[Paper Review] Motivic interpretation of Milnor $K$-groups attached to Jacobian varieties

Satoshi Mochizuki|ArXiv.org|Mar 10, 2006

Advanced Differential Equations and Dynamical Systems4 citations

TL;DR

This paper provides a motivic interpretation of Somekawa's Milnor K-groups attached to Jacobian varieties by identifying them with morphism groups in Voevodsky's category of effective motives. It establishes an isomorphism between $ K(k, ext{Jac}C_1, reak ext{Jac}C_2, reak ext{Jac}C_n) $ and $ ext{Hom}_{ ext{DM}^{ ext{eff}}_-}(k)( ext{M}( ext{Spec} hinspace k), hinspace bZ( ext{C}_1 imes ext{C}_2 imes ext{C}_n)[n]) $, thereby verifying a special case of Somekawa's conjecture in the motivic framework using reciprocity laws and motivic cohomology.

ABSTRACT

In the paper M. Somekawa, {\it{On Milnor $K$-groups attached at semi-Abelian varieties}}, K-theory, extbf{4} (1990) p.105, Somekawa conjectures that his Milnor K-group $K(k,G_1,...,G_r)$ attached to semi-abelian varieties $G_1$,...,$G_r$ over a field $k$ is isomorphic to ${ m Ext}_{\mathcal{M}_k}^r(\mathbb{Z},G_1[-1] \otimes ... \otimes G_r[-1])$ where $\mathcal{M}_k$ is a certain category of motives over $k$. The purpose of this note is to give remarks on this conjecture, when we take $\mathcal{M}_k$ as Voevodsky's category of motives ${ m DM}^{eff}_{-}(k)$ .

Motivation & Objective

• To provide a motivic interpretation of Somekawa's Milnor K-groups associated with Jacobian varieties over a perfect field.
• To verify a special case of Somekawa's conjecture that relates Milnor K-groups to Ext groups in a category of motives.
• To establish a link between motivic cohomology and classical arithmetic invariants such as the Bloch group and generalized reciprocity laws.
• To use Voevodsky's category of effective motives $ ext{DM}^{ ext{eff}}_-(k) $ as the foundational framework for this interpretation.
• To demonstrate that the motivic reciprocity law implies the classical Weil reciprocity law for Milnor K-groups.

Proposed method

• Uses Voevodsky's category of effective motives $ ext{DM}^{ ext{eff}}_-(k) $ as the category of motives $ rak{M}_k $, replacing the abstract category in Somekawa's conjecture.
• Applies the motivic cohomology theory defined via the motivic complex $ bZ(n) $, with $ bZ(n) $ represented by the $ n $-fold product of pointed curves.
• Employs the motivic reciprocity law, derived from the Gysin triangle and Thom isomorphism, to relate cycle classes to K-group elements.
• Utilizes the norm map and pairing in motivic cohomology to construct a canonical map from $ igotimes ext{Jac}C_i(L) $ to $ ext{H}^n_{ ext{M}}(k, igwedge C_i) $, which factors through the Milnor K-group.
• Constructs a natural map from motivic cohomology $ ext{H}^n_{ ext{M}}(k, igwedge C_i) $ to $ K(k, ext{Jac}C_1, reak ext{Jac}C_2, reak ext{Jac}C_n) $ using the boundary map from $ bA^1 $-homotopy invariance.
• Proves the inverse map exists by showing that the composition of the two maps is an isomorphism, using the structure of pointed curves and resolution of singularities.

Experimental results

Research questions

• RQ1 Can Somekawa's conjecture relating Milnor K-groups to motivic Ext groups be verified in the setting of Voevodsky's category of effective motives?
• RQ2 Does the motivic cohomology of products of pointed smooth curves compute the Milnor K-group $ K(k, ext{Jac}C_1, ext{Jac}C_2, ext{Jac}C_n) $?
• RQ3 How does the motivic reciprocity law relate to the classical Weil reciprocity law for Milnor K-groups?
• RQ4 What is the precise relationship between the Bloch group $ V(C) $ and motivic cohomology in the context of Jacobians?
• RQ5 Can the structure of $ ext{Hom}_{ ext{DM}^{ ext{eff}}_-}(k)( ext{M}( ext{Spec} hinspace k), bZ( ext{C}_1 imes ext{C}_2 imes ext{C}_n)[n]) $ be used to reconstruct the Milnor K-group for Jacobians?

Key findings

• The paper establishes a canonical isomorphism: $ K(k, ext{Jac}C_1, ext{Jac}C_2, ext{Jac}C_n) o ext{Hom}_{ ext{DM}^{ ext{eff}}_-}(k)( ext{M}( ext{Spec} hinspace k), bZ( ext{C}_1 imes ext{C}_2 imes ext{C}_n)[n]) $, confirming a special case of Somekawa's conjecture.
• The motivic reciprocity law, derived from the Gysin triangle and Thom isomorphism, implies the classical Weil reciprocity law for Milnor K-groups.
• The motivic cohomology group $ ext{H}^n_{ ext{M}}(k, igwedge C_i) $ is isomorphic to $ K(k, ext{Jac}C_1, ext{Jac}C_2, ext{Jac}C_n) $ via a canonical map constructed from norm maps and pairings.
• The map $ igoplus_L igotimes_{i=1}^n ext{Jac}C_i(L) o ext{H}^n_{ ext{M}}(k, igwedge C_i) $ factors through $ K(k, ext{Jac}C_1, ext{Jac}C_2, ext{Jac}C_n) $, showing compatibility with Galois descent.
• The construction relies on the assumption that $ k $ is perfect and admits resolution of singularities, ensuring the validity of the motivic cohomology machinery.
• The isomorphism is shown to be an equivalence by proving both maps (from K-group to cohomology and back) are inverse to each other, using the structure of pointed curves and $ bA^1 $-homotopy invariance.