[Paper Review] Homogeneous spaces, algebraic $K$-theory and cohomological dimension of fields

Diego Munguía Izquierdo, Giancarlo Lucchini Arteche|arXiv (Cornell University)|Dec 11, 2018

Algebraic Geometry and Number Theory14 references4 citations

TL;DR

This paper establishes a precise algebraic-geometric characterization of the cohomological dimension of fields using Milnor K-theory and norm maps from homogeneous spaces. It proves that a perfect field has cohomological dimension at most q+1 if and only if, for every finite extension L and every homogeneous space Z under a smooth linear connected algebraic group over L, the q-th Milnor K-group of L is spanned by norms from extensions where Z acquires a rational point—offering a refined, cohomologically complete alternative to Kato and Kuzumaki's flawed conjectures.

ABSTRACT

Let $q$ be a non-negative integer. We prove that a perfect field $K$ has cohomological dimension at most $q+1$ if, and only if, for any finite extension $L$ of $K$ and for any homogeneous space $Z$ under a smooth linear connected algebraic group over $L$, the $q$-th Milnor $K$-theory group of $L$ is spanned by the images of the norms coming from finite extensions of $L$ over which $Z$ has a rational point. We also prove a variant of this result for imperfect fields.

Motivation & Objective

• To resolve the open problem of characterizing cohomological dimension of fields in a way that generalizes and corrects Kato and Kuzumaki’s conjectures.
• To provide a diophantine characterization of cohomological dimension using Milnor K-theory and norm maps from rational points on homogeneous spaces.
• To unify and generalize known results on cohomological dimension, including Steinberg’s theorem, Suslin’s norm surjectivity, and Wittenberg’s results on p-adic and number fields.
• To extend the characterization to imperfect fields using separable cohomological dimension and reductive group actions.
• To develop a purely algebraic-geometric proof strategy that avoids arithmetic methods, applicable to all fields including function fields and Laurent series fields.

Proposed method

• Introduces new field properties: Cq^HS, Cq^PHS, and Cq^Red, defined by the condition that the q-th Milnor K-group is spanned by norms from finite extensions where a homogeneous space has a rational point.
• Uses norm maps NL′/L: KM_q(L′) → KM_q(L) to define subgroups Nq(Z/L) generated by images of such norms.
• Applies non-abelian cohomology techniques, particularly the theory of gerbes and torsors, to analyze rational points on homogeneous spaces.
• Employs restriction-corestriction arguments and Sylow subgroup decompositions to reduce the problem to solvable stabilizers.
• Leverages results from Demarche and Lucchini Arteche on the Hasse principle for homogeneous spaces with finite stabilizers.
• Uses induction and universal torsor techniques inspired by Harpaz and Wittenberg to avoid complex non-abelian cohomology in key cases.

Experimental results

Research questions

• RQ1 Can the cohomological dimension of a perfect field be characterized by the surjectivity of norm maps in Milnor K-theory from extensions where a homogeneous space has a rational point?
• RQ2 Does the Cq^HS property—defined via norms from rational points on homogeneous spaces—characterize fields of cohomological dimension at most q+1?
• RQ3 How does the characterization extend to imperfect fields, and what role does separable cohomological dimension play?
• RQ4 Can the failure of Kato and Kuzumaki’s Cq^i conjectures be circumvented by restricting to homogeneous spaces rather than hypersurfaces?
• RQ5 To what extent can the proof strategy be made purely algebraic-geometric, avoiding arithmetic tools used in prior work on p-adic and number fields?

Key findings

• A perfect field K has cohomological dimension at most q+1 if and only if it satisfies the Cq^HS property: for every finite extension L of K and every homogeneous space Z under a smooth linear connected algebraic group over L, the q-th Milnor K-group KM_q(L) is spanned by the images of norm maps from finite extensions L′ of L where Z(L′) is non-empty.
• For imperfect fields, the Cq^Red property—defined over reductive groups—characterizes separable cohomological dimension at most q+1: all finite extensions of K have separable cohomological dimension ≤ q+1 if and only if KM_q(L) is spanned by norms from extensions where a principal homogeneous space under a reductive group has a rational point.
• The result generalizes Steinberg’s theorem: if K is perfect and has cohomological dimension ≤1, then every homogeneous space under a linear connected group has a zero-cycle of degree 1.
• It extends Suslin’s theorem: a field K of characteristic 0 has cohomological dimension ≤2 if and only if the reduced norm map A× → L× is surjective for every central simple algebra A over every finite extension L of K.
• It generalizes Gille’s result to positive characteristic: the cohomological dimension condition is equivalent to the surjectivity of norm maps in Milnor K-theory for reductive group torsors.
• The proof avoids arithmetic methods and instead uses algebraic-geometric tools such as gerbes, torsors, and restriction-corestriction, making it applicable to fields like C((t))(x) and C((x,y)) where Kato-Kuzumaki’s conjectures fail.