Infinitesimal dilogarithm on curves over truncated polynomial rings

Let $C$ be a smooth and projective curve over the truncated polynomial ring $k_m:=k[t]/(t^m), $ where $k$ is a field of characteristic 0. Using a candidate for the motivic cohomology group ${\rm H}^{3}_{\pazocal{M}}(C,\mathbb{Q}(3))$ based on the Bloch complex of weight 3, we construct regulators to $k$ for every $m<r<2m.$ Specializing this construction, we obtain an invariant $\rho_{m,r}(f \wedge g \wedge h)$ of rational functions $f,$ $g$ and $h$ on $C.$ The current work is a twofold generalization of our work on the infinitesimal Chow dilogarithm: we sheafify the previous construction and therefore do not restrict ourselves to triples of rational functions and we construct the regulator for any $m<r<2m,$ rather than only for $m=2.$ We also define regulators of cycles, which we expect to give a complete set of invariants for the infinitesimal part of ${\rm CH}^{2}(k_{m},3). $ This generalizes Park's work, where the additive Chow cycles, namely the case of cycles close to 0, is handled for $r=m+1.$ In this paper, we generalize the reciprocity theorem to pairs of cycles which are the same modulo $(t^m)$ and for any $m<r<2m.$ We expect the theory of the paper to give regulators on categories of motives over rings with nilpotents.


Introduction
In this paper, we continue our project started in [15] and followed up in [16], which aims to give infinitesimal analogs of real analytic regulators.We first briefly explain the classical analog of our construction and then continue with explaining the contents of the paper.
If X/C is a smooth projective curve, then by the conjectural Leray-Serre spectral sequence for motivic sheaves one would expect a map Q .
1 then ρ X vanishes on the image of B 2 (C(X)) ⊗ C(X) × Q and induces the map K 3 (X) Q → R, we were looking for above.If one assumes a theory of motivic sheaves then this is the composition of ) → B 2 (C) and the Bloch-Wigner dilogarithm D 2 : B 2 (C) → R.
We will be interested in the case where C/k m is a smooth and projective curve.We denote the underlying reduced scheme of C by C. In order to state our result in the most general setting, we need an analog of the complex (1.0.1).This construction will be based on a choice P of smooth liftings of the closed points |C| of C. For different choices of liftings, we expect the complexes to be isomorphic in the derived category of complexes of sheaves.We will consider sheaves of functions on C which will satisfy certain regularity conditions with respect to c ∈ P. We will call such a function good with respect to c. Imposing this condition will allow us to define the residue of such a function along c.
For each 2 ≤ m < r < 2m, our regulator will be induced from the corresponding map on the degree 3 cohomology of the following two term complex of sheaves: [2,3].In this complex, (O C , P) × is the sheaf whose sections on an open set U are those elements of O × C,η which are c-good for c ∈ U ; B 2 (O C , P) is the sheaf associated to the presheaf whose sections on U are elements of the Bloch group on U which are c-good for c ∈ U .This construction is the precise analog of the construction of ρ X in the complex setting.Let us explain this below.
Suppose that we are given a Zariski open cover {U i } i∈I of C and a corresponding cocyle γ, given by the following data: γ i ∈ Λ 3 (O C , P) × (U i ); ε i,c ∈ B 2 (k(c)) for every c ∈ U i ; and β ij ∈ (B 2 (O C , P) ⊗ (O C , P) × )(U ij ).We will define ρ m,r (γ) ∈ k, by first making many choices and then showing that the construction is independent of all the choices.We continue with the description of this expression.
The starting point of this paper is our construction of the additive dilogarithm in [13].For a regular local Q-algebra R, letting R m := R[t]/(t m ), for every 2 ≤ m < r < 2m, we have an additive dilogarithm map ℓi m,r : B 2 (R m ) → R that satisfies all the analogous properties of the Bloch-Wigner dilogarithm function.Most importantly, the direct sums of these maps over all the possible r's give an isomorphism between the infinitesimal part of the K-group K 3 (R m ) Q and ⊕ m<r<2m R. We explain this in detail in §2 and give explicit formulas for these functions ℓi m,r .The function ℓi m,r can also be described in terms of the differential δ in the Bloch complex of B 2 (R ∞ ), with R ∞ := R[[t]], by the following commutative diagram We can then describe the first two terms in (1.0.2) as follows.For a connected, étale k malgebra (resp.k ∞ -algebra) A, there is a canonical isomorphism A ≃ k ′ m (resp.A ≃ k ′ ∞ ).Using this isomorphism for k(c), we get a canonical identification B 2 (k(c)) = B 2 (k(c) m ).Therefore, ℓi m,r (ε j,c ) ∈ k(c) is unambiguously defined using the map ℓi m,r : B 2 (k(c) m ) → k(c).Since the element γjc ∈ Λ 3 ( Ãc , c) × is assumed to be c-good, the residue res c γjc is defined as an element of Λ 2 k(c) × in the beginning of §6.Using the identification Λ 2 k(c) × = Λ 2 k(c) × ∞ and the map ℓ m,r : Λ 2 k(c) × ∞ → k(c), we define the element ℓ m,r (res c γjc ) ∈ k(c).Defining the last term res c ω m,r and proving its properties will constitute a large proportion of the paper.For any local Q-algebra R, we define a map L m,r : , by an explicit formula in (4.2.1).This should be thought of as an absolute notion and does not require that R be of dimension 1 over a field k.If R/k is a smooth, k-algebra of dimension 1, using L m,r we construct a map ω m,r : Λ 3 )}.Since we do not fix a lifting of our curve in the construction of ρ m,r , defining ω m,r on this group is not enough.More precisely, we need to extend ω m,r to the following context.Suppose that R and R ′ are smooth of relative dimension 1 over k r together with a fixed isomorphism: Ideally, we would like to extend the definition of ω m,r to a map from Λ 3 However, it turns out that for us purposes, we do not need these 1-forms themselves but only their residues and we can construct these residues independently of all the choices.Suppose that S/k m is a smooth algebra of relative dimension 1, with x a closed point and η the generic point of its spectrum.Suppose that R, R ′ /k r are liftings of S η to k r , with χ the corresponding isomorphism from R/(t m ) to R ′ /(t m ).We construct a map where k ′ is the residue field of x, which is functorial and independent of all the choices.Let χ : R → R ′ be an isomorphism of k r -algebras which is a lifting of χ.Choosing also an isomorphism R r ≃ R of k r -algebras, provides us with an identification Let ψ denote the isomorphism R → S η induced by the one from R/(t m ) to S η .Then we define res x ω m,r by the composition We prove that this composition is independent of all the choices.This statement, together with the construction of the function, takes up the whole of §4 and 5. Applying this construction in the above context, we see that γiη − δ( βjci ) and γjc are two liftings of the same object γ jc to two different generic liftings of ÔC,c .Therefore, the expression is defined.Applying traces and taking the sum over all the closed points, we obtain the expression in (1.0.2).Next we show that the sum is in fact a finite sum.The above construction involves many choices and would be completely useless if it depended on anything other than the initial data.This is the content of Theorem 7.1.1,our main theorem.Because of its basic properties that we prove below, ρ m,r deserves to be called a regulator.
In Corollary 7.2.1, we specialize to triples of functions which gives us the extension of the infinitesimal Chow dilogarithm of [13] to higher modulus.Just as in [13], this construction gives an analog of the strong reciprocity conjecture of Goncharov [6].Finally, in the last section we obtain invariants of cycles in z 2 f (k ∞ , 3) which satisfy a reciprocity property as in Theorem 7.3.2.Namely, if two cycles are the same modulo (t m ) then their invariants ρ m,r are the same.This generalizes Park's theorem [11], which was proved in the context of additive Chow groups and for r = m + 1.After the category of motives over rings with nilpotents is constructed, we expect these invariants to induce the regulators in this category.
Finally, let us describe the differences of this paper with [13], where the case m = 2 was handled.In [13], since the only possible r is 3 and hence satisfies r = m + 1, the map ω 2,3 can be defined as a map from Λ 3 (R, R ′ , χ) × → Ω 1 R/k .This is not true in general and this is why we have to pursue a different approach in this paper.Also since the formulas get quite complicated for a general m, in this paper we follow a more conceptual way of constructing res c ω m,r .We separate the construction into two parts as we described above.First we construct an absolute object L m,r which does not depend on R being of dimension 1.This is done by an explicit computation.Next in order to define ω m,r in the split case of R := R r , and with R/k smooth of relative dimension 1, we use the computation of the Milnor K-theory of truncated polynomial rings in order to express an element α ∈ im((1 The expression of ε has too constant terms so that its image under ω m,r should be 0. Therefore, we essentially use L m,r (γ) to unambiguously construct ω m,r .After showing that this construction is well-defined, we find an explicit description of it.Later in the non-split case we show that the residue of the 1-form can be unambiguously defined.The rest of the proof follows more or less along the same lines as that of [13], except that the details and the computations are more difficult.
We give an outline of the paper.In §2, we give a review of the construction in [13] of the additive dilogarithm on the Bloch group of a truncated polynomial ring.In §3, we describe the infinitesimal part of the Milnor K-theory of a local Q-algebra endowed with a nilpotent ideal, which is split, in terms of Kähler differentials.Without any doubt the results in this section are known to the experts and we do not claim originality.The reason for our inclusion of this section is first that we could not find an easily quotable statement in the full generality which we will need in our later work, and second that we found a short argument which is in line with the general setup of this paper.In §4, for a regular local Q-algebra R, we define regulators for every m < r < 2m, which vanishes on boundaries.This construction depends on the splitting of R m in an essential way.In §5, we introduce the main object of this paper: for a smooth algebra R of relative dimension 1 over k, we define regulators ω m,r : Λ 3 (R r , (t m )) × → Ω 1 R/k , for each m < r < 2m.In §6, we compute the residues of the value of ω m,r on good liftings.In §7, we use the results of the previous sections to construct the regulator from H 3 B (C, Q(3)) and specializing to triples of rational functions we obtain the infinitesimal Chow dilogarithm of higher modulus.Finally, using the infinitesimal Chow dilogarithm, we construct an invariant of codimension 2 cycles in the 3 dimensional affine space over k m .
Convention.We are interested in everything modulo torsion.Therefore, we tensor all abelian groups under consideration with Q without explicitly signifying this in the notation.For example, K M n (A) denotes Milnor K-theory of A tensored with Q etc.For an appropriate functor F, we let

Additive dilogarithm of higher modulus
In this section, we review and rephrase the theory of the additive dilogarithm over truncated polynomial rings in a manner which we will need in the remainder of the paper.Further results for this function can be found in [13].
For a Q-algebra R, let R ∞ := R[[t]], denote the formal power series over R and R m := R ∞ /(t m ) the truncated polynomial ring of modulus m over R. Since R is a Q-algebra we have the logarithm log : and 1 ≤ a then let q| a := 0≤i<a q i t i ∈ R ∞ , denote the truncation of q to the sum of the first a-terms, and t a (q) := q a , the coefficient of t a in q.If u ∈ tR ∞ and s(1 for m < r < 2m.Fixing m ≥ 2, these ℓi m,r 's, for m < r < 2m together constitute a regulator for the infinitesimal part of the K-group K 3 (R m ) (2) exactly analogous to the Bloch-Wigner dilogarithm in the complex case [1], [12], [13].
For any ring A, we let A ♭ := {a ∈ A|a(1 − a) ∈ A × }.Since every element of R ♭ ∞ can be written in the form se u as above, we can linearly extend ℓi m,r , to obtain a map from the vector space We denote this map by the same symbol.For a local Q-algebra R, let B 2 (R) is the Q-space generated by the symbols [x], with x(1−x) ∈ R × , modulo the subspace generated by computes the weight 2 motivic cohomology of R, when R is a field.When we would like to specify the δ defined on B 2 (R ∞ ) (resp.B 2 (R m )) we denote it by δ ∞ (resp.δ m ).
Let V be a free R module with basis {e i } i∈I and {e ∨ i } i∈I the dual basis of V ∨ .Given v and α = i∈I a i e i in V, we let If there is an ordering on I, we let {e i ∧ e j } i>j be the corresponding basis of Λ 2 V.Then, with the above notation, the expression (w|β), for w, β ∈ Λ 2 V, is defined.We consider tR ∞ , as a free R-module with basis {t i } 1≤i .
Let us denote the composition of and this function descends through the canonical projections With the notation ℓ i (α) := t i (log • (α)), ℓi * m,r can be rewritten as Then we have ℓi * m,r (se u ) = ℓi * m,r (se u|m ), since we know that ℓi * m,r descends to Q[R ♭ m ].We have ℓ i (se u|m ) = u i , for 1 ≤ i < m and ℓ i (se u|m ) = 0, for m ≤ i.Using this we obtain that ℓi * m,r (se Let us give a name to the essential map which constitute ℓi m,r .
Definition 2.0.2.We denote the map from Λ The additive dilogarithm above is given in terms of this function as We will use the main result from [13], there it was stated in the case when R is a field of characteristic 0, but the same proof works when R is a regular, local Q-algebra.
• computes the infinitesimal part of the weight two motivic cohomology of R m , and the map ⊕ m<r<2m ℓi m,r induces an isomorphism from the relative cyclic homology group HC • 2 (R m ) (1) to R ⊕(m−1) .

Infinitesimal Milnor K-theory of local rings
Suppose that R is a local Q-algebra and A is an R-algebra, together with a nilpotent ideal I such that the natural map R → A/I is an isomorphism.Then the Milnor K-theory K M n (A) of A, naturally splits into a direct sum In this section, we will describe this infinitesimal part K M n (A) • in terms of Kähler differentials.It is easy to find such an isomorphism using Goodwillie's theorem [7], and standard computations in cyclic homology.However, in the next section, we need an explicit description of this isomorphism in order to determine which symbols vanish in the corresponding Milnor K-group.Fortunately, determining what this isomorphism turns out to be quite easy.By the functoriality and the multiplicativity of the isomorphism, we reduce the computation to the case of K M 2 of the dual numbers over R where the computation is easy.
There is no doubt that the results in this section are well-known and we do not claim any originality.We simply have not been able to find a description of the map ϕ below which is easily quotable in the literature.Since our discussion is quite short we did not refrain from including it in the present paper.We will only need the result below for A = R m .On the other hand, in a future work we will need this result in full generality which justifies our somewhat more general discussion: Proposition 3.0.1.There exists a unique map ϕ : and this map is an isomorphism.
Proof.The uniqueness follows since the infinitesimal part of Milnor K-theory is generated by terms {α, We define a functorial map ϕ by the following composition: The first map is the multiplicative map induced by the isomorphism when n = 1, the second one is the Goodwillie isomorphism [7], and the last one is given by [9,Theorem 4.6.8].
By Nesterenko-Suslin's theorem [10], Milnor K-theory is the first obstruction to the stability of the homology of general linear groups: [10].This implies the injectivity of ϕ.It only remains to prove the property (3.0.1), since then the surjectivity of ϕ also follows.
The multiplicativity of ϕ takes the following form: for a ).We do induction on n.The statement is clear for n = 1.We show that we may assume that β i ∈ R × : Lemma 3.0.2.Suppose that we have the formula (3.0.1) for α ∈ 1 + I and β i ∈ R × , for 1 ≤ i ≤ n − 1, then we have the same formula for α ∈ 1 + I and Proof.We do induction on the number of β i which are not in R × .If all of them are in R × , the hypothesis of the lemma gives the expression.If there is at least one β i which is not in R × , without loss of generality assume that By the multiplicativity of ϕ, the formula for n = 1, and the induction hypothesis on n, we have By the induction hypothesis on the number of β i not in R × , we have Adding these two expressions, we obtain the expression we were looking for.
The above lemma shows that we may without loss of generality assume that the β i ∈ R × .The next lemma shows that we may also assume that A = R r and α = 1 + t.Lemma 3.0.3.Suppose that we have the formula (3.0.1) for α = 1 + t and ).Then we have the same formula for any A as above.
Proof.Given α ∈ 1 + I ⊆ A × and β i ∈ R × .Since α − 1 is nilpotent, we have an R-algebra morphism ψ : R r → A, for some r, such that ψ(t) = α − 1.The result then follows by the functoriality of ϕ since the map induced by ψ maps {1+t, Next we show that we can also assume that r = 2. Lemma 3.0.4.Suppose that we have the formula (3.0.1) for α = 1 + t and Then we have the same formula for any A as above.
Proof.We need to prove the result for 1 + t ∈ R r , and 1+t) , it is a product of elements of the form e at m , for 1 ≤ m < r, and a ∈ Q. Therefore it is enough to prove the formula for elements as above with α = e at m .
Since the element . Therefore, without loss of generality, we will assume that r = m + 1.Then we use the map from R 2 to R m+1 that sends t to at m .This map sends 1+t to e at m and hence maps {1+t, Therefore, again by the functoriality of ϕ, the result follows from the assumption on R 2 .
To finish the proof, we will need a special identity in K M 2 (R 3 ) : Lemma 3.0.5.We have the following relation in Proof.It is possible to give a direct computational proof of this statement.We choose to give a proof which is based on the ideas in this section.First suppose that R is a field.We know that both sides are in , is an isomorphism.
The left hand side goes to t 2 dλ λ , whereas the right hand side goes to This proves the statement when R is a field.In general, the statement for Q[x, x −1 ] implies the one for a general R by sending x to λ.Finally, if we can show that where the injectivity of ϕ was proven above.This finishes the proof of the lemma.
Finally, we prove the result for R 2 .
In the case of truncated polynomial rings, we can also describe this isomorphism as follows: given by Their sums induce an isomorphism: be given by µ i (w) := res t=0 is an isomorphism.The corollary then follows from Proposition 3.0.1.

Construction of maps from
In this section, we will deal with the absolute case, i.e. we will not assume the existence of a basefield k over which R is smooth of relative dimension 1.We will only assume that R is a regular, local Q-algebra.Note that we have a complex where the first map sends [x] to [x] ⊗ x and the second one sends [x] ⊗ y to δ(x) ∧ y.Abusing the notation, we will denote all the differentials in this complex by δ.The first group when divided by the appropriate relations is generally denoted by B 3 (R m ) and the corresponding sequence sequence obtained is a candidate for the weight 3 motivic cohomology complex generalizing the Bloch complex of weight 2. This complex and its variants are defined and studied in detail in [5].Over the dual numbers of a field, this complex and its higher weight analogs, still called the Bloch complexes, were used in [14] to construct the additive polylogarithms.
Let us put the above cohomological complex in degrees [1,3] and denote its cohomology after tensored with Q as H i (R m , Q(3)) in degrees i = 2 and 3.

4.1.
Preliminaries on the construction.In this section, we fix m and r such that 2 ≤ m < r < 2m.We let f (s, u) As in the proof of Proposition 2.0.1, we define ℓ i : R × ∞ → R, by the formula ℓ i (a) := t i (log • (a).Let us consider the expression Summing these, we find that Let Du := 1≤i du i t i and u t := ∂u ∂t .Then the last expression can be rewritten as We would like to see that the coefficient of , the last expression is 0. Therefore α j (δ(se u )) does not depend on the du i 's, and can rewrite (4.1.2) as where f s = ∂f ∂s .Lemma 4.1.2.If u = u| m and m ≤ j < r, we have Proof.The expression jt j (f ) − st j−1 (f s u t ) is equal to the above expression is equal to t j−1 ( s s−1 u t ), which is 0, under the assumption that u = Let dℓ 0 : R × ∞ → Ω 1 R be defined as dℓ 0 (α) := d log(α(0)).Note that ℓ 0 itself is not defined, even though ℓ i are defined for i > 0. There is an action of R × on the R-algebra R ∞ by scaling the parameter t.More precisely, for λ ∈ R × the corresponding automorphism is given by sending t to λt.We call this the ⋆-action following [3].For any functor Proof.That M m,r is of ⋆-weight r follows immediately from the expression for α j (δ(se u )) in Lemma 4.1.1,which shows that α j (δ(se u )) is of ⋆-weight j.
Let us now show that M m,r evaluated on by Lemma 4.1.2,since u = u| m .The final expression can be rewritten as since u = u| m .This proves the first part of the proposition.When r = m + 1, M m,r takes the form Since all the functions in this expression depend on the classes of the elements in R m , the statement easily follows.

The regulator maps from H
based on the maps M m,r in Proposition 4.1.3.The problem with M m,r is that it does not descend to a map on B 2 (R m ) ⊗ R × m in general.We will modify M m,r slightly to correct this defect but keep the other properties to obtain L m,r .
In order to simplify the notation from now on we are going to let ℓi m,m := 0. Note that ℓi m,r was previously defined only when m + 1 ≤ r ≤ 2m − 1 so this will not cause any confusion.We define β m (j), for m ≤ j < 2m − 1, by We would like to emphasize that, because of our conventions, the summand that corresponds to j = m is equal to r−m m α m ⊗ ℓ r−m exactly as in the case of M m,r , the terms corresponding to m < j are modified however.
Proof.Since all the terms in the definition of L m,r depend on the variables modulo t m , we obtain a map from Since we know that M m,r is of ⋆-weight r, in order to prove that L m,r is of ⋆-weight r, it suffices to prove the same for L m,r − M m,r .This difference is equal to the sum of Therefore the term (4.2.3) is of ⋆-weight r.
Similarly, (r This implies that the term (4.2.2) is of ⋆-weight r and finishes the proof of the lemma.
which by restriction gives the regulator map H 2 (R m , Q(3)) → Ω 1 R of ⋆-weight r we were looking for.We continue to denote these two induced maps by the same notation L m,r .
Proof.We know that M m,r vanishes on the boundary δ(se u ) of elements se u ∈ Q[R ∞ ], with u = u| m , by Proposition 4.1.3.We also know by the previous lemma that L m,r descends to a map on B 2 (R m ) ⊗ R × m .Therefore, in order to prove the statement, we only need to prove that L m,r (δ(se u )) = M m,r (δ(se u )), for u = u| m .We first rewrite L m,r as the composition of δ ⊗ id with On the other hand, recall that M m,r is the composition of δ ⊗ id with If we compare the two expressions we see that all of the terms match above except possibly the ones that correspond to the triples The following is a direct consequence of Lemma 4.2.1 which we record for later reference.
Corollary 4.2.3.We have a commutative diagram We expect that the above maps combine to give an isomorphism between the infinitesimal part of the cohomology of R m and the direct sum of the module of Kähler differentials, justifying the name of the regulator, c.f. [5,Conjecture 1.15].However, at this point, we can only prove the surjectivity: Proposition 4.2.4.Suppose that R is a regular local Q-algebra and 2 ≤ m as above.The direct sum of the L m,r induce a surjection: R .This implies the surjectivity.

Construction of the maps from
For a ring A and ideal I, let (A, I) × := {(a, b)|a, b ∈ A × , a − b ∈ I}, and let π i : (A, I) × → A × , for i = 1, 2 denote the two projections.In this section we will define a map ω m,r : Λ 3 (R r , (t m )) × → Ω 1 R/k , which will fundamentally depend on the assumption that R/k is a smooth, k-algebra of relative dimension 1.
5.1.Description of the construction.Assume that R/k is a smooth, k-algebra of relative dimension 1.Using the map L m,r , we will construct a map ω m,r : Λ r , be the map given by s The map is well-defined only after we pass to the quotient consisting of relative differentials.

5.2.
Proof that Ω m,r is well-defined.In this section, we will justify the construction we made in the above paragraph in several steps.We assume that R is a smooth local k-algebra of relative dimension 1.
The infinitesimal part of the cokernel of the map R via the map from Λ 3 R × r , whose i-th coordinate is given by res t=0 1 t i d log(y 1 ) ∧ d log(y 2 ) ∧ d log(y 3 ), (5.2.1) by Corollary 3.0.7.Further, by the assumption on smoothness of dimension 1, we conclude that the natural map and under the coordinate j maps in (5.2.1) with j = i, the image is 0. Together with the above, this shows that the map is surjective and hence proves the first statement.
For the second statement, note that if α ∈ im((1 R lands in the summand ⊕ m≤i<r t i Ω 2 R .Since by the above discussion, we also see that the composition is surjective, the second statement similarly follows.
Note that the map L m,r : obtained by composing L m,r with the projection from Ω 1 R to Ω 1 R/k are of ⋆-weight r.Since the ⋆-weights of H 2 (R r , Q(3)) are expected to be between r + 1 and 2r − 1 as we mentioned in the paragraph preceding Proposition 4.2.4,one would expect the following lemma: Lemma 5.2.2.The composition Proof.This follows immediately from Corollary 4.2.3.
Remark 5.2.3.We emphasize that in the above lemma our assumption is that δ r (γ) = 0 in Λ 3 R × r and not the weaker assumption that δ m (γ| t m ) = 0.In fact, there are elements such that δ m (γ| t m ) = 0 and L m,r (γ| t m ) = 0.
Let us compute L m,r ((α 0 ⊗ λ)| t m ).Since ℓ i (λ) = 0, for 0 < i by the formula for L m,r we see that Taking the sum of expressions such as above, we deduce that if r such that δ r (α) = α and L m,r (α| t m ) = 0. Applying this to α := δ r (γ) we deduce that there exists α Proof.Suppose that γ is as in the statement of the lemma.Fix some a ∈ Z >1 .We inductively define γ [i] as follows.Let γ [−1] = γ, and and therefore the previous lemma implies that L m,r (γ These lemmas together finish the proof that Ω m,r is well-defined as follows.Starting with α ∈ I m,r , we know, by Lemma 5.2.1, that there exists ε ∈ im((1 The following proposition gives an explicit expression for Ω m,r .
Proposition 5.2.6.Suppose that x ≥ m and x + y + z = r, with y or z possibly 0, then Here when y = 0, we let e b denote an arbitrary element of R × and yb denote 0 and db denote d log(e b ) = d(e b ) e b .Note that when y = 0, the expression b does not make sense.Similarly, for zc and dc when z = 0.
(ii) Case when y = 0 and z = 0.In this case we try to compute the image of e at x ∧ e bt y ∧ γ under Ω m,r .Here we assume that γ ∈ R × and x + y = r, with x ≥ m.By exactly the same argument as above, we deduce that there exists α ∈ B • 2 (R r ) such that δ r (α) = e at x ∧ e bt y and we have This exactly coincides with the expression in the statement of the proposition.
(iii) Case when y = 0 and z = 0. Note that, by localizing, we may assume that R is local.Moreover, since both sides of the expression are linear in a, b and c, we may assume without loss of generality that a, b, c ∈ R × .Since any element in a local ring can be written as a sum of units.
If θ := e at x ∧ e bt y then its image in x ∧ e bt y )), which only has a non-zero component in degree x + y equal to yb • da − xa • db.
If we compute the image of ϕ := x x+y e abt x+y ∧ b − y x+y e abt x+y ∧ a in the same group, we obtain the same element.Therefore θ − ϕ lies in the image of B 2 (R r ).Suppose that γ 0 ∈ B 2 (R r ) such that δ(γ 0 ) = θ − ϕ.Since e abt x+y ∧ e ct z has weight r, there is We now write By the definition of Ω m,r , we have Ω m,r (e at x ∧ e bt y ∧ e ct z ) = By the same argument, L m,r (ε In order to compute L m,r (γ 0 | t m ⊗ e ct z ), first note that, by the definition of L m,r , we have Combining all of these gives, Ω m,r (e at x ∧ e bt y ∧ e ct z ) = This finishes the proof of the proposition.
Definition 5.2.7.We define ω m,r : Λ Behaviour of ω m,r with respect to automorphisms of 2m−1 which are identity modulo (t m ).In this section, we will show the invariance of Ω m,r with respect to reparametrizations of R r that are identity on the reduction to R m .In order to do this, we will need to make an explicit computation on k ′ ((s)) ∞ , where k ′ is a finite extension on k.In order to make the formulas concise and intuitive, we will use several notational conventions as follows.If a ∈ k ′ ((s)), we let a ′ = a (1) ∈ k ′ ((s)) denote its derivative with respect to s and a (n+1) = (a (n) ) ′ .Similarly, we let e a denote an arbitrary non-zero element in k ′ ((s)) and a ′ = a (1) := (e a ) ′ e a and a (n+1) = (a (n) ) ′ .This notation is intuitive in the sense that, if one thinks of a as log(e a ) then a ′ is the logarithmic derivative of e a .With these conventions, we will state the following basic lemma.Lemma 5.3.1.Let σ be the automorphism of the k ∞ algebra k ′ ((s)) ∞ , which is the identity map modulo (t) and has the property that σ(s) = s + αt w , with w ≥ 1 and α ∈ k((s)), then σ(e at x ) = e 0≤i α i a (i) Proof.First note that such an automorphism should be identity on k ′ since k ′ /k is étale, the map is identity on k and is also identity from The proof is then separated into two cases, when x = 0 and when x = 0.In both cases, the statement follows from the Taylor expansion formula.Lemma 5.3.2.Let σ be the automorphism of the k ∞ algebra k ′ ((s)) ∞ given by σ(s) = s + αt w , with m ≤ w and identity modulo (t).Then we have, σ(e at i ∧ e bt j ∧ e ct k ) e at i ∧ e bt j ∧ e ct k = 0, Proof.Since m ≤ w, 0 < i + j + k and m < r < 2m, the weight r terms of σ(e at i ∧e bt j ∧e ct k ) e at i ∧e bt j ∧e ct k are possibly non-zero only when i + j + k + w = r and in this case they are given by e αa ′ t i+w ∧ e bt j ∧ e ct k + e at i ∧ e αb ′ t j+w ∧ e ct k + e at i ∧ e bt j ∧ e αc ′ t k+w .
By the formula in Proposition 5.2.6, the above sum is sent to Proof.This follows by the corresponding statement for Ω m,r .This in turn reduces to Lemma 5.3.2 after localizing and completing.Definition 5.3.4.If R/k r is a smooth k r -algebra of relative dimension 1.We defined the map Let τ : R r → R be a splitting, that is an isomorphism of k r -algebras which is the identity map modulo (t).By transport of structure, this gives a map R/k .Suppose that τ ′ is another such splitting which agrees with τ modulo (t m ).Applying Corollary 5.3.3 to τ ′ −1 • τ, we deduce that ω m,r,τ = ω m,r,τ ′ .Therefore, if σ : R m → R/(t m ) is a splitting of the reduction R/(t m ) of R then ω m,r,σ is unambiguously defined as ω m,r,τ , where τ is any splitting of R that reduces to σ modulo (t m ).
Proof.By the definition of ω m,r,σ , we easily reduce to the split case where R = R r .We need to prove that Ω m,r vanishes on the following two types of elements: where f , f ∈ R ♭ have the same reduction modulo (t m ) and ĝ, g ∈ R × the same reduction modulo (t m ).By the definition of Ω m,r , its value on the first and the second expressions are respectively: 5.4.Behaviour of res(ω m,r,σ ) with respect to automorphisms of R m .In order to proceed with our construction, we need an object such as the 1-form in [15] which controls the effect of changing splittings.This object in ⋆-weight r will be constructed below by using ω m,r .On the other hand, this objects does depend on the choice of splittings if these splittings are different modulo (t m ), when r > m+ 1.In the modulus m = 2 case the only possible r is 3 so this situation does not occur in [15].In the current case of higher modulus, we will see that the residues of the 1-form ω m,r is invariant under the automorphisms of R m which are identity modulo (t), which will imply that the residue can be defined independent of various choices.We will see that this will be enough for constructing the Chow dilogarithm of higher modulus.We will again start with an explicit computation on k ′ ((s)) ∞ .
Then Ω m,r σ(e at x ∧e bt y ∧e ct z ) e at x ∧e bt y ∧e ct z is equal to Otherwise, Ω m,r σ(e at x ∧e bt y ∧e ct z ) e at x ∧e bt y ∧e ct z = 0.
Proof.First note that by the above lemma t z+iw e at x ∧ e bt y ∧ e ct z and hence if r − (x + y + z) ≤ 0 or w ∤ r − (x + + z) then R does not have a component of weight r and Ω m,r (R) = 0. Suppose then that r − (x + y + z) > 0, w|(r − (x + y + z)) and let q := r−(x+y+z) w as in the statement of the proposition.In this case the weight r term of R is given as We first claim that the expression above does not depend on w.The coefficient of w k! α q ds in the same expression is Therefore Ω m,r (R) can be rewritten as The coefficient of α ′ α q−1 ds in the above expression is equal to which agrees with the coefficient of α ′ α q−1 ds in (5.4.1).Fix i 0 , j 0 , and k 0 such that i 0 + j 0 + k 0 = q.Then the coefficient of yα q a (k0) b (i0) c (j0+1) in (5.4.1) is equal to which is exactly the same as the coefficient of the same term in (5.4.2).By symmetry, we deduce the same statement for the coefficients of zα q a (k0) b (i0+1) c (j0) .This finishes the proof of the proposition.
Corollary 5.4.2.Suppose that σ and e at x ∧ e bt y ∧ e ct z are as above.If r = m + 1, then Ω m,r σ(e at x ∧e bt y ∧e ct z ) e at x ∧e bt y ∧e ct z = 0.
Proof.In this case in order to have m ≤ x and (m + 1) − (x + y + z) = r − (x + y + z) > 0, we have to have x = m and y = z = 0.In this case, (5.4.1) is equal to 0.
Corollary 5.4.3.If R/k m+1 is a smooth k m+1 -algebra of relative dimension 1 as above, then for r = m + 1, we have a well-defined map as in Definition 5.3.4,which does not depend on the choice of a splitting of R/(t m ).
Proof.This follows immediately from Corollary 5.4.2, by reducing to the case R = k ′ ((s)) m+1 , after localising and completing.
For a general r between m and 2m, the following corollary will be essential.
Corollary 5.4.4.Fix m < r < 2m, and let R/k r be a smooth algebra of relative dimension 1 as above.Let x be a closed point of the spectrum of R, k ′ its residue field, and let η be the generic point of R. Then for any two splittings σ and σ ′ of R η /(t m ), the reduction modulo (t m ) of the local ring of R at η, and for any α ∈ Λ 3 (R η , (t m )) × , the residues of ω m,r,σ (α) and ω m,r,σ ′ (α) ∈ Ω 1 R η /k at x are the same: Proof.Again by localising and completing we reduce to the case of k ′ ((s)) r .By Proposition 5.4.1, we see that the difference ω m,r,σ ′ (α) − ω m,r,σ (α) is the differential of an element in k ′ ((s)) and hence has zero residue.
Remark 5.4.5.Let R/k r be as above.Suppose that τ and σ are two splittings R m → R/(t m ).
In this case, there should be a map Moreover, hω m,r (τ, σ) should vanish on the image of B 2 (R, (t m )) ⊗ (R, (t m )) × .
In case r = m + 1, hω m,m+1 = 0 does satisfy the properties above.Let us look at the first non-trivial case when m = 3 and r = 5.Note that the reduction modulo (t 2 ) of the automorphism τ −1 •σ : R 3 → R 3 , which lifts the identity map on R, is determined by a k-derivation θ : R → R.
Define hΩ 3,5 (θ) : where a, b, ∈ R and c ∈ R × .Let hΩ 3,5 (θ) be defined as 0 on all the other type of elements in I 3.5 .Then hω 3,5 (τ, σ) : Λ 3 (R, (t 3 )) × → R defined by satisfies the desired properties above.An analog of this construction is one of the main tools in defining an infinitesimal version of the Bloch regulator in [16].
Definition 5.4.6.Let R/k r be a smooth algebra of relative dimension 1 as above.Let η be the generic point and x be a closed point of the spectrum of R. Then we have a canonical map where k ′ is the residue field of x.The map is defined by choosing any splitting σ of R η /(t m ) and letting res x ω m,r := res x ω m,r,σ .This is independent of the choice of the splitting σ, by Corollary 5.4.4.Suppose that S/k m is a smooth algebra of relative dimension 1, with x a closed point and η the generic point of its spectrum.Suppose that R, R ′ /k r are liftings of S η to k r .In other words, we have fixed isomorphisms: ψ : R/(t m ) → S η and ψ ′ : R ′ /(t m ) → S η .
Letting χ := ψ ′ −1 • ψ, we would like to construct a map where k ′ is the residue field of x.Note that (R, R ′ , χ) × consists of pairs of (p, p ′ ) with p ∈ R × and p ′ ∈ R ′ × such that ψ(p| In other words, it consists of different liftings of elements of S × η .We sometimes use the notation (R, R ′ , ψ, ψ ′ ) × to denote the same set.In order to construct this map, let χ : R → R ′ be an isomorphism of k r -algebras which is a lifting of χ.This provides us with a map (R, R ′ , χ) × χ * / / (R, (t m )) × .
Choosing a splitting σ : R m → R/(t m ), by Definition 5.3.4we obtain the map ω m,r,σ , composing this with the map induced by the reduction ψ of ψ, we obtain If ψ and ψ ′ are clear from the context, we denote this map by res x ω m,r , and (R, R ′ , ψ, ψ ′ ) × by (R, R ′ , (t m )) × .Depending on the context, we also use the notation res x ω m,r (χ) : Λ 3 (R, R ′ , χ) → k ′ for the same map, with χ = ψ ′−1 • ψ.
With these definitions, we have the following corollary.
Corollary 5.5.3.Suppose that R and R ′ are smooth k r -algebras of dimension 1 as above which are liftings of the generic local ring S η of a smooth k m -algebra S. Let χ : R/(t m ) → R ′ /(t m ) be the corresponding isomorphism of k m -algebras.Let x be a closed point of S. Then the map Proof.Follows from Proposition 5.3.5.
6.The residue of ω m,r on good liftings.
Suppose that R/k r is as above.Moreover, we assume that the reduction R of R modulo (t) is a discrete valuation ring with x being the closed point.We let x/k r be a lifting of x to R.
By this what we mean is as follows.Let s be a uniformizer at x, and let s be any lifting of s to R, we call s also a uniformizer at x on R. The associated scheme x, which is smooth over k r , is what we call a lifting of x.In other words a lifting of x is a 0-dimensional closed subscheme x of R such that its ideal is generated by a single element which reduces to a uniformizer on the closed fiber.Note that if we are given x, then s is determined up to a unit in R. Sometimes we will abuse the notation and write (s) instead of x.Let η denote the generic point of R. We let (R, x) × := {α ∈ R × η |α = us n , for some u ∈ R × and n ∈ Z}.We say that an element α ∈ R × η is good with respect to x, if α ∈ (R, x) × .Note that this property depends only on x, and not on s.The importance of this notion for us is that for wedge products of good liftings, we can define their residue along (s) as in [15, §2.4.5].Namely, there is a map with the properties that it vanishes on Λ n R × and s ∧ α Suppose that R ′ /k r is another such ring, and x′ a lifting of the closed point of R ′ .Suppose that there is an isomorphism χ : R/(t m ) → R ′ /(t m ) which transfers x/(t m ) to x′ /(t m ).Then we let Note that clearly (R, R ′ , x, x′ , χ) × ⊆ (R, R ′ , χ) × .In case R ′ = R with χ the identity map, we denote the corresponding group by (R, x, (t m )) × .Denote the natural maps (R, R ′ , x, x′ , χ) × → (R, x) × and (R, R ′ , x, x′ , χ) × → (R ′ , x′ ) × by π 1 and π 2 .
In this section, we would like to compute res x ω m,r (χ)(α) for α ∈ Λ 3 (R, R ′ , x, x′ , χ) × in terms of the value of ℓ m,r on the residue of α.The main result of this section is Proposition 6.0.3.We will first start with certain explicit computations on the formal power series rings and then finally reduce our general statement to these special cases.Let us immediately remark that in order to compute the residues, we immediately reduce to the case when R and R ′ are complete with respect to the ideal which correspond to their closed points.
We will first consider the case of R = k ′ [[s]] and that of the same uniformizer on both of the liftings as follows.
Note that where α and β are the images of α and β under the natural projection R × → (R/(s) Similarly for p ′ .Lemma 6.0.1.
Proof.By Definition 5.2.7, we see that ω m,r (p, Let us compute the residue at s = 0 of an expression of the type Ω m,r (s ∧ e at i ∧ e bt j ), with a, b ∈ k[[s]] and i ≥ m, such that if j = 0, we use the convention in Proposition 5.2.6.Since Ω m,r (s ∧ e at i ∧ e bt j ) = jab ds s , when i + j = r and is 0 otherwise, by Proposition 5.2.6, we conclude that its residue is equal to ja(0)b(0) if i + j = r, and is 0 otherwise.Since, for i ≥ m, ℓ m,r (res (s) (s ∧ e at i ∧ e bt j )) = ℓ m,r (e a(0)t i ∧ e b(0)t j ) is equal to ja(0)b(0) if i + j = r, and is 0 otherwise, we conclude that res s=0 (Ω m,r (s ∧ e at i ∧ e bt j )) = ℓ m,r (res (s) (s ∧ e at i ∧ e bt j )).(6.0.1)On the other hand, since α ′ | t m = α| t m , s ∧ α α ′ ∧ β is a sum of terms of the above type, and the linearity of both sides of (6.0.1) imply that (6.0.1) is also valid for s ∧ α α ′ ∧ β.By linearity of ℓ m,r and res (s) , we have , which together with the above proves the lemma.
Let us now try to prove the same formula when the choice of the uniformizer is not the same.In other words, with notation as above let s ′ ∈ R such that s ′ | t m = s| t m .For simplicity, let us temporarily use the notation (R, (s), (s ′ ), (t m )) Lemma 6.0.2.With notation as above, the residue of ω m,r (p, p ′ ) at the closed point of k ′ [[s]] is given by the following formula: Therefore in order to prove the lemma we may assume without loss of generality that s ′ = s+ at i , with a ∈ k ′ [[s]] and m ≤ i.Note that in R, we have s+at i = se if i + j + k = r and 0 otherwise, with the usual conventions if j or k is 0.
On the other hand, res (s) (p) = e b(0)t j ∧ e c(0)t k and By the linearity of ℓ m,r , the right hand side of the expression in the statement of the lemma is then equal to The last summand is equal to 0 since ℓ m,r is of weight r and i + j + i + k ≥ 2i ≥ 2m > r.For the same reason, the first two summands are 0 if i + j + k = r and if i + j + k = r, then the total expression is equal to −a(0)c ′ (0)jb(0) + a(0)b ′ (0)kc(0), which agrees with the formula (6.0.2) for the residue of Ω m,r .Since α and β are sums of the terms of the above type, this proves the lemma.
Proposition 6.0.3.Suppose that R, R ′ /k r are local algebras which are smooth of relative dimension 1 over k r , together with liftings x, x′ of their closed points and a k m -isomorphism χ : R/(t m ) → R ′ /(t m ) which maps the reductions of x and x′ to each other.Then for q ∈ Λ 3 (R, R ′ , x, x′ , χ) × , we have the following formula for the at the closed point x, Note that it does not make sense to require that γiη be a good lifting since in this context there is no a fixed specialization of the generic point.Similarly, we cannot require that βjci,η be a good lifting, since we know that δ( βjci,η ) is a lifting of δ(β jci,η ) = γ i − γ jc and even this last expression need not be good at c as γ i need not be good at c.
We define the value of the regulator ρ m,r on the above element by the expression For the second term, note that, as above, there is a canonical isomorphism k(c) ≃ k(c) m of k m algebras using which we can view ε jc,c ∈ B 2 (k(c) m ).Applying ℓi m,r : B 2 (k(c) m ) → k(c) to this element gives ℓi m,r (ε jc,c ) ∈ k(c).For the last term, note that γiη − δ( βjci,η ) is a lifting of γ iη − δ(β jci,η ) = γ jc to Λ 3 Ã× η and so is γjc a lifting of γ jc to Λ 3 Ã× c .Using the theory of §5.5, we see that the last term res c ω m,r (γ iη − δ( βjci,η ), γjc ) ∈ k(c) is unambiguously defined.Letting Tr k denote the normalized trace to k, the summands above are defined.
In order to show that the sum makes sense, we also need to show that the sum is finite.Below we will show that the sum is independent of all the choices, therefore it will be enough to show that the sum is finite for a particular choice.First by shrinking U i if necessary, and choosing a refinement of the cover, we will assume that γ i ∈ Λ 3 O × C (U i ).Similarly, by shrinking U i even further, we will assume that the lifting γi is good on U i .Therefore, for c ∈ U i , we can choose j c = i and γjc = γi .Since for these c, β jci = 0 we can choose βjci = 0.In order to show that the sum in (7.1.1)is finite, we can concentrate on c ∈ U i , as |C| \ |U i | is finite.For c ∈ U i , res c γjc = res c γi = 0, since γ i is invertible on U i by assumption.Also for the residues we have res c ω m,r (γ iη − δ( βjci,η ), γjc ) = res c ω m,r (γ i , γi ) = 0 since i = j c , γjc = γi and βjci = 0. Therefore the summand, for c ∈ U i , is equal to Tr k (−ℓi m,r (ε jc,c )) = Tr k (−ℓi m,r (ε i,c )).Since ε i,c = 0, for all but a finite number of c ∈ U i , we are done.
We now show that the expression makes sense and is independent of all the choices.Note that there are many of them.Proof.We first show the independence of the definition from the various choices.For readability, we separate these into parts.
Independence of the choice of j c and the liftings βjci and γjc .Suppose that we choose a different j ′ c with c ∈ U j ′ c ; a different lifting Ã′ c of ÔC,c , together with c′ as above; a c′ -good lifting γ′ , where s is a choice of a uniformizer associated to c and similarly for Ã′ c , we choose and fix a k ∞ -algebra isomorphism between Ãc and Ã′ c which is identity modulo (t m ) and which sends s to s′ .This last condition is possible to impose since both s and s′ lift s by assumption.Below we identify these two algebras using this isomorphism.
We need to compare the two expressions of [15, §4], we do not go into the details and explain certain constructions in a slighty alternate way.
First, let us recall the definition of cubical higher Chow groups over a smooth k-scheme X/k [2].Let k := P 1 k \ {1} and n k the n-fold product of k with itself over k, with the coordinate functions y 1 , • • • , y n .For a smooth k-scheme X, we let n X := X × k n k .A codimension 1 face of n X is a divisor F a i of the form y i = a, for 1 ≤ i ≤ n, and a ∈ {0, ∞}.A face of n X is either the whole scheme n X or an arbitrary intersection of codimension 1 faces.Let z q (X, n) be the free abelian group on the set of codimension q, integral, closed subschemes Z ⊆ n X which are admissable, i.e. which intersect each face properly on n X .For each codimension one face F a i , and irreducible Z ∈ z q (X, n), we let ∂ a i (Z) be the cycle associated to the scheme Z ∩ F a i .We let ∂ := n i=1 (−1) n (∂ ∞ i − ∂ 0 i ) on z q (X, n), which gives a complex (z q (X, •), ∂).Dividing this complex by the subcomplex of degenerate cycles, we obtain Bloch's higher Chow group complex whose homology CH q (X, n) := H n (z q (X, •)) is the higher Chow group of X.
In order to work with a candidate for Chow groups of cycles on k m , we need to work with cycles over k ∞ which have a certain finite reduction property.Let k := P 1 k , n k , the n-fold product of k with itself over k, and f (k ∞ , 3) → k as the composition l m,r • ∂.Exactly as in [15], one proves that the regulator above vanishes on boundaries and products, is alternating and has the same value on cycles which are congruent modulo (t m ).We state only this last property, which is the most important one, in detail.Suppose that Z i for i = 1, 2 are two irreducible cycles in z 2 f (k ∞ , 3).We say that Z 1 and Z 2 are equivalent modulo t m if the following condition (M m ) holds: (i) Z i /k ∞ are smooth with (Z i ) s ∪ (∪ j,a |∂ a j Z i |) a strict normal crossings divisor on Z i .and (ii) Then we have: 3), for i = 1, 2, satisfy the condition (M m ), for some m ≥ 2, then they have the same infinitesimal regulator value: ρ m,r (Z 1 ) = ρ m,r (Z 2 ), for every m < r < 2m.
Proof.The proof is exactly as in [15] and is based on Corollary 7.2.1.
As we remarked above, we expect the invariants ρ m,r for m < r < 2m to give a full set of invariants in the infinitesimal part of a yet to be defined Chow group CH 2 (k m , 3).

Proposition 4 . 1 . 3 .
The map M m,r defined as

Lemma 4 . 2 . 1 .
With the above definition, L m,r defines a map from B 2 (a, b, c) with 1 ≤ a, b, c, a + b + c = r, and m ≤ a or m ≤ b.By anti-symmetry, we may assume without loss of generality that m ≤ a.We need to compare the coefficients of the terms dℓ a ∧ ℓ b ⊗ ℓ c , ℓ a ∧ dℓ b ⊗ ℓ c , and ℓ a ∧ ℓ b ⊗ dℓ c , subject to the above constraints, in L m,r and M m,r .The coefficient of dℓ a ∧ ℓ b ⊗ ℓ c in L m,r and M m,r are both equal to − cb a+b .The coefficient of ℓ a ∧ dℓ b ⊗ ℓ c in L m,r is −cb a+b and in M m,r , it is ca a+b .Finally, the coefficient of ℓ a ∧ ℓ b ⊗ dℓ c in L m,r is b, whereas in M m,r it is 0. We finally note that the values of ℓ a ∧ dℓ b ⊗ ℓ c and ℓ a ∧ ℓ c ⊗ dℓ b on δ(se u ) ⊗ se u are the same when u = u| m .Then the equality −cb a+b + c = ca a+b finishes the proof.
in the part of kernel of the δ • which is of ⋆-weight r.By Theorem 2.0.3, this part is isomorphic to R via the restriction of the map ℓi m,r .Computing the value of L m,r on α ⊗ b, for b ∈ R × , we see that L m,r (α ⊗ b) = ℓi m,r (α) db b .Since α ⊗ b is in the kernel of δ, we see that the image of L m,r above is the additive group generated by the set Rd log

Corollary 5 . 3 . 3 .
Let σ be any automorphism of R r as a k r -algebra, which reduces to identity on R m , then ω m,r • Λ 3 σ = ω m,r .

5. 5 .
Variant of the residue map for different liftings.For the construction of the infinitesimal Chow dilogarithm, we need a variant of Definition 5.4.6.Fortunately, we do not need to do extra work, Corollary 5.3.3 and Proposition 5.4.1 will still be sufficient to give us what we are looking for.Suppose that A is a ring with an ideal I and B and B ′ are two A-algebras together with an isomorphism χ : B/IB ≃ B ′ /IB ′ of A-algebras.We let(B, B ′ , χ) × := {(p, p ′ )|p ∈ B × and p ′ ∈ B ′× s.t.χ(p| I ) = p ′ | I },where p| I denotes the image of p in (B/IB) × .Similarly, we define (B, B ′ , χ) ♭ and B 2 (B, B ′ , χ) and obtain maps,B 2 (B, B ′ , χ) → Λ 2 (B, B ′ , χ) × and B 2 (B, B ′ , χ)⊗(B, B ′ , χ) × → Λ 3 (B, B ′ , χ) × .We will use these definitions below with A = k ∞ and I = (t m ).In fact the following variant will be essential in what follows.

.5. 1 ) 1 . 5 . 5 . 2 .
Proposition 5.5.1.The map(5.5.1)  above is independent of the choices of the lifting χ of χ and the choice of the splitting σ of R/(t m ).Proof.That the composition is independent of the choice of χ follows from Corollary 5.3.3 and Definition 5.3.4.That it is independent of the choice of the splitting σ follows from Proposition 5.4.Definition We denote the composition (5.5.1) above by a s t i , since r < 2m.Letting α = e bt j and β = e ct k , we can rewrite ω m,r (p, p ′ ) as Ω m,r (p − p ′ ) = Ω m,r (e − a s t i ∧ e bt j ∧ e ct k ) = − a s (jb • dc − kc • db) by Proposition 5.2.6, if i + j + k = r and 0 otherwise.Its residue is −a(0)(jb(0)c ′ (0) − kc(0)b ′ (0)) (6.0.2)

η whose closure p in 2 k∞
as the subgroup generated by integral, closed subschemes Z ⊆ n k∞ which are admissible in the above sense and have finite reduction, i.e.Z intersects each s × F properly on n k∞ , for every face F of n k∞ .Here s denotes the closed point of the spectrum of k ∞ and for a subscheme Y ⊆ n k∞ , Y denotes its closure in n k∞ .Modding out by degenerate cycles, we have a complexz q f (k ∞ , •).Fix 2 ≤ m < r < 2m.Let η denote the generic point of the spectrum of k ∞ .An irreducible cycle p in z 2 f (k ∞ , 2) is given by a closed point p η of 2 does not meet ({0, ∞}× k∞ )∪( k∞ ×{0, ∞}).Let p denote the normalisation of p and T denote the underlying set of the closed fiber p × k∞ s of p.For every s ′ ∈ T, and 1 ≤ i, define ℓ p,s ′ ,i : Ô× p,s ′ → k(s ′ ) by the formula:ℓ p,s ′ ,i (y) := 1 i res p,s ′ 1 t i d log(y).Let l m,r (p) := s ′ ∈T Tr k 1≤i≤r−m i • (ℓ p,s ′ ,r−i ∧ ℓ p,s ′ ,i )(y 1 ∧ y 2 ).(7.3.1)Note the similarity with Definition 2.0.2.Definition 7.3.1.We define the regulator ρ m,r : z 2 by the same symbol.
denoted by the same notation.