# Pairing between cyclic cohomology and $K$-theory: the odd case

I would like to understand the proof of Proposition 15 (see page 70 in this link ). More precisely: I would like to understand a particular step in the proof namely:

Why $\frac{d}{dt}(\varphi \# Tr)(U_t^{-1},U_t,...,U_t^{-1},U_t)=0$?

Maybe it would take too much place to explain everything what happened in the proof so far: I will restrict myself to some general remarks which one cannot find in the original paper. Here one deals with an abstract algebra, without topology therefore the presence of differentials is rather surprising. However the matrices $U_t$ are defined as product of scalar matrices (depending on $t$) and $A$-valued matrices which are constants. Therefore even on the level of $M_2(A)$ the derivative has a natural meaning. But even if we don't want to speak about differentiation in $M_2(A)$, we apply the cyclic cocycle to our expression thus $t \mapsto (\varphi \# Tr)(U_t^{-1},U_t,...,U_t^{-1},U_t)$ is scalar function which can be easily shown to be differentiable. Note also that we would like to prove that our pairing is defined on the level of algebraic $K$-theory. However this theory is defined as the quotient of invertible matrices by the commutator subgroup: from the fact that $\frac{d}{dt}(\varphi \# Tr)(U_t^{-1},U_t,...,U_t^{-1},U_t)=0$ we infer that $(\varphi \# Tr)(uv,(uv)^{-1},...,(uv)^{-1},uv)=(\varphi \# Tr)(u \oplus v,(u \oplus v)^{-1},...,(u \oplus v)^{-1},u \oplus v)$. The expression $(\varphi \# Tr)(u,u^{-1},...,u^{-1},u)$ is additive with respect to direct sums and lands in the commutative ring (in $\mathbb{C}$) thus we conclude that $(\varphi \# Tr)(u^{-1},u,...,u^{-1},u)$ vanishes on if $u$ is of the form $xyx^{-1}y^{-1}$. Thus indeed $$\frac{d}{dt}(\varphi \# Tr)(U_t^{-1},U_t,...,U_t^{-1},U_t)=0$$ would finish the proof.