Pro-representability of $K^M$-cohomology in weight 3 generalizing a result of Bloch

TL;DR

This paper extends Bloch's pro-representability results for Milnor K-cohomology groups at the identity to weight 3, under specific Hodge number vanishing conditions, for smooth proper varieties over algebraic fields.

Contribution

It generalizes pro-representability of Milnor K-cohomology to weight 3, broadening the scope of Bloch's original results for algebraic varieties.

Findings

Proves pro-representability of a specific K-cohomology functor under Hodge number conditions.

Establishes a connection between K-cohomology and deformation theory of algebraic varieties.

Extends known results from weight 2 to weight 3 in K-theoretic cohomology.

Abstract

We generalize a result, on the pro-representability of Milnor $K$-cohomology groups at the identity, that's due to Bloch. In particular, we prove, for $X$ a smooth, proper, and geometrically connected variety defined over an algebraic field extension $k/\mathbb{Q}$, that the functor \[\mathscr{T}_{X}^{i,3}(A)=\ker\left(H^i(X_A,\mathcal{K}_{3,X_A}^M)\rightarrow H^i(X,\mathcal{K}_{3,X}^M)\right),\] defined on Artin local $k$-algebras $(A,\mathfrak{m}_A)$ with $A/\mathfrak{m}_A\cong k$, is pro-representable provided that certain Hodge numbers of $X$ vanish.