Lemma 10.13.3. Let $R$ be a ring. Let $M$ be an $R$-module. Let $x_ i$, $i \in I$ be a given system of generators of $M$ as an $R$-module. Let $n \geq 2$. There exists a canonical exact sequence

\[ \bigoplus _{1 \leq j_1 < j_2 \leq n} \bigoplus _{i_1, i_2 \in I} \text{T}^{n - 2}(M) \oplus \bigoplus _{1 \leq j_1 < j_2 \leq n} \bigoplus _{i \in I} \text{T}^{n - 2}(M) \to \text{T}^ n(M) \to \wedge ^ n(M) \to 0 \]

where the pure tensor $m_1 \otimes \ldots \otimes m_{n - 2}$ in the first summand maps to

\begin{align*} \underbrace{ m_1 \otimes \ldots \otimes x_{i_1} \otimes \ldots \otimes x_{i_2} \otimes \ldots \otimes m_{n - 2} }_{\text{with } x_{i_1} \text{ and } x_{i_2} \text{ occupying slots } j_1 \text{ and } j_2 \text{ in the tensor}} \\ + \underbrace{ m_1 \otimes \ldots \otimes x_{i_2} \otimes \ldots \otimes x_{i_1} \otimes \ldots \otimes m_{n - 2} }_{\text{with } x_{i_2} \text{ and } x_{i_1} \text{ occupying slots } j_1 \text{ and } j_2 \text{ in the tensor}} \end{align*}

and $m_1 \otimes \ldots \otimes m_{n - 2}$ in the second summand maps to

\[ \underbrace{ m_1 \otimes \ldots \otimes x_ i \otimes \ldots \otimes x_ i \otimes \ldots \otimes m_{n - 2} }_{\text{with } x_{i} \text{ and } x_{i} \text{ occupying slots } j_1 \text{ and } j_2 \text{ in the tensor}} \]

There is also a canonical exact sequence

\[ \bigoplus _{1 \leq j_1 < j_2 \leq n} \bigoplus _{i_1, i_2 \in I} \text{T}^{n - 2}(M) \to \text{T}^ n(M) \to \text{Sym}^ n(M) \to 0 \]

where the pure tensor $m_1 \otimes \ldots \otimes m_{n - 2}$ maps to

\begin{align*} \underbrace{ m_1 \otimes \ldots \otimes x_{i_1} \otimes \ldots \otimes x_{i_2} \otimes \ldots \otimes m_{n - 2} }_{\text{with } x_{i_1} \text{ and } x_{i_2} \text{ occupying slots } j_1 \text{ and } j_2 \text{ in the tensor}} \\ - \underbrace{ m_1 \otimes \ldots \otimes x_{i_2} \otimes \ldots \otimes x_{i_1} \otimes \ldots \otimes m_{n - 2} }_{\text{with } x_{i_2} \text{ and } x_{i_1} \text{ occupying slots } j_1 \text{ and } j_2 \text{ in the tensor}} \end{align*}

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