equal
deleted
inserted
replaced
173 and this cancels exactly with the term indexed by $\rest(i)$ (with a value of $m$ off by one) in the third row. |
173 and this cancels exactly with the term indexed by $\rest(i)$ (with a value of $m$ off by one) in the third row. |
174 The second row gives |
174 The second row gives |
175 \begin{align*} |
175 \begin{align*} |
176 & \sum_{m=0}^k \sum_{\most(i) \in \{1, \ldots, k\}^{m-1} \setminus \Delta} \sum_{\substack{i_m = 1 \\ i_1 \not\in \most(i)}}^{k} (-1)^{\sigma(\most(i) i_m) + m} \ev\left(\restrict{\phi_{\most(i)\!\downarrow_{i_m}(b_{i_m})}}{x_0 = 0}\tensor b_{\most(i) i_m}\right) \\ |
176 & \sum_{m=0}^k \sum_{\most(i) \in \{1, \ldots, k\}^{m-1} \setminus \Delta} \sum_{\substack{i_m = 1 \\ i_1 \not\in \most(i)}}^{k} (-1)^{\sigma(\most(i) i_m) + m} \ev\left(\restrict{\phi_{\most(i)\!\downarrow_{i_m}(b_{i_m})}}{x_0 = 0}\tensor b_{\most(i) i_m}\right) \\ |
177 & \quad = \sum_{m=0}^{k-1} \sum_{q=1}^k \sum_{i \in \{1, \ldots, k-1\}^m\setminus \Delta} (-1)^{\sigma(i)+q+1} \ev\left(\restrict{\phi_{i(b_{q})}}{x_0 = 0}\tensor (b_q)_i\right) \\ |
177 & \quad = \sum_{m=0}^{k-1} \sum_{q=1}^k \sum_{i \in \{1, \ldots, k-1\}^m\setminus \Delta} (-1)^{\sigma(i)+q+1} \ev\left(\restrict{\phi_{i(b_{q})}}{x_0 = 0}\tensor (b_q)_i\right) \\ |
178 \intertext{(here we've used Equation \eqref{eq:sigma(ab)} and renamed $i_m$ to $q$ and $most(i)$ to $i$, as well as shifted $m$ by one), which is just} |
178 \intertext{(here we've used Equation \eqref{eq:sigma(ab)} and renamed $i_m$ to $q$ and $\most(i)$ to $i$, as well as shifted $m$ by one), which is just} |
179 & \quad = \sum_{q=1}^k (-1)^{q+1} s(b_q) \\ |
179 & \quad = \sum_{q=1}^k (-1)^{q+1} s(b_q) \\ |
180 & \quad = s(\bdy b). |
180 & \quad = s(\bdy b). |
181 \end{align*} |
181 \end{align*} |
182 |
182 |
183 Finally, the calculation that $\bdy h+h \bdy=i\circ s - \id$ is very similar, and we omit it. |
183 Finally, the calculation that $\bdy h+h \bdy=i\circ s - \id$ is very similar, and we omit it. |