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The Wikipedia page for Borel-Moore homology states that for a locally compact set $X$ and a closed subset $Z$, if we write $U = X \setminus Z$ we have the following long exact sequence $$\cdots \to H^ …

Let $f: X \to Y$ be a locally trivial fibration between locally compact spaces with fiber $F$. It is well known that for a constant sheaf $A_X$ on $X$, the higher direct images $R^n f_* A_X$ are local …

With the information added in the last update, the result is already proved once you realize that the unit of the adjunction $$ id \to a_{B,*} a_B^* $$ is actually an isomorphism on the category of ab …

Let $L \subset \mathbb{C}^n$ be a hyperplane and let $i:L \to \mathbb{C}^n$ be the inclusion. Since $i$ is proper, we have induced maps $i^*: H^k_c(\mathbb{C}^n) \to H^k_c(L)$, and these maps are zero …