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

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