Having trouble finding an exact sequence.

Question

Find an exact sequence: $\{0\} \rightarrow \mathbb{V} \xrightarrow{{\alpha}} \mathbb{U} \xrightarrow{{\phi}} W \rightarrow \{0\} $, such that $$\dim(\mathbb{V}) = \dim(\mathbb{U}) = \infty$$ $$\dim(\mathbb{W}) < \infty$$

Answer 1

Let $M$ be a closed smooth manifold, and $\Omega^p(M)$ the $\mathbb R$ vector space of $p\leq \dim M$ forms, and $d$ the exterior derivative. Then: $$0\longrightarrow \operatorname{im} d\subset \Omega^p(M)\longrightarrow \ker d\subset \Omega^{p}(M)\longrightarrow H^p_{dR}(M)\longrightarrow 0$$ is a short exact sequence where $\ker d$, $\operatorname{im}d$ are infinite dimensional vector spaces, and $H^p_{dR}(M)$ is finite dimensional since $M$ is closed.

Since all the exact forms (i.e. those of the form $d\omega$ for $\omega\in \Omega^{p-1}$) are automatically in the kernel of $d$, the first map is the inclusion, and automatically injective. Since every equivalence class $[\omega]\in H^p_{dR}(M)$ is represented by a closed form (i.e. something that satisfies $d\eta=0$ for $\eta\in \Omega^p(M)$), we have that that map is automatically surjective. Since ever exact form is zero in $H^p_{dR}(M)$ we thus have that the sequence is exact.

Answer 2

Consider $0 \to \mathbb{R}^\infty \overset{\alpha}{\to} \mathbb{R}^\infty \to \mathbb{R} \to 0$ where $\alpha(x_1, \ldots) = (0, x_1, \ldots)$ shifts the input to the right by one place (here $\mathbb{R}^\infty = \bigoplus_{n \in \mathbb{N}} \mathbb{R}$).