Having the same wild ramification is preserved by the direct image

TL;DR

This paper introduces a notion of wild ramification equivalence for constructible sheaves on schemes over a henselian DVR and proves its invariance under most of Grothendieck's six operations.

Contribution

It defines wild ramification equivalence in the context of Grothendieck groups and shows its preservation under key functorial operations.

Findings

Wild ramification equivalence is preserved by four of Grothendieck's six operations.

The notion applies to constructible sheaves on schemes over an excellent henselian DVR.

The derived tensor product and hom do not preserve this equivalence.

Abstract

We introduce the notion that two elements of Grothendieck groups of constructible sheaves on a separated scheme over an excellent henselian discrete valuation ring have the same wild ramification. We prove that this condition is preserved by four of Grothendieck's six operations except the derived tensor product and the derived hom.