---- > [!definition] Definition. ([[exterior algebra]]) > Let $M$ be a [[module]] over a [[commutative ring]] $R$. We define the **exterior algebra** of $M$ as the [[graded algebra|graded]] $R$-[[algebra]] $\Lambda^{\bullet}_{R}(M):=\bigoplus_{\ell \geq 0} \Lambda_{R}^{\ell}(M),$ where multiplication is defined in terms of [[exterior power|pure alternating tensors]] by $(m_{1} \wedge \dots \wedge m_{i}) \cdot (n_{1} \wedge \dots \wedge n_{j}):=m_{1} \wedge \dots \wedge m_{i} \textcolor{Thistle}{\wedge} n_{1} \wedge \dots \wedge n_{j}$ and in defined general via the linear extension of this. > When $M=V$ is a [[vector space]], the elements of $\Lambda_{R}^{\bullet}(M)$ are sometimes called **multivectors**. In this case, multiplication in $\Lambda^{{\bullet}}(M)$ is sometimes called the **wedge product of multivectors**. [[the exterior algebra of a dual vector space is canonically identified with that of alternating multilinear forms|There is a canonical identification]] $\Lambda_{R}^{\bullet}(V^{*}) \cong \text{Alt}^{\bullet}(V)$ of the exterior algebra on the [[dual vector space]] $V^{*}$ with that on [[vector space of alternating k-tensors on a vector space|the space]] of [[alternating multilinear map|alternating]] forms, under which the the wedge product of multivectors and the [[algebra of alternating multilinear forms]] agree. ^definition > [!basicproperties] > - basis and dimension > - graded skew-commutativity ^properties ---- #### ---- #### 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 > ```