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

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.