A note on polyhedral cones and toric polylogarithms

Abstract

We extend some methods of our previous work on special elements in Milnor K-theory of algebraic tori, exhibiting in particular a ${\mathrm{GL}}_n(\mathbb{Q})$-equivariant isomorphism between a chain complex of simplicial cones, computing the homology of $S^{n-1}$, and the trace-fixed part of the weight-n Gersten complex for the Milnor K- theory of ${\mathbb{G}}_m^n$ over $\mathbb{Q}$. Via a relationship between graded pieces of algebras of cones and Steinberg modules, this refines a result of Charlton-Radchenko-Rudenko.