Tangent to Bloch-Suslin and Grassmannian Complexes over the dual numbers

TL;DR

This paper extends classical cross-ratio identities to dual number rings, computes related ratios, and establishes relations between tangent and Grassmannian complexes, advancing algebraic K-theory and polylogarithm studies.

Contribution

It generalizes cross-ratio identities over dual numbers and links tangent complexes with Grassmannian complexes through new morphisms.

Findings

Extended Siegel's cross-ratio identity to dual numbers.

Computed cross-ratios and triple-ratios in dual number rings.

Verified a key five-term relation in the tangent group.

Abstract

In this article, we extend Siegel's cross-ratio identity for $2\times2$ determinants over the truncated polynomial ring $F[\varepsilon]_\nu:=F[\varepsilon]/\varepsilon^\nu$. We compute cross-ratios and Goncharov's triple-ratios in $F[\varepsilon]_2$ and $F[\varepsilon]_3$ and use them extensively in our computations for the tangential complexes. We also verify a "projected five-term" relation in the group $T\mathcal{B}_2(F)$ which is crucial to prove one of our central statements that describe the morphisms between tangent complex and Grassmannian complex