It is expected by Beilinson-Soule vanishing conjecture that negative motivic cohomology groups are zero so the motivic complexes are supported on nonnegative degrees. My question is about characteristic zero. Is there any conjectural constructions of the motivic complex that is naturally supported on non-negative degrees?

After studying the work of Goncharov I realized he has conjectured a construction of motivic complex for regular schemes that is supported on the positive degrees. Below I will explain construction. My question is:

• Why is it defined the way it is (assuming for fields it is the correct motivic complex)? Is it obvious that this complex satisfies excision?

Goncharov conjectures a complex of the following form (omitting some details on how to define the differentials) rationally coincides with the weight $n$ motivic complex for all infinite fields $F$:

$$\mathcal{B}_n(F)\xrightarrow{\delta} \mathcal{B}_{n-1}\otimes F^*\xrightarrow{\delta}\mathcal{B}_{n-2}\otimes \wedge^2F^*\rightarrow \ldots \rightarrow \mathcal{B}_2(F)\otimes \wedge^{n-2}F^*\xrightarrow{\delta}\wedge^nF^*$$

Here $\mathcal{B}_i(F)$ are generalization of Bloch group and are defined by explicit generator and relations. The group $\mathcal{B}_n(F)$ is located at degree 1. More details can be found in "Geometry of Configurations, Polylogarithms and Motivic Cohomology" by Goncharov.

Now here is the puzzling part. It further is conjectured in the paper that for a regular scheme $X$ the weight $n$ motivic complex should be the total complex associated to the following bi-Complex (for clear reasons according to the paper):

$$\Gamma_{F(X)}(n)\xrightarrow{\partial}\bigoplus_{x\in X^1}\Gamma_{F(x)}(n-1)[-1]\xrightarrow{\partial} \ldots \rightarrow \bigoplus_{x\in X^{n}}\Gamma_{F(x)}(0)[-n]$$

Here $X^i$ means codimension $i$ points and $F(x)$ is the residue field at the point $x$. The complex $\Gamma_{F(x)}(i)$ is simply the weight $i$ complex for the field $F(x)$ defined above.

The morphisms $\partial$ are induced by residue morphisms of the following form($F$ is a field with discrete valuation $v$ with residue field $F_v$):

$$\partial_v: \Gamma_F(n) \rightarrow \Gamma_{F_v}(n-1)[-1]$$

This is very similar to the way the residue morphism is defined for the Milnor $K$-theory.

It is expected by Beilinson-Soule vanishing conjecture that negative motivic cohomology groups are zero so the motivic complexes are supported on nonnegative degrees. My question is about characteristic zero. Is there any conjectural constructions of the motivic complex that is naturally supported on non-negative degrees?

After studying the work of Goncharov I realized he has conjectured a construction of motivic complex for regular schemes that is supported on the positive degrees. Note that his construction is a complex of abelian groups that the cohomology of the complex coincides with the motivic cohomology (conjecturally). So it is not a complex of sheaves (at least the way it is presented below). Below I will explain construction. My question is:

• Why is it defined the way it is (assuming for fields it is the correct motivic complex)? Is it obvious that this complex satisfies excision?

Goncharov conjectures a complex of the following form (omitting some details on how to define the differentials) that is expected to rationally coincide with the weight $n$ motivic complex for all infinite fields $F$:

$$\mathcal{B}_n(F)\xrightarrow{\delta} \mathcal{B}_{n-1}\otimes F^*\xrightarrow{\delta}\mathcal{B}_{n-2}\otimes \wedge^2F^*\rightarrow \ldots \rightarrow \mathcal{B}_2(F)\otimes \wedge^{n-2}F^*\xrightarrow{\delta}\wedge^nF^*$$

Here $\mathcal{B}_i(F)$ are generalization of Bloch group and are defined by explicit generator and relations. The group $\mathcal{B}_n(F)$ is located at degree 1. More details can be found in "Geometry of Configurations, Polylogarithms and Motivic Cohomology" by Goncharov.

Now here is the puzzling part. It further is conjectured in the paper that for a regular scheme $X$ the weight $n$ motivic complex should be the total complex associated to the following bi-Complex (for clear reasons according to the paper):

$$\Gamma_{F(X)}(n)\xrightarrow{\partial}\bigoplus_{x\in X^1}\Gamma_{F(x)}(n-1)[-1]\xrightarrow{\partial} \ldots \rightarrow \bigoplus_{x\in X^{n}}\Gamma_{F(x)}(0)[-n]$$

Here $X^i$ means codimension $i$ points and $F(x)$ is the residue field at the point $x$. The complex $\Gamma_{F(x)}(i)$ is simply the weight $i$ complex for the field $F(x)$ defined above.

The morphisms $\partial$ are induced by residue morphisms of the following form($F$ is a field with discrete valuation $v$ with residue field $F_v$):

$$\partial_v: \Gamma_F(n) \rightarrow \Gamma_{F_v}(n-1)[-1]$$

This is very similar to the way the residue morphism is defined for the Milnor $K$-theory.

It is expected by Beilinson-Soule vanishing conjecture that negative motivic cohomology groups are zero so the motivic complexes are supported on nonnegative degrees. My question is about characteristic zero. Is there any conjectural constructions of the motivic complex that is naturally supported on non-negative degrees?

What is considered natural is a little bit subjective but for example one could say that just truncate the motivic complex so it is only includes non-negative degrees, I don't consider this natural.

In char $p$ there is something like this which is given by the Tate conjecture. You can find the conjecture 9.6 here. On the left side we have something motivic and the right side we have a cohomology theory that is known to be zero below certain range.

There are conjectural constructions of motivic complexes only for fields that are positively supported (Conjectures due to Goncharov and Rognes for two different constructions).

There is Beilinson's first conjecture on the regulator map from motivic cohomology to Deligne cohomology. It is only true for projective varieties and also isomorphism is only true in a certain range so it does not give what I am looking for.

It is expected by Beilinson-Soule vanishing conjecture that negative motivic cohomology groups are zero so the motivic complexes are supported on nonnegative degrees. My question is about characteristic zero. Is there any conjectural constructions of the motivic complex that is naturally supported on non-negative degrees?

After studying the work of Goncharov I realized he has conjectured a construction of motivic complex for regular schemes that is supported on the positive degrees. Below I will explain construction. My question is:

• Why is it defined the way it is (assuming for fields it is the correct motivic complex)? Is it obvious that this complex satisfies excision?

Goncharov conjectures a complex of the following form (omitting some details on how to define the differentials) rationally coincides with the weight $n$ motivic complex for all infinite fields $F$:

$$\mathcal{B}_n(F)\xrightarrow{\delta} \mathcal{B}_{n-1}\otimes F^*\xrightarrow{\delta}\mathcal{B}_{n-2}\otimes \wedge^2F^*\rightarrow \ldots \rightarrow \mathcal{B}_2(F)\otimes \wedge^{n-2}F^*\xrightarrow{\delta}\wedge^nF^*$$

Here $\mathcal{B}_i(F)$ are generalization of Bloch group and are defined by explicit generator and relations. The group $\mathcal{B}_n(F)$ is located at degree 1. More details can be found in "Geometry of Configurations, Polylogarithms and Motivic Cohomology" by Goncharov.

Now here is the puzzling part. It further is conjectured in the paper that for a regular scheme $X$ the weight $n$ motivic complex should be the total complex associated to the following bi-Complex (for clear reasons according to the paper):

$$\Gamma_{F(X)}(n)\xrightarrow{\partial}\bigoplus_{x\in X^1}\Gamma_{F(x)}(n-1)[-1]\xrightarrow{\partial} \ldots \rightarrow \bigoplus_{x\in X^{n}}\Gamma_{F(x)}(0)[-n]$$

Here $X^i$ means codimension $i$ points and $F(x)$ is the residue field at the point $x$. The complex $\Gamma_{F(x)}(i)$ is simply the weight $i$ complex for the field $F(x)$ defined above.

The morphisms $\partial$ are induced by residue morphisms of the following form($F$ is a field with discrete valuation $v$ with residue field $F_v$):

$$\partial_v: \Gamma_F(n) \rightarrow \Gamma_{F_v}(n-1)[-1]$$

This is very similar to the way the residue morphism is defined for the Milnor $K$-theory.