> [!definition] > > Let $R$ be a [[Ring|ring]] and $E$ and $F$ be $R$-[[Module|modules]]. A [[Multilinear Map|multilinear map]] $\lambda: E^k \to F$ is **alternating** if the following equivalent conditions are satisfied: > - $\lambda \circ \sigma = \sgn(\sigma) \cdot \lambda$ for any $\sigma \in S_k$. > - For any $v \in E^k$, if there exists $(i, j)$ such that $i \ne j$ and $v_i = v_j$, $\lambda v = 0$. > > *Proof*. The first condition directly implies the second. Suppose that the second condition holds, and only consider two coordinates for simplicity: > $ > \begin{align*} > \lambda(x + y, x + y) &= \lambda(x, x + y) + \lambda(y, x + y) \\ > &= \lambda(x, x) + \lambda(x, y) + \lambda(y, x) + \lambda(y, y) \\ > &= \lambda(x, y) + \lambda(y, x) = 0 > \end{align*} > $