Borel-Moore homology for resolution of singularities

Question

Let $X$ be a singular projective variety. Denote by $Z$ the singular locus of $X$. Consider the resolution of singularities $$\pi: \widetilde{X} \to X$$ Denote by $E$ the exceptional divisor. We know that there is a long exact sequence in cohomology given by: $$... \to H^{i-1}(E) \to H^i(X) \to H^i(\widetilde{X}) \oplus H^i(Z) \to H^i(E) \to ...$$ Is there a similar exact sequence using Borel-Moore homology, instead of cohomology? If necessary, assume that $Z$ is a union of finitely many points.

Answer 1

Borel-Moore homology is actually the easiest variant of (co)homology to prove this in. More generally let $\pi \colon Y \to X$ be a proper morphism which induces an isomorphism away from a closed subscheme $Z \subset X$. Write $E := \pi^{-1}(Z)$. Then we have the distinguished triangles $$ C^{BM}_*(Z) \to C^{BM}_*(X) \to C^{BM}_*(X-Z) \to $$ $$ C^{BM}_*(E) \to C^{BM}_*(Y) \to C^{BM}_*(Y-E) \to $$ There is a map of distinguished triangles, from bottom to top, given by proper pushforward, which induces a quasi-isomorphism on the cones/cofibers. So the square $$\require{AMScd}\begin{CD} C^{BM}_*(E) @>>> C^{BM}_*(Y) \\ @V{\pi_*}VV @V{\pi_*}VV \\ C^{BM}_*(Z) @>>> C^{BM}_*(X) \end{CD}$$ is homotopy coCartesian as it induces quasi-isomorphisms on the cones of the horizontal maps. Equivalently there is a distinguished triangle $$ C^{BM}_*(E) \to C^{BM}_*(Y) \oplus C^{BM}_*(Z) \to C^{BM}_*(X) \to $$ which gives rise to the long exact sequence you asked for.

When $X$ is proper, you can then rewrite this using homology or cohomology.