On singular supports in mixed characteristic

TL;DR

This paper develops a new notion of singular support for constructible étale sheaves over regular schemes in mixed characteristic, proving its existence under certain conditions using Radon transform techniques.

Contribution

It introduces a relative singular support in mixed characteristic and establishes its existence via Beilinson's Radon transform method, extending the theory of singular supports.

Findings

Defined a variant of singular support relative to a base scheme

Proved existence of a saturated relative singular support

Deduced existence of the usual singular support under specific conditions

Abstract

We fix an excellent regular noetherian scheme $S$ over ${\mathbf Z}_{(p)}$ satisfying a certain finiteness condition. For a constructible \'etale sheaf ${\cal F}$ on a regular scheme $X$ of finite type over $S$, we introduce a variant of the singular support relatively to $S$ and prove the existence of a saturated relative variant of the singular support by adopting the method of Beilinson using the Radon transform. We may deduce the existence of the singular support itself, if we admit an expected property on the micro support of tensor product and if the scheme $X$ is sufficiently ramified over the base $S$.