3�:

Note that for M1 we choose L1 as the special L-face and for M2 and M3 wechoose L0 as usual. Hence we are in Case (ii) of x 4.2. After this choice of specialL-faces one sees that ½M1jL1;0;2;M0;2;3�, ½M1jL1;2;3;M0;2;3� and ½L12jL0;3;M0;2�are degenerate.

6.2. Examples

As applications of our general theory presented in the previous subsection let us�nd the images of the coproduct map on the multiple polylogarithmic pairs ofsimplices of weight 3.

Example 6.1 (Li1;1;1). The triple logarithm is de�ned as

Li1;1;1ðx; y; zÞ ¼X

0<l<m<n

xlymzn

lmnfor jxj; jyj; jzj < 1:

It can be extended to a meromorphic function on C3 by iterated integrals (cf. [16])by setting ðx; y; zÞ ¼ ða2=a1; a3=a2; 1=a3Þ 6¼ ð0; 0; 0Þ:

Li1;1;1ðx; y; zÞ ¼ �ð10

dt

t� a1� dt

t� a2� dt

t� a3:

When x, y and z are real this becomes a 3-dimensional integral

�ð0<t1<t2<t3<1

dt1t1 � a1

dt2t2 � a2

dt3t3 � a3

: ð30Þ

Here we have assumed that ai 6¼ 0. By substitution ti ! ti þ ai and takingL1 ¼ ft2 ¼ 0g, L2 ¼ ft1 ¼ 0g, L3 ¼ ft3 ¼ 0g and L0 ¼ ft0 ¼ 0g as the in�nite planewe see that the above iterated integral corresponds to an admissible pair of tetrahedraðL;MðaÞÞ where a ¼ ½1; a1; a2; a3� in P3

C (see Figure 7). In aQne coordinates the regionbounded by the tetrahedronMðaÞ is a translation of the region 0 < t1 < t2 < t3 < 1 bytranslating ð0; 0; 0Þ to the coordinate ð�a1;�a2;�a3Þ.

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 347

For arbitrary �eldF we canputL as the standard simplexwithL1 andL2 exchanged(because of the negative sign in equation (30)) and de�neM ¼ ðM0; . . . ;M3Þ in P3

F asfollows: writemi as the vertex facingMi and set

m0 ¼ ½1; a1; a2; a3 þ 1�; m1 ¼ ½1; a1 þ 1; a2 þ 1; a3 þ 1�;m2 ¼ ½1; a1; a2 þ 1; a3 þ 1�; m3 ¼ ½1; a1; a2; a3�:

We see that the following conditions are satis�ed:

L02 �M1; L03 �M3; L013 2M03; L012 2M12; L023 2M13:

Because ðALð01Þ;M½30�Þ ¼ 0 (note that L0 \M30 ¼ L1 \M30) and

ðLi;M½32�Þ þ ðLi;M½31�Þ þ ðLi;M½30�Þ ¼ d�12 ððLi;M3ÞÞ for i ¼ 0; 1;

we see that

G1ðL;MðaÞÞ ¼ ha ¼ ðALð02Þ;M½30�Þ; b ¼ ðALð03Þ;M½21�Þ;c ¼ ðALð01Þ;M½31�Þ ¼ �ðALð01Þ;M½32�Þ mod imðd�12 Þi:

Hence rkG1ðL;MðaÞÞ ¼ 3. Now we can de�ne

’01 : a 7�! ½M30jL0;2;M2;1� ¼ ½1; 0; 1� a1;�a1�;b 7�! ½M21jL0;3;M0;3� ¼ �½1; 0;�a3; 1� a3�;c 7�! ½M31jL0;1;M0;2� ¼ ½1; 0; 1� a2;�a2�;

’13 : a 7�! �½L2jL0;1;3;M2;3;0�;b 7�! ½L3jL0;1;2;M0;2;1�;c 7�! �

�½L1 \M3jL0;2;M2;1� � ½L1 \M1jL0;3;M3;0�Þ¼ ½1; 0; a2 � a1;�a1� � ½1; 0; 1� a3; a2 � a3�:

We get ’13ðcÞ by cutting MðaÞ into two parts along M02. Note that ½M12jL0;1;M0;3�

Figure 7. Triple logarithmic pairs of simplices.

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO348

and ½Mi3jL0;3; . . .� for i ¼ 1; 2 are degenerate. For L2 we choose M2 as the specialM-face since both ½M13jL0;2;M2;0� and ½M01jL0;2;M2;3� are degenerate.

By using equations (29) and

L012 2M1 ¼) ðALð012Þ;M1Þ ¼ 0 ð31Þ

it is not too hard to �nd

G2ðL;MðaÞÞ ¼ hðALð023Þ;M2Þ ¼ �ðALð023Þ;M0Þ;ðALð013Þ;M2Þ ¼ �ðALð013Þ;M1Þ;ðALð012Þ;M3Þ ¼ �ðALð012Þ;M0Þi:

So rkG2ðL;MðaÞÞ ¼ 3 and we can de�ne

’02 : ðALð023Þ;M2Þ 7�!�½M2jL0;2;3;M0;1;3�;ðALð013Þ;M2Þ 7�! ½M2jL0;1;3;M1;0;3�;ðALð012Þ;M3Þ 7�! ½M3jL0;1;2;M0;1;2�;

’23 : ðALð023Þ;M2Þ 7�! ½L23jL0;1;M0;2�;ðALð013Þ;M2Þ 7�!�½L13jL0;2;M1;2�;ðALð012Þ;M3Þ 7�! ½L12jL0;3;M0;3�:

Note that we use M1 as the special M-face when we de�ne ’23ðALð013Þ;MkÞbecause when k ¼ 0; 3 they are degenerate due to the inclusion L013 2M03. Similarlyone can show that ’02 sends all other possible candidates of the basis of G2 to zero:½MkjL0;1;2; :::� for k ¼ 1; 2 and ½MkjL0;2;3; . . . � for k ¼ 1; 3 are degenerate.

The coproducts produced by the above maps ’i;j using formula (22) agree withthe variations of the mixed Hodge structures related to Li1;1;1 obtained by using adiIerent technique (see [15]).

Example 6.2 (Li1;2). If we let a3 ¼ 0 in the preceding example we getLi1;2. There is only one more condition M012 2 L3. This does not aIect G2 sorkG2 ¼ 3. But

M012 2 L3 ¼) ðALð03Þ;M½12�Þ =2C 02 ;

so that

G1ðL;M Þ ¼ ha ¼ ðALð02Þ;M½30�Þ; c ¼ ðALð01Þ;M½31�Þi:Example 6.3 (Li2;1). If we let a2 ¼ 0 in Example 6.1 we will get Li2;1. The

one more restriction is M12 � L1 which does not aIect G1 as given in Example6.2. But from equation (28), ðALð013Þ;M1Þ =2 ker d0

4 , one has

G2ðL;MðaÞÞ ¼ hðALð023Þ;M2Þ ¼ �ðALð023Þ;M0Þ;ðALð012Þ;M3Þ ¼ �ðALð012Þ;M0Þi:

So rkG1 ¼ rkG2 ¼ 2.

Example 6.4 (Li3). If we let a2 ¼ a3 ¼ 0 in Example 6.1 we get Li3, theclassical trilogarithm. The additional restrictions are M12 � L1, M012 2 L13 andL13 �M0 (see Figure 8).

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 349

We see that b of Example 6.1 is not in ker d02 since, by equation (25),

M012 2 L3 ¼) ðL3;M½ij�Þ =2C02 for 06 i; j6 2:

Also c of Example 6.1 is not in ker d02 because, by equation (27),

M12 � L1 ¼) ðL1;M½i3�Þ =2C 02 for i ¼ 1; 2:

Hence one has

G1ðL;MðaÞÞ ¼ ha ¼ ðALð02Þ;M½30�Þi:

Now from equation (29), L13 �M0 implies that ðALð132Þ;M0Þ ¼ 0, and from(31), L13 �M0 implies that

ðALð012Þ;M0Þ ¼ ðALð032Þ;M0Þ ¼ �ðALð023Þ;M0Þ:

Therefore G2ðL;MðaÞÞ is generated by

hðALð012Þ;M3Þ ¼ �ðALð012Þ;M0Þ ¼ ðALð023Þ;M0Þ ¼ �ðALð023Þ;M2Þi:

The upshot is that rkG1 ¼ rkG2 ¼ 1 and

�2;1ð½L;M�Þ ¼ ’13ðaÞ � ’01ðaÞ¼ ð�½L2jL0;1;3;M2;3;0�Þ � ½M30jL0;2;M2;1�¼ 1

2�ð½1; 0; 1; a1� � ½1; 0; 1; a1�Þ � ½1; 0; 1� a1;�a1�;�1;2ð½L;M�Þ ¼ ’23ððALð012Þ;M3ÞÞ � ’02ððALð012Þ;M3ÞÞ

¼ ½1; 0; 1; a1� � ½M3jL0;1;2;M0;1;2�:

Here we have used bary-center subdivision of M to �nd

’23ððALð012Þ;M3ÞÞ ¼ ½1; 0; 1; a1�:

Let U3ðxÞ ¼ ðL;M Þ be as given in Figure 8. It represents the element in A3ðF Þ whichcorresponds to Li3ðxÞ when F ¼ C. Then we see that ½M3jL0;1;2;M0;1;2� ¼ �U2ðxÞ

Figure 8. Trilogarithmic pairs of simplices.

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO350

which is the dilogarithmic pair (see Figure 4) and

�2;1ðU3ðxÞÞ ¼ � ½12 x2 � ð1� xÞ�;

�1;2ðU3ðxÞÞ ¼ � ½x� U2ðxÞ�;

under the identi�cation A1 ffi F ( by the cross ratio.The coproducts are consistent with the well-known variation matrix of the

mixed Hodge structures related to the trilogarithm (see [11])

M3ðxÞ ¼

1 0 0 0� logð1� xÞ 1 0 0Li2ðxÞ log x 1 0Li3ðxÞ 1

2ðlog xÞ2 log x 1

0BB@1CCAdiag½1; 25i; ð25iÞ2; ð25iÞ3�:

7. Admissible pairs of simplices in P4F

Using a procedure similar to the one adopted in the previous section one can de�nethe coproduct on A4 without too much diQculty. Nevertheless, to make theexposition shorter we content ourselves with only two examples in this case. To de�nethe coproduct on An (for n> 5) in general is conceivably much more diQcult.

EXAMPLE 7.1 (Part of the tetralogarithm Li4). When x is real, thetetralogarithm Li4ðxÞ can be de�ned by the 4-dimensional integral

Li4ðxÞ ¼ �ð0<1�t1<t2<t3<t4<x

dt1t1^ dt2t2^ dt3t3^ dt4t4:

Take L to be the standard simplex under the odd permutation:

L0 �! L3 �! L1 �! L4 �! L0:

Then the region of the above integration is de�ned by the pair of simplicesU4ðxÞ ¼ ðL;M Þ where M is given by:

M0 : ft4 ¼ t3g; M1 : ft3 � t4 ¼ t2g; M2 : ft2 ¼ t1g;M3 : ft1 ¼ t0g; M4 : ft0 ¼ xt3g:

Recall that for any ðL;M Þ 2 A4 we may apply additivity on both M and Lusing a method similar to bary-center subdivision such that ðL;M Þ is decomposedinto reduced pairs. The inclusion conditions of the subdivisions of U4ðxÞ aremostly similar to the ones we encountered in A3, except the following:

M0123 2 L012 � L01 �M1:

In what follows we show how to de�ne the coproduct on this piece of subdivisionof ðL;M Þ.

One has

M0123 2 L012 ¼) ðL0;M½123�Þ ¼ ðL1;M½123�Þ ¼ ðL2;M½123�Þand

L01 �M1 ¼) ðALð01Þ;M½1kl�Þ ¼ 0; for 26 k < l6 4:

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 351

Hence

G1 ¼ hðALð01Þ;M½234�Þ; ðALð02Þ;M½jkl�Þ; ðj; k; lÞ 6¼ ð1; 2; 3Þ;ðALð0iÞ;M½jkl�Þ; i ¼ 3; 4; 16 j < k < l6 4

:

Thus rkG1 ¼ 12. Note that ½L2j . . . ;M0;1;2;3� and ½M1kljL0;1; . . .� (for 26 k < l6 4)are degenerate.

We can see that

M0123 2 L012 ¼) ðALð012Þ;M½kl�Þ ¼ 0; for all 06 k; l6 3; ð32Þand

L01 �M1 ¼) ðALð01iÞ;M½1k�Þ ¼ 0; for 26 j; k6 4:

These yield

G2 ¼ hðALð012Þ;M½k4�Þ; k ¼ 2; 3;

ðALð01iÞ;M½kl�Þ; i ¼ 3; 4; 26 k < l6 4;

ðALð0ijÞ;M½kl�Þ; 26 i < j6 4; 16 k < l6 4i:

Thus rkG2 ¼ 26. Note that ½M23jL0;1;2; . . .� is degenerate and for any 26 i; k6 4,½M1kjL0;1;i; . . .� are degenerate too.

To treat G3ðL;M Þ one has

M0123 2 L012 ¼) ðALð012kÞ;MlÞ =2 ker d06 ; for all 06 l6 3:

It is clear that

L01 �M1 ¼) ðALð01jkÞ;M1Þ ¼ 0:

Thus

G3 ¼ hðALð0234Þ;M1Þ; ðALð0ijkÞ;M4Þ; 16 i < j < k6 4;

ðALð3ij4Þ;MlÞ; 06 i < j6 2; l ¼ 2; 3i:

Thus rkG3 ¼ 11. Note that ½L012j . . . ;M0;l� (for l ¼ 2; 3) and ½M1jL01jk; . . .� aredegenerate. Here for �xed M2 and M3 we used L3 as the special L-face.

Example 7.2. Suppose ðL;M Þ is an admissible pair of simplices in P4F with

only the following non-generic conditions:

M0123 2 L012 �M01 � L1: ð33Þ

Together with its dual condition (L and M exchanged), such a con�guration isessentially the only obstacle to expressing Aomoto tetralogarithms as linearcombinations of the classical tetralogarithms together with products of poly-logarithms of lower weights, because no such condition as (33) appears in thecon�gurations corresponding to polylogarithms or their products. We sketch theproof here. We know that A2 is generated by dilogarithmic pairs U2ðxÞ andsquares [2] and in [14, x 4.7] we proved that A3 is generated by trilogarithmic pairsU3ðyÞ and prisms. So we only need to look into the following �ve cases:

(i) U4ðxÞ (Example 7.1),(ii) �ðx� U3ðyÞÞ,(iii) �ðU2ðxÞ � U2ðyÞÞ,(iv) �ðx� y� U2ðzÞÞ and(v) �ðx� y� z� wÞ

PLMS 1470---10/2/2004---SRUMBAL---89152

JIANQIANG ZHAO352

for x, y , z and w in F ( . Then we can easily check that the condition (33) cannotappear in any of the above �ve cases.

Now we determine the group G1ðL;M Þ. From

M01 � L1 ¼) ðL1;M½0kl�Þ =2C 02 ; ðL1;M½1kl�Þ =2 ker d0

2

(but ðL1;M½0kl�Þ þ ðL1;M½1kl�Þ 2 C 02 Þ and

M0123 2 L012 ¼) ðLi;M½123�Þ =2 ker d02 ; for i ¼ 0; 1; 2;

one has

G1 ¼ hðALð34Þ;M½123�Þ; ðALð0iÞ;M½234�Þ; 16 i6 4;

ðALð0iÞ;M½1k4�Þ; i ¼ 2; 3; 4; k ¼ 2; 3i:

Thus rkG1 ¼ 11. Note that ½L1j . . . ;M0;1;k;4� ðfor k ¼ 2; 3Þ, and ½Lij . . . ;M0;1;2;3� (fori ¼ 0; 1; 2) are degenerate. Here for �xed M1;2;3 we used L3 as the special L-face.

To determine G2ðL;M Þ and G3ðL;M Þ one can use relation (32) and

M01 � L1 ¼) ðALð1ijÞ;M½0k�Þ =2 ker d04 ; ðALð1ijÞ;M½1k�Þ =2 d0

4 ;

but

ðALð1ijÞ;M½0k�Þ þ ðALð1ijÞ;M½1k�Þ 2 ker d04 :

Then

G2 ¼ hðALð012Þ;M½k4�Þ; k ¼ 2; 3;

ðALð01iÞ;M½kl�Þ; i ¼ 3; 4; 26 k < l6 4;

ðALð0ijÞ;M½kl�Þ; 26 i < j6 4; 16 k < l6 4i:

Thus rkG2 ¼ 26. Note that ½M23jL0;1;2; . . .� is degenerate and for any 26 i; k6 4,½L1ij . . . ;M0;1;k� are degenerate too.

We now turn to G3ðL;M Þ. Because

M0123 2 L012 ¼) ðALð012iÞ;MlÞ =2 ker d06 ; for 06 l6 3;

and

M01 � L1 ¼) ðALð1ijkÞ;MlÞ =2 ker d06 ; for l ¼ 0; 1;

one has

G3 ¼ hðALð0234Þ;M1Þ; ðALð0ijkÞ;M4Þ; 16 i < j < k6 4;

ðALð3ij4Þ;MlÞ; 06 i < j6 2; l ¼ 2; 3i:

Thus rkG3 ¼ 11. Note that ½L1jkj . . . ;M0;1� (for 26 j < k6 4) and ½L012j . . . ;M0;l�ðfor l ¼ 2; 3Þ are degenerate. Here for �xed M2 and M3 we use L3 as the specialL-face. So we are in Case (ii) of x 4.2.

References

1. K. AOMOTO, ‘Addition theorem of Abel type for hyperlogarithms’, Nagoya Math. J. 88(1982) 55--71.

2. A. A. BEILINSON, A. B. GONCHAROV, V. V. SCHECHTMAN and A. N. VARCHENKO, ‘Aomotodilogarithms, mixed Hodge structures and motivic cohomology of pairs of triangles inthe plane’, Grothendieck Festschrift II (ed. P. Cartier), Progress in Mathematics 87(Birkh€aauser, Boston, 1991) 78--131.

3. A. A. BEILINSON, R. MACPHERSON and V. V. SCHECHTMAN, ‘Notes on motivic cohomology’,Duke Math. J. 54 (1987) 679--710.

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES 353

4. A. A. BEILINSON, A. N. VARCHENKO, A. B. GONCHAROV and V. V. SCHECHTMAN,‘Projective geometry and algebraic K-theory’, Leningrad Math. J. 2 (1991) 523--576.

5. K.-T. CHEN, ‘Iterated path integrals’, Bull. Amer. Math. Soc. 83 (1977) 831--879.6. P. DELIGNE, ‘ThTeeorie de Hodge, II’, Inst Hautes TEEtudes Sci. Publ. Math. 40 (1972) 5--57.7. P. DELIGNE, ‘ThTeeorie de Hodge, III’, Inst Hautes TEEtudes Sci. Publ. Math. 44 (1975) 5--77.8. A. B. GONCHAROV, ‘The double logarithm and Manin’s complex for modular curves’, Math.

Res. Lett. 4 (1997) 617--636.9. A. B. GONCHAROV, ‘Mixed elliptic motives’, Galois representations in arithmetic algebraic

geometry (ed. A. J. Scholl and R. L. Taylor), London Mathematical Society Lecture NoteSeries 254 (Cambridge University Press, 1998) 147--221.

10. A. B. GONCHAROV, ‘Geometry of trilogarithm and the motivic Lie algebra of a �eld’,Regulators in analysis, geometry and number theory (ed. A. Reznikov andN. Schappacher), Progress in Mathematics 171 (Birkh€aauser, Boston, 2000) 127--165.

11. R. HAIN, ‘Classical polylogarithms’, Motives (ed. U. Jannsen, S. Kleiman and J.-P. Serre),Proceedings of Symposia in Pure Mathematics 55, part 2 (American MathematicalSociety, Providence, RI, 1994) 3--42.

12. R. HARTSHORNE, Algebraic geometry, Graduate Texts in Mathematics 52 (Springer, Berlin,1977).

13. J. ZHAO, ‘Remarks on a Hopf algebra for de�ning motivic cohomology’, Duke Math. J. 103(2000) 445--458.

14. J. ZHAO, ‘Motivic complexes of weight three and pairs of simplices in projective 3-space’,Adv. in Math. 161 (2001) 141--208.

15. J. ZHAO, ‘Multiple polylogarithms: analytic continuation, monodromy, and variations ofmixed Hodge structures’, Contemporary trends in algebraic geometry and algebraictopology (ed. S. S. Chern, L. Fu and R. Hain), Nankai Tracts in Mathematics 5 (WorldScienti�c, River Edge, NJ, 2002) 167--193.

16. J. ZHAO, ‘Analytic continuation of multiple polylogarithms’, Preprint, Eckerd College, 2003.

Jianqiang ZhaoDepartment of MathematicsEckerd CollegeSt PetersburgFL 33711USA

zhaoj@eckerd.edu

PLMS 1470---10/2/2004---SRUMBAL---89152

MOTIVIC COHOMOLOGY OF PAIRS OF SIMPLICES354