Proposition 2.5.7.8 (00RP)

Proposition 2.5.7.8. Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups. Then the unnormalized shuffle product

\[ \bar{\triangledown }: \mathrm{C}_ p(A) \times \mathrm{C}_ q(B) \rightarrow \mathrm{C}_{p+q}(A \otimes B) \]

carries the subsets $\mathrm{D}_{p}(A) \times \mathrm{C}_{q}(B)$ and $ \mathrm{C}_{p}(A) \times \mathrm{D}_{q}(B)$ into the subgroup $\mathrm{D}_{p+q}(A \otimes B) \subseteq \mathrm{C}_{p+q}( A \otimes B)$.

Proof. Let $a \in A_ p$ and $b \in B_{q}$ be simplices of $A_{\bullet }$ and $B_{\bullet }$, respectively. We wish to show that if either $a$ belongs to $\mathrm{D}_{p}(A)$ or $b$ belongs to $\mathrm{D}_{q}(B)$, then the unnormalized shuffle product $a \bar{\triangledown } b$ belongs to $\mathrm{D}_{p+q}(A \otimes B)$. Without loss of generality, we may assume that $a$ belongs to $\mathrm{D}_{p}(A)$. Decomposing $a$ into summands, we can further assume that $a = s^{p-1}_ i(a')$ for some $0 \leq i \leq p-1$ and some $a' \in A_{p-1}$. Let $\sigma = (\sigma _{-}, \sigma _{+})$ be a $(p,q)$-shuffle. Then there exists a unique integer $0 \leq j < p+q$ satisfying $\sigma _{-}( j) = i$ and $\sigma _{-}(j+1) = i+1$. It then follows that both $\sigma _{-}^{\ast }(a)$ and $\sigma _{+}^{\ast }(b)$ are fixed points of the composite maps

\[ A_{p+q} \xrightarrow {d^{p+q}_ j} A_{p+q-1} \xrightarrow {s^{p+q-1}_ j} A_{p+q} \quad \quad B_{p+q} \xrightarrow {d^{p+q}_ j} B_{p+q-1} \xrightarrow {s^{p+q-1}_ j} B_{p+q}, \]

so that $\sigma _{-}^{\ast }(a) \otimes \sigma _{+}^{\ast }(b)$ is a degenerate simplex of $(A \otimes B)_{\bullet }$. Allowing $\sigma $ to vary, we deduce that the shuffle product

\[ \sum _{\sigma } (-1)^{\sigma } \sigma _{-}^{\ast }(a) \otimes \sigma _{+}^{\ast }(b) \]

belongs to $\mathrm{D}_{p+q}( A \otimes B)$. $\square$