Microlocal homology

Abstract

Let $Z$ be an l.c.i. scheme over $\mathbb{C}$. In this paper, we introduce a Kashiwara--Schapira-style functor of derived microlocalization, which we use to define a perverse sheaf $\mu_{Z}$ on the $-1$-shifted cotangent bundle, $T^*[-1]Z$. The sheaf $\mu_{Z}$ is designed to be a refinement of the microlocal homology of $Z$: a family of invariants introduced by Nadler that interpolates between the singular cohomology and Borel--Moore homology of $Z$. Our main result is an equivalence between $\mu_{Z}$ and the DT sheaf $\varphi_{T^*[-1]Z}$ on $T^*[-1]Z$. This provides an alternative construction for the DT sheaf in the case of a shifted cotangent bundle. The main step of our argument, which may be of independent interest, is a local computation -- closely related to one obtained recently by Kinjo using different methods -- providing a description of the classical microlocalization functor in terms of vanishing cycles.