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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