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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.

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    $\begingroup$ The key word here might be: coniveau spectral sequence, e.g. see here: webusers.imj-prg.fr/~bruno.kahn/preprints/bo.pdf The first page is a Cousin complex, and the Beilinson-Soulé vanishing conjecture will make this spectral sequence degenerate to the point where it provides a resolution of motivic cohomology. In other words, you can mimick what is done in this paper numdam.org/item/AFST_2014_6_23_3_591_0 replacing Weil cohomologies by motivic cohomology (and working in the heart of the putative motivic t-structure). $\endgroup$
    – D.-C. Cisinski
    Commented Sep 21, 2022 at 8:36
  • $\begingroup$ @D.-C.Cisinski Thanks. Indeed the construction is designed in a way so the cohomologies satisfy Gersten type resolution. This is super interesting it seems to me that homotopy invariance is the only obstruction to prove this conjecture. The rest of follows from a Geisser-Levine type argument using polyrelative versions of the cohomology. Even the construction of a map from this cohomology to the motivic one follows from it. One also needs to carefully check the groups in Goncharov's complex applied to $\hat{\Delta}^{\bullet}$ (simplices semi-localized at the vertices) is exact (it seems to be) $\endgroup$
    – user127776
    Commented Sep 22, 2022 at 8:00

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