A Hochschild-cyclic approach to additive higher Chow cycles

A HOCHSCHILD-CYCLIC APPRO A CH TO ADDITIVE HIGHER CHO W CYCLES JINHYUN P ARK Abstract. Ov er a field of characte r istic zero, we i ntroduce t wo motivic op- erations on additiv e higher Chow cycles: analogu es of the Connes boundary B operator and t he shuffle product on Ho c hschild complexes. The former allows us to apply the formalism of mixed complexes to additive Cho w com- plexes buildi ng a bridge betw een additive hi gher Chow theo ry and additiv e K -theory . The latter induces a wedg e pro duct on additiv e Chow groups for which we sho w that the Connes operator is a graded deriv ation f or the w edge product using a v ariation of a T otaro’s cy cle. Hence, the additive higher Cho w groups wi th the wedge pro duct and the Connes operator form a comm utativ e differen tial graded algebra. On zero-cycles, they induce the wedge pro duct and the exterior deri v ation on the absolute K¨ ahler differentials, answering a question of S. Blo ch and H. Esnault. Introduction There have bee n so far a t least tw o pro blems with which the a dditive higher Chow theory , defined fir st in [3, 4], had bee n shown to have connections with: motives ov er k [ x ] / ( x m ) and the Euclidea n s cissors congruence (see [5, 8, 15, 16]). This pap er provides the third such area, in particular, reg a rding the adjectiv e “additive” that originally comes fro m the a dditiv e K -theo ry . Throughout this pap er, we always suppose that k is a field of characteristic ze r o. Under this assumption the additive K -theo r y is the cyclic ho mology o f A. Connes ([7, 10]) s o that this third interpretation in this pap er should inv o lve c yclic ob jects, or more generally mixed co mplex es ( c.f. [11]) a nd their related formalisms. This pap er shows how the a dditiv e Chow cycles form such o b jects with a remark that it could be nice if we ca n rename them as Ho chschild Chow cycles, or maybe additive nonc ommutative Chow cycles to reserve the name additive Chow cycles for a new cycle complex theo ry . In § 1, w e review some relev ant prop erties of K -theor ies: Quillen K -theor y , Ho c hschild homo lo gy , a nd cyclic homolog y . The a uthor expla ins briefly why he prop oses the a bove new no menc la ture, although he still used the name add itive in this pap er. The § 2 extends the definition in [15] o f the additive Chow complexes in such a w ay that for each Artin lo cal k -algebr a ( A, m ) with A/ m ≃ k , we asso ciate a cyc le c omplex Z ∗ ( X × ♦ ∗ ( A )) for a k -v ar iet y X (see Definition 2.5). When A = k [ x ] / ( x m ), we recov er the additive Chow g roups with mo dulus considered in Date : F ebruary 5, 2008. This r esearc h was partially s upp orted by Purdue Universit y in USA and Institut des Hautes Etudes Scientifique s in F rance. I would like to thank b oth the institutions for their supports. 1 2 JINHYUN P ARK [16, 17]. This construction is necess ary for our discussion of the shuffle pro duct structure defined in § 5. Although it is not used in the sequel we show that it is cov ariant functor ial in A so that it for ms a functor Z ∗ ( X × ♦ ∗ ( − )) : ( Ar t/k ) → K om − ( Ab ). Combined with the homology functor at Z q ( X × ♦ n ( − )), we deduce a cov ar iant functor AC H q ( X, n ; − ) : ( Ar t/k ) → ( Ab ) . Thu s, the a dditiv e Cho w theo ry gives a deformation functor. ( c.f. [19]) The § 3 a nd b eyond discuss the Ho c hs ch ild-cyclic ar g umen t o n additive higher Chow cycles. The § 3 defines a motivic analogue of the Co nnes b oundar y op erator B ( c.f. [11]) deno ted by δ on the a dditive Chow complex es for A = k [ x ] / ( x m ): δ : Z p ( X, n ; A ) → Z p ( X, n + 1; A ) . Unfortunately this C o nnes op er a tor δ do esn’t b ehav e w ell with the intersection bo undary map ∂ on cy cles. After using a moving lemma argument as in [2], we obtain a qua si-isomor phic sub complex  ( Z ∗ ( X, ∗ ; A ) 0 , ∂ ′ = ∂ 0 last ) ⊂ ( Z ∗ ( X, ∗ ; A ) , ∂ )  . W e ca ll it the re duc e d additive higher Cho w complex. The opera tor δ descends to this reduced one, and works well with the bo undary map: Prop ositio n 0 .1. ∂ ′ δ = δ ∂ ′ . F or Z ( n ) := L p ≥ 0 Z p ( X, n ; A ) 0 , thu s we hav e Theorem 0.2. The r e duc e d additive higher Chow c omplex ( Z ( ∗ ) , ∂ ′ ) is a mixe d c omplex with a Connes op er ator δ : . . . ∂ ′   . . . ∂ ′   . . . ∂ ′   . . . ∂ ′   Z (3) ∂ ′   Z (2) δ o o ∂ ′   Z (1) δ o o ∂ ′   Z (0) δ o o Z (2) ∂ ′   Z (1) δ o o ∂ ′   Z (0) δ o o Z (1) ∂ ′   Z (0) δ o o Z (0) The total complex will b e called the cyclic higher Chow complex, and its homol- ogy gr oups will b e denoted b y C C H p ( X, n ; A ), the cyclic higher Chow groups. The usual formalis m of mixed co mplexes then g ive Theorem 0.3. AC H and C C H form the fol lowing Connes p erio dicity long exact se quenc e · · · B → AC H p ( n ) I → C C H p ( n ) S → C C H p − 1 ( n − 2) B → AC H p − 1 ( n − 1) I → · · · , ADDITIVE CHO W GROUPS 3 wher e AC H p ( n ) := AC H p ( X, n ; A ) and C C H p ( n ) := C C H p ( X, n ; A ) . The maps I , S , B have bide gr e es (0 , 0) , ( − 1 , − 2) , (0 , +1) in ( p, n ) , r esp e ctively. This result with the calculation C C H 0 ( k , n ; k [ x ] / ( x 2 )) ≃ Ω n − 1 k/ Z /d Ω n − 2 k/ Z (see The- orem 4.5) supp orts the author ’s p oint rega rding the usage of the w or d additive . Suppo se X = Spec ( k ). In § 5, under the iden tifica tio n A e 1 ×  r 1 × A e 2 ×  r 2 ≃ A e 1 + e 2 ×  r 1 + r 2 we firs t define the concatena tion × and the shuffle pro duct × sh of additive Chow cycles. Sp ecifically for given tw o Artin lo ca l k -algebr as A 1 and A 2 , we hav e × , × sh : Z p ( k , r 1 ; A 1 ) ⊗ Z q ( k , r 2 ; A 2 ) → Z d ( k , n ; A 1 ⊗ k A 2 ) , where d := p + q and n := r 1 + r 2 . The intersection b oundary ∂ is a gr aded deriv ation for × and × sh so that for × · = × or × sh , we have ∂ ( x × · y ) = ( ∂ x ) × · y + ( − 1) r 1 x × · ( ∂ y ) . These pro duct maps induce the homomorphisms on the a dditiv e Chow gr oups: × ·∗ : AC H p ( k , r 1 ; A 1 ) ⊗ AC H q ( k , r 2 ; A 2 ) → AC H d ( k , n ; A 1 ⊗ k A 2 ) , where × ·∗ = × ∗ or sh ∗ . When A 1 = k [ x ] / ( x m 1 ) a nd A 2 = k [ x ] / ( x m 2 ), using the pr o duct µ : G m × G m → G m , we obtain a well-defined m ultiplication µ ∗ on the image of sh ∗ in Z d ( k , n ; A 1 ⊗ k A 2 ). W e define the wedge pro duct ∧ := µ ∗ ◦ sh ∗ that gives a homomorphism ∧ : AC H p ( k , r 1 ; k [ x ] / ( x m 1 )) ⊗ AC H q ( k , r 2 ; k [ x ] / ( x m 2 )) → AC H d ( k , n ; k [ x ] / ( x m )) , where m = min { m 1 , m 2 } . When A := A 1 = A 2 = k [ x ] / ( x m ), the most interesting r esult b etw een the Connes map δ ∗ and the wedge pro duct ∧ is the following: Theorem 0.4. F or cycles ξ ∈ Z p ( k , r 1 ; A ) 0 , η ∈ Z q ( k , r 2 ; A ) 0 with ∂ ′ ξ = 0 , ∂ ′ η = 0 , we have δ ∗ ( ξ ∧ η ) − ( δ ∗ ξ ) ∧ η − ( − 1) r 1 ξ ∧ ( δ ∗ η ) = − ∂ ( ξ ∧ ′ η ) , for some cycle ξ ∧ ′ η (se e Definition 6.5), c al le d the cyclic shuffle pr o duct. That is, the Connes map δ ∗ is a gr ade d derivation for the we dge pr o duct ∧ on the additive higher Chow gr oups. Corollary 0. 5. The triple ( AC H ∗ ( k , ∗ ; A ) , ∧ , δ ∗ ) is a CDGA. In particular, when A = k [ x ] / ( x 2 ) and for 0-cycles , the CDGA (Ω ∗ k/ Z , ∧ , d ) is motivic. This answers the Challenge 5.3 of B lo c h a nd Esnault in [3] that asks for a mo tivic description of the wedge pro duct and the exterio r pr o duct o n the K¨ ahler differen- tials. 1. Additive Cho w theor y and ad ditive K -theor y Several r esults on v ar ious K -theor ies presented in this se c tio n guide our s tudies, where we use a para digm in [12] on the classification of v ario us K -theories: noncommutativ e commutativ e m ultiplicative Leibniz K -theory? Quillen K -theory additive Ho c hschild homolog y cyclic homolo g y 4 JINHYUN P ARK F or instance, we think of the Hochschild homology theory as the additive no n- commutativ e K -theory . W e will compar e the Q uillen K -theory , Ho chsc hild homol- ogy , and cyclic homology . First, recall the following res ult: Theorem 1. 1 ([14, 18, 20]) . The gr ou p homolo gy of GL ( k ) has t he fol lowing list of pr op erties: Q1) stability: H n ( GL n ( k ); Q ) ≃ → H n ( GL n +1 ( k ); Q ) ≃ → · · · ≃ → H n ( GL ( k ); Q ) Q2) 1st obstruction to stability: The fol lowing se quenc e is exact: H n ( GL n − 1 ( k ); Q ) → H n ( GL n ( k ); Q ) → K M n ( k ) → 0 Q3) primitive p art: H ∗ ( GL ( k ) , Q ) is a gr ade d Hopf algebr a with its primitive p art P r im ∗ ( H ∗ ( GL ( k ) , Q )) ≃ K ∗ ( k ) Q . Q4) 0 -cycles: The higher Chow gr oups give the obstruction gr oups in Q2) : C H 0 ( k , n ) ≃ K M n ( k ) . When w e work with the Lie algebra g l ( k ) and its Lie alge bra homo logy H n ( g l ( k )) instead of the group GL ( k ) a nd its g roup homology , w e g e t t he additive K -the ory K + n ( k ). This gro up K + n ( k ) ≃ H C n − 1 ( k ), th us the additive K -theor y is in fact the cyclic homo logy , with a shift of the degree. This theory to o enjoys analogous prop erties: Theorem 1.2 ([7, 11, 10]) . The Lie algebr a homolo gy of g l ( k ) has the fol lowing list of pr op erties: C1) stability: H n ( g l n ( k ); k ) ≃ → H n ( g l n +1 ( k ); k ) ≃ → · · · ≃ → H n ( g l ( k ); k ) C2) 1st obstruction to stability: The fol lowing se qu enc e is exact: H n ( g l n − 1 ( k ); k ) → H n ( g l n ( k ); k ) → Ω n − 1 k/ Q /d Ω n − 2 k/ Q → 0 . C3) primitive p art: H ∗ ( g l ( k ); k ) is a Hopf algebr a with its primitive p art P r im ∗ ( H ∗ ( g l ( k ); k )) ≃ K + ∗ ( k ) the additive K -the ory of F eigin and Tsygan, which is isomorphic to the cyclic homolo gy H C ∗− 1 ( k ) over Q . F rom the comparison of the Quillen K -theory (Theor em 1 .1) a nd the a dditiv e K -theor y (Theor e m 1.2), we can call the g roup Ω n − 1 k/ Q /d Ω n − 2 k/ Q the Milnor- cyclic homology H C M n − 1 ( k ). ( c.f. 10.3 .3 in [11]) It is eas y to see that for char( k ) = 0, we hav e Ω n − 1 k/ Q = Ω n − 1 k/ Z . One problem we see here is that C4) is missing from Theorem 1.2, a nd des pite the name additive higher Chow group, its ze r o-cycle gro up AC H 0 ( k , n − 1) do es not pro duce the o bstruction g roup H C M n − 1 ( k ). Tha t mea ns, althoug h the gro ups AC H are imp ortant in the calculation of the motivic cohomo logy gro ups o f the fat po in t Sp ec  k [ ǫ ] /ǫ 2  as seen in [15], the a dditiv e Chow theory is no t quite the right additive cycle theo ry for the additive K - theory . But, the Ho chsc hild homolo gy a nd the theory of Leibniz homolog y of Cuvier and Lo day ([6, 11]) show that the naming of AC H was very close; the Leibniz ADDITIVE CHO W GROUPS 5 homology is a noncommutativ e vesion o f the Lie algebra homo logy in the sense that we sys tematically ignore the antisymmetry axiom of Lie a lgebras, where we replace the Ja cobi identit y by the Leibniz rule, and w e replace the w edge pro duct ∧ by the tensor pro duct ⊗ in the Chev alley -Eilenberg r esolution of the Lie algebr a. (See p. 302 and p. 326 in [6]) Theorem 1.3 ([4, 6, 11]) . The L eibniz homolo gy H L ∗ ( g l ( k )) of g l ( k ) has t he fol- lowing list of pr op erties: H1) stability: H L n ( g l n ( k )) ≃ → H L n ( g l n +1 ( k )) ≃ → · · · ≃ → H L n ( g l ( k )) H2) 1st obstruction to stability: The fol lowing se quen c e is exact: H L n ( g l n − 1 ( k )) → H L n ( g l n ( k )) → Ω n − 1 k/ Q → 0 . H3) primitive p art: H L ∗ ( g l ( k )) is a Hopf algebr a with its primitive p art P r im ∗ ( H L ∗ ( g l ( k ))) ≃ H H ∗− 1 ( k ) , the Ho chschild homolo gy of k over Q . H4) 0 -cycles: The additive higher Chow gr oups give the obstruction gr oups in H2) : AC H 0 ( k , n − 1 ) ≃ Ω n − 1 k/ Q . W e can call the gro up Ω n − 1 k/ Q the Milnor-Ho chsc hild homology g roup H H M n − 1 ( k ) of k over Q us ing the analogy b etw een Theor em 1.1 and Theorem 1.3 as b efore. ( c.f. p. 337 in [11]) Thu s, we may susp ect that the gr oup AC H should a ctually be called the a dditiv e nonc ommutative higher Chow groups. Since the Ho chsc hild homolo gy H H n and the cyclic ho mology H C n satisfy the Connes per io dicity sequence · · · → H H n ( k ) I → H C n ( k ) S → H C n − 2 ( k ) B → H H n − 1 ( k ) → · · · , the absence of C4) suggests the following new question: Question 1.4. Can we find a c ycle co mplex, to be named as the additive c ommu- tative higher Chow c omplex or cyclic higher Chow c omplex , tha t sa tisfies at lea st the following tw o prop erties: (i) its ho mology fits into a Connes p er io dic ity exact sequence as the cyclic part, whe r e the AC H fits into the sequence as the Hochsc hild part. (ii) its zer o-cycle g roup gives the righ t obstruction g roup o f stability: Ω n − 1 k/ Z /d Ω n − 2 k/ Z . The ans wer to this que s tion is g iven in the Theorems 4.4 a nd 4.5. 2. Additive Cho w groups associa ted to A r tin local rings Let q , n ≥ 0 be in tegers, and let X b e an equi-dimensional quasi-pr o jective k - v a riety . Let ( Ar t/k ) b e the categor y of Artin lo cal k -alg ebras with the res idue field k . F or an o b ject A in ( Art/ k ), we denote by m A its unique maximal ideal. F or tw o rings A, B ∈ ( Art/ k ), a morphism A → B is a k -algebra homomorphism such that f ( m A ) = m B . F or e ach integer e ≥ 0, denote by X e , the set of l indeterminates { x 1 , · · · , x e } . 6 JINHYUN P ARK The sectio n extends the definition of the additive higher Chow groups with mo d- ulus in [1 5, 17] to the category ( Art/ k ). In this way , we obtain a functor AC H q ( X, n ; − ) : ( Ar t/k ) → ( Ab ) , where ( Ab ) is the categ ory of ab elian groups. When the Artin lo ca l ring A is k [ x ] / ( x m ) where m ≥ 2, we recover the notio n of additive Chow groups with mo dulus. This functor AC H q ( X, n ; − ) is a fun ctor of Artin rings in the sense of Schlessinger [1 9], which is sometimes ca lle d a deformation functor . The essential point of the construction is to use the em be dding dimension (s e e Definition 2.1) of a g iven Artin ring A to construct an affine space on which o ur algebraic cycles r eside. 2.1. Presen tations of Artin lo cal k -algebras. Let ( A, m A ) b e an Artin ring in ( Art/k ). Definition 2.1 . The emb e dding dimension of A is the dimensio n of the k -vector space m A / m 2 A . W e denote it by edim( A ). The following is a p olynomial version of the Cohen s tructure theo rem for Artin lo cal k -alg ebras. Lemma 2.2. L et ( A, m A ) b e an Artin ring in ( Art/k ) with emb e dding dimension e . Then ther e is a p olynomial pr esentation 0 → J → k [ X e ] → A → 0 such t hat the ide al J of k [ X e ] is c ont aine d in the ide al m 2 , wher e m is the ide al gener ate d by X e in k [ X e ] . This e is the minimal numb er of indeterminates for which A has a p olynomial pr esent ation. Pr o of. Our pro of uses the Co hen’s theo rem applied to A , that alr eady g ives us a presentation by a formal p ow er se ries ring 0 → b J → k [[ X e ]] → A → 0 (2.1) where b m is the ideal gener ated by X e in k [[ X e ]] and b J is an ideal in k [[ X e ]] satis fying b J ⊂ b m 2 . W e c ho os e a presentation with the minimal v alue of e . Since A is Artinian, there is an integer N such that b m N ⊂ b J ⊂ b m 2 . Let I N be the image of b J in k [ X e ] / m N under the is o morphism of k - algebras k [[ X e ]] / b m N ∼ → k [ X e ] / m N . Then the seq uenc e (2.1) gives a n exact s equence 0 → I N → k [ X e ] / m N → A → 0 . Let J b e the ideal in k [ X e ] whose image in k [ X e ] / m N is I N , thus we obtain a commutativ e diagra m 0 / / I N / / k [ X e ] / m N / / A / / 0 0 / / J O O / / k [ X e ] O O that immediately yields a desired pres en tation. That e is a minimal such integer is obvious.  ADDITIVE CHO W GROUPS 7 Let P resF in ( k ) b e a categor y defined in the following fashion. Its ob jects are pa irs ( k [ X e ] , J ), where J is an ideal o f the p olynomia l ring k [ X e ] such that k [ X e ] /J ∈ ( Art/k ). F or t wo ob jects ( k [ X e 1 ] , J 1 ) and ( k [ X e 2 ] , J 2 ), and tw o k -a lgebra homomorphisms φ, ψ : k [ X e 1 ] → k [ X e 2 ] such that φ ( J 1 ) ⊂ J 2 , ψ ( J 1 ) ⊂ J 2 , w e define an equiv ale nc e φ ∼ ψ if the induced homomor phisms ¯ φ, ¯ ψ : k [ X e 1 ] /J 1 → k [ X e 2 ] /J 2 are equal. Mor phisms in P resF in ( k ) a r e the collec tio n of all such k -alge br a ho mo- morphisms mo dulo the ab ove equiv alence. Lemma 2. 3. L et A 1 , A 2 b e two Artin lo c al k -algebr as in ( Ar t/k ) with pr esent ations P 1 = ( k [ X e 1 ] , J 1 ) , P 2 = (( k [ X e 2 ] , J 2 ) given by the L emma 2.2. L et f : A 1 → A 2 b e a morphism in ( Ar t/k ) . T hen, it induc es a un ique morphism ˜ f : P 1 → P 2 in P r esF i n ( k ) . F urthermor e, ther e is a natur al bije ction hom ( Ar t/k ) ( A 1 , A 2 ) ∼ → hom P r esF in ( k ) ( P 1 , P 2 ) . Pr o of. W e construct e f that fits into the diagram: 0 / / J 1   / / k [ X e 1 ] q 1 / / e f   A 1 / / f   0 0 / / J 2 / / k [ X e 2 ] q 2 / / A 2 / / 0 This is easy: fo r each i ∈ { 1 , · · · , e } , pick y i ∈ q − 1 2 f ( q 1 ( x i )) and define e f ( x i ) = y i . It determines e f . F or g ∈ J 1 , since q 1 ( g ) = 0 we have e f ( g ) ∈ q − 1 2 (0) = J 2 . Suppo se we choose a differen t s et o f y ′ i ∈ q − 1 2 ( f ( q 1 ( x i ))) and the co rresp onding e f ′ . Then, y i − y ′ i ∈ J 2 so tha t q 2 ( y i ) = q 2 ( y ′ i ) = f ( q 1 ( x i )). Hence e f a nd e f ′ induce the same map f : A 1 → A 2 . Hence in the categor y P re sF in ( k ), they are the equal morphisms. The bijection betw een the t wo sets ob v ious.  Lemma 2. 4. F or two obje cts P i = ( k [ X e i ] , J i ) , i = 1 , 2 , if ther e is an isomorph ism P 1 ≃ P 2 in P r esF i n ( k ) , then the c orr esp onding Artin rings A 1 := k [ X e 1 ] /J 1 and A 2 := k [ X e 2 ] /J 2 ar e isomorphic under the indu c e d isomorph ism. Pr o of. Suppo se that e g ◦ e f = Id P 1 and e f ◦ e g = Id P 2 for morphisms e f : P 1 → P 2 and e g : P 2 → P 1 . The maps e f and e g induce ma ps f : A 1 → A 2 and g : A 2 → A 1 . That e g ◦ e f = Id P 1 means g ◦ f = Id A 1 . Similarly we ha e f ◦ g = Id A 2 . Thus A 1 ≃ A 2 .  Thu s, we hav e an e quiv ale nce of catego r ies ι : ( Art/ k ) → P r esF in ( k ) (2.2) that s ends a n Artin loca l k -algebra A to its pres e n ta tio n P , and a morphism f : A 1 → A 2 of Artin loca l k -alge br as to a morphism e f : P 1 → P 2 defined as above. 2.2. Additiv e higher Cho w com plex. Let X b e a n irr educible qua si-pro jective v a riety ov er a field k , and let p, q , n ≥ 0 b e int egers. Let  = P 1 − { 1 } , and let P = ( X e , J ) b e an ob ject in P res F in ( k ). Define an affine space ♦ n ( P ) := A e ×  n . Let b ♦ n ( P ) = A e ×  P 1  n . F or an ir reducible closed subv ar ie t y Z ⊂ X × ♦ n ( P ), le t b Z b e its Za riski clos ure in X × b ♦ n ( P ) a nd let ν : Z → b Z b e its normalization. Let I ( P ) be the sheaf of ide a ls of O b ♦ n ( P ) generated b y X e . Let V 0 := V ( I ( P )) be the 8 JINHYUN P ARK closed subv a riety of X × b ♦ n ( P ) a sso ciated to the ideal sheaf I ( P ). F or the Artin ring A := k [ X e ] /J , we also denote then by I ( A ) and V ( I ( A )). F or ea ch i ∈ { 1 , · · · , n } and j ∈ { 0 , ∞} we have co dimension one “nondegener- ate” faces  F j i : X × ♦ n − 1 ֒ → X × ♦ n ( y , x, t 1 , · · · , t n − 1 ) 7→ ( y , x, t 1 , · · · , t i − 1 , j, t i , · · · , t n − 1 ) and the “ degenerate” face V 0 ֒ → X × ♦ n . V ar ious higher co dimension face s are obta ine d b y in ter secting the a bove faces a s well. W e asso cia te to each ob ject P = ( k [ X d ] , J ) a complex of ab elian gro ups Z q ( X × ♦ ∗ ( P )) ∈ Ko m − ( Ab ) as fo llows: Definition 2.5. Let c 0 ( X, n ; P ) be the free ab elian gro up on the set of 0-dimensional irreducible reduced closed points in X × ♦ n ( P ) not intersecting the face s . F or p ≥ 1, let c p ( X, n ; P ) b e the free a belia n g roup on the set of p -dimensional irreducible closed subv ar ie ties Z ⊂ X × ♦ n ( P ) s atisfying (1) b Z intersects all lower dimensional face s prop erly . (2) F o r each clo sed p oint p ∈ ν ∗ V 0 , there is an integer 1 ≤ i ≤ n such that 1 − t i ∈ ( J ) · O Z ,p . This second condition is called the mo dulus c ondition in t i . Let Z p ( X, n ; P ) be the group c p ( X, n ; P ) modulo the subgroup o f degenera te cycles, i.e. those obtained by pulling back cyc les on X × ♦ n − 1 ( P ) via v ar ious pro jections. The cycles in Z ∗ ( X, ∗ ; P ) are said to b e admiss ible. F or i ∈ { 1 , · · · , n } and j ∈ { 0 , ∞} , let ∂ i : Z p ( X, n ; P ) → Z p − 1 ( X, n − 1; P ) b e the int e r section pro duct with the face F j i . Let ∂ := P n i =1 ( − 1) i  ∂ 0 i − ∂ ∞ i  . It is easy to s ee that ∂ 2 = 0. Thus, we hav e a co mplex · · · ∂ → Z p +1 ( X × ♦ n +1 ( P )) ∂ → Z p ( X × ♦ n ( P )) ∂ → Z p − 1 ( X × ♦ n − 1 ( P )) ∂ → · · · called the a dditive Chow complex as so ciated to P . I f p + q = dim X + e + n , then we also write Z q ( X × ♦ n ( P )) in terms of c o dimensions. Definition 2.6. The homolog y gro up a t Z p ( X, n ; P ), deno ted by AC H p ( X, n ; P ), is called the additive Chow gr oup asso ciate d to P . F or an Artin loca l k -a lgebra in ( Art/k ), choos e a pr esentation P for A , and define AC H p ( X, n ; A ) := AC H p ( X, n ; P ). If p + q = dim X + e + n , then w e define AC H q ( X, n ; P ) = AC H p ( X, n ; P ). Lemma 2.7. Supp ose we have two isomorphic obje cts P 1 ≃ P 2 in P re sF in ( k ) . Then, the c omplex ( Z q ( X × ♦ ∗ ; P 1 ) , ∂ ) is isomorphi c t o ( Z q ( X × ♦ ∗ ; P 2 ) , ∂ ) . Pr o of. Obvious.  Lemma 2.8. When A = k , the gr oup Z p ( X × ♦ n ( k )) = 0 . In p articular, its homolo gy gr oup AC H p ( X, n ; k ) = 0 . Pr o of. The r ing k has its embedding dimension 0, thus its co rresp onding prese n ta - tion is P = ( k , 0) with m k = 0. Thus, the set V 0 is the whole X × ♦ n ( k ) = X ×  n . Hence if Z ⊂ X × ♦ n ( k ) is a n a dmissible irreducible closed subv ar iety with the normalizatio n ν : Z → b Z , then ν ∗ V 0 = Z . But, the mo dulus condition r equires ADDITIVE CHO W GROUPS 9 that for all p ∈ Z = ν ∗ V 0 we m ust hav e 1 − t i ∈ (0 ) · O Z , p , i.e. 1 = t i at mathf r ak p . Hence, we must ha ve ν ( p ) ∈ b Z \ Z for all p . But this is imp ossible unless Z = ∅ . Thu s Z p ( X, n ; k ) = 0 so that AC H p ( X, n ; k ) = 0.  2.3. F unctoriality. In this sectio n, for tw o ob jects P 1 , P 2 of P res F in ( k ) and a morphism P 1 → P 2 , we define a homomorphism Z q ( X × ♦ n ( P 1 )) → Z q ( X × ♦ n ( P 2 )). Then we s how that it giv es a functor Z q ( X × ♦ ∗ ( − )) : ( Art/k ) → Kom − ( Ab ) . Let A 1 , A 2 be tw o Artin lo cal k -algebras in ( Ar t/k ) with embedding dimensions e 1 , e 2 , resp ectively . W e define Z q ( X × ♦ n ( f )) for each f : A 1 → A 2 . It will be denoted by f ∗ when no confusion arises. There cor resp onds a k - algebra homomor- phism e f : k [ X e 1 ] → k [ X e 2 ] on the level of their minima l presentations. It induces a morphism f # : X × A e 2 ×  n → X × A e 1 ×  n . F or an irr educible clos ed subv ariety Z ⊂ X × A e 1 ×  n , cons ide r the fibre squa re Z × X × ♦ n ( A 1 ) X × ♦ n ( A 2 ) / /   X × ♦ n ( A 2 ) f #   Z / / X × ♦ n ( A 1 ) and define f ∗ ( Z ) := Z q ( X × ♦ n ( f ))( Z ) := [ Z × X × ♦ n ( A 1 ) X × ♦ n ( A 2 )], where [ ] means the a sso ciated cycle. Lemma 2. 9. F or f : A 1 → A 2 and an irr e ducible r e duc e d admissible close d subvariety Z ∈ Z q ( X × ♦ n ( A 1 )) , the cycle f ∗ ( Z ) is admissible, thus b elongs t o Z q ( X × ♦ n ( A 2 )) . Pr o of. Cho ose presentations P i = ( k [ X e i ] , J i ) for A i , i = 1 , 2. Let W ⊂ Z × X × ♦ n ( A 1 ) X × ♦ n ( A 2 ) b e an ir reducible reduced co mp onent. Consider a co mm utative diagr am W ν 1 / / φ   c W   / / X × b ♦ n ( A 2 ) f #   Z ν / / b Z / / X × b ♦ n ( A 1 ) where c W and b Z are the Zaris k i closur es and ν 1 and ν are normalization maps. The morphism φ is given by the univ er sal prop erty of the norma liz a tion ν . T o c heck the mo dulus condition, let p ∈ ν ∗ 1 V ( I ( P 2 )) b e a closed p oint. Then, φ ( p ) ∈ φ ( ν ∗ 1 V ( I ( P 2 )) = ν ∗ ( f # ( V ( I ( P 2 )))) = ν ∗ V ( I ( P 1 )) . Thu s, by the modulus condition at φ ( p ), there exis ts 1 ≤ i ≤ n such tha t 1 − t i ∈ ( J 1 ) · O Z ,φ ( p ) . Since we hav e a natural map O Z ,φ ( p ) → O W , p for which the ide a l J 1 is mapp ed int o J 2 , we have a natural map ( J 1 ) · O Z ,φ ( p ) → ( J 2 ) · O W , p . Thus, w e have 1 − t i ∈ ( J 2 ) · O W , p as desired. This pr ov es the mo dulus condition. Prop er intersection with faces is obvious since the map f # is a closed immersion whose image is at w or st one of the faces. Hence f ∗ sends a dmissible cycles to admissible cycles .  10 JINHYUN P ARK Notice that in terms of dimensions, wher e p + q = dim X + e 1 + n , the induced map f ∗ maps Z p ( X × ♦ n ( A 1 )) into Z p +( e 2 − e 1 ) ( X × ♦ n ( A 2 )). Lemma 2.10. F or n ≥ 0 , the maps f ∗ ar e c omp atible with the b oundary maps. In other wor ds, f ∗ : Z q ( X × ♦ ∗ ( A 1 )) → Z q ( X × ♦ ∗ ( A 2 )) is a morphism in Kom − ( Ab ) . Pr o of. Obvious.  The las t proper t y is the functoriality . Lemma 2.11. Consider two m orphisms A 1 f 1 → A 2 f 2 → A 3 in ( Art/k ) . Then, the induc e d morphisms of c omplexes satisfy ( f 2 ◦ f 1 ) ∗ = f 2 ∗ ◦ f 1 ∗ . In other wor ds, we have a c ovariant functor Z q ( X × ♦ ∗ ( − )) : ( Art/k ) → Kom − ( Ab ) . Pr o of. The transitivity of the fibre pro ducts implies the functoriality .  Thu s, b y comp osing the above functor with the homolog y functor H n : Ko m − ( Ab ) → ( Ab ), we o btain the following coro llary: Corollary 2.12. W e hav e the additiv e higher Chow functor AC H q ( X, n ; − ) : ( Ar t/k ) → ( Ab ) . A deta ile d study of this deformatio n functor is not pursued in this pap er, and should b e tre a ted elsewhere. 3. A motivic Connes boundar y opera tor δ 3.1. K¨ ahler differen ti als. Recall the following use ful result on the a bsolute K¨ ahler differentials: Lemma 3. 1 (Lemma 4.1 in [3]) . We have an isomorphism of k -ve ctor sp ac es φ : k ⊗ ∧ n k × R ∼ → Ω n k/ Z a ⊗ b 1 ∧ · · · ∧ b n 7→ ad log b 1 ∧ · · · ∧ d log b n , wher e R is the ve ctor subsp ac e sp anne d by al l elements of typ e ( a ⊗ a + (1 − a ) ⊗ (1 − a )) ∧ b 1 ∧ · · · ∧ b n − 1 for a ∈ k , b i ∈ k × . R emark 3.2 . What do exa ct for ms in Ω n k/ Z lo ok like? F or a generator ad log b 1 ∧ · · · ∧ d log b n − 1 ∈ Ω n − 1 k/ Z , obse r ve that d ( ad log b 1 ∧ · · · ∧ b n − 1 ) = da ∧ d log b 1 ∧ · · · ∧ d log b n − 1 = ad log a ∧ d log b 1 ∧ · · · ∧ d lo g b n − 1 = φ ( a ⊗ a ∧ b 1 ∧ · · · ∧ b n − 1 ) . This simple observ ation combined with Lemma 3.1 implies the following result. ADDITIVE CHO W GROUPS 11 Corollary 3.3. The map φ induces an isomorphism of k -vector spaces φ : k ⊗ ∧ n k × R ′ ∼ → Ω n k/ Z /d Ω n − 1 k/ Z , where R ′ is the vector subspa c e spanned by a ll elements o f type a ⊗ a ∧ b 1 ∧ · · · ∧ b n − 1 , for a ∈ k , b i ∈ k × . R emark 3.4 . F ro m the pro of of Theorem 6.4 in [4] that AC H 0 ( k , n ) ≃ Ω n k/ Z , w e know that the generators of R ′ corres p ond to the closed p oints  1 a , a, b 1 , · · · , b n − 1  ∈ A 1 ×  n − 1 . Our motivic op e rator δ in the next sectio n is motiv ated by this p oint. 3.2. The op erator δ . Let A = k [ x ] / ( x m ) for some m ≥ 2. F o r in tege r s 1 ≤ k ≤ n + 1, define rational maps δ k : A 1 ×  n · · · → A 1 ×  n +1 ( x, t 1 , · · · , t n ) 7→  x, t 1 , · · · , t k − 1 , 1 x , t k , · · · , t n  . These ratio na l maps naturally induce ho momorphisms δ k : Z p ( X × ♦ n ( A )) → Z p ( X × ♦ n +1 ( A )) . Lemma 3. 5. F or i, j ∈ { 1 , · · · , n + 1 } , we have  δ i δ j = δ j +1 δ i if i ≤ j, δ i δ j = δ j δ i − 1 if i > j. Pr o of. Straightforward.  Define the opera tor (3.1) δ := n +1 X k =1 ( − 1) k δ k : Z p ( X × ♦ n ( A )) → Z p ( X × ♦ n +1 ( A )) . Corollary 3.6. δ 2 = 0 . Thus, ( Z ∗ ( X × ♦ ∗ )( A ) , δ ) is a complex. Pr o of. It immediately fo llows from Le mma 3.5.  W e w a n t to know how δ k int e r acts with the face maps ∂ j i . Lemma 3. 7. We have the fol lowing identities: (3.2)    ∂ j i δ k = δ k − 1 ∂ j i if i < k , ∂ j i δ k = 0 if i = k , ∂ j i δ k = δ k ∂ j i − 1 if i > k . Equivalently, (3.3)  δ k ∂ j i = ∂ j i +1 δ k if k ≤ i, δ k ∂ j i = ∂ i δ j k +1 if k ≥ i. In p articular, ∂ i +1 δ i = ∂ i δ i +1 . Pr o of. T e dio us but stra ightf o rward calculations show them easily .  12 JINHYUN P ARK Unfortunately these ident ities show that ∂ and δ do not int eract very nicely . F or instance, the r eader can eas ily see that (3.4)  ∂ δ 1 = − δ 1 ∂ , ∂ δ last = δ last ∂ , but for intermediate δ k , we do not really have v e ry enlight ening interactions. T o remedy this situation, w e r eplace the additive higher Cho w co mplex ( Z ∗ ( X × ♦ ∗ ( A )) , ∂ ) by a q uasi-isomor phic sub complex. W e us e the idea s given in Lemma 4 .2.3 a nd Theorem 4.4.2 in [2]: Definition 3. 8. Define Z p ( X × ♦ n ( A )) 0 := n − 1 \ i =1 ker( ∂ 0 i ) ∩ ker( ∂ ∞ i ) ! ∩ ker( ∂ ∞ n ) . Note that ∂ 0 n = ∂ 0 last is the only no n tr ivial face map. Let’s denote this map by ∂ ′ . Cer tainly ∂ ′ 2 = 0. Then, Lemma 3.9 . The inclusion ( Z ∗ ( X × ♦ ∗ ( A )) 0 , ∂ ′ ) ⊂ ( Z ∗ ( X × ♦ ∗ ( A )) , ∂ ) is a ho- motopy e quivalenc e. Pr o of. The same arg uments a s in Lemma 4.2.3, Theorem 4.4.2 in [2] s how this lemma.  Let’s call ( Z ∗ ( X × ♦ ∗ ( A )) 0 , ∂ ′ ) t he r e duc e d additiv e higher Chow c omplex asso- ciated to A . Lemma 3. 10. δ k ( Z p ( X × ♦ n ( A )) 0 ) ⊂ Z p ( X × ♦ n +1 ( A )) 0 for al l 1 ≤ k ≤ n + 1 . Pr o of. Let x ∈ Z p ( X × ♦ n ( A )) 0 so that (3.5) ∂ j i ( x ) = 0 for  1 ≤ i ≤ n − 1 , i = n, j = ∞ . If i = n + 1 a nd j = ∞ , then we a lways hav e i ≥ k so that ∂ ∞ n +1 δ k ( x ) =  0 if i = k , δ k ∂ ∞ n ( x ) = 0 if i > k , by (3.2) and (3.5). If 1 ≤ i ≤ n , then bo th ∂ j i ( x ) and ∂ j i − 1 ( x ) ar e zer o. Thus, by (3.2) ∂ j i δ k ( x ) =    δ k − 1 ∂ j i ( x ) = 0 if i < k , 0 if i = k , δ k ∂ j i − 1 ( x ) = 0 if i > k . This finishes the pro of.  Corollary 3.11. The o per ator δ desc ends to the s ubgroups: δ : Z p ( X × ♦ n ( A )) 0 → Z p ( X × ♦ n +1 ( A )) 0 . On the level of these subgroups, the b oundar y ∂ ′ int e r acts very nicely with δ : Theorem 3. 12. The op er ator δ and ∂ ′ = ∂ 0 last satisfy δ ∂ ′ = ∂ ′ δ. ADDITIVE CHO W GROUPS 13 In p articular, t he r e duc e d additive higher Chow c omplexes form a bic omplex with two b oundary maps ( ∂ ′ , δ ) : (3.6) . . . ∂ ′   . . . ∂ ′   · · · Z p +1 ( n + 2) 0 ∂ ′   δ o o Z p +1 ( n + 1) 0 ∂ ′   δ o o · · · δ o o · · · Z p ( n + 1) 0 ∂ ′   δ o o Z p ( n ) 0 ∂ ′   δ o o · · · δ o o . . . . . . wher e Z p ( n ) 0 := Z p ( X × ♦ n ( A )) 0 . Pr o of. It follows fr om the (3.2) and (3.3). Indeed, for an y x ∈ Z r +1 ( n + 1), δ ∂ ′ ( x ) = δ ∂ 0 n +1 ( x ) = n +1 X k =1 ( − 1) k δ k ∂ 0 n +1 ( x ) = n +1 X k =1 ( − 1) k ∂ 0 n +2 δ k ( x ) = ∂ 0 n +2 n +1 X k =1 ( − 1) k δ k ! ( x ) = ∂ 0 n +2 ( δ − ( − 1) n +2 δ n +2 )( x ) . But by (3.2) we hav e ∂ 0 n +2 δ n +2 = 0 . Thus, the last expres sion is ∂ ′ δ ( x ), as desir ed.  Corollary 3.13. The δ induces a map δ ∗ : AC H p ( X, n ; A ) → AC H p ( X, n + 1; A ) . In particular, when X = Sp ec( k ), A = k [ x ] / ( x 2 ), and p = 0, the map δ ∗ : Ω n k/ Z → Ω n +1 k/ Z is identical to ( n + 1) d , wher e d is the exterior der iv atio n. Pr o of. The firs t asser tion is ob v ious. The second ass ertion follows from Theor em 6.4 of [4], Lemma 3.1, Remark 3.2, and Remark 3.4.  Recall that the maps δ k came fro m rationa l maps b etw een v ar ieties. Thus, we prov ed that Corollary 3.14. The ex terior deriv ation is motivic. 14 JINHYUN P ARK 4. Cyclic higher Chow theor y In this section we prop ose a cycle complex that b ehav es like the cyclic homolog y out of the bicomplex Z r ( n ) o f (3.6) . Note that ∂ ′ decreases bo th p a nd n by 1, whereas δ incr e a ses n by 1 but doe s not change p . If we let Z ( n ) := L p Z p ( n ) 0 , then the bicomplex Z ( n ) look s as follows: (4.1) . . . ∂ ′   . . . ∂ ′   . . . ∂ ′   . . . ∂ ′   Z (3) ∂ ′   Z (2) δ o o ∂ ′   Z (1) δ o o ∂ ′   Z (0) δ o o Z (2) ∂ ′   Z (1) δ o o ∂ ′   Z (0) δ o o Z (1) ∂ ′   Z (0) δ o o Z (0) Let B Z b e this bicomplex. A p erceptive rea der will notice that this is actually a mixed complex in the sense of A. Connes ( c.f. p. 79 in [11]). W e apply the us ual formalism of mixed co mplex es to B Z . By definition, its homology H n ( B Z ) is the homology of the first co lumn, i.e. the additive hig her Chow g roup AC H ∗ ( X, n ; A ). Its cyclic homolo gy H C n ( B Z ) is the homolog y H n ( T ot ( B Z )) o f the total complex. Notice that the bico mplex (3.6) itself is not a mixed complex, but since B Z is the direct sum of these, the gr oups H n ( B Z ) and H C n ( B Z ) have natural decomp osi- tions. Definition 4.1. The cyclic (or, add itive c ommu tative ) Chow gr o up C C H ∗ ( X, n ; A ) is the cyclic homology H C n ( B Z ) of the bicomplex B Z . The g roup C C H p ( X, n ; A ) is the dir e ct summa nd of C C H ∗ ( X, n ; A ) tha t comes from the diagonal of (3.6) that contains Z p ( n ) in the first column. R emark 4.2 . Notice that, despite the indices ( p, n ) of the gr oup C C H p ( X, n ; A ), by definition this group contains cycles not just from Z p ( n ) 0 , but from min { p, ⌊ n 2 ⌋} M i ≥ 0 Z p − i ( n − 2 i ) 0 . R emark 4.3 . If we work with the additive higher Cho w c omplex, not the re duce d one, then w e ha ve serious difficulties due to the absence of the co mm utativity of δ and ∂ . F ollowing the formalism of mixed complexe s , we hav e the long exact sequence of complexes 0 → ( Z ( ∗ ) , ∂ ′ ) I → T ot ( B Z ) S → T ot ( B Z [1 , 1 ]) → 0 . ADDITIVE CHO W GROUPS 15 Notice that T ot ( B Z [1 , 1]) = ( T ot ( B Z )) [2]. Th us , w e obta in the homology long exact s e q uence, whic h is the Connes p erio dicity exact s equence, that deco mpos es as follows: Theorem 4.4 . We have the Connes p erio dicity exact se qu enc e involving AC H and C C H : · · · B → AC H p ( n ) I → C C H p ( n ) S → C C H p − 1 ( n − 2) B → AC H p − 1 ( n − 1) I → · · · , wher e AC H p ( n ) := AC H p ( X, n ; A ) and C C H p ( n ) := C C H p ( X, n ; A ) . The maps I , S , B have bide gr e es (0 , 0) , ( − 1 , − 2) , (0 , +1) in ( p, n ) , r esp e ctively. Using the functoriality of the additive higher Chow complex for pro jective mor - phisms f : X → Y and for flat morphisms f : X → Y o f tw o v arieties X , Y of finite type ov er k (Lemma 3.6 and Lemma 3.7 of K r ishna and Levine [9]), up to a po ssible shift of indices, we can see that the above Connes per io dicit y sequence is functorial for a pr o jectiv e mo r phism and a flat morphism, up to a p o ssible shift o f degrees. Now we can provide the missing C4) of the section § 1: Theorem 4. 5. We have an isomorph ism C C H 0 ( k , n ) := C C H 0 ( k , n ; k [ x ] / ( x 2 )) ≃ Ω n k/ Z /d Ω n − 1 k/ Z . Pr o of. By definition, C C H 0 ( k , n ) = Z 0 ( k , n ) 0 ∂ ′ Z 1 ( k , n + 1 ) 0 + δ Z 0 ( k , n − 1 ) 0 . By Theo rem 6.4 in [4], we hav e Z 0 ( k , n ) 0 /∂ ′ Z 1 ( k , n + 1) 0 ≃ Ω n k/ Z , whereas by Remark 3.2, Corollary 3.3, and Remar k 3.4, elements of the gr oup δ Z 0 ( k , n − 1 ) 0 are exact for ms. This finishes the pr o of.  5. The shuffle product and the wedge pr o duct In this section w e define the shuffle pro duct str uc tur e on the cla sses of additiv e higher Cho w groups. T og e ther with the m ultiplication of the algebr aic gro up G m , we define the wedge pro duct for the additive hig he r Chow gro ups with modulus. 5.1. P ermutations. 5.1.1. Definitions. W e use the following notations: • F or in teg e rs r ≥ 0, let P erm r be the group o f p ermutations on { 1 , 2 , · · · , r } . • F or a n integer s ≥ 1 and integers p 1 , p 2 , · · · , p s ≥ 0, a ( p 1 , · · · , p s )-shuffle is a permutation σ ∈ P erm p 1 + ··· + p s such that                σ (1) < σ (2) < · · · < σ ( p 1 ) , . . . σ ( p 1 + · · · + p i + 1) < σ ( p i + 2) < · · · < σ ( p 1 + · · · + p i + p i +1 ) , . . . σ ( p 1 + · · · + p s − 1 + 1) < σ ( p 1 + · · · + p s − 1 + 2) < · · · < σ ( p 1 + · · · + p s − 1 + p s ) . W e denote by P erm ( p 1 , ··· ,p s ) the set of a ll ( p 1 , · · · , p s )-shuffles in P erm p 1 + ··· + p s . 16 JINHYUN P ARK Note that | P erm ( p 1 , ··· ,p s ) | = ( p 1 + ··· + p s )! p 1 ! ··· p s ! . Also , P erm r = P erm ( 1 , · · · , 1 | {z } r ) W e will use the double shuffles P erm ( r,s ) and triple shuffles P er m (1 ,r,s ) in this pap er. 5.1.2. Permutation actions. L e t A ∈ ( Ar t/k ) with edim( A ) = e ≥ 1. Co nsider the asso ciated space ♦ r ( A ) = A e ×  r . F or k -r ational clos ed p oints, a p ermutation σ ∈ P erm r acts natura lly via σ · ( x, t 1 , · · · , t r ) :=  x, t σ − 1 (1) , · · · , t σ − 1 ( r )  . This action naturally generalizes to all closed sub v ar ieties of ♦ r ( A ) as w ell. F ur- thermore, it se nds admissible cycles to admissible cycles in ♦ r ( A ). 5.1.3. Some pr op erties of multiple shuffles. The following lemmas on double shuffles and triple shuffles will play imp ortant r oles in the Prop o sition 6.6. It is no t nece s sary to read them now. Lemma 5.1. Permutations τ in P erm (1 ,n ) ar e in one to one c orr esp ondenc e with the numb ers { 1 , · · · , n } , wher e t he c orr esp ondenc e is given by τ ↔ τ (1) . Pr o of. Obvious.  Definition 5. 2. F or p ermutations σ ∈ P erm n and τ ∈ P erm (1 ,n ) with τ (1) = i ∈ { 1 , · · · , n } , define the permutation σ τ = σ [ i ] ∈ P erm n +1 by sending j ∈ { 1 , · · · , n + 1 } 7→    σ ( j ) if j < i, j if j = i, σ ( j − 1) if j > i . Lemma 5.3. L et σ ∈ P erm ( r,s ) and τ ∈ P erm (1 ,r + s ) . Then, the pr o duct σ τ · τ in P erm r + s +1 is a (1 , r, s ) -shuffle, i.e. σ τ · τ ∈ P erm (1 ,r,s ) . F u rthermor e, the set- the or etic map φ 1 : P er m ( r,s ) × P erm (1 ,r + s ) → P er m (1 ,r,s ) ( σ , τ ) 7→ σ τ · τ is a bije ction. Pr o of. The first statement is obvious. F or the sec o nd statement, the surjectivity part is ob vio us by keeping track of where 1 is sen t. But since b oth sides hav e the cardinality ( r + s )! r ! s ! ( r + s +1)! ( r + s )! = ( r + s +1)! r ! s ! , the map φ 1 m ust b e bijectiv e.  Lemma 5. 4. In the gr oup ring Z [ P er m r + s +1 ] , we have X σ ∈ P erm ( r,s ) (sgn( σ ))   X τ ∈ P erm (1 ,r + s ) (sgn( τ )) σ τ · τ   = X ν ∈ P e rm (1 ,r,s ) (sgn( ν )) ν . Pr o of. Note that sgn( σ τ · τ ) = s gn( σ )sgn( τ ). Thus, toge ther with the Lemma 5.3, we get the desired result.  Lemma 5.5. F or σ ∈ P erm ( r +1 ,s ) and τ ∈ P er m (1 ,r ) , t he p ermutation σ · ( τ × Id s ) is in P erm (1 ,r,s ) . F urthermor e, the set-the or etic map φ 2 : P erm ( r +1 ,s ) × P erm (1 ,r ) → P er m (1 ,r,s ) ( σ , τ ) 7→ σ · ( τ × Id s ) is a bije ction. ADDITIVE CHO W GROUPS 17 Pr o of. The first statement is obvious. F or the sec o nd statement, the surjectivity part is ob vio us by keeping track of where 1 is sen t. But since b oth sides hav e the cardinality ( r + s +1)! ( r +1)! s ! ( r +1)! r ! = ( r + s +1)! r ! s ! , the map φ 2 m ust b e bijective.  Lemma 5. 6. In the gr oup ring Z [ P er m r + s +1 ] , we have   X σ ∈ P erm ( r +1 ,s ) (sgn( σ )) σ     X τ ∈ P erm (1 ,r ) (sgn( τ ))( τ × Id s )   = X ν ∈ P e rm (1 ,r,s ) (sgn( ν )) ν. Pr o of. It follo w s immediately from Lemma 5 .5 tog ether with the observ atio n that (sgn( σ ))(sgn ( τ )) = (sg n( σ · ( τ × Id s ))) .  Lemma 5.7. F or σ ∈ P erm ( r,s +1) and τ ∈ P erm (1 ,r + s ) , the p ermu t ation σ · (Id r × τ ) is in P erm (1 ,r,s ) . F urthermor e, the set-the or etic map φ 3 : P er m ( r,s +1) × P erm (1 ,r + s ) → P er m (1 ,r,s ) ( σ , τ ) 7→ σ · (Id r , × τ ) is a bije ction. Pr o of. The pro of is similar to Le mma 5.5.  Lemma 5. 8. In the gr oup ring Z [ P er m r + s +1 ] , we have ( − 1) r   X σ ∈ P erm ( r,s +1) (sgn( σ )) σ     X τ ∈ P erm (1 ,r + s ) (sgn( τ ))(Id r × τ )   = X ν ∈ P e rm (1 ,r,s ) (sgn( ν )) ν. Pr o of. It follo w s immediately from Lemma 5 .7 tog ether with the observ atio n that ( − 1) r (sgn( σ ))(sgn(Id r × τ )) = (sg n( σ · (Id r × τ ))) .  5.2. The shuffle pro duct. Although we can develop our theory more gener ally , for simplicity of nota tions we w o rk with X = Sp ec( k ). Let ( A 1 , m 1 ) , ( A 2 , m 2 ) ∈ ( Art/k ) be tw o Ar tin lo cal k -alg ebras with edim( A i ) = e i ≥ 1. W e will always ident ify the pro duct ♦ r 1 ( A 1 ) × ♦ r 2 ( A 2 ) = A e 1 ×  r 1 × A e 2 ×  r 2 with A e 1 + e 2 ×  r 1 + r 2 . Lemma 5 .9. L et Z 1 ⊂ ♦ r 1 ( A 1 ) , Z 2 ⊂ ♦ r 2 ( A 2 ) b e admissible irr e ducible close d subvarieties. Then, their pr o duct Z 1 × Z 2 ⊂ A e 1 + e 2 ×  r 1 + r 2 is also an admissible close d subvariety in ♦ r 1 + r 2 ( A 1 ⊗ k A 2 ) . Pr o of. Out of the requirements for admiss ibility , only the mo dulus condition is less trivial. Fix presentations A 1 ≃ k [ X 1 ] /J 1 , A 2 ≃ k [ X 2 ] /J 2 as in § 2. The ring ( A 1 ⊗ k A 2 , h m 1 , m 2 i ) is in ( Ar t/k ), and A 1 ⊗ k A 2 ≃ k [ X 1 ∪ X 2 ] / h J 1 , J 2 i . Note that the em b edding dimension of A 1 ⊗ k A 2 is e 1 + e 2 . Let c Z i ⊂ A e i × ( P ) r i be the Zarisk i closure s of Z i , where i = 1 , 2. Let ν i : Z i → c Z i be their norma lizations. Recall ( c.f. Lemma 3 .1 in [9]) that the pr o duct o f tw o reduced norma l finite type k -schemes is aga in normal ov er p erfect fields. Thus, ν = ν 1 × ν 2 : Z 1 × Z 2 → c Z 1 × c Z 2 = \ Z 1 × Z 2 ⊂ A e 1 + e 2 ×  r 1 + r 2 18 JINHYUN P ARK is a nor malization of \ Z 1 × Z 2 . W e will identif y Z 1 × Z 2 with Z 1 × Z 2 . F or a closed po int p ∈ Supp ( ν ∗ V ( h m 1 , m 2 i )) in Z 1 × Z 2 , there corre spo nd closed po in ts p i ∈ Supp ( ν ∗ 1 V ( m i )) in Z i for i = 1 , 2. Thus b y the mo dulus conditions for Z 1 and Z 2 , there exist indices j 1 ∈ { 1 , · · · , r 1 } a nd j 2 ∈ { r 1 + 1 , · · · , r 1 + r 2 } such that 1 − t j 1 ∈ ( J 1 ) · O Z 1 , p 1 , 1 − t j 2 ∈ ( J 2 ) · O Z 2 , p 2 . Via the natur al maps ( J i ) · O Z i , p i → ( J i ) · O Z 1 × Z 2 , p → h J 1 , J 2 i · O Z 1 × Z 2 , p for i = 1 , 2 , we ha ve the both 1 − t j 1 , 1 − t j 2 ∈ h J 1 , J 2 i · O Z 1 × Z 2 , p . This prov es the mo dulus condition.  Definition 5.10. Extend the ab ov e pr o duct Z -bilinearly to obtain the c onc atena- tion pr o duct × = × ( r 1 ,r 2 ) : Z p ( ♦ r 1 ( A 1 )) ⊗ Z q ( ♦ r 2 ( A 2 )) → Z p + q ( ♦ r 1 + r 2 ( A 1 ⊗ k A 2 )) . Lemma 5. 11. F or x ∈ Z p ( ♦ r 1 ( A 1 )) and y ∈ Z q ( ♦ r 2 ( A 2 )) , we have ∂ ( x × y ) = ( ∂ x ) × y + ( − 1 ) r 1 x × ( ∂ y ) . Pr o of. This is ob vio us fro m the definition of ∂ .  Definition 5.12. Let d ≥ 0 be an integer. An admissible irreducible closed sub- v a riety Z ∈ Z d ( ♦ n ( A 1 ⊗ A 2 )) is said to be ( A 1 , A 2 ) -de c omp osable if for some per - m utation σ ∈ P e rm n and integers r 1 , r 2 ≥ 1 such tha t r 1 + r 2 = n , the v ar iet y σ · Z is in the imag e of the concatenation pro duct × ( r 1 ,r 2 ) . The subgroup generated b y ( A 1 , A 2 )-decomp osable cycles will b e denoted by Z d ( ♦ n ( A 1 ⊗ k A 2 )) dec ( A 1 ,A 2 ) , o r just Z d ( ♦ n ( A 1 ⊗ k A 2 )) dec if refere nc e to ( A 1 , A 2 ) is unneces s ary . Definition 5.13. Let p, q ≥ 0 b e integers. Let Z i ⊂ ♦ r i ( A i ), i = 1 , 2, be a dmissible irreducible closed s ub v ar ieties o f dimension p and q , resp ectively . Let d = p + q and n = r 1 + r 2 . Define the ( r 1 , r 2 ) -shuffle pr o duct × sh ( r 1 ,r 2 ) by Z 1 × sh ( r 1 ,r 2 ) Z 2 := X σ ∈ P erm ( r 1 ,r 2 ) sgn( σ ) σ · ( Z 1 × Z 2 ) ∈ Z d ( ♦ n ( A 1 ⊗ k A 2 )) . When the reference to ( r 1 , r 2 ) is unnecessary , we may write × sh instead of × sh ( r 1 ,r 2 ) . By extending it Z -bilinearly , we hav e the ( r 1 , r 2 )-shuffle pro duct × sh = × sh ( r 1 ,r 2 ) : Z p ( ♦ r 1 ( A 1 )) ⊗ Z q ( ♦ r 2 ( A 2 )) → Z d ( ♦ n ( A 1 ⊗ k A 2 )) . R emark 5.1 4 . Note that in gene r al we do not ha ve x × y = ( − 1 ) r 1 r 2 y × x , but we do have x × sh y = ( − 1) r 1 r 2 y × sh x . The pro of just follows fro m the definition of the double shuffles and the shuffl e pro duct. Prop ositio n 5.15 . F or the in ters ection b oundary map ∂ : Z d ( ♦ n ( A 1 ⊗ k A 2 )) → Z d − 1 ( ♦ n − 1 ( A 1 ⊗ k A 2 )) and tw o admiss ible c y cles x ∈ Z p ( ♦ r 1 ( A 1 )) and y ∈ Z q ( ♦ r 2 ( A 2 )) with p + q = d and r 1 + r 2 = n , w e have ∂ ( x × sh y ) = ( ∂ x ) × sh y + ( − 1) r 1 x × sh ( ∂ y ) . In other w or ds, the b oundar y map ∂ is a graded der iv atio n for the shuffl e pro duct × sh . Pr o of. It follows from Lemma 5 .11. This is just a matter o f r ewriting the definition of ∂ and × sh carefully midning signs.( c.f. Prop os itio n 4.2.2. on p. 123 in [11])  ADDITIVE CHO W GROUPS 19 Generally for a fixed integer d ≥ 0 and n ≥ 0, we can define the total shuffle pr o duct a s the sum of all p os s ible shuffle pr o ducts: × sh := X r 1 + r 2 = n × sh ( r 1 ,r 2 ) : M r 1 + r 2 = n M p + q = d Z p ( ♦ r 1 ( A 1 )) ⊗ Z q ( ♦ r 2 ( A 2 )) → Z d ( ♦ n ( A 1 ⊗ k A 2 )) . F or the tensor pr o duct of t wo total additive Chow complexes Z ( ♦ ∗ ( A i )) = L p ≥ 0 Z p ( ♦ ∗ ( A i )), b y the Pro po s ition 5.15 we have a homo morphism of complexes × sh : ( Z ( ♦ ∗ ( A 1 )) ⊗ Z ( ♦ ∗ ( A 2 )) , ∂ ⊗ Id + ( − 1) ∗ Id ⊗ ∂ ) → ( Z ( ♦ ∗ ( A 1 ⊗ k A 2 )) , ∂ ) . Corollary 5.16. Let d = p + q and n = r 1 + r 2 . The shuffle pro duct × sh induces homomorphisms sh ∗ : AC H p ( k , r 1 ; A 1 ) ⊗ AC H q ( k , r 2 ; A 2 ) → AC H d ( k , n ; A 1 ⊗ k A 2 ) , sh ∗ : AC H ∗ ( k , ∗ ; A 1 ) ⊗ AC H ∗ ( k , ∗ ; A 2 ) → AC H ∗ ( k , ∗ ; A 1 ⊗ k A 2 ) . R emark 5.17 . One may wonder if this sh ∗ is an iso morphism. Unfortunately to ap- ply the argument of the pr o of of the Eilenber g-Zilb er theorem ( c.f. Theor em 4.2.5. in [11] and Theorem 8 .1. in [1 3]), one needs an analo gue of the deconcatenatio n op eration on tenso r coalge br as, which is an unlikely one for cycles beca use we do not in general has a natural wa y o f decomp osing a c lo sed subv ar iety Z ⊂ A e ×  n int o the concatenation of tw o Z 1 ⊂ A e 1 ×  r and Z 2 ⊂ A e − e 1 ×  n − r even up to bo undary . 5.3. The w ed g e pro duct. Let A 1 , A 2 ∈ ( Art/k ) be Artin lo cal k -algebr as of edim( A i ) = 1. Thus, A i ≃ k [ x ] / ( x m i ) for so me m i ≥ 1 . The gro ups AC H p ( X, n ; A i ) are equal to the additive C how groups with mo dulus m i in [15, 1 7]. F or an inde- terminate w , define a k -algebr a ho momorphism φ : k [ w ] → A 1 ⊗ k A 2 ≃ k [ x, y ] / ( x m 1 , y m 2 ) , w 7→ xy . Notice that ker φ = ( w m ) where m = min { m 1 , m 2 } . Definition 5.18. Define min { A 1 , A 2 } = min { k [ x ] / ( x m 1 ) , k [ x ] / ( x m 2 ) } := k [ w ] / ( w m ). Recall tha t the linear alg e braic gro up G m has the multip lication µ : G m × G m → G m . This extends to µ : A 1 × A 1 → A 1 , and it gives a morphism of v arieties µ : A 2 ×  n → A 1 ×  n . F or a k -rationa l p oint ( x, y , t 1 , · · · , t n ) ∈ A 2 ×  n = ♦ n ( A 1 ⊗ k A 2 ), this µ induces an action µ ∗ ( x, y , t 1 , · · · , t n ) := ( xy , t 1 , · · · , t n ) ∈ A 1 ×  n . F or a general admissible clo sed sub v ar iety Z ⊂ A 2 ×  n = ♦ n ( A 1 ⊗ k A 2 ), since µ is not a clo s ed morphism (but it is an o pen morphis m), the set µ ( Z ) is not necessar y closed. Thus, we define µ ∗ ( Z ) := cl ( µ ( Z )) ⊂ A 1 ×  n where cl ( · ) means the Zar iski clo s ure in the space. O ne difficulty is that µ ∗ do es not alwa ys r e s pect the admissibility conditio ns , esp ecially the mo dulus condition. Lemma 5.19. L et Z ⊂ ♦ n ( A 1 ⊗ k A 2 ) b e an admissible irr e ducible close d subvariety. L et ( x, y , t 1 , · · · , t n ) ∈ A 2 ×  n = ♦ n ( A 1 ⊗ k A 2 ) b e the c o or dinates. L et A = min { A 1 , A 2 } . If Z is a ( A 1 , A 2 ) -de c omp osable cycle, then µ ∗ ( Z ) is admissible in ♦ n ( A ) . 20 JINHYUN P ARK Pr o of. W e may ass ume that Z is ir reducible a nd Z = Z 1 × Z 2 for some irreducible Z 1 ∈ Z p ( ♦ r 1 ( A 1 )) and Z 2 ∈ Z q ( ♦ r 2 ( A 2 )), where d = p + q a nd n = r 1 + r 2 . Note that the Zar iski closure of cl ( µ ( Z )) in A 1 ×  P 1  n is equal to the Zarisk i clo sure of µ ( Z ) in the same spa ce. Let [ µ ( Z ) b e the Zar iski clos ure, and le t ν : µ ( Z ) → [ µ ( Z ) ⊂ A 1 ×  P 1  n be its no rmalization. Cons ide r the nor malization ν ′ : Z → b Z of the Zar iski closur e of Z in A 2 ×  P 1  n . By the universality of normalization ν , we hav e a map µ : Z → µ ( Z ) that fits in to the commutativ e diag ram: Z ν ′ / / µ   b Z µ   / / A 2 ×  P 1  n µ   µ ( Z ) ν / / [ µ ( Z ) / / A 1 ×  P 1  n Let ( w , t 1 , · · · , t n ) ∈ A 1 ×  P 1  n be the co or dinates. Let p ∈ Supp ( ν ∗ { w = 0 } ) b e a closed point on µ ( Z ). Then, µ − 1 ( p ) lies in Supp( ν ′ ∗ { x = 0 } + ν ′ ∗ { y = 0 } ). Pick any p oint q ∈ µ − 1 ( p ). Since Z is ( A 1 , A 2 )- decomp osable, we have either 1 − t j 1 ∈ ( x m 1 ) · O Z 1 ,π 1 ( q ) for some 1 ≤ j 1 ≤ r 1 , or 1 − t j 2 ∈ ( y m 2 ) · O Z 2 ,π 2 ( q ) for some r 1 + 1 ≤ j 2 ≤ n, where π i are the pro jections from ♦ n ( A 1 ⊗ k A 2 ) to ♦ r i ( A i ) for i = 1 , 2. But, either case, we hav e maps ( x m 1 ) · O Z 1 ,π 1 ( q ) ( ( R R R R R R R R R R R R R ( x m 1 , y m 2 ) · O Z , q / / ( w m ) · O µ ( Z ) , p ( y m 2 ) · O Z 2 ,π 2 ( q ) 6 6 l l l l l l l l l l l l l where m = min { m 1 , m 2 } , so that the ima ge of 1 − t j 1 for some 1 ≤ j 1 ≤ r 1 or 1 − t j 2 for so me r 1 + 1 ≤ j 2 ≤ n lies in ( w m ) · O µ ( Z ) , p . This proves the mo dulus condition. Pr op er in ter sections with faces are obvious.  Lemma 5.20. The pr o duct µ ∗ : Z d ( ♦ n ( A 1 ⊗ k A 2 )) dec ( A 1 ,A 2 ) → Z d ( ♦ n ( A )) is ∂ - e quivariant, wher e A = min { A 1 , A 2 } . Thus, µ ∗ induc es a map µ ∗ : AC H d ( ♦ n ( A 1 ⊗ k A 2 )) dec ( A 1 ,A 2 ) → AC H d ( ♦ n ( A )) . Pr o of. Obvious.  Prop ositio n 5.21. Let A ∈ ( Ar t/k ) b e an Artin local k -algebr a with edim( A ) = 1. Then the map ∧ := µ ∗ ◦ sh ∗ makes AC H ∗ ( k , ∗ ; A ) a co mm utative graded alge bra. Pr o of. W e a pply the a bove discussion with A 1 = A 2 = A and consider the shuffle pro duct sh ∗ : AC H ∗ ( k , ∗ ; A ) ⊗ AC H ∗ ( k , ∗ ; A ) → AC H ∗ ( k , ∗ ; A ⊗ k A ) . ADDITIVE CHO W GROUPS 21 The map µ ∗ is not necessar ily defined on all of AC H ∗ ( k , ∗ ; A ⊗ k A ), but by Lemma 5.19, it is defined on the image of sh ∗ so that ∧ is well-defined: ∧ = µ ∗ ◦ sh ∗ : AC H ∗ ( k , ∗ ; A ) ⊗ AC H ∗ ( k , ∗ ; A ) → AC H ∗ ( k , ∗ ; A ) , where min { A, A } = A . That this is commutativ e in gra ded sense follows from Remark 5.14.  6. CDGA of additive Cho w groups In § 3, w e defined the motivic Connes b oundar y op erator δ on the re duce d a dditive Chow complex that induces δ ∗ : AC H p ( k , n ; A ) → AC H p ( k , n + 1 ; A ) , with δ 2 ∗ = 0 , when e dim( A ) = 1. O n the o ther hand, in § 5 we prov ed that ( AC H ∗ ( k , ∗ ; A ) , ∧ ) is a commutativ e gra ded alg e bra. In this section, we show that δ ∗ is actually a deriv ation fo r ∧ thus ( AC H ∗ ( k , ∗ ; A ) , ∧ , δ ∗ ) is the CDGA (commutative differential graded a lgebra). This result generalize s R ¨ ulling’s theorem [1 7] that AC H 0 ( k , ∗ ; A ) is a CDGA of generalize d de Rham-Witt complex. The central result is the construction o f the “cyclic s huffle pro duct” that gives Prop osition 6.6. This construc tio n is motiv ated by Chapter 4 of [1 1], but unlike this r eference we do not a ctually use what Loday ca lls the cyclic shuffl e pro duct, although we use this na me for a psychological r eason. In our co nstruction w e use triple sh uffles in P erm (1 ,r 1 ,r 2 ) . Recall the follo wing result from Lemma 2.5 in [16]: ( c.f. Pr op o sition 6.3 in [4]. This is a v ariatio n of a cycle of B. T ota ro in [20].) Lemma 6. 1. F or a ∈ k , b 1 , b 2 ∈ k × , define a 1 -cycles in Z 1 ( ♦ 2 ( k [ x ] / ( x 2 ))) C a, ( b 1 ,b 2 ) 2 := ( n 1 a , t, b 1 t − b 1 b 2 t − b 1 b 2  | t ∈ k o if a 6 = 0 , 0 if a = 0 . Then, ∂ C a, ( b 1 ,b 2 ) 2 =  1 a , b 1  +  1 a , b 2  −  1 a , b 1 b 2  , wher e the symb ol  1 a , b  is interpr et e d as 0 if a = 0 . The cycle C a, ( b 1 ,b 2 ) 2 is in fact in Z 1 ( ♦ 2 ( k [ x ] / ( x m ))) for all m ≥ 2. Definition 6.2. Let A = k [ x ] / ( x m ) and let p, q , r 1 , r 2 ≥ 0 b e integers. Let n = r 1 + r 2 and d = p + q . F or tw o admiss ible irreducible closed subv ar ieties Z 1 ∈ Z p ( ♦ r 1 ( A 1 )) and Z 2 ∈ Z q ( ♦ r 2 ( A 2 )), motiv ated by the ab ov e Lemma, we define the extr a-de gener ate c onc atenation Z 1 × ′ Z 2 as follows: co nsider a lo ca lly closed spa ce M ⊂ G m × G m × A 1 ×  2 defined by the collection of curves parametrized by ( x, y ) ∈ G m × G m M =  ( x, y ) ×  xy , t, y t − 1 xy t − 1  | x, y ∈ k × , t ∈ k  where we have tw o natura l pro jections M π α z z t t t t t t t t t π β $ $ I I I I I I I I I G m × G m A 1 ×  2 . 22 JINHYUN P ARK W e hav e a cross -section of π α : C : G m × G m · · · → M ( x, y ) 7→ the curve  xy , t, y t − 1 xy t − 1  | t ∈ k  . It induces C ′ : G m × G m ×  n C × ( Id  n ) − → M ×  n π β × Id  n − → A 1 ×  n +2 Define Z 1 × ′ Z 2 := the Za riski closure o f C ′ ( Z 1 × Z 2 ) in A 1 ×  n +2 . In general, w e ha ve dim( Z 1 × ′ Z 2 ) = dim Z 1 + dim Z 2 + 1. Lemma 6.3. F or t he ab ove Z 1 and Z 2 , the ext r a-de gener ate c onc atenation Z 1 × ′ Z 2 is admissible in ♦ n +2 ( A 1 ⊗ k A 2 ) Pr o of. The prop er in tersection condition is o bvious. The modulus condition follows from the co nditions for Z 1 and Z 2 .  R emark 6.4 . On k - rational p oints ( x, t 1 , · · · , t r 1 ) ∈ Z 1 and ( y , t r 1 +1 , · · · , t n ) ∈ Z 2 , we hav e ( x, t 1 , · · · , t r 1 ) × ′ ( y , t r 1 +1 , · · · , t n ) = C 1 xy , ( 1 x , 1 y ) 2 × ( t 1 , · · · , t r 1 , t r 1 +1 , · · · , t n ) . This is a 1 -cycle in Z 1 ( ♦ n +2 ( A )). Definition 6.5. Under the ab ov e assumptions, define the cyclic shuffle pr o duct ∧ ′ by Z 1 ∧ ′ Z 2 := X ν ∈ P e rm (1 ,r 1 ,r 2 ) (sgn( ν )) ν · ( Z 1 × ′ Z 2 ) ∈ Z d +1 ( ♦ n +2 ( A )) , where ν ∈ P erm (1 ,r 1 ,r 2 ) acts o n the s et { (1 , 2) , 3 , 4 , · · · , n + 2 } of ( n + 1 )-ob jects treating (1 , 2 ) as a single blo ck. W e extend it Z -bilinea rly . Prop ositio n 6.6. Let A = k [ x ] / ( x m ), and le t p , q , r 1 , r 2 ≥ 0 b e in teger s. Let d = p + q a nd n = r 1 + r 2 . Let ξ ∈ Z p ( ♦ r 1 ( A )) 0 and η ∈ Z q ( ♦ r 2 ( A )) 0 such that ∂ ′ ( ξ ) = 0 a nd ∂ ′ ( η ) = 0. Then, δ ∗ ( ξ ∧ η ) − ( δ ∗ ξ ) ∧ η − ( − 1) r 1 ξ ∧ ( δ ∗ η ) = − ∂ ( ξ ∧ ′ η ) in Z d ( ♦ n +1 ( A )) . R emark 6 .7 . Befor e we prov e the Pro po sition, observe that the map δ ∗ can b e written a s a sum ov er the set P er m (1 ,n ) of do uble shuffle p ermutations. Indeed, for a k -ra tional po in t ( x, t 1 , · · · , t n ), we have δ ∗ ( x, t 1 , · · · , t n ) = n +1 X i =1 ( − 1) i ( x, t 1 , · · · , t i − 1 , 1 x |{z} i th , t i , · · · , t n ) = − X τ ∈ P erm (1 ,n ) (sgn( τ )) τ ·  x, 1 x , t 1 , · · · , t n  . But, it is no t an element of Z [ P erm n +1 ]. Pr o of of Pr op osition 6.6. It is enough to check the identit y for k -rational p oints. Let ξ = ( x, t 1 , · · · , t r 1 ) , η = ( y , t r 1 +1 , · · · , t n ). Thus, ( ξ × η = ( x, y , t 1 , · · · , t n ) , ξ × ′ η = n xy , t, y t − 1 xy t − 1  t ∈ k o × ( t 1 , · · · , t n ) . Let’s compute eac h ter m. ADDITIVE CHO W GROUPS 23 δ ∗ ( ξ ∧ η ) = δ ∗ µ ∗ ( ξ × sh η ) = δ ∗   X σ ∈ P erm ( r 1 ,r 2 ) (sgn( σ )) σ · µ ∗ ( ξ × η )   = δ ∗   X σ ∈ P erm ( r 1 ,r 2 ) (sgn( σ )) σ · ( xy , t 1 , · · · , t n )   = X σ ∈ P erm ( r 1 ,r 2 ) (sgn( σ )) δ ∗ ( xy , t σ − 1 (1) , · · · , t σ − 1 ( n )) ) = − X σ ∈ P erm ( r 1 ,r 2 ) (sgn( σ )) X τ ∈ P erm (1 ,n ) (sgn( τ )) τ · ( xy , 1 xy , t σ − 1 (1) , · · · , t σ − 1 ( n ) ) = − X σ ∈ P erm ( r 1 ,r 2 ) (sgn( σ )) X τ ∈ P erm (1 ,n ) (sgn( τ ))( σ τ · τ ) · ( xy , 1 xy , t 1 , · · · , t n ) = −   X ν ∈ P e rm (1 ,r 1 ,r 2 ) (sgn( ν )) ν   · ( xy , 1 xy , t 1 , · · · , t n ) , where the last e quality follows fro m Lemma 5.4. ( δ ∗ ξ ) ∧ η = µ ∗   X σ ∈ P erm ( r 1 +1 ,r 2 ) (sgn( σ )) σ · (( δ ∗ ξ ) × η )   = − µ ∗ X σ ∈ P erm ( r 1 +1 ,r 2 ) (sgn( σ )) σ ·   X τ ∈ P erm (1 ,r 1 ) (sgn( τ )) τ · ( x, 1 x , t 1 , · · · , t r 1 )   × ( y , t r 1 +1 , · · · , t n ) = −   X σ ∈ P erm ( r 1 +1 ,r 2 ) (sgn( σ )) σ   ·   X τ ∈ P erm (1 ,r 1 ) (sgn( τ ))( τ × Id r 2 )   · ( xy , 1 x , t 1 , · · · , t n ) = −   X ν ∈ P e rm (1 ,r 1 ,r 2 ) (sgn( ν )) ν   · ( xy , 1 x , t 1 , · · · , t n ) , where the last e quality follows fro m Lemma 5.6. 24 JINHYUN P ARK Similarly , ( − 1) r 1 ξ ∧ ( δ ∗ η ) = −   X σ ∈ P erm ( r 1 ,r 2 +1) (sgn( σ )) σ   ·   X τ ∈ P erm (1 ,r 2 ) (sgn( τ ))(Id r 1 × τ )   · ( xy , 1 y , t 1 , · · · , t n ) = −   X ν ∈ P e rm (1 ,r 1 ,r 2 ) (sgn( ν )) ν   · ( xy , 1 y , t 1 , · · · , t n ) , where the last e quality follows fro m Lemma 5.8. Thus, δ ∗ ( ξ ∧ η ) − ( δ ∗ ξ ) ∧ η − ( − 1) r 1 ξ ∧ ( δ ∗ η ) = −   X ν ∈ P e rm (1 ,r 1 ,r 2 ) (sgn( ν )) ν   ·  ( xy , 1 x ) + ( xy , 1 y ) − ( xy , 1 xy )  × ( t 1 , · · · , t n ) . On the other hand, − ∂ ( ξ ∧ ′ η ) = − ∂   X ν ∈ P e rm (1 ,r 1 ,r 2 ) (sgn( ν )) ν   · C 1 xy , ( 1 x , 1 y ) 2 × ( t 1 , · · · , t n ) . Since ∂ j i  C 1 xy , ( 1 x , 1 y ) 2 × ( t 1 , · · · , t n )  = 0 for i ≥ 3 and j = 0 , ∞ , to prov e the equality of the Pr o po sition, we may assume that r 1 = r 2 = n = 0, in which case the set P er m (1 ,r 1 ,r 2 ) is a singleton. Thus, it rema ins to chec k that  xy , 1 x  +  xy , 1 y  −  xy , 1 xy  = ∂ C 1 xy , ( 1 x , 1 y ) 2 . But, this is indeed true by Lemma 6 .1. This finishes the pro o f.  Corollary 6.8. The Connes boundary map δ induces δ ∗ which is a deriv ation for ∧ in AC H ∗ ( k , ∗ ; A ). Thu s, we prov ed that Theorem 6.9. The t riple ( AC H ∗ ( k , ∗ ; A ) , ∧ , δ ∗ ) is a CDGA, wher e ∧ and δ ∗ ar e induc e d by algebr aic cycles. In p articular, on 0 - cycles, the we dge pr o duct and the exterior derivation for Ω ∗ k/ Z ar e motivic. In other wor ds, the CDGA  Ω ∗ k/ Z , ∧ , d  is motivic. Ac kno wl edgement I would like to thank Donu Arapur a, Spenc e r Blo ch, C.-Y. Jean Chan, Alain Co nnes, William Heinzer , Maxim K o nt s evich, Ja mes McClur e , and Bernd Ulrich for generously sha ring their time during this work. ADDITIVE CHO W GROUPS 25 References 1. Blo ch, S., Algebr aic cycles and higher K -the ory , Adv. Math. 61 (3) (1986) 267–304. 2. Blo ch, S., http://www.math .uch i cago.edu/˜ bloch/cubical c how.pdf 3. Blo ch, S. and Esnault, H., A n additive version of higher Chow gr oups , Ann. Scient. ´ Ec. Norm. Sup. 4 e s´ erie, t. 36, (2003) 463–477. 4. Blo ch, S. and Esnault, H., The additive dilo garithm , Kazuy a Kato’s fif tieth birthday , Doc. Math. (2003) Extra V ol. 131–155. 5. Cathelineau, J.-L., R emar ques sur les diff´ er entiel le des p olylo garit hmes uniformes , Ann. Inst. F ourier, Grenoble 46 (1996) no. 5, 1327–1347. 6. Cuvier, C., A lg` ebr es de L eibnitz: D´ efinitions, Pr opri ´ e t´ es , Ann. Scien t. ´ Ec. Norm. Sup., 4 e s´ erie, t. 27, (1994) 1–45. 7. F eigin, B., Ts ygan, B., A dditive K -theo ry in K -theory , arithmetic and geometry , (Mosco w, 1984–198 6), Lecture Notes in M ath. 1289, Springer, Berlin (1987), 67–209. 8. Gonc harov, A . B., Euclide an scissor c ongruenc e gr oups and mixe d T ate motives over dual numb ers , Math. Res. Lett. 11 (2004), no. 5-6, 771–784. 9. Krishna, A., Levine, M., A dditive higher Chow gro ups of schemes , ar Xiv:math.AG/0 702138 10. Lo day , J.-L. , Quillen, D., Cy clic homolo gy and t he L ie algebr a homolo gy of matric es , Comm . Math. Helv. 59 (1984), no. 4, 569–591. 11. Lo day , J.-L., Cyclic homolo gy Appendix E by Mar ´ ıa O. Ronco. Second edition. Chapter 13 b y the author in collaboration wi th T eimuraz Pirashvili. Grundlehren der Mathematisc hen Wissensc haften, 301. Springer-V erlag, Berlin, 1998. xx+513 pp. 12. Lo day , J.- L., Algebr aic K-the ory and the c onje ct ur al L eibniz K-the ory , K-theory 30 (2003) 105–127. 13. M ac Lane, S., Homolo gy , Reprint of the 1975 edition. Classics i n M athematics. Springer- V erlag, Berlin, 1995. x+422 pp. 14. N esterenk o, Y u., Susli n, A., H omolo gy of the ful l line ar gr oup over a lo c al ring and Mi lnor’s K -the ory , Izv. Ak ad. Nauk SSSR, Ser. Mat. 53, no. 1, (1989) 121-146 (Russian); English translation: M ath. USSR, Izv. 34, no. 1 (1990) 121–145. 15. Park, J., R e gulators on additive higher Chow gr oups , ar Xiv:math.AG/ 0605702v2 16. Park, J., Algebr aic cycles and additive dilo garithm , In t. Math. Res. Not. IMRN V ol 2007, no. 18, 19 pp. doi:10.1093/imrn/rnm067 17. R ¨ ulling, K., The gener alize d de Rham-Witt c omplex over a field is a co mplex of ze r o- cycles , J. Alg. Geom. 16 (2007) no. 1, 109–169 18. Susl in, A . , Homolo g y of GL n , char acteristi c classes and Milnor K -the ory , Lecture Notes in Math. 1046, Springer, Berlin (1984) 357–375. 19. Schlessinger, M., F unctor of Artin rings , T rans. Am er. Math. Soc. 130 (1968) 208–222 . 20. T otaro, B., M ilnor K -theo ry is the simplest p art of algebr aic K - the ory , K - theory 6, no. 2 (1992) 177–189 . Dep ar tment of M a thema tics, Purdue University, West Laf a y ette, Indiana 47907, USA E-mail addr ess : jinhyun@m ath.purd ue.edu; park.jinh yun@gmai l.com