Order reduction of $\Lambda$-marked monomial ideals and weak resolutions

TL;DR

This paper develops an order reduction technique for $\Lambda$-marked monomial ideals within $\Lambda$-schemes, leading to a weak resolution of singularities inspired by $\mathbb{F}_1$-geometry and Witt vector extensions.

Contribution

It introduces a novel order reduction method for $\Lambda$-marked ideals and establishes a weak resolution of singularities in the context of $\Lambda$-schemes.

Findings

Established existence of weak resolution of singularities for $\Lambda$-schemes.

Developed order reduction techniques for $\Lambda$-marked monomial ideals.

Connected $\Lambda$-geometry with classical resolution methods.

Abstract

Borger's theory of $\Lambda$-spaces imbues algebraic spaces, which include schemes, with an additional structure defined by an extension of the Witt vector functor. Motivated by $\mathbb{F}_1$-geometry, we prove the existence of a weak resolution of singularities in the category of $\Lambda$-schemes. Our arguments are based on standard arguments in characteristic $0$ using the order reduction of an ideal marked with $\Lambda$-equivariant data. This paper is based on work from the author's PhD thesis.