Microlocal homology

We introduce a functor of microlocalization along quasi-smooth closed immersions of derived schemes. By microlocalizing along an embedding of a quasi-smooth Z into a smooth X, we obtain a perverse sheaf µ [subscript Z/X] on the derived conormal bundle N [∨ over subscript Z/X], which refines Nadler’s microlocal homology of singular spaces in the case when Z is presented as the derived zero fiber of a regular function on X. Using the canonical inclusion T [superscript ∗] [−1]Z⊂N [∨ over subscript Z/X], we show that µ [subscript Z/X] is in fact equivalent to the canonical perverse sheaf of twisted vanishing cycles on T [superscript ∗] [−1]Z categorifying Donaldson–Thomas invariants. This equivalence proves that µ [subscript Z/X] is actually intrinsic to Z, which allows us to formulate a proposed theory of singular support, as a subset of T [superscript ∗] [−1]Z, for Borel–Moore chains on the underlying classical scheme of Z equipped with the analytic topology.