[proofplan] We compute $H_c^i(\mathbb{R}^d; R)$ using the [Direct Limit Description of Compactly Supported Cohomology](/theorems/2287). The closed balls $\{n \overline{B}(0,n)\}_{n \geq 1}$ form a cofinal family in $\mathcal{K}(\mathbb{R}^d)$, so $H_c^*(\mathbb{R}^d; R) \cong \varinjlim_n H^*(\mathbb{R}^d, \mathbb{R}^d \setminus n\overline{B}(0,n); R)$. We compute each $H^i(\mathbb{R}^d, \mathbb{R}^d \setminus n\overline{B}(0,n); R)$ using the long exact sequence of the pair and the contractibility of $\mathbb{R}^d$, then show the transition maps are isomorphisms, so the direct limit stabilizes. [/proofplan]

[step:Reduce to a cofinal directed system of closed balls]By the [Direct Limit Description of Compactly Supported Cohomology](/theorems/2287), \begin{align*} H_c^i(\mathbb{R}^d; R) \cong \varinjlim_{K \in \mathcal{K}(\mathbb{R}^d)} H^i(\mathbb{R}^d, \mathbb{R}^d \setminus K; R). \end{align*} The closed balls $K_n := \overline{B}(0, n) = \{x \in \mathbb{R}^d : |x| \leq n\}$ for $n = 1, 2, 3, \ldots$ form a cofinal family in $\mathcal{K}(\mathbb{R}^d)$: every compact set $K \subseteq \mathbb{R}^d$ is bounded, so $K \subseteq K_n$ for all sufficiently large $n$. Since the direct limit over a cofinal subsystem equals the direct limit over the full system: \begin{align*} H_c^i(\mathbb{R}^d; R) \cong \varinjlim_{n} H^i(\mathbb{R}^d, \mathbb{R}^d \setminus K_n; R). \end{align*}[/step]

[guided]Why is cofinality useful? Computing a direct limit over all compact subsets of $\mathbb{R}^d$ is unwieldy, but the result depends only on the "tail" behavior of the directed system. A cofinal subsystem $(K_n)_n$ suffices: for every compact $K$ there exists $n$ with $K \subseteq K_n$, so the map $H^i(\mathbb{R}^d \mid K; R) \to H^i(\mathbb{R}^d \mid K_n; R)$ connects every element of the original directed system to our subsystem. The closed balls are a natural cofinal choice because every compact subset of $\mathbb{R}^d$ is bounded.[/guided]

[step:Compute $H^i(\mathbb{R}^d, \mathbb{R}^d \setminus K_n; R)$ via the long exact sequence] Consider the long exact sequence of the pair $(\mathbb{R}^d, \mathbb{R}^d \setminus K_n)$: \begin{align*} \cdots \to H^{i-1}(\mathbb{R}^d; R) \to H^{i-1}(\mathbb{R}^d \setminus K_n; R) \xrightarrow{\partial} H^i(\mathbb{R}^d, \mathbb{R}^d \setminus K_n; R) \to H^i(\mathbb{R}^d; R) \to \cdots \end{align*} Since $\mathbb{R}^d$ is contractible, $H^i(\mathbb{R}^d; R) = 0$ for all $i \geq 1$ and $H^0(\mathbb{R}^d; R) \cong R$. Therefore for $i \geq 2$, the long exact sequence gives isomorphisms \begin{align*} \partial: H^{i-1}(\mathbb{R}^d \setminus K_n; R) \xrightarrow{\;\sim\;} H^i(\mathbb{R}^d, \mathbb{R}^d \setminus K_n; R). \end{align*} Now $\mathbb{R}^d \setminus K_n$ is homotopy equivalent to $S^{d-1}$: the radial projection $x \mapsto n \cdot x/|x|$ provides a deformation retraction of $\mathbb{R}^d \setminus K_n$ onto the sphere $\partial K_n = S^{d-1}(0, n) \cong S^{d-1}$. By the [Homology of Spheres](/theorems/1945), $H^j(S^{d-1}; R) \cong R$ for $j = 0$ and $j = d-1$, and is zero otherwise. Therefore: \begin{align*} H^i(\mathbb{R}^d, \mathbb{R}^d \setminus K_n; R) \cong H^{i-1}(\mathbb{R}^d \setminus K_n; R) \cong H^{i-1}(S^{d-1}; R) \cong \begin{cases} R & i = d, \\ 0 & i \geq 2, \; i \neq d. \end{cases} \end{align*} For $i = 1$: the relevant portion of the long exact sequence is \begin{align*} H^0(\mathbb{R}^d; R) \xrightarrow{j^*} H^0(\mathbb{R}^d \setminus K_n; R) \xrightarrow{\partial} H^1(\mathbb{R}^d, \mathbb{R}^d \setminus K_n; R) \to H^1(\mathbb{R}^d; R) = 0. \end{align*} Since $\mathbb{R}^d \setminus K_n$ is path-connected (for $d \geq 2$), $H^0(\mathbb{R}^d \setminus K_n; R) \cong R$, and the restriction $j^*: R \cong H^0(\mathbb{R}^d; R) \to H^0(\mathbb{R}^d \setminus K_n; R) \cong R$ sends the constant function $1$ to the constant function $1$, so $j^*$ is an isomorphism. Exactness gives $H^1(\mathbb{R}^d, \mathbb{R}^d \setminus K_n; R) = 0$ for $d \geq 2$. For $d = 1$: $\mathbb{R} \setminus [-n, n]$ consists of two path components $(-\infty, -n)$ and $(n, \infty)$, so $H^0(\mathbb{R} \setminus K_n; R) \cong R^2$. The map $j^*: R \to R^2$ sends $1$ to $(1, 1)$, so $\operatorname{coker}(j^*) \cong R$ and $H^1(\mathbb{R}, \mathbb{R} \setminus K_n; R) \cong R$. For $i = 0$: the sequence begins $0 \to H^0(\mathbb{R}^d, \mathbb{R}^d \setminus K_n; R) \to H^0(\mathbb{R}^d; R) \to H^0(\mathbb{R}^d \setminus K_n; R)$. The map $H^0(\mathbb{R}^d; R) \to H^0(\mathbb{R}^d \setminus K_n; R)$ is injective (it is restriction of locally constant functions from a connected space to a nonempty subspace), so $H^0(\mathbb{R}^d, \mathbb{R}^d \setminus K_n; R) = 0$. Combining all cases: $H^i(\mathbb{R}^d, \mathbb{R}^d \setminus K_n; R) \cong R$ if $i = d$, and $0$ otherwise. [/step]

[step:Show the transition maps are isomorphisms]For $n \leq m$, the inclusion $K_n \subseteq K_m$ gives a transition map \begin{align*} \iota_{n,m}: H^i(\mathbb{R}^d, \mathbb{R}^d \setminus K_n; R) \to H^i(\mathbb{R}^d, \mathbb{R}^d \setminus K_m; R). \end{align*} We claim this is an isomorphism for all $n, m$. The inclusion $\mathbb{R}^d \setminus K_m \hookrightarrow \mathbb{R}^d \setminus K_n$ induces the transition map on relative cohomology. Consider the commutative diagram arising from the naturality of the long exact sequence of pairs, applied to the inclusion of pairs $(\mathbb{R}^d, \mathbb{R}^d \setminus K_m) \hookrightarrow (\mathbb{R}^d, \mathbb{R}^d \setminus K_n)$. The radial projection provides a homotopy equivalence $\mathbb{R}^d \setminus K_n \simeq S^{d-1}(0,n)$ and $\mathbb{R}^d \setminus K_m \simeq S^{d-1}(0,m)$, and the inclusion $\mathbb{R}^d \setminus K_m \hookrightarrow \mathbb{R}^d \setminus K_n$ is a homotopy equivalence (both spaces deformation retract onto concentric spheres, and the inclusion is compatible with these retractions). Therefore the induced map $H^{i-1}(\mathbb{R}^d \setminus K_n; R) \to H^{i-1}(\mathbb{R}^d \setminus K_m; R)$ is an isomorphism for all $i$. By naturality of the connecting homomorphism and the [Five Lemma](/theorems/1938), the transition map $\iota_{n,m}$ is an isomorphism.[/step]

[guided]The homotopy equivalence $\mathbb{R}^d \setminus K_m \hookrightarrow \mathbb{R}^d \setminus K_n$ is the essential geometric input. Removing a larger ball from $\mathbb{R}^d$ does not change the homotopy type of the complement — both complements retract onto any sphere $S^{d-1}(0, r)$ with $r > n$ (or $r > m$). The transition maps in the directed system are therefore all isomorphisms, which means the direct limit is simply isomorphic to any single term in the system. The limit stabilizes, and \begin{align*} H_c^i(\mathbb{R}^d; R) \cong H^i(\mathbb{R}^d, \mathbb{R}^d \setminus K_1; R) \end{align*} for any $n \geq 1$.[/guided]

[step:Conclude the computation] Since all transition maps in the directed system are isomorphisms, the direct limit is isomorphic to any single term: \begin{align*} H_c^i(\mathbb{R}^d; R) \cong \varinjlim_n H^i(\mathbb{R}^d, \mathbb{R}^d \setminus K_n; R) \cong H^i(\mathbb{R}^d, \mathbb{R}^d \setminus K_1; R) \cong \begin{cases} R & i = d, \\ 0 & \text{otherwise.} \end{cases} \end{align*} This completes the computation. [/step]