Lemma 10.13.4. Let $A \to B$ be a ring map. Let $M$ be a $B$-module. Let $n > 1$. The kernel of the $A$-linear map $M \otimes _ A \ldots \otimes _ A M \to \wedge ^ n_ B(M)$ is generated as an $A$-module by the elements $m_1 \otimes \ldots \otimes m_ n$ with $m_ i = m_ j$ for $i \not= j$, $m_1, \ldots , m_ n \in M$ and the elements $m_1 \otimes \ldots \otimes bm_ i \otimes \ldots \otimes m_ n - m_1 \otimes \ldots \otimes bm_ j \otimes \ldots \otimes m_ n$ for $i \not= j$, $m_1, \ldots , m_ n \in M$, and $b \in B$.

**Proof.**
Omitted.
$\square$

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