---- > [!definition] Definition. ([[alternating multilinear map]]) > Let $R$ be a [[ring]] and let $M_{1},\dots,M_{\ell}, P$ be $R$-[[module|modules]]. An $R$-[[multilinear map|multilinear map]] $\varphi: M_{1} \times \dots \times M_{\ell} \to P$ is called **alternating** if $\varphi(m_{1},\dots,m_{\ell})=0 \text{ whenever } m_{i}=m_{j} \text{ for some }i \neq j.$ > > In the special case $P=R$, $\varphi$ is called an **alternating (multilinear) form** or **alternating tensor**. If we further have $\ell=2$, it is called an **alternating bilinear form** or a **skew-symmetric [[bilinear map|bilinear form]]**. > > The set of all $R$-alternating-multilinear maps $M_{1} \times \dots \times M_{\ell} \to P$ has a clear $R$-[[module]] structure and is denoted $A(M_{1},\dots,M_{\ell}; P)$. We in particular denote $A(V , \dots, V; \mathbb{F})$ by $\text{Alt}_{\mathbb{F}}^{\ell}(V)$ or just $\text{Alt}^{\ell}(V)$ for $V$ a finite-dimensional [[vector space]] over $\mathbb{F}$. Indeed, $A(M_{1},\dots,M_{\ell};P)$ arises as the [[quotient module|quotient]] of $L(M_{1},\dots,M_{\ell};P)$ by the [[group action|action]] of the [[alternating group]]. ^definition - [ ] todo substantiate that last comment (we have done it for $\text{Mult}^{\ell}(V)$ and $\text{Alt}^{\ell}(V)$ specifically) > [!note] Remark. > The name 'alternating' comes from the following equivalence for base rings not having characteristic $2$: ^note > [!equivalence] > If $\varphi$ is alternating, then for all $\sigma \in S_{\ell}$, and all $m_{1},\dots,m_{\ell}$, $\varphi(m_{\sigma(1)},\dots,m_{\sigma(\ell)})=\text{sgn}(\sigma) \ \varphi(m_{1},\dots,m_{\ell}).$ > If $2$ is a [[unit]] in $R$, the converse holds as well. ^equivalence > [!proof] Proof of Equivalence. > For the first statement, it suffices to show that swapping any two arguments switches the sign (since transpositions generate $S_{\ell}$). Since the other arguments have no effect on this operation, this reduces the question to the case $\ell=2$ — therefore, we just have to show that if $\varphi(m , m)=0$ for all $m \in M$, then antisymmetry holds: $\varphi(m_{2},m_{1})=-\varphi(m_{1},m_{2}).$ This is straightforward to see:$\begin{align} 0 = & \varphi(m_{1}+m_{2}, m_{1} + m_{2}) \\ = & \varphi(m_{1}, m_{1}+ m_{2}) + \varphi(m_{2}, m_{1} + m_{2}) \\ = & \cancel{\varphi(m_{1}, m_{1})} + \varphi(m_{1}, m_{2}) + \varphi(m_{2}, m_{1}) + \cancel{\varphi(m_{2}, m_{2})}. \end{align}$ For the second statement, we again reduce to the case $\ell=2$. If $m_{1}=m_{2}=m$, then the assumption $\varphi(m_{2},m_{1})=-\varphi(m_{2},m_{1})$ says $2\varphi(m,m)=0$. If $2$ is a [[unit]] in $R$, this implies $\varphi(m,m)=0$ as desired. ^proof ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```