[proofplan]
We derive the Mayer-Vietoris sequence for compactly supported cohomology from the ordinary Mayer-Vietoris sequence for local cohomology (cohomology relative to the complement of a compact set) by passing to a direct limit. For each pair of compact sets $K \subseteq A$ and $L \subseteq B$, excision identifies local cohomology groups with those in the ambient manifold, and a Mayer-Vietoris sequence relates $H^n(M \mid K \cup L)$ to $H^n(M \mid K)$, $H^n(M \mid L)$, and $H^n(M \mid K \cap L)$. Taking the direct limit over all such compact pairs and using the [Direct Limit Description of Compactly Supported Cohomology](/theorems/2287) yields the desired long exact sequence.
[/proofplan]
[step:Establish the Mayer-Vietoris sequence for local cohomology of a compact pair]
Fix compact sets $K \subseteq A$ and $L \subseteq B$. Consider the open cover $M = A \cup B$ and the compact set $K \cup L \subseteq M$. We have:
\begin{align*}
M \setminus (K \cup L) &= (M \setminus K) \cap (M \setminus L), \\
M \setminus K &\supseteq M \setminus (K \cup L), \\
M \setminus L &\supseteq M \setminus (K \cup L).
\end{align*}
The cochain complexes $C^*(M, M \setminus K; R)$, $C^*(M, M \setminus L; R)$, $C^*(M, M \setminus (K \cup L); R)$, and $C^*(M, M \setminus (K \cap L); R)$ fit into a short exact sequence of cochain complexes:
\begin{align*}
0 \to C^n(M, M \setminus (K \cup L); R) \xrightarrow{\;(\iota_K, \iota_L)\;} C^n(M, M \setminus K; R) \oplus C^n(M, M \setminus L; R) \xrightarrow{\;j_K - j_L\;} C^n(M, M \setminus (K \cap L); R) \to 0.
\end{align*}
Here $\iota_K$ and $\iota_L$ are the natural inclusions induced by $M \setminus (K \cup L) \subseteq M \setminus K$ and $M \setminus (K \cup L) \subseteq M \setminus L$, while $j_K$ and $j_L$ are the restrictions from $M \setminus K$ and $M \setminus L$ to $M \setminus (K \cap L)$.
[guided]
The exactness of this short exact sequence at the cochain level requires verification. Injectivity on the left: if a cochain $\phi$ vanishes on chains in $M \setminus K$ and vanishes on chains in $M \setminus L$, then $\phi$ vanishes on chains in $(M \setminus K) \cup (M \setminus L) = M \setminus (K \cap L)$. Wait -- we need the other direction. A cochain in $C^n(M, M \setminus (K \cup L); R)$ vanishes on chains in $M \setminus (K \cup L) = (M \setminus K) \cap (M \setminus L)$; this is a stronger condition than vanishing on $M \setminus K$ alone.
The injectivity on the left says: if $\phi \in C^n(M, M \setminus (K \cup L); R)$ maps to zero in both direct summands, meaning $\phi = 0$ as an element of $C^n(M, M \setminus K; R)$ and as an element of $C^n(M, M \setminus L; R)$, then $\phi = 0$ as a cochain on $M$. This holds because a relative cochain in $C^n(M, M \setminus (K \cup L); R)$ is already a cochain on $M$, and if it is zero in $C^n(M, M \setminus K; R) \cong C^n(M; R) / C^n(M \setminus K; R)$, that means it lies in $C^n(M \setminus K; R)$; but it also lies in $C^n(M, M \setminus (K \cup L); R)$, meaning it vanishes on $M \setminus (K \cup L)$. The two conditions together force $\phi = 0$.
Surjectivity on the right uses the fact that $M \setminus K$ and $M \setminus L$ together cover $M \setminus (K \cap L)$, so a cochain vanishing on $M \setminus (K \cap L)$ can be expressed as a difference of cochains vanishing on $M \setminus K$ and $M \setminus L$ respectively. This is the cochain-level analogue of the Mayer-Vietoris argument for open covers.
[/guided]
The associated long exact sequence in cohomology is the Mayer-Vietoris sequence for local cohomology:
\begin{align*}
\cdots \to H^n(M \mid K \cup L; R) \to H^n(M \mid K; R) \oplus H^n(M \mid L; R) \to H^n(M \mid K \cap L; R) \xrightarrow{\partial} H^{n+1}(M \mid K \cup L; R) \to \cdots
\end{align*}
[/step]
[step:Apply excision to identify the local cohomology groups]
Since $K \subseteq A$ and $A$ is open, the closed set $M \setminus A$ satisfies $M \setminus A \subseteq M \setminus K = \operatorname{int}(M \setminus K)$ (because $M \setminus K$ is open, being the complement of the compact set $K$ in the Hausdorff space $M$). By excision applied to the pair $(M, M \setminus K)$ with the closed set $M \setminus A$:
\begin{align*}
H^n(M, M \setminus K; R) \cong H^n(A, A \setminus K; R),
\end{align*}
i.e., $H^n(M \mid K; R) \cong H^n(A \mid K; R)$. Similarly, $H^n(M \mid L; R) \cong H^n(B \mid L; R)$, and since $K \cap L \subseteq A \cap B$ with $A \cap B$ open:
\begin{align*}
H^n(M \mid K \cap L; R) \cong H^n(A \cap B \mid K \cap L; R).
\end{align*}
Under these identifications, the Mayer-Vietoris sequence from Step 1 becomes:
\begin{align*}
\cdots \to H^n(M \mid K \cup L; R) \to H^n(A \mid K; R) \oplus H^n(B \mid L; R) \to H^n(A \cap B \mid K \cap L; R) \xrightarrow{\partial} H^{n+1}(M \mid K \cup L; R) \to \cdots
\end{align*}
[/step]
[step:Take the direct limit over compact pairs to obtain the compactly supported sequence]
The Mayer-Vietoris sequence in Step 2 is natural in the compact pair $(K, L)$: for $K \subseteq K'$ and $L \subseteq L'$, the inclusions induce a morphism of long exact sequences. The directed set of pairs $(K, L)$ with $K \in \mathcal{K}(A)$ and $L \in \mathcal{K}(B)$ is directed under the product ordering $(K, L) \leq (K', L')$ iff $K \subseteq K'$ and $L \subseteq L'$.
This directed set is cofinal in $\mathcal{K}(M)$ for the term $H^n(M \mid K \cup L; R)$: given any compact $C \subseteq M$, set $K = C \cap \overline{A'}$ and $L = C \cap \overline{B'}$ for appropriate closed subsets; more directly, every compact subset of $M$ is contained in $K \cup L$ for some $K \in \mathcal{K}(A)$ and $L \in \mathcal{K}(B)$ (take $K := C \setminus B$ need not be compact, but we can use a partition of unity argument: let $\lambda_A + \lambda_B = 1$ subordinate to $\{A, B\}$, and set $K = C \cap \operatorname{supp}(\lambda_A) \subseteq A$, $L = C \cap \operatorname{supp}(\lambda_B) \subseteq B$; then $K, L$ are compact and $C \subseteq K \cup L$).
Taking the direct limit over all such pairs $(K, L)$ and applying the [Direct Limit Description of Compactly Supported Cohomology](/theorems/2287), the four types of terms become:
\begin{align*}
\varinjlim_{K \cup L} H^n(M \mid K \cup L; R) &\cong H_c^n(M; R), \\
\varinjlim_K H^n(A \mid K; R) &\cong H_c^n(A; R), \\
\varinjlim_L H^n(B \mid L; R) &\cong H_c^n(B; R), \\
\varinjlim_{K \cap L} H^n(A \cap B \mid K \cap L; R) &\cong H_c^n(A \cap B; R).
\end{align*}
Since the direct limit functor over a directed set is exact, the long exact sequence passes to the limit:
\begin{align*}
\cdots \to H_c^n(A \cap B; R) \to H_c^n(A; R) \oplus H_c^n(B; R) \to H_c^n(M; R) \xrightarrow{\partial} H_c^{n+1}(A \cap B; R) \to \cdots
\end{align*}
[guided]
The cofinality argument for the direct limit is the most delicate point. We need the compact sets of the form $K \cup L$ (with $K \in \mathcal{K}(A)$, $L \in \mathcal{K}(B)$) to be cofinal in $\mathcal{K}(M)$. The partition of unity argument works: given $C \in \mathcal{K}(M)$ and a partition of unity $\{\lambda_A, \lambda_B\}$ subordinate to $\{A, B\}$, the sets $K := C \cap \operatorname{supp}(\lambda_A)$ and $L := C \cap \operatorname{supp}(\lambda_B)$ are compact (as closed subsets of the compact set $C$), satisfy $K \subseteq \operatorname{supp}(\lambda_A) \subseteq A$ and $L \subseteq \operatorname{supp}(\lambda_B) \subseteq B$, and $K \cup L = C \cap (\operatorname{supp}(\lambda_A) \cup \operatorname{supp}(\lambda_B)) = C$ since the supports cover $M$.
Similarly, for the $A \cap B$ term, the sets $K \cap L$ for $K \in \mathcal{K}(A)$ and $L \in \mathcal{K}(B)$ are cofinal in $\mathcal{K}(A \cap B)$: given compact $C' \subseteq A \cap B$, take $K = L = C'$.
The exactness of direct limits over directed sets then guarantees that the limiting sequence inherits exactness from the individual Mayer-Vietoris sequences.
[/guided]
[/step]
