On multilinearity and skew-symmetry of certain symbols in motivic cohomology of fields

ON MUL TILINEARITY AND SKEW-SYMMETR Y OF CER T AIN SYMBOLS IN MOTIVIC COHOMOLOGY OF FIELDS SUNG MYUNG Abstract. The purp ose of the present article is to sho w the multilin earit y fo r symbols in Goo dwilli e-Lic hten baum complex i n tw o cases. The first case shown is where the degree is equal to the weigh t. In this case, the motivi c cohomology groups of a field are isomorphi c to the Mil nor’s K -groups as shown by N esterenk o-Suslin, T otaro and Suslin- V o ev odsky for v arious motivic complexes, b ut w e give an expli cit isomo rphism for Goo dwilli e-Lic hten baum complex in a f orm which visi bly carries multili n earit y of Milnor’s sym b ols to our m ultili nearit y of motivic symbols. Next, we establish m ultili nearit y and sk ew-symmetry for irredu cible Go odwil l ie-Lich ten baum symbols in H l − 1 M ` Sp ec k, Z ( l ) ´ . These prop erties hav e been exp ected to hol d from th e author’s construction of a bilin ear form of dilogarithm in case k i s a subfield of C and l = 2. The multili nearit y of sym b ols ma y be view ed as a generalization of the w ell-known formula det( AB ) = d et( A ) det( B ) for commuting matrices. 1. Introducti on When R is a comm utat ive ri ng, the gr oup K 1 ( R ) is an ab elian group generated by inv ertible matrices with entries in R . In particular, when R i s a field, it is well-kno wn that the determinan t map det : K 1 ( R ) → R × is an isomorphism. An important consequence of th is fact is that ( AB ) = ( A ) + ( B ), i.e., the pro duct AB of tw o inv ertible matrices A and B r epresen ts the element obtained b y adding tw o elemen ts in K 1 ( R ), whic h are represen ted by the matrices A and B , resp ectively , since det „ A 0 0 B « = det A det B . In the presen t article, we ende a v or to generalize this prop ert y to the case of commut ing matri ces in terms of m otivic cohomology . The motivic c hain complex prop osed by Go odwi llie and Lich ten b aum as follo ws wil l be p erfectly suitable for our purp ose. In [4], a c hain complex for motivic cohomology of a regular lo cal ring R , by Goo dwillie and Lich t en baum, is define d to be the c hain complex asso ciated to the simplicial abelian group d 7→ K 0 ( R ∆ d , G ∧ t m ), together with a shift of degree by − t . Here, K 0 ( R ∆ d , G ∧ t m ) is the Grothendiec k group of the exact cate gory of pro jectiv e R -mo dules with t comm uting automorphisms factored b y the subgroup generated b y classes of the ob jects one of whose t automorphisms is the iden tit y map. The motivic cohomology of a r egular sch eme X i s give n b y hypercohomology of the sheafific ation of the complex ab o v e. W alker sho w ed, in Theorem 6.5 of [15], that it agrees with motivic cohomology give n by V o ev odsky and th us v arious other definitions of motivic cohomology for smo oth sc hemes o v er an algebraically closed field. In [4], Grayson sho we d that a related ch ain complex Ω − t | d 7→ K ⊕ 0 ( R ∆ d , G ∧ t m ) | , whic h uses direct-sum Grothendiec k gr oups instead, arises as the consecutiv e quotients in K - theory space K ( R ) when R is a regular no etherian r ing and so gives rise to a sp ectral sequenc e conv erging to K -theory . Susli n, in [ 11], show ed that Gr ayson’s motivic cohomology complex is equiv a len t to the other definitions of motivic complex and conseq uen tly settled the problem of a motivic sp ectral sequence. See also [5] for an ov e rview. The main results of this article are multilinearit y and skew-symmetry prop erties for the symbols of Go odwil lie and Lich ten bau m in motivic cohomology . First, we establish the m for H n M ` Spec k , Z ( n ) ´ of a field k in Corollary 2.4 . W e also gi ve a direct pr o of of Nesterenk o-Suslin’s theorem ([9]) that the motivic cohomology of a field k , when the degree is equal to the weigh t, is equal to the M ilnor’s K - group K M n ( k ) for this version of m otivic complex in Theorem 2.11. Even though Nesterenk o-Suslin’s theorem hav e already appeared in several articles i ncluding [9], [14] and [13], we b elieve that the theorem is a cen tral one in the related sub jects and it is worth wh ile to Key wor ds and phr ase s. motivi c cohomology , Milnor’s K-theory . 1 2 S. Myung ha v e another pro of of it. M oreo v er, multilinearity and skew-symmetry prop erties for the symbols of Go odwil l ie and Lic h ten baum motivic cohomology H n M ` Spec k , Z ( n ) ´ and the similar properties for the symbols in Milnor’ s K - groups are visibly compatible through our isomorphism. Secondly , we establish m ultilinearity and sk ew-symmetry of the irreducible sym bols f or H l − 1 M ` Spec k , Z ( l ) ´ in Theorem 3.3 and Proposition 3.7. These results are particularly interesting b ecause these are the prop erties whi c h hav e been expected through the construction of the author’s regulator map in [8] in case k is a subfield of the field C of complex n umbers and l = 2. These properties ma y prov ide the Go odwilli e-Lic h ten baum complex wi th a p oten tial to b e one of the b etter descriptions of m otivic cohomology of fields. 2. Mul tili nearity for Goodwil lie-Lichtenbaum motivic complex and Milnor ’s K -groups F or a ri ng R , l et P ( R, G l m ) b e the exact category each of whose ob jects ( P , θ 1 , . . . , θ l ) consists of a finitely generate d pro jective R -mo dule P and commu ting automorphisms θ 1 , . . . , θ l of P . A morphism fr om ( P , θ 1 , . . . , θ l ) to ( P ′ , θ ′ 1 , . . . , θ ′ l ) i n this category is a homomorphism f : P → P ′ of R -mo dules suc h that f θ i = θ ′ i f for eac h i . Let K 0 ( R, G l m ) b e the Grothendiec k group of this category and let K 0 ( R, G ∧ l m ) b e the quot ien t of K 0 ( R, G l m ) by the subgroup generat ed b y those ob jects ( P , θ 1 , . . . , θ l ) where θ i = 1 for some i . F or each d ≥ 0, let R ∆ d be the R -algebra R ∆ d = R [ t 0 , . . . , t d ] / ( t 0 + · · · + t d − 1) . It is isomorphic to a polynomial ring with d i ndete rminates o v e r R . W e den ote by Ord the category of finite nonempt y ordered sets and b y [ d ] where d is a nonnegativ e integer the ob ject { 0 < 1 < · · · < d } . Given a map ϕ : [ d ] → [ e ] in Ord , the map ϕ ∗ : R ∆ e → R ∆ d is defined by ϕ ∗ ( t j ) = P ϕ ( i )= j t i . The map ϕ ∗ give s us a simplicial ring R ∆ • . By applying the functor K 0 ( − , G ∧ l m ), we get the simplicial abelian gr oup [ d ] 7→ K 0 ( R ∆ d , G ∧ l m ) . The asso ciated (normalized) c hain complex, shif ted cohomologically by − l , is called the motivic complex of Go odwil lie and Lich ten baum of w eig ht l . F or each ( P , θ 1 , . . . , θ l ) in K 0 ( R, G ∧ l m ), there exists a pro jective m odule Q such that P ⊕ Q is free ov er R . Then ( P ⊕ Q, θ 1 ⊕ 1 Q , . . . , θ l ⊕ 1 Q ) represents the same elemen t of K 0 ( R, G ∧ l m ) as ( P , θ 1 , . . . , θ l ). Th us K 0 ( R ∆ d , G ∧ l m ) can b e explicitly presen ted with generators and relations inv olving l -tuples of commuting matrices in GL n ( R ∆ d ) , n ≥ 0. F or a regular lo cal ring R , the motivic cohomology H q M ` Spec R, Z ( l ) ´ will b e the ( l − q )-th homology group of the Goo dwillie-Lich tenbau m complex of weigh t l . In particular, when k is any field, H q M ` Spec k , Z ( l ) ´ = π l − q | d 7→ K 0 ( k ∆ d , G ∧ l m ) | . K 0 ( k ∆ d , G ∧ l m ) ( l ≥ 1) may be considered as the abeli an group generated by l -tuples of the form ( θ 1 ( t 1 , . . . , t d ) , . . . , θ l ( t 1 , . . . , t d )) and certain explicit rel ations, where θ 1 ( t 1 , . . . , t d ) , . . . , θ l ( t 1 , . . . , t d ) are commuting matrices in GL n ( k [ t 1 , . . . , t d ]) for v arious n ≥ 1. When d = 1, w e set t = t 1 and the b oundary map ∂ on the motivic complex sends ( θ 1 ( t ) , . . . , θ l ( t )) in K 0 ( k ∆ 1 , G ∧ l m ) to ( θ 1 (1) , . . . , θ l (1)) − ( θ 1 (0) , . . . , θ l (0)) i n K 0 ( k ∆ 0 , G ∧ l m ). W e wi ll denote by the same notation ( θ 1 , . . . , θ l ) the elemen t i n K 0 ( k ∆ 0 , G ∧ l m ) /∂ K 0 ( k ∆ 1 , G ∧ l m ) = H l M ` Spec k , Z ( l ) ´ represen ted by ( θ 1 , . . . , θ l ), b y abuse of notat i on, whenev er θ 1 , . . . , θ l are commuting matrices in GL n ( k ). Lemma 2.1. L et a 1 , a 2 , . . . , a n and b 1 , b 2 , . . . , b n b e elements in ¯ k (an algebr aic closur e of k ) not e qual to either 0 or 1. Supp ose also that a 1 a 2 · · · a n = b 1 b 2 · · · b n and (1 − a 1 )(1 − a 2 ) · · · (1 − a n ) = (1 − b 1 )(1 − b 2 ) · · · (1 − b n ) . If al l the elementary sy mmet ric functions evaluate d at a 1 , a 2 , . . . , a n and b 1 , b 2 , . . . , b n ar e in k , then ther e is a matrix θ ( t ) in GL n ( k [ t ]) such t hat 1 n − θ ( t ) i s also invertible and the eigenvalues of θ (0) and θ (1) ar e a 1 , a 2 , . . . , a n and b 1 , b 2 , . . . , b n , r e sp ectively. Pr o of. Let p ( λ ) = (1 − t ) n Y i =1 ( λ − a i ) + t n Y i =1 ( λ − b i ) multilinearit y and skew-symmetry 3 be a p olynomial in λ with coefficients in k [ t ]. It i s a monic p olynomial with the constan t term equal to ( − 1) n a 1 a 2 · · · a n . It has ro ots b 1 , b 2 , . . . , b n and a 1 , a 2 , . . . , a n when t = 1 and t = 0, resp ectively . Now let θ ( t ) b e i ts companion matrix in GL n ( k [ t ]). Then det (1 n − θ ( t )) = p (1) since det ( λ 1 n − θ ( t ) ) = p ( λ ). But p (1) = (1 − a 1 )(1 − a 2 ) · · · (1 − a n ) = (1 − b 1 )(1 − b 2 ) · · · (1 − b n ) i s in k × , and so 1 n − θ ( t ) i s inv ertible. It i s clear that the eigenv alues of θ ( t ) are a 1 , a 2 , . . . , a n and b 1 , b 2 , . . . , b n when t = 0 and t = 1, resp ectiv ely .  Definition 2 . 2. F or l ≥ 2 , let Z b e the sub gr oup of K 0 ( k ∆ 1 , G ∧ l m ) gener ate d by the elements of the fol lowing typ es for various n ≥ 1 : ( Z 1 ) ( θ 1 , . . . , θ l ) , wher e θ 1 , . . . , θ l ∈ GL n ( k [ t ]) c ommute and θ i is in GL n ( k ) for some i ; ( Z 2 ) ( θ 1 , . . . , θ l ) , wher e θ i = θ j ∈ GL n ( k [ t ]) for some i 6 = j ; ( Z 3 ) ( θ 1 , . . . , θ l ) , wher e θ i = 1 n − θ j ∈ GL n ( k [ t ]) for some i 6 = j . Lemma 2.3. L et ∂ Z denote the image of Z under the bou ndary homomorp hism ∂ : K 0 ( k ∆ 1 , G ∧ l m ) → K 0 ( k ∆ 0 , G ∧ l m ) when l ≥ 2 . Then ∂ Z c ontains al l element s of the f ol lowing forms: (i) ( ϕψ, θ 2 , . . . , θ l ) − ( ϕ, θ 2 , . . . , θ l ) − ( ψ , θ 2 , . . . , θ l ) , f or al l c ommuting ϕ, ψ, θ 2 , . . . , θ l ∈ GL n ( k ) ; Similarly, ( θ 1 , . . . , θ i − 1 , ϕψ, θ i +1 , . . . , θ l ) − ( θ 1 , . . . , θ i − 1 , ϕ, θ i +1 , . . . , θ l ) − ( θ 1 , . . . , θ i − 1 , ψ, θ i +1 , . . . , θ l ) for al l c ommuting ϕ, ψ, θ 1 , . . . , θ i − 1 , θ i +1 , . . . , θ l ∈ GL n ( k ) ; (ii) ( θ 1 , . . . , θ i , . . . , θ j , . . . , θ l )+( θ 1 , . . . , θ j , . . . , θ i , . . . , θ l ) , for al l c ommuting θ 1 , . . . , θ l ∈ GL n ( k ) ; (iii) ( θ 1 , . . . , θ i , . . . , θ j , . . . , θ l ) , when θ i = − θ j for c ommuting θ 1 , . . . , θ l ∈ GL n ( k ) ; (iv) ( c 1 , . . . , b, . . . , 1 − b, . . . , c l ) − ( c 1 , . . . , a, . . . , 1 − a, . . . , c l ) , for a, b ∈ k − { 0 , 1 } and c i ∈ k × for e ach appr opriate i . Pr o of. ( i ) W e first observe the f ollo wi ng ident i ties of matrices: „ 1 n 0 ψ 1 n « „ ψ 1 n 0 ϕ « „ 1 n 0 − ψ 1 n « = „ 0 1 n − ϕψ ϕ + ψ « , (1) „ 1 n 0 1 n 1 n « „ 1 n 1 n 0 ϕψ « „ 1 n 0 − 1 n 1 n « = „ 0 1 n − ϕψ 1 n + ϕψ « . (2) Let Θ( t ) be the 2 n × 2 n matrix „ 0 1 n − ϕψ t (1 n + ϕψ ) + (1 − t )( ϕ + ψ ) « . Then, Θ( t ) is i n GL 2 n ( k [ t ]), ` Θ( t ) , θ 2 ⊕ θ 2 , . . . , θ l ⊕ θ l ´ is i n Z by Definition 2. 2 ( Z 1 ) and the boundary of ` Θ( t ) , θ 2 ⊕ θ 2 , . . . , θ l ⊕ θ l ´ is, b y (1) and by (2), (1 n ⊕ ϕψ , θ 2 ⊕ θ 2 , . . . , θ l ⊕ θ l ) − ( ϕ ⊕ ψ, θ 2 ⊕ θ 2 , . . . , θ l ⊕ θ l ) = ( ϕψ, θ 2 , . . . , θ l ) − ( ϕ, θ 2 , . . . , θ l ) − ( ψ , θ 2 , . . . , θ l ). The proof is si milar for other cases. ( ii ) W e let Θ( t ) b e the matrix „ 0 1 n − θ i θ j t (1 n + θ i θ j ) + (1 − t )( θ i + θ j ) « . Then ` θ ⊕ 2 1 , . . . , Θ( t ) , . . . , Θ( t ) , . . . , θ ⊕ 2 l ´ is in Z by Definition 2.2 ( Z 2 ) and the b oundary of ` θ ⊕ 2 1 , . . . , Θ( t ) , . . . , Θ( t ) , . . . , θ ⊕ 2 l ´ is ( θ 1 , . . . , θ i θ j , . . . , θ i θ j , . . . , . . . , θ l ) − ( θ 1 , . . . , θ i , . . . , θ i , . . . , θ l ) − ( θ 1 , . . . , θ j , . . . , θ j , . . . , θ l ) = ` ( θ 1 , . . . , θ i , . . . , θ i , . . . , θ l ) + (( θ 1 , . . . , θ i , . . . , θ j , . . . , θ l ) + ( θ 1 , . . . , θ j , . . . , θ i , . . . , θ l ) + ( θ 1 , . . . , θ j , . . . , θ j , . . . , θ l ) ´ − ( θ 1 , . . . , θ i , . . . , θ i , . . . , θ l ) − ( θ 1 , . . . , θ j , . . . , θ j , . . . , θ l ) = ( θ 1 , . . . , θ j , . . . , θ i , . . . , θ l ) + ( θ 1 , . . . , θ j , . . . , θ i , . . . , θ l ) modulo ∂ Z by ( i ) . ( iii ) W e note that „„ θ 1 0 0 θ 1 « , . . . , „ − θ 0 0 − θ « , . . . , „ 0 1 n − θ t ( θ + 1 n ) « . . . , „ θ l 0 0 θ l «« is an elemen t of Z by Definition 2.2 ( Z 1 ). So i ts boundary 4 S. Myung „„ θ 1 0 0 θ 1 « , . . . , „ − θ 0 0 − θ « , . . . , „ 0 1 n − θ θ + 1 n « , . . . , „ θ l 0 0 θ l «« − „„ θ 1 0 0 θ 1 « , . . . , „ − θ 0 0 − θ « , . . . , „ 0 1 n − θ 0 « , . . . , „ θ l 0 0 θ l «« = „„ θ 1 0 0 θ 1 « , . . . , „ − θ 0 0 − θ « , . . . , „ θ 1 n 0 1 n « , . . . , „ θ l 0 0 θ l «« − „„ θ 1 0 0 θ 1 « , . . . , „ − θ 0 0 − θ « , . . . , „ 0 1 n − θ 0 « , . . . , „ θ l 0 0 θ l «« = ( θ 1 , . . . , − θ , . . . , , θ , . . . , θ l ) − „„ θ 1 0 0 θ 1 « , . . . , „ − θ 0 0 − θ « , . . . , „ 0 1 n − θ 0 « , . . . , „ θ l 0 0 θ l «« is in ∂ Z . Thus it suffices to prov e that „„ θ 1 0 0 θ 1 « , . . . , „ − θ 0 0 − θ « , . . . , „ 0 1 n − θ 0 « , . . . , „ θ l 0 0 θ l «« is in ∂ Z . But it i s equal to „ θ 1 0 0 θ 1 « , . . . , „ 0 1 n − θ 0 « 2 , . . . , „ 0 1 n − θ 0 « , . . . , „ θ l 0 0 θ l « ! = 2 „„ θ 1 0 0 θ 1 « , . . . , „ 0 1 n − θ 0 « , . . . , „ 0 1 n − θ 0 « , . . . , „ θ l 0 0 θ l «« , which is i n ∂ Z by ( ii ) ab o ve. ( iv ) Apply Lemma 2.1 to a 1 = a, a 2 = √ b, a 3 = − √ b, b 1 = − √ a, b 2 = √ a, b 3 = b to get θ ( t ) ∈ GL 3 ( k [ t ]) with the prop erties stated in the lemma. Then z = 2 ` c ⊕ 3 1 , . . . , θ ( t ) , . . . , 1 3 − θ ( t ) , . . . , c ⊕ 3 l ´ is in Z by Definition 2.2 ( Z 3 ). But, by the theory of r ational cano nical form, we ha ve ∂ z = 2 „ ( c 1 , . . . , b, . . . , 1 − b, . . . , c l ) + „„ c 1 0 0 c 1 « , . . . , „ 0 1 a 0 « , . . . , „ 1 − 1 − a 1 « , . . . , „ c l 0 0 c l ««« − 2 „ ( c 1 , . . . , a, . . . , 1 − a, . . . , c l ) + „„ c 1 0 0 c 1 « , . . . , „ 0 1 b 0 « , . . . , „ 1 − 1 − b 1 « , . . . , „ c l 0 0 c l ««« = − 2( c 1 , . . . , a, . . . , 1 − a, . . . , c l ) + 2( c 1 , . . . , b, . . . , 1 − b, . . . , c l ) − „ c 1 0 0 c 1 « , . . . , „ 0 1 b 0 « 2 , . . . , „ 1 − 1 − b 1 « , . . . , „ c l 0 0 c l « ! + „ c 1 0 0 c 1 « , . . . , „ 0 1 a 0 « 2 , . . . , „ 1 − 1 − a 1 « , . . . , „ c l 0 0 c l « ! = „„ c 1 0 0 c 1 « , . . . , „ b 0 0 b « , . . . , „ 1 − b 0 0 1 − b « , . . . , „ c l 0 0 c l «« − „„ c 1 0 0 c 1 « , . . . , „ b 0 0 b « , . . . , „ 1 − 1 − b 1 « , . . . , „ c l 0 0 c l «« − „„ c 1 0 0 c 1 « , . . . , „ a 0 0 a « , . . . , „ 1 − a 0 0 1 − a « , . . . , „ c l 0 0 c l «« + „„ c 1 0 0 c 1 « , . . . , „ a 0 0 a « , . . . , „ 1 − 1 − a 1 « , . . . , „ c l 0 0 c l «« = „ c 1 0 0 c 1 « , . . . , „ b 0 0 b « , . . . , „ 1 − b 0 0 1 − b « „ 1 − 1 − b 1 « − 1 , . . . , „ c l 0 0 c l « ! − „ c 1 0 0 c 1 « , . . . , „ a 0 0 a « , . . . , „ 1 − a 0 0 1 − a « „ 1 − 1 − a 1 « − 1 , . . . , „ c l 0 0 c l « ! = „„ c 1 0 0 c 1 « , . . . , „ b 0 0 b « , . . . „ 1 1 b 1 « , . . . , „ c l 0 0 c l «« − „„ c 1 0 0 c 1 « , . . . , „ a 0 0 a « , . . . , „ 1 1 a 1 « , . . . , „ c l 0 0 c l «« multilinearit y and skew-symmetry 5 = 0 @ „ c 1 0 0 c 1 « , . . . , „ b 0 0 b « , . . . , − b 1 − b 1 1 − b 0 1 ! „ 1 1 b 1 « − b 1 − b 1 1 − b 0 1 ! − 1 , . . . , „ c l 0 0 c l « 1 A − „ c 1 0 0 c 1 « , . . . , „ a 0 0 a « , . . . , „ − a 1 − a 1 1 − a 0 1 « „ 1 1 a 1 « „ − a 1 − a 1 1 − a 0 1 « − 1 , . . . , „ c l 0 0 c l « ! = „„ c 1 0 0 c 1 « , . . . , „ b 0 0 b « , . . . , „ 0 1 b − 1 2 « , . . . , „ c l 0 0 c l «« − „„ c 1 0 0 c 1 « , . . . , „ a 0 0 a « , . . . , „ 0 1 a − 1 2 « , . . . , „ c l 0 0 c l «« . By taking the b oundary of the element „„ c 1 0 0 c 1 « , . . . , „ b 0 0 b « , . . . , „ 0 1 b − 1 (2 − b ) t + 2(1 − t ) « , . . . , „ c l 0 0 c l «« − „„ c 1 0 0 c 1 « , . . . , „ a 0 0 a « , . . . , „ 0 1 a − 1 (2 − a ) t + 2(1 − t ) « , . . . , „ c l 0 0 c l «« , which is i n Z by Definition 2.2 ( Z 1 ), we see that ∂ z = „„ c 1 0 0 c 1 « , . . . , „ b 0 0 b « , . . . , „ 0 1 b − 1 2 − b « , . . . , „ c l 0 0 c l «« − „„ c 1 0 0 c 1 « , . . . , „ a 0 0 a « , . . . , „ 0 1 a − 1 2 − a « , . . . , „ c l 0 0 c l «« = „„ c 1 0 0 c 1 « , . . . , „ b 0 0 b « , . . . , „ 1 − b 0 0 1 « , . . . , „ c l 0 0 c l «« b y (1) , which then i s equal to − „„ c 1 0 0 c 1 « , . . . , „ a 0 0 a « , . . . , „ 1 − a 0 0 1 « , . . . , „ c l 0 0 c l «« = ( c 1 , . . . , b, . . . , 1 − b, . . . , c l ) − ( c 1 , . . . , a, . . . , 1 − a, . . . , c l ) in K 0 ( k ∆ 0 , G ∧ l m ) /∂ Z . Therefore, ( iv ) lies in ∂ Z .  Coroll ary 2. 4. (Multiline arity and Skew-symmetry for H l M ` Spec k , Z ( l ) ´ ) (i) ( θ 1 , . . . , θ i − 1 , ϕψ, θ i +1 , . . . , θ l ) = ( θ 1 , . . . , θ i − 1 , ϕ, θ i +1 , . . . , θ l )+( θ 1 , . . . , θ i − 1 , ψ, θ i +1 , . . . , θ l ) in H l M ` Spec k , Z ( l ) ´ , for al l c ommuting ϕ, ψ, θ 1 , . . . , θ i − 1 , θ i +1 , . . . , θ l ∈ GL n ( k ) (ii) ( θ 1 , . . . , θ i , . . . , θ j , . . . , θ l ) = − ( θ 1 , . . . , θ j , . . . , θ i , . . . , θ l ) i n H l M ` Spec k , Z ( l ) ´ for al l c om- muting θ 1 , . . . , θ l ∈ GL n ( k ) If θ 1 , . . . , θ l and θ ′ 1 , . . . , θ ′ l are commuting matrices in GL n ( k ) and GL m ( k ), resp ectiv ely , then ( θ 1 , . . . , θ l ) + ( θ ′ 1 , . . . , θ ′ l ) = ( θ 1 ⊕ θ ′ 1 , . . . , θ l ⊕ θ ′ l ) in H l M ` Spec k , Z ( l ) ´ . Therefor e, we obtain the following result from Corollary 2.4. Coroll ary 2 . 5. Every element i n H l M ` Spec k , Z ( l ) ´ c an b e writte n as a single symb ol ( θ 1 , . . . , θ l ) , wher e θ 1 , . . . , θ l ar e c ommuting matric es in GL n ( k ) . Thanks to Lemma 2.3, we can construct a map from Milnor’s K -groups to the motivic coho- mology groups. Prop o sition 2. 6. F or any field k , the assignment { a 1 , a 2 , . . . , a l } 7→ ( a 1 , a 2 , . . . , a l ) for e ach Steinb er g symb ol { a 1 , a 2 , . . . , a l } g i ves a wel l-define d homomorphism ρ l fr om the Milnor’s K - gr oup K M l ( k ) to H l M ` Spec k , Z ( l ) ´ . Pr o of. This prop osition turns out to be straightforw ar d when l = 1. So we assume that l ≥ 2. By Corollary 2.4 ( i ), the m ultili nearit y is satisfied b y our symbol ( , . . . , ). Therefore all w e need to sho w is that for every α ∈ k − { 0 , 1 } and c r ∈ k × for 1 ≤ r ≤ l , r 6 = i, j , ( c 1 , . . . , α, . . . , 1 − α, . . . , c l ) is in ∂ K 0 ( k ∆ 1 , G ∧ l m ). W e will actually show that it is cont ai ned i n ∂ Z . The prop osition is i mmediate for a prime field F p because K M l ( F p ) = 0 for l ≥ 2. So we may assume that there exists an element e ∈ k suc h that e 3 − e 6 = 0. By Lemma 2.3 ( iv ) wi th a = e, b = 1 − e , w e hav e ( c 1 , . . . , e, . . . , 1 − e, . . . , c l ) − ( c 1 , . . . , 1 − e, . . . , e, . . . , c l ) = 2( c 1 , . . . , e, . . . , 1 − e, . . . , c l ) = 0 mo dulo ∂ Z . With a = − e, b = 1 + e , w e ha ve 2( c 1 , . . . , e, . . . , 1 + e, . . . , c l ) = 6 S. Myung 2( c 1 , . . . , − e, . . . , 1 + e, . . . , c l ) = 0. Hence, ( c 1 , . . . , e 2 , . . . , 1 − e 2 , . . . , c l ) = 2( c 1 , . . . , e, . . . , 1 − e, . . . , c l ) + 2( c 1 , . . . , e, . . . , 1 + e , . . . , c l ) = 0. On the other hand, b y Lemma 2.3 ( iv ) with a = e 2 , b = α , we see that − ( c 1 , . . . , e 2 , . . . , 1 − e 2 , . . . , c l ) + ( c 1 , . . . , α, . . . , 1 − α, . . . , c l ) is in ∂ Z and we’re done. More explicitly , let z = 2 ` c ⊕ 3 1 , . . . , θ ( t ) , . . . , 1 − θ ( t ) , . . . , c ⊕ 3 l ´ ∈ Z , where θ ( t ) = 0 @ 0 1 0 0 0 1 − e 2 α ( e 2 − α ) t + α ( α − e 2 ) t + e 2 1 A . This matrix θ ( t ) is constructed wi th Lemma 2.1 with a 1 = e 2 , a 2 = √ α, a 3 = − √ α, b 1 = − e, b 2 = e, b 3 = α . Hence, by the computation we hav e done in the pro of of Lemma 2.3 ( iv ), ∂ z = 2( c 1 , . . . , − e, . . . , 1 + e, . . . , c l ) + 2( c 1 , . . . , e, . . . , 1 − e, . . . , c l ) + 2( c 1 , . . . , α, . . . , 1 − α, . . . , c l ) − 2( c 1 , . . . , e 2 , . . . , 1 − e 2 , . . . , c l ) − 2 „„ c 1 0 0 c 1 « , . . . , „ 0 1 α 0 « , . . . , „ 1 − 1 − α 1 « . . . , „ c l 0 0 c l «« = − ( c 1 , . . . , e 2 , . . . , 1 − e 2 , . . . , c l ) + ( c 1 , . . . , α, . . . , 1 − α, . . . , c l ) = (( c 1 , . . . , α, . . . , 1 − α, . . . , c l ) .  F or Goo dwillie-Li c hten baum motivic complex, there is a straigh tforward f unctorial definition of the norm map for the motivic cohomology for any finite exte nsi on k ⊂ L . Definition 2.7. If θ 1 , . . . , θ l ar e c ommuting automorphisms on a finitely gener ate d pr oje ct ive L ∆ d -mo dule P , t hen by i dentifying L ∆ d as a fr e e k ∆ d -mo dule of finite r ank, we may c onsider P as a finitely gene r ated pr oje cti ve k ∆ d -mo dule and θ 1 , . . . , θ l as c ommuting automorph i sms on it. This gives a simplicial map K 0 ( L ∆ d , G ∧ l m ) → K 0 ( k ∆ d , G ∧ l m ) . The r esulting homomo rphism N L/k : H q M ` Spec L, Z ( l ) ´ → H q M ` Spec k , Z ( l ) ´ is c al le d the norm map. W e summ ar ize some basic results for the norm in the foll o wing lemma. Lemma 2.8. (i) N L ′ /L ◦ N L/k = N L ′ /k whenever we have a tower of finite fie ld exte nsions k ⊂ L ⊂ L ′ . (ii) If [ L : k ] = d , the c omp osition H q M ` Spec k , Z ( l ) ´ i L/k / / H q M ` Spec L, Z ( l ) ´ N L/k / / H q M ` Spec k , Z ( l ) ´ , wher e i L/k is induc e d by the inclusion of the fields k ⊂ L , is multiplic ation by d . (iii) F or α 1 , . . . , α l ∈ k × and β ∈ L × , N L/k ( α 1 , . . . , α l , β ) = ` α 1 , . . . , α l , N L/k ( β ) ´ in H l +1 M ` Spec k , Z ( l + 1) ´ , wher e N L/k ( β ) ∈ k × is the image of β under the usual norm map N L/k : L × → k × . Pr o of. ( i ) and ( ii ) ar e i mmediate from Definition 2.7 . ( iii ) follows f rom the observ ation that, in H 1 M ` Spec k , Z (1) ´ , the t wo elemen ts represen ted b y t wo matri ces with same dete r minan ts are equal since any matrix with determinant 1 is a pr o duct of elementary matrices and an elemen t represen ted b y an elementary matrix v anis hes in H 1 M ` Spec k , Z (1) ´ .  W e also hav e the norm maps N L/K : K M l ( L ) → K M l ( k ) f or the M i lnor’s K -groups whenever L/k is a finite field extension, whose definition we r ecall br i efly as follows. (See [1] or [6] § 1.2) F or each discrete v aluation v of the field K = k ( t ) of rational functions o ver k , let π v be a uniformizing parameter and k v = R v / ( π v ) b e the residue field of the v aluation ri ng R v = { r ∈ K | v ( r ) ≥ 0 } . Then we define the tame symbol ∂ v : K M l +1 ( K ) → K M l ( k v ) to b e the epimorphism suc h that ∂ ( { u 1 , . . . , u l , y } ) = v ( y ) { u 1 , . . . , u l } whenev er u 1 , . . . , u l are units of the v aluation ri ng R v . Let v ∞ be the v aluation on K = k ( t ), which v anishes on k , s uch that v ∞ ( t ) = − 1. Every simple algebraic extension L of k i s isomorphic to k v for some discrete v aluation v 6 = v ∞ which corresponds to a prime ideal p of k [ t ]. The norm maps N v : K M l ( k v ) → K M l ( k ) are the unique homomorphisms suc h that, f or every w ∈ K M l +1 ( k ( t )) , X v N v ( ∂ v w ) = 0 w here the s um is taken multilinearit y and skew-symmetry 7 o ver all di screte v aluations, including v ∞ on k ( t ), v anishing on k . This equality is called the W eil recipro cit y law. Note that we take N v ∞ = Id for v = v ∞ . Kato ([6] § 1.7) has shown that these maps, if defined as comp ositions of norm maps for si mple extensions for a given to wer of simpl e exte nsi ons, dep end only on the field extension L/k , i.e., that it enjoys functoriality . See also [12]. It also enjoys a pro j ect ion f orm ula simi lar to ( iii ) of Lemma 2.8. T he f ollo wi ng ke y lemma shows the compatibility b et ween these tw o types of norm maps. Lemma 2.9. F or every finite field extension k ⊂ L , we have the fol lowing co mmutative diagr am, wher e the vertic al maps ar e the norm maps and the horizontal maps ar e the homo morphisms in Pr op osition 2.6: K M l ( L ) ρ l / / N L/k   H l M ` Spec L, Z ( l ) ´ N L/k   K M l ( k ) ρ l / / H l M ` Spec k , Z ( l ) ´ Pr o of. W e wi ll follow the s ame pro cedure whic h is used in [7] for the pro of. Because of the func- toriality property of the norm maps, we ma y assume that [ L : k ] is a pri me num b er p . First, l et us assume that k has no extensions of degree prime to p . By Lemma (5.3) in [1], K M l ( L ) is generated b y the sym b ols of the f orm x = { x 1 , . . . , x l − 1 , y } where x i ∈ k and y ∈ L . Then, b y the pr o jec- tion formula for Milnor’s K -gr oups, ρ l N L/k ( { x 1 , . . . , x l − 1 , y } ) = ρ l ` { x 1 , . . . , x l − 1 , N L/k ( y ) } ´ = ` x 1 , . . . , x l − 1 , N L k ( y ) ´ . W e also ha ve N L/k ρ l ( { x 1 , . . . , x l − 1 , y } ) = N L/k ` ( x 1 , . . . , x l − 1 , y ) ´ = ` x 1 , . . . , x l − 1 , N L k ( y ) ´ b y ( iii ) of Lemma 2.8 and so we’re done i n this case. Next, for the general case, let k ′ be a maximal prime-to- p extension of k . Then, b y the previous case applied to k ′ and b y ( i ) of Lemma 2.8, we see that z = N L/k ρ l ( x ) − ρ l N L/k ( x ), which is in the k er nel of i k ′ /k : H l M ` Spec k , Z ( l ) ´ → H l M ` Spec k ′ , Z ( l ) ´ , is a torsion element of H l M ` Spec k , Z ( l ) ´ of exp onen t pri me to p . In particular , if L/k is a purely i nseparable extension of degree p , then y p ∈ k and so z is clearly kill ed by p , i .e., z = 0. Hence we ma y assume that L/k is s eparable. Since the ke r nel of i L/k : H l M ` Spec k , Z ( l ) ´ → H l M ` Spec L, Z ( l ) ´ has exp onen t p , it suffices to prov e that i L/k ( z ) = 0 to conclude z = 0. No w L ⊗ k L is a finite pr oduct of fields L i with [ L i : L ] < p and we hav e the following commutat i v e diagrams. K M l ( L ) ⊕ i L i /L / / N L/k   ⊕ i K M l ( L i ) P i N L i /L   K M l ( k ) i L/k / / K M l ( L ) H l M ` Spec L, Z ( l ) ´ ⊕ i L i /L / / N L/k   ⊕ i H l M ` Spec L i , Z ( l ) ´ P i N L i /L   H l M ` Spec k , Z ( l ) ´ i L/k / / H l M ` Spec L, Z ( l ) ´ The left diagram is the diagram (15) in p.387 of [1] and the ri gh t diagram follows easily fr om Defi- nition 2.7. B y induction on p , we hav e i L/k ( z ) = ⊕ N L i /L ρ l ( i L i /L ( x )) − ⊕ ρ l N L i /L ( i L i /L ( x )) = 0 and the pro of is complete.  Lemma 2.10. F or any field k , ther e is a homomo rphism φ l : H l M ` Spec k , Z ( l ) ´ → K M l ( k ) such that, for e ach element z ∈ H l M ` Spec k , Z ( l ) ´ , ther e is an expr ession z = r X j =1 N L i /k ` ( α 1 j , . . . , α lj ) ´ wher e L 1 , . . . , L r ar e finite field extensions of k , α ij ∈ GL 1 ( L j ) = L × j ( 1 ≤ i ≤ l , 1 ≤ j ≤ r ) and an equa lity φ l ( z ) = X j N L j /k ` { α 1 j , . . . , α lj } ´ in K M l ( k ) Pr o of. F or a tuple z = ( θ 1 , θ 2 , . . . , θ l ), where θ 1 , θ 2 , . . . , θ l are commuting matrices in GL n ( k ), consider the vector space E = k n as an R = k [ t 1 , t − 1 1 . . . , t l , t − 1 l ]-mo dule, on which t i acts as θ i . Since E is of finite rank o v er k , it has a comp osition series 0 = E 0 ⊂ E 1 ⊂ · · · ⊂ E r = E with simple factors L j = E j /E j − 1 ( j = 1 , . . . , r ). Then, there exists a maximal ideal m j of R such that L j ≃ R/ m j . So we see that L j is a finite field ext ension of k , and z = r X j =1 ( θ 1 | L j , . . . , θ l | L j ), where θ i | L j is the automorphism on L j induced by θ i . 8 S. Myung Let us denote by α ij the elemen t of L × j which corresponds to t i (mod m j ) for i = 1 , . . . , l , then ( θ 1 | L j , . . . , θ l | L j ) = N L j /k ` ( α 1 j , . . . , α lj ) ´ . Since these factors L j are unique up to an order and a M ilnor sym b ol v anishes if one of its coordinates is 1, the assignment ( θ 1 , θ 2 , . . . , θ l ) 7→ P j N L j /k ` { α 1 j , . . . , α lj } ´ give s us a well- defined homomorphism from K 0 ( k, G ∧ l m ) to K M l ( k ). It remains to sho w that this homomorphism v anishes on ∂ K 0 ( k ∆ 1 , G ∧ l m ). Let A 1 ( t ) , . . . , A l ( t ) be comm uting matrices in GL n ( k [ t ]), where t is an indet erminate. Then M = k ( t ) n can b e considered as an S = k ( t )[ t 1 , t − 1 1 , . . . , t l , t − 1 l ]-mo dule, on which t i acts as A i ( t ). The n find a composition series 0 = M 0 ⊂ M 1 ⊂ · · · ⊂ M s = M with si m ple S -mo dules Q j = M j / M j − 1 ( j = 1 , . . . , s ) and maximal ideals n j of S suc h that Q j ≃ S/ n j . W e also denot e by β ij the element of Q × j which corresp onds to t i (mod n j ) for i = 1 , . . . , l and j = 1 , . . . , s . Each Q j is a finite extension field of k ( t ) and let x = P s j =1 N Q j /k ( t ) ( { β 1 j , . . . , β lj } ) ∈ K M l ( k ( t )). No w consider the elemen t y = { x, ( t − 1) /t } in K M l +1 ( k ( t )), where the symbol { x, ( t − 1) /t } denotes P u { x 1 u , . . . , x lu , ( t − 1) /t } if x = P u { x 1 u , . . . , x lu } in K M l ( k ( t )). Then ∂ v ( y ) = − φ l ` ( A 1 (0) , . . . , A l (0)) ´ if π v = t and ∂ v ( y ) = φ l ` ( A 1 (1) , . . . , A l (1)) ´ if π v = t − 1. Also, the image ∂ v ( y ) is zero unless v is the v aluation ass o ciated with either π v = t − 1 or π v = t . Hence w e ha ve φ l ` ( A 1 (0) , . . . , A l (0)) ´ = φ l ` ( A 1 (1) , . . . , A l (1)) ´ b y the W eil recipro cit y law f or the M ilnor’s K -groups.  The isomorphism in the following theorem w as first giv en b y Nesterenk o and Suslin ([9]) for Blo c h’s higher Chow groups. T otaro, in [14], ga ve another pro of of the theorem. Suslin and V o ev o dsky , in Chapter 3 of [13], gav e a pro of of it for their motivic cohomology . Here, we present another version of it for the Go odwill ie-Lic hten baum motivic complex such that the isomorphism is given explicitly in the f orm whic h transforms the multilinearit y of the the s ym b ols of M ilnor int o the corresp onding prop erties of the symbols of Go odwill ie and Li c hten baum. Theorem 2.11. F or any field k and l ≥ 1 , the assignment { a 1 , a 2 , . . . , a l } 7→ ( a 1 , a 2 , . . . , a l ) for e ach Steinb er g sy mb ol { a 1 , a 2 , . . . , a l } gives rise to an isomorphism K M l ( k ) ≃ H l M ` Spec k , Z ( l ) ´ . Pr o of. The case l = 1 is straigh tfor w ard and w e assume l ≥ 2. By Prop osition 2.6, the assignmen t { a 1 , a 2 , . . . , a l } 7→ ( a 1 , a 2 , . . . , a l ) gives ri se to a homomorphism ρ l from the Milnor’ s K -group K M l ( k ) to the motivic cohomology group H l M ` Spec k , Z ( l ) ´ . W e also hav e a w ell- define d map φ l : H l M ` Spec k , Z ( l ) ´ → K M l ( k ) in Lemma 2.10 and i t suffices to sho w that they are the inv ers es to eac h other. It is clear that φ l ◦ ρ l is the identit y map on K M l ( k ) since eac h Steinberg symbol is fixed by it. On the other hand, f or each z ∈ H l M ` Spec k , Z ( l ) ´ , z = r X j =1 N L j /k ` ( α 1 j , . . . , α lj ) ´ for some finite field exten s ions L 1 , . . . , L r of k and α ij ∈ L j (1 ≤ i ≤ l , 1 ≤ j ≤ r ). Then ( ρ l ◦ φ l )( z ) = ρ l 0 @ X j N L j /k ` { α 1 j , . . . , α lj } ´ 1 A = X j N L j /k ` ρ l ` { α 1 j , . . . , α lj } ´´ = X j N L j /k `` α 1 j , . . . , α lj ´´ = z b y Lemma 2.9. Therefore, ρ l ◦ φ l is also the iden tity map and the pro of is complete.  3. Mul tilinearity a nd Skew-symmetr y for H l − 1 M ` Spec k , Z (l) ´ In [8], the author construct ed a dil ogarithm map D : H 1 M ` Spec k , Z (2) ´ → R whenev er k is a subfield of C suc h that D satisfies certain bil inearit y and skew-symmetry . (See Lemm a 4.8 in [8]). Since D can detect all the torsion-free elemen ts of the motivic cohomology group , e.g., if k is a n umber field ([3], [2]), we ha ve expected that bilinearit y and skew-symmetry for symbols should hold for D : H 1 M ` Spec k , Z (2) ´ → R in such cases. In this section, we extend multilinearit y and sk ew-symmetry r esults of the previous section to the symbols in the motivic cohomology groups H l − 1 M ` Spec k , Z ( l ) ´ when k is a field. K 0 ( k ∆ 1 , G ∧ l m ) ( l ≥ 1) can b e iden tified with the abeli an group generated b y l -tuples ( θ 1 , . . . , θ l ) (= ( θ 1 ( t ) , . . . , θ l ( t ))) and certain explicit r elations, where θ 1 , . . . , θ l are commuting matrices in GL n ( k [ t ]) for v ar i ous n ≥ 1. K 0 ( k ∆ 2 , G ∧ l m ) is iden tified with the ab elian group generated by the sym b ols ( θ 1 ( x, y ) , . . . , θ l ( x, y )) with commuting θ 1 ( x, y ) , . . . , θ l ( x, y ) ∈ GL n ( k [ x, y ]) and cer- tain relations, and the b oundary map ∂ on the motivic complex sends ( θ 1 ( x, y ) , . . . , θ l ( x, y )) to ( θ 1 (1 − t, t ) , . . . , θ l (1 − t, t )) − ( θ 1 (0 , t ) , . . . , θ l (0 , t )) + ( θ 1 ( t, 0) , . . . , θ l ( t, 0)) in K 0 ( k ∆ 1 , G ∧ l m ). The multilinearit y and skew-symmetry 9 same s ymbol ( θ 1 , . . . , θ l ) will denote the elemen t in K 0 ( k ∆ 1 , G ∧ l m ) /∂ K 0 ( k ∆ 2 , G ∧ l m ) represent ed b y ( θ 1 , . . . , θ l ), b y abuse of notation. The motivic cohomology group H l − 1 M ` Spec k , Z ( l ) ´ is a subgroup of this quot ien t group, which consists of the elements kill ed by ∂ . Lemma 3.1. In H l − 1 M ` Spec k , Z ( l ) ´ , we have t he fol lowing two simple r elations of symb ols for any c ommuting matric e s θ 1 , . . . , θ l and any other c ommuting matric es ψ 1 , . . . , ψ l in GL n ( k [ t ]) : − ( θ 1 ( t ) , . . . , θ l ( t )) = ( θ 1 (1 − t ) , . . . , θ l (1 − t )) ( θ 1 ( t ) , . . . , θ l ( t )) + ( ψ 1 ( t ) , . . . , ψ l ( t )) = ( θ 1 ( t ) ⊕ ψ 1 ( t ) , . . . , θ l ( t ) ⊕ ψ l ( t )) . Pr o of. The second relation is immediate f rom definition of the motivic complex. The first relation can b e sho wn by applying the boundary m ap ∂ to the elemen t ( θ 1 ( x ) , . . . , θ l ( x )) regarded as in K 0 ( k ∆ 2 , G ∧ l m ) and by noting that ( θ 1 , . . . , θ l ) = 0 i n H l − 1 M (Sp ec k, Z ( l )) when θ 1 , . . . , θ l are constan t matrices. T he fact that ( θ 1 , . . . , θ l ) = 0 for consta n t m atri ces θ 1 , . . . , θ l is obta i ned simpl y b y applying the b oundary map ∂ to the element ( θ 1 , . . . , θ l ) regarded as in K 0 ( k ∆ 2 , G ∧ l m ).  Coroll ary 3.2. Any element of the c ohomo lo gy gr oup H l − 1 M ` Spec k , Z ( l ) ´ c an b e r epr ese nte d b y a single expr ession ( θ 1 , . . . , θ l ) , wher e θ 1 , . . . , θ l ar e c ommuting matric es in GL n ( k [ t ]) for some nonne gative inte ger n . W e remark that the symbol ( θ 1 ( t ) , . . . , θ l ( t )) represen ts an element in H l − 1 M ` Spec k , Z ( l ) ´ only when its i m age under the b oundary map ∂ v anishes in K 0 ( k ∆ 0 , G ∧ l m ). A tuple ( θ 1 ( t ) , . . . , θ l ( t )) where θ 1 , . . . , θ l are comm uting matrices in GL n ( k [ t ]) is called i rre- ducible i f k [ t ] n has no nont r ivial prop er submodule when regarded as a k [ t, x 1 , x − 1 1 . . . , x l , x − 1 l ]- module where x i acts on k [ t ] n via θ i ( t ) for each i = 1 , . . . , l . Note that i f k ( t ) n is r egarded as a k ( t ) [ x 1 , x − 1 1 . . . , x l , x − 1 l ]-mo dule with the same actions and i f M is a nontrivial pr op er submo dule of k ( t ) n , then M ∩ k [ t ] n is a nont r ivial proper k [ t, x 1 , x − 1 1 . . . , x l , x − 1 l ]-submo dule of k [ t ] n . There- fore, k ( t ) n is irreducible as a k ( t )[ x 1 , x − 1 1 . . . , x l , x − 1 l ]-mo dule if ( θ 1 ( t ) , . . . , θ l ( t )) i s ir r educible. It can b e easil y c hec ked that, if t wo matri ces A, B ∈ GL n ( k ) commute and A is a blo c k m atrix of the form A = „ I 0 0 C « where I is a m atri x whose c haracteristic p olynomial is a p o wer of x − 1 and C do es not ha ve 1 as an eigenv alue, then B must be a blo c k matrix B = „ B 1 0 0 B 2 « , where the blo ck s B 1 and B 2 are of compatible sizes with the blocks I and C of A . Therefore, we may easily relax the notion of ir reduciblit y of a symbol ( θ 1 ( t ) , . . . , θ l ( t )) as an element in K 0 ( k ∆ 1 , G ∧ l m ) b y declaring it irreducible when its restriction to the largest submodule V ⊂ k [ t ] n , where none of the r estrictions of θ 1 ( t ) , . . . , θ l ( t ) has 1 as an eigenv alue, is i rreducible. Theorem 3.3. (Multiline arity) Supp ose that ϕ ( t ) , ψ ( t ) and θ 1 ( t ) , . . . , θ l ( t ) (with θ i ( t ) omitted) ar e c ommuting matric e s in GL n ( k [ t ]) such t hat t he sy mb ol r e pr ese nt e d by one of these matric es is irr e ducible in K 0 ( k ∆ 1 , G ∧ 1 m ) . Assume further that the symb ols ` θ 1 ( t ) , . . . , ϕ ( t ) , . . . , θ l ( t ) ´ and ` θ 1 ( t ) , . . . , ψ ( t ) , . . . , θ l ( t ) ´ r epr esent elements in H l − 1 M ` Spec k , Z ( l ) ´ . Then ` θ 1 ( t ) , . . . , ϕ ( t ) ψ ( t ) , . . . , θ l ( t ) ´ r epr esents an element in H l − 1 M ` Spec k , Z ( l ) ´ and ` θ 1 ( t ) , . . . , ϕ ( t ) , . . . , θ l ( t ) ´ + ` θ 1 ( t ) , . . . , ψ ( t ) , . . . , θ l ( t ) ´ = ` θ 1 ( t ) , . . . , ϕ ( t ) ψ ( t ) , . . . , θ l ( t ) ´ in H l − 1 M ` Spec k , Z ( l ) ´ . Pr o of. F or simplicity of notation, we ma y assume that i = 1 and pr o ve the multilinearit y on the first v ar i able, i.e., we will wan t to sho w that ` ϕ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ + ` ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ = ` ϕ ( t ) ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ . In this pr oof, all equalities are in K 0 ( k ∆ 1 , G ∧ l m ) /∂ K 0 ( k ∆ 2 , G ∧ l m ) unless men tioned otherwise and 1 denotes the identit y matrix 1 n of r ank n whenev er appropriate. Let p ( t ) and q ( t ) b e matrices wi th en tri es i n k [ t ] such that p ( t ) is inv ertible and p ( t ) , q ( t ) and θ 2 ( t ) , . . . , θ l ( t ) commu te. Then the b oundary of the elemen t „„ 0 1 − p ( y ) xy q ( y ) « , „ θ 2 ( y ) 0 0 θ 2 ( y ) « , . . . , „ θ l ( y ) 0 0 θ l ( y ) «« 10 S. Myung of K 0 ( k ∆ 2 , G ∧ l m ) v anishes in H l − 1 M ` Spec k , Z ( l ) ´ b y the definition of the cohomology group. Hence we hav e 0 = „„ 0 1 − p ( t ) (1 − t ) tq ( t ) « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« − „„ 0 1 − p ( t ) 0 « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« + „„ 0 1 p (0) 0 « , „ θ 2 (0) 0 0 θ 2 (0) « , . . . , „ θ l (0) 0 0 θ l (0) «« . But, as in the pro of of Lemma 3.1, the l ast term, whi ch is a tuple of constan t matri ces, is 0 and we hav e (3) „„ 0 1 − p ( t ) (1 − t ) tq ( t ) « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« = „„ 0 1 − p ( t ) 0 « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« . Next, b y taking the boundary of „„ 0 1 − p ( y ) ( x + y ) q ( y ) « , „ θ 2 ( y ) 0 0 θ 2 ( y ) « , . . . , „ θ l ( y ) 0 0 θ l ( y ) «« , we get (4) „„ 0 1 − p ( t ) q ( t ) « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« = „„ 0 1 − p ( t ) tq ( t ) « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« − „„ 0 1 − p (0) tq (0) ´ « , „ θ 2 (0) 0 0 θ 2 (0) « , . . . , „ θ l (0) 0 0 θ l (0) «« . If p ( t ), q ( t ) and θ 2 ( t ) , . . . , θ l ( t ) are replaced b y p (1 − t ) ,(1 − t ) q (1 − t ) and θ 2 (1 − t ) , . . . , θ l (1 − t ) resp ectively i n (4), then we obtain (5) „„ 0 1 − p (1 − t ) (1 − t ) q (1 − t ) « , „ θ 2 (1 − t ) 0 0 θ 2 (1 − t ) « , . . . , „ θ l (1 − t ) 0 0 θ l (1 − t ) «« = „„ 0 1 − p (1 − t ) t (1 − t ) q (1 − t ) « , „ θ 2 (1 − t ) 0 0 θ 2 (1 − t ) « , . . . , „ θ l (1 − t ) 0 0 θ l (1 − t ) «« − „„ 0 1 − p (1) tq (1) ´ « , „ θ 2 (1) 0 0 θ 2 (1) « , . . . , „ θ l (1) 0 0 θ l (1) «« . If we apply Lemma 3.1 to the firs t term, the r igh t hand side of the equalit y (4) can be written as − „„ 0 1 − p (1 − t ) (1 − t ) q (1 − t ) « , „ θ 2 (1 − t ) 0 0 θ 2 (1 − t ) « , . . . , „ θ l (1 − t ) 0 0 θ l (1 − t ) «« − „„ 0 1 − p (0) tq (0) ´ « , „ θ 2 (0) 0 0 θ 2 (0) « , . . . , „ θ l (0) 0 0 θ l (0) «« . By applying (5) to the first term and by (4), we ha ve „„ 0 1 − p ( t ) q ( t ) « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« = − „„ 0 1 − p (1 − t ) t (1 − t ) q (1 − t ) « , „ θ 2 (1 − t ) 0 0 θ 2 (1 − t ) « , . . . , „ θ l (1 − t ) 0 0 θ l (1 − t ) «« + „„ 0 1 − p (1) tq (1) ´ « , „ θ 2 (1) 0 0 θ 2 (1) « , . . . , „ θ l (1) 0 0 θ l (1) «« − „„ 0 1 − p (0) tq (0) ´ « , „ θ 2 (0) 0 0 θ 2 (0) « , . . . , „ θ l (0) 0 0 θ l (0) «« multilinearit y and skew-symmetry 11 = „„ 0 1 − p ( t ) t (1 − t ) q ( t ) « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« + „„ 0 1 − p (1) tq (1) ´ « , „ θ 2 (1) 0 0 θ 2 (1) « , . . . , „ θ l (1) 0 0 θ l (1) «« − „„ 0 1 − p (0) tq (0) ´ « , „ θ 2 (0) 0 0 θ 2 (0) « , . . . , „ θ l (0) 0 0 θ l (0) «« = „„ 0 1 − p ( t ) 0 « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« + „„ 0 1 − p (1) tq (1) « , „ θ 2 (1) 0 0 θ 2 (1) « , . . . , „ θ l (1) 0 0 θ l (1) «« − „„ 0 1 − p (0) tq (0) ´ « , „ θ 2 (0) 0 0 θ 2 (0) « , . . . , „ θ l (0) 0 0 θ l (0) «« . The second equality is obtained by applying Lemma 3.1 to the first term and the l ast equality is b y (3). No w by setting p ( t ) = ϕ ( t ) ψ ( t ) and q ( t ) = ϕ ( t ) + ψ ( t ) in the ab o ve equalit y , we hav e (6) „„ 0 1 − ϕ ( t ) ψ ( t ) ϕ ( t ) + ψ ( t ) « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« = „„ 0 1 − ϕ ( t ) ψ ( t ) 0 « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« + „„ 0 1 − ϕ (1) ψ (1) t ` ϕ (1) + ψ (1) ´ « , „ θ 2 (1) 0 0 θ 2 (1) « , . . . , „ θ l (1) 0 0 θ l (1) «« − „„ 0 1 − ϕ (0) ψ (0) t ` ϕ (0) + ψ (0) ´ « , „ θ 2 (0) 0 0 θ 2 (0) « , . . . , „ θ l (0) 0 0 θ l (0) «« . Similarly , with p ( t ) = ϕ ( t ) ψ ( t ) and q ( t ) = 1 + ϕ ( t ) ψ ( t ) this time, we get (7) „„ 0 1 − ϕ ( t ) ψ ( t ) 1 + ϕ ( t ) ψ ( t ) « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« = „„ 0 1 − ϕ ( t ) ψ ( t ) 0 « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« + „„ 0 1 − ϕ (1) ψ (1) t ` 1 + ϕ (1) ψ (1) ´ « , „ θ 2 (1) 0 0 θ 2 (1) « , . . . , „ θ l (1) 0 0 θ l (1) «« − „„ 0 1 − ϕ (0) ψ (0) t ` 1 + ϕ (0) ψ (0) ´ « , „ θ 2 (0) 0 0 θ 2 (0) « , . . . , „ θ l (0) 0 0 θ l (0) «« . The first terms on the r igh t of (6) and (7) are the same, so b y subtracting (7) from (6), we obtain „„ 0 1 − ϕ ( t ) ψ ( t ) ϕ ( t ) + ψ ( t ) « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« − „„ 0 1 − ϕ ( t ) ψ ( t ) 1 + ϕ ( t ) ψ ( t ) « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« = „„ 0 1 − ϕ (1) ψ (1) t ` ϕ (1) + ψ (1) ´ « , „ θ 2 (1) 0 0 θ 2 (1) « , . . . , „ θ l (1) 0 0 θ l (1) «« − „„ 0 1 − ϕ (0) ψ (0) t ` ϕ (0) + ψ (0) ´ « , „ θ 2 (0) 0 0 θ 2 (0) « , . . . , „ θ l (0) 0 0 θ l (0) «« − „„ 0 1 − ϕ (1) ψ (1) t ` 1 + ϕ (1) ψ (1) ´ « , „ θ 2 (1) 0 0 θ 2 (1) « , . . . , „ θ l (1) 0 0 θ l (1) «« + „„ 0 1 − ϕ (0) ψ (0) t ` 1 + ϕ (0) ψ (0) ´ « , „ θ 2 (0) 0 0 θ 2 (0) « , . . . , „ θ l (0) 0 0 θ l (0) «« . Now we state our claim: Claim: The right hand side of the ab ove e q uality is e qual to 0. 12 S. Myung Once the claim is prov ed, we obtain the following equality . (8) „„ 0 1 − ϕ ( t ) ψ ( t ) ϕ ( t ) + ψ ( t ) « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« = „„ 0 1 − ϕ ( t ) ψ ( t ) 1 + ϕ ( t ) ψ ( t ) « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« . T o prov e the clai m , note first that, by our assumption, one of ϕ ( t ), ψ ( t ) and θ 2 ( t ) , . . . , θ l ( t ), denoted θ ( t ), is irreducible on the largest submodule V ⊂ k [ t ] n , where none of the restric- tions of θ 1 ( t ) , . . . , θ l ( t ) has 1 as an eigen v alue. W e may easily assume that V = k [ t ] n since all the symbols under our inte r est v anish on the complemen t of V in k [ t ] n . Then all of ϕ ( t ), ψ ( t ) and θ 2 ( t ) , . . . , θ l ( t ) can b e wri tten as p olynomials of θ ( t ) with coefficients in k ( t ). Since ` ϕ (0) , θ 2 (0) , . . . , θ l (0) ´ = ` ϕ (1) , θ 2 (1) , . . . , θ l (1) ´ in K 0 ( k, G ∧ l m ) by our assumption, it foll o ws that S ϕ (0) S − 1 = ϕ (1), S θ i (0) S − 1 = θ i (1) for eve r y legitimate i , for some S ∈ GL n ( k ). No w, it is immediate that, i n K 0 ( k ∆ 1 , G ∧ l m ), „„ 0 1 − ϕ (1) ψ (1) t ` ϕ (1) + ψ (1) ´ « , „ θ 2 (1) 0 0 θ 2 (1) « , . . . , „ θ l (1) 0 0 θ l (1) «« = „„ 0 1 − ϕ (0) ψ (0) t ` ϕ (0) + ψ (0) ´ « , „ θ 2 (0) 0 0 θ 2 (0) « , . . . , „ θ l (0) 0 0 θ l (0) «« and „„ 0 1 − ϕ (1) ψ (1) t ` 1 + ϕ (1) ψ (1) ´ « , „ θ 2 (1) 0 0 θ 2 (1) « , . . . , „ θ l (1) 0 0 θ l (1) «« = „„ 0 1 − ϕ (0) ψ (0) t ` 1 + ϕ (0) ψ (0) ´ « , „ θ 2 (0) 0 0 θ 2 (0) « , . . . , „ θ l (0) 0 0 θ l (0) «« . Therefore, the pro of of the claim is complete. Thanks to the iden tities (1) and (2), w e hav e, by (8), „„ ψ ( t ) 1 0 ϕ ( t ) « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« = „„ 0 1 − ϕ ( t ) ψ ( t ) ϕ ( t ) + ψ ( t ) « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« = „„ 0 1 − ϕ ( t ) ψ ( t ) 1 + ϕ ( t ) ψ ( t ) « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« = „„ 1 1 0 ϕ ( t ) ψ ( t ) « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« . Hence ` ϕ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ + ` ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ = ` ϕ ( t ) ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ , as required.  The irreducibility assumption i n Theorem 3.3 is used only to justif y the clai m in the pro of of the theorem. The v arious conditions in the fol l o wing corollary can replace the ir reducibilit y assumption in the theorem. W e s tat e the multilinearity of symbols only i n the first co ordinate to simplify the notation, but a simi lar statemen t in another coordinate holds obviously . Coroll ary 3.4. Supp ose t hat ϕ ( t ) , ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ar e c ommuting matric es in GL n ( k [ t ]) and that the symb ols ` ϕ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ and ` ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ r epr esent elements in H l − 1 M ` Spec k , Z ( l ) ´ . Then ` ϕ ( t ) ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ r epr esents an element in H l − 1 M ` Spec k , Z ( l ) ´ and ` ϕ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ + ` ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ = ` ϕ ( t ) ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ in H l − 1 M ` Spec k , Z ( l ) ´ if one of the fol lowing assumptions is satisfie d: (i) The symb ol ` ϕ ( t ) , ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ is irr ed uci b le and k is a field of char acteristic 0 or n < c har ( k ) . (ii) Ther e exists a filtr ation 0 = V 0 ⊂ V 1 ⊂ · · · ⊂ V n = k [ t ] n of k [ t, x 0 , x − 1 0 . . . , x l , x − 1 l ] - mo dules wher e x 0 and x 1 act v ia ϕ ( t ) and ψ ( t ) and x i acts via θ i ( t ) for i ≥ 2 such that the r estriction of the symb ol ` ϕ ( t ) , ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ to ea c h V i +1 /V i ( i = 0 , . . . , n − 1 ) is irr e- ducible and k is of char acterist ic 0 or n < ch ar ( k ) . (iii) One of t he matric es ϕ ( t ) , ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) has a char acterist ic p olynomial e qual to it s minimal p olynomial. This is the c ase, for example, when one of the matric es is a co mp anion matrix of a p olynomial with c o efficient s in k [ t ] and c onstant term in k × . multilinearit y and skew-symmetry 13 Pr o of. ( i ) k ( t ) n as a k ( t ) [ x 0 , x − 1 0 . . . , x l , x − 1 l ]-mo dule, where x 0 and x 1 act via ϕ ( t ) and ψ ( t ) and x i acts via θ i ( t ) for i ≥ 2, i s ir reducible. Therefore, it is a field extension of k ( t ) of degree n . By our assumption on the field k , it is generated b y a primitive element , say θ ( t ), and all of ϕ ( t ) , ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) can b e written as p olynomials of θ ( t ) with coefficients in k ( t ). So the claim in the proof of Theorem 3.3 holds and we obtain the multilinearity . ( ii ) is an obvious consequence of ( i ). ( iii ) it true since an y matrix whi c h commutes with a give n companion matrix of a p olynomial can b e written as a p olynomial of the companion matrix. (Theorem 5 of Chapter 1 in [10])  In the following corollary , we don’t r equire the commutativit y of ϕ ( t ) and ψ ( t ). Coroll ary 3.5. Su pp ose that θ 2 ( t ) , . . . , θ l ( t ) ar e co mmuting matric es in GL n ( k [ t ]) which c om- mute also wit h ϕ ( t ) , ψ ( t ) ∈ GL n ( k [ t ]) and t hat the symb ols ` ϕ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ and ` ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ r epr esent elements in H l − 1 M ` Spec k , Z ( l ) ´ . Then ` ϕ ( t ) ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ r epr esents an element in H l − 1 M ` Spec k , Z ( l ) ´ and ` ϕ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ + ` ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ = ` ϕ ( t ) ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ in H l − 1 M ` Spec k , Z ( l ) ´ if one of the fol lowing assumptions is satisfie d: (i) ϕ (0) = ϕ (1) , ψ (0) = ψ (1) and θ i (0) = θ i (1) for i = 2 , . . . , l as matrices in GL n ( k ) . (ii) θ i (0) or θ i (1) has n distinct eigenvalues for some i = 2 , . . . , l . Pr o of. ( i ) clearly guarantees the claim in the pro of of Theorem 3. 3. ( ii ) W e may assume that none of ϕ (0) , ψ (0) , θ 2 (0) , . . . , θ l (0) has 1 as an eigen v alue. If θ i (0) has n di stinct eigenv al ues for some i , then θ i (1) also has the same n distinct eigen v alues since ( θ i (0)) = ( θ i (1)) in K 0 ( k, G ∧ 1 m ) by the assumption that ` ϕ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ belongs to H l − 1 M ` Spec k , Z ( l ) ´ . Also, eac h of ϕ (0) , ψ (0) , θ 2 (0) , . . . , θ l (0) are diagonalizable b y the same similar it y matrix by the comm utativity of the matrices with θ i (0). Let us denote the tuples of joint eigen v alues by ( a i , b i , c 2 i , . . . , c li ) for i = 1 , . . . , n . A similar statemen t is true for ϕ (1) , ψ (1) , θ 2 (1) , . . . , θ l (1) and their joi nt eigen v alues are denoted by ( a ′ i , b ′ i , c ′ 2 i , . . . , c ′ li ) for i = 1 , . . . , n . By p ermuting the indices i if necessary , we may assume that a i = a ′ i , b i = b ′ i , c j i = c ′ j i for j = 2 , . . . , l and i = 1 , . . . , n . Then the claim i n the pro of of Theorem 3.3 holds since „„ 0 1 − ϕ (1) ψ (1) t ` ϕ (1) + ψ (1) ´ « , „ θ 2 (1) 0 0 θ 2 (1) « , . . . , „ θ l (1) 0 0 θ l (1) «« = n X i =1 „„ 0 1 − a i b i t ` a i + b i ´ « , „ c 2 i 0 0 c 2 i « , . . . , „ c li 0 0 c li «« = „„ 0 1 − ϕ (0) ψ (0) t ` ϕ (0) + ψ (0) ´ « , „ θ 2 (0) 0 0 θ 2 (0) « , . . . , „ θ l (0) 0 0 θ l (0) «« and similarly „„ 0 1 − ϕ (1) ψ (1) t ` 1 + ϕ (1) ψ (1) ´ « , „ θ 2 (1) 0 0 θ 2 (1) « , . . . , „ θ l (1) 0 0 θ l (1) «« = „„ 0 1 − ϕ (0) ψ (0) t ` 1 + ϕ (0) ψ (0) ´ « , „ θ 2 (0) 0 0 θ 2 (0) « , . . . , „ θ l (0) 0 0 θ l (0) «« .  Remark 3.6. (i) In The or em 3.3, the c ommutativity of ϕ ( t ) and ψ ( t ) would not have b e en ne c es- sary if we wante d merely to define the symb ols ` ϕ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ and ` ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ . But, if we do not insist the c ommutativity of these two matrices, then ` ϕ ( t ) ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ do es not have to r e pr ese nt an element in H l − 1 M ` Spec k , Z ( l ) ´ even if the symb ols ` ϕ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ and ` ψ ( t ) , θ 2 ( t ) , . . . , θ l ( t ) ´ do. F or example, take l = 2 and let a, b ∈ k − { 0 , 1 } be two distinct numb e rs and take any c ∈ k − { 0 , 1 } . L et ϕ ( t ) = ( a + b ) t ( a + b ) 2 ab t (1 − t ) − 1 ab ( a + b )(1 − t ) ! , ψ ( t ) = „ a 0 0 b « , θ ( t ) = „ c 0 0 c « Then the bo undaries of b oth ` ψ ( t ) , θ ( t ) ´ and ` ϕ ( t ) , θ ( t ) ´ ar e 0, but the b oundary of ` ϕ ( t ) ψ ( t ) , θ ( t ) ´ is not 0 in K 0 ( k ∆ 0 , G ∧ 2 m ) . 14 S. Myung (ii) The irr e ducibi lity c ondition in The or em 3.3 or other similar assumptions in Cor ol lary 3.4 and 3.5 ar e e sse nt ial. F or example, take l = 1 and let a, b ∈ k − { 0 , 1 } b e two distinct elements. Find any distinct c, d ∈ k − { 0 , ± 1 } such that the set { a, acd, bc, bd } is not equa l to { ac, ad, b, acd } . Consider A ( t ) = 0 B B @ a 0 0 0 0 a 0 0 0 0 b 0 0 0 0 b 1 C C A , B ( t ) = 0 B B @ 0 − cd 0 0 1 ( c + d ) t + (1 + cd )(1 − t ) 0 0 0 0 0 − c d 0 0 1 ( c + d )(1 − t ) + (1 + cd ) t 1 C C A . Then A (0) = A (1) and ( B (0)) = (1) + ( cd ) + ( c ) + ( d ) = ( B (1)) in K 0 ( k, G ∧ 1 m ) . But, ( A (0) B (0)) = ( a ) + ( acd ) + ( bc ) + ( bd ) 6 = ( ac ) + ( ad ) + ( b ) + ( bcd ) = ( A (1) B (1)) in K 0 ( k, G ∧ 1 m ) and thus ( A ( t ) B ( t )) do es not r epr esent an element in H 0 M ` Spec k , Z (1) ´ Prop o sition 3.7. (Skew-Sy mmetry) Supp ose t hat θ 1 ( t ) , . . . , θ l ( t ) ∈ GL n ( k [ t ]) c ommute and one of the symb ols r epr esent e d by θ 1 ( t ) , . . . , θ l − 1 ( t ) or θ l ( t ) is irr e ducible. If ` θ 1 ( t ) , . . . , θ l ( t ) ´ r epr esents an element in H l − 1 M ` Spec k , Z ( l ) ´ ( l ≥ 2 ), then ` θ 1 ( t ) , . . . , θ i ( t ) , . . . , θ j ( t ) , . . . , θ l ( t ) ´ = − ` θ 1 ( t ) , . . . , θ j ( t ) , . . . , θ i ( t ) , . . . , θ l ( t ) ´ in H l − 1 M ` Spec k , Z ( l ) ´ . Pr o of. F or simplicity of notations, we assume that i = 1 and j = 2. Let ϕ = θ 1 and ψ = θ 2 . An argumen t si milar to the one utilized in the pro of of Theorem 3.3 can b e used to prov e that (9) „„ 0 1 − ϕ ( t ) ψ ( t ) ϕ ( t ) + ψ ( t ) « , „ 0 1 − ϕ ( t ) ψ ( t ) ϕ ( t ) + ψ ( t ) « , „ θ 3 ( t ) 0 0 θ 3 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« = „„ 0 1 − ϕ ( t ) ψ ( t ) 1 + ϕ ( t ) ψ ( t ) « , „ 0 1 − ϕ ( t ) ψ ( t ) 1 + ϕ ( t ) ψ ( t ) « , „ θ 3 ( t ) 0 0 θ 3 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« . Just replace „„ 0 1 − p ( t ) q ( t ) « , „ θ 2 ( t ) 0 0 θ 2 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« b y „„ 0 1 − p ( t ) q ( t ) « , „ 0 1 − p ( t ) q ( t ) « , „ θ 3 ( t ) 0 0 θ 3 ( t ) « , . . . , „ θ l ( t ) 0 0 θ l ( t ) «« and make similar r eplaceme nts throughout the course of the pro of of the claim in the pro of of Theorem 3.3. Then note that „„ 0 1 − ϕ (0) ψ (0) t ` ϕ (0) + ψ (0) ´ « , „ 0 1 − ϕ (0) ψ (0) t ` ϕ (0) + ψ (0) ´ « , „ θ 3 (0) 0 0 θ 3 (0) « , . . . , „ θ l (0) 0 0 θ l (0) «« = „„ 0 1 − ϕ (0) ψ (0) t ` 1 + ϕ (0) ψ (0) ´ « , „ 0 1 − ϕ (0) ψ (0) t ` 1 + ϕ (0) ψ (0) ´ « , „ θ 3 (0) 0 0 θ 3 (0) « , . . . , „ θ l (0) 0 0 θ l (0) «« to show that the righ t-hand side of an equalit y simil ar to the one as in the claim in the proof of Theorem 3.3 v anishes. T hi s prov es (9). F rom (9), we ha ve, using (1) and (2), ` ϕ ( t ) ψ ( t ) , ϕ ( t ) ψ ( t ) , θ 3 ( t ) , . . . , θ ( l ) ´ = ` ϕ ( t ) , ϕ ( t ) , θ 3 ( t ) , . . . , θ ( l ) ´ + ` ψ ( t ) , ψ ( t ) , θ 3 ( t ) , . . . , θ ( l ) ´ . On the other hand, by Theorem 3.3, we also hav e ` ϕ ( t ) ψ ( t ) , ϕ ( t ) ψ ( t ) , θ 3 ( t ) , . . . , θ ( l ) ´ = ` ϕ ( t ) , ϕ ( t ) , θ 3 ( t ) , . . . , θ ( l ) ´ + ` ϕ ( t ) , ψ ( t ) , θ 3 ( t ) , . . . , θ ( l ) ´ + ` ψ ( t ) , ϕ ( t ) , θ 3 ( t ) , . . . , θ ( l ) ´ + ` ψ ( t ) , ψ ( t ) , θ 3 ( t ) , . . . , θ ( l ) ´ . The equalit y of the r igh t hand sides of these tw o i den tities leads to the skew-symmetry .  The irr educibilit y assumption in Proposition 3.7 can be r eplaced by an assumption si milar to one of the conditions in Corollary 3.4 or 3.5. F or example, it is enough to require that the symbol ` θ 1 ( t ) , . . . , θ l ( t ) ´ is irreducible if the field k i s of cha r acteristic 0. multilinearit y and skew-symmetry 15 References [1] H. Bass and J. T ate. The Mil nor ring of a global field. In Algebr aic K -the ory, II: “Classic al” algebr aic K -the ory and c onne ctions with arithmetic (Pr o c. Conf., Se att le, Wash., Battel le Memorial Inst., 1972) , pages 349–446. Lecture Notes in Math., V ol. 342. Springer, Berlin, 1973. [2] Spencer J. Blo c h. H i gher r e gulators, algebr aic K -the ory, and zeta functions of el liptic curves . American M athemat i cal Society , Pr o vidence, RI, 2000. [3] Armand Borel. Cohomologie de sl n et v aleurs de fonctions zeta aux poi n ts entiers. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) , 4(4):613–636, 1977. [4] Daniel R. Gra yson. W eight filtrations via commut ing automorphisms. K -The ory , 9:139–172, 1995. [5] Daniel R. Grayson. The motivic spectral sequence. In Handb o ok of K - t he ory. V ol. 1, 2 , pages 39–69. Springer, Berl i n, 2005. [6] Kazuy a K ato. A generalization of lo cal class field theory by using K - gr oups. I I. J. F ac. Sc i. Univ. T okyo Se ct. IA Math. , 27(3):603–683, 1980. [7] Carlo Mazza, Vl adim ir V o ev o dsky , and Charles W eib el. Le ctur e notes on motivic co homolo gy , v ol ume 2 of Clay Mathematics M ono gr aphs . A merican Mathematical So ciet y , Providenc e, RI, 2006. [8] Sung Myung. A bilinear form of dilogarithm and motivic regulator map. A dv. Math. , 199(2):331 –355, 2006. [9] Y u. P . Nesterenk o and A. A. Susl i n. Homology of the general linear group ov er a l ocal ring, and Milnor’s K -theory . Izv. Akad. Nauk SSSR Ser. Mat. , 53(1):121–146, 1989. [10] D. A. Suprunenk o and R. I. Ty ˇ ske viˇ c Commutative Matric es . A cade mic Press, 1968. [11] A. Susli n. On the Grayson s p ectral sequence. T r. M at. Inst. Steklova , 241(T eor. Chisel, Algebra i Algebr. Geom.):218–253, 2003. [12] A. A . Suslin. Mennic ke symbols and their applications i n the K -theory of fields. In Algebr aic K -the ory, Part I (Oberwo lfach, 1980) , volume 966 of Le ctur e N ote s in Math. , pages 334–356. Springer, Berlin, 1982. [13] Andrei Suslin and Vladimir V o ev o dsky . Blo c h-Kato conjecture and motivic cohomology with finite coefficients. In The arithmetic and ge ometry of algebr aic cycles (Banff, AB, 1998) , v ol ume 548 of NA TO Sci. Ser. C Math. Phys. Sci. , page s 117–189 . Kluwe r Acad. Publ., Dordrech t, 2000. [14] Burt T otaro. Mi lnor K -theory i s the si mplest part of algebraic K -theory . K -The ory , 6(2):177– 189, 1992. [15] Mark E. W alker. Thomason’s theorem for v arieties ov er algebraically closed fields. T r ans. Am er. Math. So c. , 356(7):2569–2648 (elect r onic), 2004. Dep artment of Mathema tics Educa tion , Inha Un iversity, 2 53 Yonghyu n-dong, Nam-gu, Incheon, 402-7 51 Korea E-mail addr ess : s-myung 1@inha.ac.kr