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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-ratiosand Goncharov's triple-ratios in $F[\varepsilon]_2$ and $F[\varepsilon]_3$ anduse them extensively in our computations for the tangential complexes. We alsoverify a "projected five-term" relation in the group $T\mathcal{B}_2(F)$ whichis crucial to prove one of our central statements that describe the morphismsbetween tangent complex and Grassmannian complex

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