Example 14.34.6. In Example 14.34.4 we have $X_ n(M) = R[R[\ldots [M]\ldots ]]$ with $n + 1$ brackets. We describe the maps constructed above using a typical element

\[ \xi = \sum \nolimits _ i r_ i\left[\sum \nolimits _ j r_{ij}[m_{ij}]\right] \]

of $X_1(M)$. The maps $d_0, d_1 : R[R[M]] \to R[M]$ are given by

\[ d_0(\xi ) = \sum \nolimits _{i, j} r_ ir_{ij}[m_{ij}] \quad \text{and}\quad d_1(\xi ) = \sum \nolimits _ i r_ i\left[\sum \nolimits _ j r_{ij}m_{ij}\right]. \]

The maps $s_0, s_1 : R[R[M]] \to R[R[R[M]]]$ are given by

\[ s_0(\xi ) = \sum \nolimits _ i r_ i\left[\left[\sum \nolimits _ j r_{ij}[m_{ij}]\right]\right] \quad \text{and}\quad s_1(\xi ) = \sum \nolimits _ i r_ i\left[\sum \nolimits _ j r_{ij}[[m_{ij}]]\right]. \]

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