Cotangent bundle and micro-supports in mixed characteristic case

TL;DR

This paper introduces the FW-cotangent bundle for regular schemes in mixed characteristic and defines micro-supports for constructible complexes, providing computations of singular supports in specific cases.

Contribution

It defines the FW-cotangent bundle in mixed characteristic and establishes a framework for micro-supports of constructible complexes on schemes.

Findings

Defined the FW-cotangent bundle on schemes with prime p

Introduced micro-support conditions for constructible complexes

Computed singular supports in particular cases

Abstract

For a regular scheme and a prime number $p$, we define the FW-cotangent bundle as a vector bundle on the closed subscheme defined by $p=0$, under a certain finiteness condition. For a constructible complex on the etale site of the scheme, we introduce the condition to be micro-supported on a closed conical subset in the FW-cotangent bundle. At the end of the article, we compute the singular supports in some cases.