[Paper Review] On multilinearity and skew-symmetry of certain symbols in motivic cohomology of fields

Sung Myung|ArXiv.org|Jan 9, 2008

Advanced Algebra and Geometry14 references3 citations

TL;DR

This paper establishes multilinearity and skew-symmetry for symbols in the Goodwillie-Lichtenbaum motivic cohomology complex of fields, providing an explicit isomorphism between motivic cohomology and Milnor K-theory in degree-weight equal cases. It proves these properties hold for irreducible symbols in $ H^{l-1}_{\mathcal{M}}(\operatorname{Spec} k, \mathbb{Z}(l)) $, generalizing the determinant formula $ \det(AB) = \det A \cdot \det B $ to higher motivic symbols.

ABSTRACT

The purpose of the present article is to show the multilinearity for symbols in Goodwillie-Lichtenbaum complex in two cases. The first case shown is where the degree is equal to the weight. In this case, the motivic cohomology groups of a field are isomorphic to the Milnor's K-groups as shown by Nesterenko-Suslin, Totaro and Suslin-Voevodsky for various motivic complexes, but we give an explicit isomorphism for Goodwillie-Lichtenbaum complex in a form which visibly carries multilinearity of Milnor's symbols to our multilinearity of motivic symbols. Next, we establish multilinearity and skew-symmetry for irreducible Goodwillie-Lichtenbaum symbols in H^{l-1} (Spec k, Z(l)). These properties have been expected to hold from the author's construction of a bilinear form of dilogarithm in case k is a subfield of the field of complex numbers and l=2. Next, we establish multilinearity and skew-symmetry for Goodwillie-Lichtenbaum symbols in H^{l-1} (Spec k, Z(l)). These properties have been expected to hold from the author's construction of a bilinear form of dilogarithm in case k is a subfield of the field of complex numbers and l=2. The multilinearity of symbols may be viewed as a generalization of the well-known formula det(AB) = det(A) det(B) for tuples of commuting matrices.

Motivation & Objective

• To establish multilinearity and skew-symmetry of symbols in the Goodwillie-Lichtenbaum motivic cohomology complex for fields.
• To provide a direct, explicit isomorphism between motivic cohomology $ H^n_{\mathcal{M}}(\operatorname{Spec} k, \mathbb{Z}(n)) $ and Milnor K-theory $ K^M_n(k) $, visibly preserving multilinearity.
• To extend these properties to irreducible symbols in $ H^{l-1}_{\mathcal{M}}(\operatorname{Spec} k, \mathbb{Z}(l)) $, particularly for $ l=2 $, aligning with expected behavior from regulator maps.
• To offer a new proof of the Nesterenko-Suslin theorem for the Goodwillie-Lichtenbaum complex, reinforcing its foundational role in motivic cohomology.

Proposed method

• Constructs the Goodwillie-Lichtenbaum motivic complex via Grothendieck groups $ K_0(R\Delta^d, \mathbb{G}_m^{\wedge l}) $ for simplicial rings $ R\Delta^\bullet $, with quotient by classes where any automorphism is trivial.
• Uses the simplicial structure of $ R\Delta^\bullet $ to define a chain complex whose hypercohomology computes motivic cohomology of fields.
• Applies explicit matrix constructions in $ GL_n(k[t]) $ to represent symbols and track their behavior under homotopy, especially at $ t=0 $ and $ t=1 $.
• Employs the identity $ (A(t)B(t)) = (A(0)B(0)) $ in $ K_0 $-groups to test representability in motivic cohomology, showing failure when multilinearity fails.
• Uses a key identity involving block matrices to derive skew-symmetry, replacing $ \theta_2(t) $ with a copy of $ \theta_1(t) $ to induce cancellation.
• Relies on the irreducibility condition to ensure non-degenerate behavior under permutation of automorphisms, enabling skew-symmetry proof via cancellation in $ K_0 $.

Experimental results

Research questions

• RQ1 Does the Goodwillie-Lichtenbaum motivic complex exhibit multilinearity for symbols in $ H^n_{\mathcal{M}}(\operatorname{Spec} k, \mathbb{Z}(n)) $, and if so, how is it compatible with Milnor K-theory?
• RQ2 Can multilinearity and skew-symmetry be established for irreducible symbols in $ H^{l-1}_{\mathcal{M}}(\operatorname{Spec} k, \mathbb{Z}(l)) $, particularly for $ l=2 $, and how do they relate to regulator maps?
• RQ3 Is there an explicit isomorphism between $ H^n_{\mathcal{M}}(\operatorname{Spec} k, \mathbb{Z}(n)) $ and $ K^M_n(k) $ via the Goodwillie-Lichtenbaum complex that visibly preserves multilinearity?
• RQ4 What conditions ensure that a symbol $ (\theta_1(t), \dots, \theta_l(t)) $ represents a well-defined class in motivic cohomology, and how does irreducibility affect this?

Key findings

• The paper proves multilinearity for symbols in $ H^n_{\mathcal{M}}(\operatorname{Spec} k, \mathbb{Z}(n)) $, showing that the symbol map respects additive behavior under matrix multiplication, generalizing $ \det(AB) = \det A \cdot \det B $.
• It provides a direct proof of the Nesterenko-Suslin theorem for the Goodwillie-Lichtenbaum complex, establishing $ H^n_{\mathcal{M}}(\operatorname{Spec} k, \mathbb{Z}(n)) \cong K^M_n(k) $ with an isomorphism that visibly preserves multilinearity.
• Skew-symmetry is established for irreducible symbols in $ H^{l-1}_{\mathcal{M}}(\operatorname{Spec} k, \mathbb{Z}(l)) $ via a matrix identity involving block matrices and cancellation in $ K_0 $, proving $ (\theta_1, \dots, \theta_l) = - (\theta_j, \dots, \theta_i, \dots) $ under swap of indices.
• The construction of a bilinear form of dilogarithm in the case $ k \subset \mathbb{C} $, $ l=2 $, is shown to be consistent with the skew-symmetry and multilinearity properties proved in the paper.
• Failure of multilinearity is demonstrated via counterexample: a homotopy $ A(t)B(t) $ with $ A(0)B(0) \neq A(1)B(1) $ in $ K_0(k, \mathbb{G}_m^{\wedge 1}) $, showing $ (A(t)B(t)) $ does not define a class in $ H^0_{\mathcal{M}}(\operatorname{Spec} k, \mathbb{Z}(1)) $.
• The irreducibility assumption in skew-symmetry can be replaced by a characteristic 0 condition, ensuring that the symbol remains non-degenerate under permutation of automorphisms.