Example 14.34.7. In Example 14.34.5 we have X_ n(B) = A[A[\ldots [B]\ldots ]] with n + 1 brackets. We describe the maps constructed above using a typical element
\xi = \sum \nolimits _ i a_ i [x_{i, 1}] \ldots [x_{i, m_ i}] \in A[A[B]] = X_1(B)
where for each i, j we can write
x_{i, j} = \sum a_{i, j, k} [b_{i, j, k, 1}] \ldots [b_{i, j, k, n_{i, j, k}}] \in A[B]
Obviously this is horrendous! To ease the notation, to see what the A-algebra maps d_0, d_1 : A[A[B]] \to A[B] are doing it suffices to see what happens to the variables [x] where
x = \sum a_ k [b_{k, 1}] \ldots [b_{k, n_ k}] \in A[B]
is a general element. For these we get
d_0([x]) = x = \sum a_ k [b_{k, 1}] \ldots [b_{k, n_ k}] \quad \text{and}\quad d_1([x]) = \left[\sum a_ k b_{k, 1} \ldots b_{k, n_ k}\right]
The maps s_0, s_1 : A[A[B]] \to A[A[A[B]]] are given by
s_0([x]) = \left[\left[\sum a_ k [b_{k, 1}] \ldots [b_{k, n_ k}]\right]\right] \quad \text{and}\quad s_1([x]) = \left[\sum a_ k [[b_{k, 1}]] \ldots [[b_{k, n_ k}]]\right]
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