---- > [!definition] Definition. ([[orientation of a vector space]]) > Let $R$ be a (assume [[commutative ring|commutative]]) [[ring]]. Let $V$ be a $d$-dimensional [[vector space]]. Put $V^{\sharp}:=V-\{ 0 \}$. > [[the rank theorem for free modules|We can]] choose an [[isomorphism]] $V \xrightarrow{}\mathbb{R}^{d}$. After making this choice, we know that[^1] $H^{i}(V, V^{\sharp}; R) \cong H^{i}(\mathbb{R}^{d}, \mathbb{R}^{d}- \{ 0 \}; R) = \begin{cases} R & i=d \\ 0 & \text{otherwise.} \end{cases}$ Because it is [[isomorphism|isomorphic]] as an $R$-[[module]] to $R$, $H^{d}(V, V^{\sharp}; R)$ is [[submodule generated by a subset|generated by]] a single element. But there is no canonical generator, as we had to pick an [[isomorphism]] $V \xrightarrow{\sim}\mathbb{R}^{d}$. > We call a choice of generator of $H^{d}(V, V^{\sharp}; R)$, or equivalently a choice of [[isomorphism]] $R \to H^{d}(V, V^{\sharp}; R)$, an **$R$-orientation** of $V$. ^definition ---- #### [^1]: Indeed: [[exact sequence|The]] [[long exact sequence for relative singular homology|LES for the]] [[topological pair|pair]] $(\mathbb{R}^{d}, \mathbb{R}^{d}-\{ 0 \})$ gives [[relative singular homology|a]] natural [[isomorphism]]$H^{d-1}(\mathbb{R}^{d}-\{ 0 \}; R) \xrightarrow[\cong]{\partial^{*}} H^{d}(\mathbb{R}^{d}, \mathbb{R}^{d}-\{ 0 \}; R).$ Per [[(co)homology of the spheres]], [[homotopy invariance of singular homology|this is]] [[isomorphism|isomorphic]] as an $R$-[[module]] to $R$. (Explicitly, consider $H^{d-1}(\mathbb{R}^{d};R) \xrightarrow{\iota^{*}} H^{d-1}(\mathbb{R}^{d}-\{ 0 \};R) \xrightarrow{\partial^{*}} H^{d}(\mathbb{R}^{d}, \mathbb{R}^{d}- \{ 0 \};R ) \xrightarrow{q^{*}}H^{d}(\mathbb{R}^{d}; R).$) ---- #### 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 > ```