[Paper Review] Higher Chow groups with modulus and relative Milnor K-theory

Kay Rülling, Shuji Saito|arXiv (Cornell University)|Apr 10, 2015

Algebraic Geometry and Number Theory4 references4 citations

TL;DR

This paper constructs an explicit cycle map from the Nisnevich motivic complex of a smooth variety $X$ with a simple normal crossing divisor $D$ to the relative Milnor $K$-sheaf, proving an isomorphism between motivic cohomology and cohomology of the relative Milnor $K$-sheaf for degrees $i \geq \dim X$. The result generalizes the classical isomorphism for $D=0$ and establishes a Zariski descent property for motivic cohomology of $({\mathbb{A}}^1_k, (m+1)\{0\})$, linking it to additive Chow groups and the big de Rham-Witt complex.

ABSTRACT

Let X be a smooth variety over a field k and D an effective divisor whose support has simple normal crossings. We construct an explicit cycle map from the r-th Nisnevich motivic complex of the pair (X,D) to a shift of the r-th relative Milnor K-sheaf of (X,D). We show that this map induces an isomorphism for all i greater or equal the dimension of X between the motivic Nisnevich cohomology of (X,D) in bidegree (i+r,r) and the i-th Nisnevich cohomology of the r-th relative Minor K-sheaf of (X,D). This generalizes the well-known isomorphism in the case D=0. We use this to prove a certain Zariski descent property for the motivic cohomology of the pair (\A^1_k, (m+1){0}).

Motivation & Objective

• To extend the classical isomorphism between motivic cohomology and Milnor K-theory to pairs $(X,D)$ with a divisor $D$ having simple normal crossings.
• To define and study the relative Milnor $K$-sheaf ${\mathcal{K}}^{M}_{r,X|D}$ as a subsheaf of the absolute Milnor $K$-sheaf.
• To establish a cycle map from the Nisnevich motivic complex $\mathbb{Z}(r)_{X|D,\mathrm{Nis}}$ to the shifted relative Milnor $K$-sheaf, and prove its isomorphism in high degrees.
• To prove a Zariski descent property for the motivic cohomology of $({\mathbb{A}}^1_k, (m+1)\{0\})$, linking it to additive Chow groups and the big de Rham-Witt complex.

Proposed method

• Constructs the relative Milnor $K$-sheaf ${\mathcal{K}}^{M}_{r,X|D}$ as a subsheaf of ${\mathcal{K}}^{M}_{r,X}$ via a filtration induced by the divisor $D$, using the structure of Milnor $K$-sheaves on schemes with modulus.
• Defines the Nisnevich motivic complex $\mathbb{Z}(r)_{X|D,\mathrm{Nis}}$ as a shift of the cubical cycle complex with modulus, satisfying a modulus condition on cycles.
• Constructs an explicit cycle map $\phi^{r}_{X|D,\mathrm{Nis}}: \tau_{\geq r}\mathbb{Z}(r)_{X|D,\mathrm{Nis}} \to {\mathcal{K}}^{M}_{r,X|D,\mathrm{Nis}}[-r]$ in the derived category of Nisnevich sheaves.
• Proves the cycle map induces an isomorphism $H^{i+r}_{{\mathcal{M}},\mathrm{Nis}}(X|D,\mathbb{Z}(r)) \cong H^i(X_{\mathrm{Nis}}, {\mathcal{K}}^{M}_{r,X|D,\mathrm{Nis}})$ for all $i \geq \dim X$, generalizing the classical case $D=0$.
• Uses homotopy invariance and dimension arguments to prove vanishing of certain motivic cohomology groups, particularly for $({\mathbb{A}}^1_k, (m+1)\{0\})$, to establish the Zariski descent property.
• Relates the motivic cohomology of $({\mathbb{A}}^1_k, (m+1)\{0\})$ to additive Chow groups and the big de Rham-Witt complex via the isomorphism and vanishing results.

Experimental results

Research questions

• RQ1 Does there exist a natural cycle map from the motivic cohomology of a pair $(X,D)$ with $D$ a simple normal crossing divisor to the relative Milnor $K$-sheaf, generalizing the classical isomorphism for $D=0$?
• RQ2 Can the motivic cohomology of $({\mathbb{A}}^1_k, (m+1)\{0\})$ be described in terms of relative Milnor $K$-sheaves and additive Chow groups?
• RQ3 What is the behavior of motivic cohomology under Zariski descent for pairs $({\mathbb{A}}^n_k, D)$ with $D$ a divisor of modulus?
• RQ4 How do the relative Milnor $K$-sheaves ${\mathcal{K}}^{M}_{r,X|D}$ relate to the big de Rham-Witt complex in the context of additive K-theory?
• RQ5 What vanishing theorems hold for motivic cohomology of $({\mathbb{A}}^n_k, D)$ in high degrees, and how do they imply Zariski descent?

Key findings

• The cycle map $\phi^{r}_{X|D,\mathrm{Nis}}: \tau_{\geq r}\mathbb{Z}(r)_{X|D,\mathrm{Nis}} \to {\mathcal{K}}^{M}_{r,X|D,\mathrm{Nis}}[-r]$ is constructed and proven to be an isomorphism in the derived category of Nisnevich sheaves when $D_{\mathrm{red}}$ is a simple normal crossing divisor.
• The isomorphism $H^{i+r}_{{\mathcal{M}},\mathrm{Nis}}(X|D,\mathbb{Z}(r)) \cong H^i(X_{\mathrm{Nis}}, {\mathcal{K}}^{M}_{r,X|D,\mathrm{Nis}})$ holds for all $i \geq \dim X$, generalizing the classical isomorphism for $D=0$.
• The motivic cohomology of $({\mathbb{A}}^1_k, (m+1)\{0\})$ is shown to satisfy a Zariski descent property, with the cohomology group $H^{r+d+n}_{{\mathcal{M}},\mathrm{Nis}}(X\times{\mathbb{A}}^n|p^*D+E_{\mathfrak{m}}, \mathbb{Z}(r))$ vanishing for $n \geq 2$ and $\mathfrak{m} \in ({\mathbb{N}}_{\geq 1})^n$.
• The group $\mathrm{CH}^{r}(X\times{\mathbb{A}}^n|p^*D+E_{\mathfrak{m}}, r-(d+n))$ vanishes for $n \geq 2$, implying that the natural map to motivic cohomology is an isomorphism.
• The relative Milnor $K$-sheaf ${\mathcal{K}}^{M}_{r,X|D}$ is shown to be a successive extension of sheaves $\omega^{r-1}_{\mathfrak{n},\nu}/B^{r-1}_{s+1,\mathfrak{n},\nu}$, which allows the use of spectral sequence arguments for cohomological vanishing.
• The vanishing of $H^{d+1}(X_{\nu}\times{\mathbb{A}}^1, \omega^{r-1}_{\mathfrak{n},\nu})$ is established via the Leray spectral sequence and dimension reasons, proving the key vanishing result for the descent property.